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We study the classical XY (plane rotator) model at the Kosterlitz-Thouless phase transition. We simulate the model using the single cluster algorithm on square lattices of a linear size up to L=2048.We derive the finite size behaviour of the second moment correlation length over the lattice size xi_{2nd}/L at the transition temperature. This new prediction and the analogous one for the helicity modulus are confronted with our Monte Carlo data. This way beta_{KT}=1.1199 is confirmed as inverse transition temperature. Finally we address the puzzle of logarithmic corrections of the magnetic susceptibility chi at the transition temperature. Comment: Monte Carlo results for xi/L in table 1 and 2 corrected. Due to a programming error,these numbers were wrong by about a factor 1+1/L^2. Correspondingly the fits with L_min=64 and 128 given in table 5 and 6 are changed by little.The central results of the paper are not affected. Wrong sign in eq.(52) corrected. Appendix extended

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... The codes are written with PyTorch library and open on GitHub. [53] In the following, we demonstrate the results of the = 16 case for thermodynamics calculation. ...

... The thermodynamic observables of the 2D XY model have been computed with the MCMC method at a high accuracy. [40,42,53,54] To validate our method, we compare the results from our methods with computations from MCMC. Figure 2 shows the energy per site as a function of obtained from CANs, CANs-IS and MCMC with 1000 configurations. Here, the MCMC calculation was implemented with a classical algorithm, [55,56] and its details can be found in the Supplementary Materials. ...

... In the continuum limit, it is s = [ 2 ( )/ ( ) 2 ]| =0 . In practice, we use the related quantity, i.e., helicity modulus ( ), [40,42,53,54] which is equivalent to s in the limit of → 0. It can be expressed as ...

We develop deep autoregressive networks with multi channels to compute many-body systems with continuous spin degrees of freedom directly. As a concrete example, we demonstrate on the two-dimensional XY model with the continuous-mixture networks and rediscover the Kosterlitz-Thouless (KT) phase transition on a periodic square lattice. Vortices characterizing the quasi-long range order are accurately detected by the generative model. By learning the microscopic probability distributions from the macroscopic thermal distribution, the networks are trained as an efficient physical sampler which can approximate the free energy and estimate thermodynamic obervables unbiasedly with importance sampling. As a more precise evaluation, we compute the helicity modulus to determine the KT transition temperature. Although the training process becomes more time-consuming with larger lattice sizes, the training time remains unchanged around the KT transition temperature. The continuous-mixture autoregressive networks(CANs) we developed thus can be potentially used to study other many-body systems with continuous degrees of freedom.

... Beyond its link to the XY model, the Villain Hamiltonian has drown considerable attention per se, proving interesting both from the theoretical [10][11][12][13][14][15] and numerical [16][17][18][19][20] point of view, with applications running from the study of quantum-phase transitions [21] and superconductivity [22] to lattice gauge theories [8,9,[23][24][25], deconfinement [26] and duality [27] in high-energy physics. It is thus customary in the literature to refer to the Villain approximation when the coupling is adjusted to reproduce in the best way the XY model free energy, and to the Villain model when the model is studied by itself [8]. ...

... In this section we derive the effective interaction between topological charges m z in Eq. (19). To this aim, we define the Fourier transformed variables ...

... In this Section we see how some of the results obtained through the lattice calculations, starting from the partition function (19), can be understood in terms of its continuum limit (9) within the field theory formalism. ...

The nearest-neighbor Villain, or periodic Gaussian, model is a useful tool to understand the physics of the topological defects of the two-dimensional nearest-neighbor $XY$ model, as the two models share the same symmetries and are in the same universality class. The long-range counterpart of the two-dimensional $XY$ model has been recently shown to exhibit a non-trivial critical behavior, with a complex phase diagram including a range of values of the power-law exponent of the couplings decay, $\sigma$, in which there are a magnetized, a disordered and a critical phase (arXiv:2104.13217). Here we address the issue of whether the critical behavior of the two-dimensional $XY$ model with long-range couplings can be described by the Villain counterpart of the model. After introducing a suitable generalization of the Villain model with long-range couplings, we derive a set of renormalization-group equations for the vortex-vortex potential, which differs from the one of the long-range $XY$ model, signaling that the decoupling of spin-waves and topological defects is no longer justified in this regime. The main results are that for $\sigma<2$ the two models no longer share the same universality class. Remarkably, within a large region of its phase diagram, the Villain model is found to behave similarly to the one-dimensional Ising model with $1/r^2$ interactions.

... The critical temperature was numerically measured in numerous works. Probably the most precise result was reported by Hasenbusch, who obtained β c ≡ 1/T c = 1.1199(1) [8]. ...

... (the small exponential correction to the jump height was discovered in Ref. [18]). This formula refers to infinite volume, and for the standard Hamiltonian (1) the convergence towards this value for increasing L is very slow: even at L = 2048, T = T c , it is still 6.6% too high [8]. The constraint Hamiltonian behaves much better: at L = 64, δ = δ c , it already agrees with the theory to a precision below 1% [14], which was the first convincing numerical confirmation of the prediction (9). ...

... The constraint Hamiltonian behaves much better: at L = 64, δ = δ c , it already agrees with the theory to a precision below 1% [14], which was the first convincing numerical confirmation of the prediction (9). In Fig. 3 we show new simulation results with the standard Hamiltonian, which are consistent with Ref. [8], though we are limited to L ≤ 512. ...

The Berezinskii-Kosterlitz-Thouless (BKT) essential phase transition in the 2d XY model is revisited. Its mechanism is usually described by the (un)binding of vortex--anti-vortex (V--AV) pairs, which does, however, not provide a clear-cut quantitative criterion for criticality. Known sharp criteria are the divergence of the correlation length and a discontinuity of the helicity modulus. Here we propose and probe a new criterion: it is based on the concepts of semi-vortices and cluster vorticity, which are formulated in the framework of the multi-cluster algorithm that we use to simulate the 2d XY model.

... On the other hand, it is numerically challenging to precisely identify such multiplicative logarithmic correction with a very small exponent. Much numerical effort has been devoted to measuring r in the two-dimensional (2D) XY model and related models undergoing the BKT transition [10][11][12][13][14][15][16][17][18][19][20][21][22]. While early estimates of r vary from positive to negative values (see Table 4 in Ref. [23]), later large-scale Monte Carlo (MC) simulations showed improved agreement with the RG prediction. ...

... (10) from the FSS analysis of the susceptibility in the critical region for system sizes up to L = 512. Later, Hasenbusch [20] examined an alternative scaling ansatz of the susceptibility, ...

... at the fixed value of C = ln 16 in the FSS analysis of the susceptibility. The parameter C effectively includes subleading-order corrections that may decay rather slowly with increasing L [20]. Setting C = 0 provided smaller values of r = −0.0406 ...

We study the logarithmic correction to the scaling of the first Lee-Yang (LY) zero in the classical $XY$ model on square lattices by using tensor renormalization group methods. In comparing the higher-order tensor renormalization group (HOTRG) and the loop-optimized tensor network renormalization (LoopTNR), we find that the entanglement filtering in LoopTNR is crucial to gaining high accuracy for the characterization of the logarithmic correction, while HOTRG still proposes empirical bounds associated with the different bond-merging algorithms. Using the LoopTNR data computed up to the system size of $L=1024$ in the $L \times L$ lattices, we estimate the logarithmic correction exponent $r = -0.0643(9)$ from the extrapolation of the finite-size-evaluated effective exponent, which is comparable to the renormalization group prediction of $r = -1/16$.

... This is worsened by logarithmic corrections to asymptotic behavior due to a marginal operator at the fixed point predicted by perturbation renormalization-group (RG) calculations [3,19,[33][34][35]. Even subleading logarithmic corrections were found to contribute appreciably [22,26]. The KT phase transition is found in a lot of systems [11,. ...

... As such, similar behavior ought to appear above the critical point. Consequently, the relaxation time for the correlation length behaves as [78] τ ξ ξ z ln(ξ/ξ 0 ), (22) rather than the usual ...

... For the sake of completeness, we also present the results for t transforming according to Eq. (24) for the anomalous relaxation time Eq. (22). In this case, upon assuming again R(b) = Rb r , Eqs. ...

We propose a series of scaling theories for Kosterlitz-Thouless (KT) phase transitions on the basis of the hallmark exponential growth of their correlation length. Finite-size scaling, finite-entanglement scaling, short-time critical dynamics, and finite-time scaling, as well as some of their interplay, are considered. Relaxation times of both a normal power-law and an anomalous power-law with a logarithmic factor are studied. Finite-size and finite-entanglement scaling forms somehow similar to but different from a frequently employed ansatz are presented. The Kibble-Zurek scaling of topological defect density for a linear driving across the KT transition point is investigated in detail. An implicit equation for a rate exponent in the theory is derived, and the exponent varies with the distance from the critical point and the driving rate consistent with relevant experiments. To verify the theories, we utilize the KT phase transition of a one-dimensional Bose-Hubbard model. The infinite time-evolving-block-decimation algorithm is employed to solve numerically the model for finite bond dimensions. Both a correlation length and an entanglement entropy in imaginary time and only the entanglement entropy in real-time driving are computed. Both the short-time critical dynamics in imaginary time and the finite-time scaling in real-time driving, both including the finite bond dimension, for the measured quantities are found to describe the numerical results quite well via surface collapses. The critical point is also estimated and confirmed to be 0.302(1) at the infinite bond dimension on the basis of the scaling theories.

... These include the degree to the transition is well defined [7,8] as well as its relation to other critical phenomena like glassiness [9]. Somewhat more concretely, one issue is its relation to the physics of phase ordering kinetics, i.e. the extent to which an underlying, but under a given dynamics inaccessible, ordered state and its concomitant topological defects plays a fundamental role [10][11][12][13][14]. ...

... ξ 2nd (L) = 1 2 sin(π/L) S(q s )/S(q s + 2π Lx ) − 1. (10) To leading order, ξ 2nd /L becomes independent of system size in the QLRO phase and takes a universal value in the thermodynamic limit at the KT transition [11]. Turning to the chiral phase transition, we measure analogous quantities to Eqs. 8, 10, and 9, with the chirality (Eq. ...

... Anomalous critical exponent, Binder cumulant, and correlation length for near the transition. (a) Spin observables, with horizontal lines indicating universal values for the KT universality class[10,11]. (b) Chiral observables, with horizontal lines indicating values for the Ising universality class on the triangular lattice[12]. ...

We study the dynamics of hardcore spin models on the square and triangular lattice, obtained by analogy to hard spheres, where the translational degrees of freedom of the spheres are replaced by orientational degrees of freedom of spins on a lattice and the packing fraction as a control parameter is replaced by an exclusion angle. In equilibrium, both models exhibit a Kosterlitz-Thouless transition at an exclusion angle $\Delta_{\rm KT}$. We devise compression protocols for hardcore spins and find that any protocol that changes the exclusion angle nonadiabatically, if endowed with only local dynamics, fails to compress random initial states beyond a jamming angle $\Delta_{\rm J}> \Delta_{\rm KT}$. This coincides with a doubly algebraic divergence of the relaxation time of compressed states towards equilibrium. We identify a remarkably simple mechanism underpinning jamming: topological defects involved in the phase ordering kinetics of the system become incompatible with the hardcore spin constraint, leading to a vanishing defect mobility as $\Delta\rightarrow\Delta_{\rm J}$.

... If the general conjecture extends to systems with global Abelian symmetries, we expect this model to have the same critical behavior as the O(2)-invariant XY lattice model. Therefore, for N c ≥ 3, 2D lattice SOðN c Þ gauge models with two scalar flavors may undergo a finite-temperature Berezinskii-Kosterlitz-Thouless (BKT) transition [12][13][14][15][16][17][18][19][20], between the high-temperature disordered phase and a low-temperature spin-wave phase characterized by quasi-long range order (QLRO) with vanishing magnetization. We recall that BKT transitions are characterized by an exponentially divergent correlation length ξ at a finite critical temperature. ...

... where ψ x are complex phase variables, jψ x j ¼ 1, associated with each site of the square lattice. This model undergoes a BKT transition at β c ¼ 1.1199ð1Þ [16,19], with a low-temperature phase that shows QLRO with vanishing magnetization. ...

... In particular, it allows us to compute the universal asymptotic relation between the ratio R ξ and the exponent η. Results for square lattices with periodic boundary conditions are reported in Refs. [19,27] (see, in particular, the formulas reported in Appendix B of Ref. [27]). The exponent η characterizes the temperature-dependent power-law decay of the two-point function in the QLRO phase, ...

We study the phase diagram and critical behavior of a two-dimensional lattice SO(Nc) gauge theory (Nc≥3) with two scalar flavors, obtained by partially gauging a maximally O(2Nc) symmetric scalar model. The model is invariant under local SO(Nc) and global O(2) transformations. We show that, for any Nc≥3, it undergoes finite-temperature Berezinskii-Kosterlitz-Thouless (BKT) transitions, associated with the global Abelian O(2) symmetry. The transition separates a high-temperature disordered phase from a low-temperature spin-wave phase where correlations decay algebraically (quasi-long range order). The critical properties at the finite-temperature BKT transition and in the low-temperature spin-wave phase are determined by means of a finite-size scaling analysis of Monte Carlo data.

... where J 0 is the coupling constant and the sum is on all the pairs of nearest-neighbor sites of a 2D square lattice, on which the variables θ i are defined. The temperature T BKT of the XY model (1) has been the subject of considerable work, and the value determined by Monte Carlo simulations is given by k B T BKT /J 0 = 0.893 ± 0.001 [18][19][20][21][22][23], with k B the Boltzmann constant. We will put, as usual, β = 1/k B T , with T the temperature. ...

... Eq. (20) is the desired relation between the couplingsJ(r) of the optimizing model and the couplings J(r) of the initial model. In solving it, one can actually look for a solution such that the couplingsJ depend only on r. ...

... In solving it, one can actually look for a solution such that the couplingsJ depend only on r. Eq. (20) has been derived for the 2D XY model, but the same calculations can be extended for different dimensions. The same structure of Eq. (20) is found for O(n) models as well. ...

We derive the self-consistent harmonic approximation for the $2D$ XY model with non-local interactions. The resulting equation for the variational couplings holds for any form of the spin-spin coupling as well as for any dimension. Our analysis is then specialized to power-law couplings decaying with the distance $r$ as $\propto 1/r^{2+\sigma}$ in order to investigate the robustness, at finite $\sigma$, of the Berezinskii-Kosterlitz-Thouless (BKT) transition, which occurs in the short-range limit $\sigma \to \infty$. We propose an ansatz for the functional form of the variational couplings and show that for any $\sigma>2$ the BKT mechanism occurs. The present investigation provides an upper bound for the lower critical threshold $\sigma^\ast=2$, above which the traditional BKT transition persists in spite of the LR couplings.

... In the XY model, the superfluid density can be calculated from the helicity modulus (the spin stiffness) in the spin representation [5] or the mean-squared winding number in the flow representation [6] which is similar to the case of the Bose-Hubbard model. In 2D systems, the superfluid density has a sudden jump from zero to a universal value at the BKT point [7] and this property has been used to numerically determine the BKT point [8][9][10][11][12][13][14]. Besides, the magnetic susceptibility is divergent at the BKT point as well as in the whole superfluid phase, which is referred as the critical region. ...

... For finite system sizes, this exponential divergency introduces logarithmic corrections around the BKT point, and dramatically increases the difficulty for high-precision determination of the BKT point by numerical means because of the need of large system sizes and sophisticate finite-size scaling (FSS) terms. Even though, recent Monte Carlo (MC) simulations can provide precise estimates for the coupling strength K BKT = 1.119 96 (6) [11,13,15], in agreement with the high-temperature expansions [16]. It is nevertheless noted that these estimates depend on assumptions about the logarithmic finite-size corrections, and different extrapolations can lead to somewhat different values of the BKT point. ...

... This type of divergence for the correlation length leads to the logarithmic correction [4,8,37,38], that brings notorious difficulties for numerical study of the BKT transition. Even though, in recent years, the estimates of K BKT have been significantly improved by extensive MC simulations [11,13,14] and by tensor network algorithms [39][40][41]. The most precise estimate of K BKT for the 2D XY model, obtained by a large-scale MC simulation with system sizes up to L = 65536, is K BKT = 1.119 96(6) [13], which slightly deviates from the other MC result K BKT = 1.119 2(1) [14]. ...

We simulate the two-dimensional XY model in the flow representation by a worm-type algorithm, up to linear system size $L=4096$, and study the geometric properties of the flow configurations. As the coupling strength $K$ increases, we observe that the system undergoes a percolation transition $K_{\rm perc}$ from a disordered phase consisting of small clusters into an ordered phase containing a giant percolating cluster. Namely, in the low-temperature phase, there exhibits a long-ranged order regarding the flow connectivity, in contrast to the qusi-long-range order associated with spin properties. Near $K_{\rm perc}$, the scaling behavior of geometric observables is well described by the standard finite-size scaling ansatz for a second-order phase transition. The estimated percolation threshold $K_{\rm perc}=1.105 \, 3(4)$ is close to but obviously smaller than the Berezinskii-Kosterlitz-Thouless (BKT) transition point $K_{\rm BKT} = 1.119 \, 3(10)$, which is determined from the magnetic susceptibility and the superfluid density. Various interesting questions arise from these unconventional observations, and their solutions would shed lights on a variety of classical and quantum systems of BKT phase transitions.

... If the general conjecture extends to systems with global Abelian symmetries, we expect this model to have the same critical behavior as the O(2)invariant XY lattice model. Therefore, for N c ≥ 3, 2D lattice SO(N c ) gauge models with two scalar flavors may undergo a finite-temperature Berezinskii-Kosterlitz-Thouless (BKT) transition [12][13][14][15][16][17][18][19][20], between the hightemperature disordered phase and a low-temperature spin-wave phase characterized by quasi-long range order (QLRO) with vanishing magnetization. We recall that BKT transitions are characterized by an exponentially divergent correlation length: we have ξ ∼ exp(c/ √ T − T c ) approaching the BKT critical temperature T c from the high-temperature phase. ...

... where ψ x are complex phase variables, |ψ x | = 1, associated with each site of the square lattice. This model undergoes a BKT transition at β c = 1.1199(1) [16,19], with a low-temperature phase that shows QLRO with vanishing magnetization. The correspondence can be justified using the arguments presented in Ref. [8]. ...

... Under this mapping, the order parameter q f g x (which has only two independent real components) is mapped onto the complex field ψ x of the XY model. Therefore, the critical behavior of the correlation function of the operator Q x , defined in Eq. (5), is expected to correspond to that of the two-point function We also report the universal asymptotic large-L curve (full line) computed in the spin-wave theory, for a system with square geometry and periodic boundary conditions [19,27]. ...

We study the phase diagram and critical behavior of a two-dimensional lattice SO($N_c$) gauge theory ($N_c \ge 3$) with two scalar flavors, obtained by partially gauging a maximally O($2N_c$) symmetric scalar model. The model is invariant under local SO($N_c$) and global O(2) transformations. We show that, for any $N_c \ge 3$, it undergoes finite-temperature Berezinskii-Kosterlitz-Thouless (BKT) transitions, associated with the global Abelian O(2) symmetry. The transition separates a high-temperature disordered phase from a low-temperature spin-wave phase where correlations decay algebraically (quasi-long range order). The critical properties at the finite-temperature BKT transition and in the low-temperature spin-wave phase are determined by means of a finite-size scaling analysis of Monte Carlo data.

... Given a model whose real-space correlation function decays as a power law (Eq. 8), the susceptibility on an L × L lattice scales as L 2−η , plus sub-leading corrections [32]. Performing a linear fit to log χ vs. log L provides an estimate of η. ...

... If the transition is of a Kosterlitz-Thouless type, then this definition of ∆ c should be consistent with that estimated via other metrics. One of these is the second moment correlation length, ξ 2nd (L), is defined as [32]: This quantity takes its name from the fact that it is the second moment with respect to the Fourier-transformed correlation function, G(k). In the paramagnet, ξ 2nd (L) is independent of system size, whereas in the critical phase, it scales linearly with system size L. Thus, the rescaled correlation length, ξ 2nd (L)/L, is independent of L up to the critical angle, up to subleading corrections at small system sizes. ...

... In the paramagnet, ξ 2nd (L) is independent of system size, whereas in the critical phase, it scales linearly with system size L. Thus, the rescaled correlation length, ξ 2nd (L)/L, is independent of L up to the critical angle, up to subleading corrections at small system sizes. In the thermodynamic limit at the KT transition, this rescaled length takes the value [32]: ...

A system of hard spheres exhibits physics that is controlled only by their density. This comes about because the interaction energy is either infinite or zero, so all allowed configurations have exactly the same energy. The low density phase is liquid, while the high density phase is crystalline, an example of "order by disorder" as it is driven purely by entropic considerations. Here we introduce and study a family of hard spin models, which we call hardcore spin models, where we replace the translational degrees of freedom of hard spheres with the orientational degrees of freedom of lattice spins. Their hardcore interaction serves analogously to divide configurations of the many spin system into allowed and disallowed sectors. We present detailed results on the square lattice in $d=2$ for a set of models with $\mathbb{Z}_n$ symmetry which generalize Potts models and their $U(1)$ limits---for ferromagnetic and antiferromagnetic senses of the interaction which we refer to as exclusion and inclusion models. As the exclusion/inclusion angles are varied, we find a Kosterlitz-Thouless phase transition between a disordered phase and an ordered phase with quasi-long ranged order, which is the form order by disorder takes in these systems. These results follow from a set of height representations, an ergodic cluster algorithm, and transfer matrix calculations.

... is just the classical XY Hamiltonian (here the sum runs over all the links of the lattice, or, equivalently, all nearest neighbours pairs). This system exhibits a BKT transition at a critical temperature T KT ≈ 0.89 [36,37]. The distribution of natural frequencies introduces quench disorder in the XY model. ...

... in its Langevin version, or, equivalently, using single-spin flip Monte Carlo dynamics [33,34,36,37,40]. However, in non-equilibrium conditions, the specific features of the dynamics might affect the resulting large scale behavior at long times. ...

We consider the two-dimensional (2D) noisy Kuramoto model of synchronization with short-range coupling and a Gaussian distribution of intrinsic frequencies, and investigate its ordering dynamics following a quench. We consider both underdamped (inertial) and over-damped dynamics, and show that the long-term properties of this intrinsically out-of-equilibrium system do not depend on the inertia of individual oscillators. The model does not exhibit any phase transition as its correlation length remains finite, scaling as the inverse of the standard deviation of the distribution of intrinsic frequencies. The quench dynamics proceeds via domain growth, with a characteristic length that initially follows the growth law of the 2D XY model, although is not given by the mean separation between defects. Topological defects are generically free, breaking the Berezinskii-Kosterlitz-Thouless scenario of the 2D XY model. Vortices perform a random walk reminiscent of the self-avoiding random walk, advected by the dynamic network of boundaries between synchronised domains; featuring long-time super-diffusion, with the anomalous exponent α = 3/2.

... is just the classical XY Hamiltonian (here the sum runs over all the links of the lattice, or, equivalently, all nearest neighbours pairs). This system exhibits a BKT transition at a critical temperature T KT ≈ 0.89 [36,37]. The distribution of natural frequencies introduces quench disorder in the XY model. ...

... in its Langevin version, or, equivalently, using single-spin flip Monte Carlo dynamics [34,33,36,37,40]. ...

We consider the two-dimensional (2D) noisy Kuramoto model of synchronization with short-range coupling and a Gaussian distribution of intrinsic frequencies, and investigate its ordering dynamics following a quench. We consider both underdamped (inertial) and over-damped dynamics, and show that the long-term properties of this intrinsically out-of-equilibrium system do not depend on the inertia of individual oscillators. The model does not exhibit any phase transition as its correlation length remains finite, scaling as the inverse of the standard deviation of the distribution of intrinsic frequencies. The quench dynamics proceeds via domain growth, with a characteristic length that initially follows the growth law of the 2D XY model, although is not given by the mean separation between defects. Topological defects are generically free, breaking the Berezinskii-Kosterlitz-Thouless scenario of the 2D XY model. Vortices perform a random walk reminiscent of the self-avoiding random walk, advected by the dynamic network of boundaries between synchronised domains; featuring long-time super-diffusion, with the anomalous exponent $\alpha=3/2$.

... In both models, ECMC lowers the scaling of relaxation times with respect to local reversible MCMC. Although more efficient algorithms are available [82][83][84] for these particular models, the comparison of ECMC with reversible MCMC illustrates what can be achieved through non-reversible local Markov chains in real-world applications. ...

... Although it possesses no long-range spin order at finite temperatures, the 2D XY model famously undergoes a phase transition [82,85] between a low-temperature phase rigorously described by spin waves with bound vortex-antivortex pairs [77,78], and a high-temperature phase where these topological excitations are free. ...

This review treats the mathematical and algorithmic foundations of non-reversible Markov chains in the context of event-chain Monte Carlo (ECMC), a continuous-time lifted Markov chain that employs the factorized Metropolis algorithm. It analyzes a number of model applications and then reviews the formulation as well as the performance of ECMC in key models in statistical physics. Finally, the review reports on an ongoing initiative to apply ECMC to the sampling problem in molecular simulation, i.e., to real-world models of peptides, proteins, and polymers in aqueous solution.

... A sharp change in the behavior of the quasi-long-range order phase and the disordered phase occurs at the critical temperature (T /J ) BKT . This critical temperature is previously estimated using finite-size scaling methods of large-scale numerical Monte Carlo data as (T /J ) BKT ≈ 0.8929 [54][55][56] or (T /J ) BKT ≈ 0.8935 [57]. While this phase transition has been explored in both supervised [8,58] and unsupervised [6,7,[14][15][16] machine learning methods, the interpretability of the topological aspects of spin configurations is lacking. ...

... The transition (yellow points) in the proportion of diagrams belonging to each cluster emerges at T /J 0.89, which is in line with the well-known phase transition point (T /J ) BKT in Refs. [54][55][56][57]. ...

The study of phase transitions using data-driven approaches is challenging, especially when little prior knowledge of the system is available. Topological data analysis is an emerging framework for characterizing the shape of data and has recently achieved success in detecting structural transitions in material science, such as the glass-liquid transition. However, data obtained from physical states may not have explicit shapes as structural materials. We thus propose a general framework, termed "topological persistence machine," to construct the shape of data from correlations in states, so that we can subsequently decipher phase transitions via qualitative changes in the shape. Our framework enables an effective and unified approach in phase transition analysis. We demonstrate the efficacy of the approach in detecting the Berezinskii-Kosterlitz-Thouless phase transition in the classical XY model and quantum phase transitions in the transverse Ising and Bose-Hubbard models. Interestingly, while these phase transitions have proven to be notoriously difficult to analyze using traditional methods, they can be characterized through our framework without requiring prior knowledge of the phases. Our approach is thus expected to be widely applicable and will provide practical insights for exploring the phases of experimental physical systems.

... In both models, ECMC lowers the relaxation rates with respect to local reversible MCMC. Although more efficient algorithms are available [31,32,92] for these particular models, the comparison of ECMC with reversible MCMC may illustrate what can be achieved through non-reversible MCMC in real-world applications where, as discussed in Section 1, customized a priori move sets remain unavailable. ...

... Although it possesses no long-range spin order at finite temperatures, the two-dimensional XY model famously undergoes a phase transition [47,31] between a low-temperature phase rigorously described by spin waves with bound vortex-antivortex pairs [89,26], and a high-temperature phase where these topological excitations are free. ...

This review treats the mathematical and algorithmic foundations of non-reversible Markov chains in the context of event-chain Monte Carlo (ECMC), a continuous-time lifted Markov chain that employs the factorized Metropolis algorithm. It analyzes a number of model applications, and then reviews the formulation as well as the performance of ECMC in key models in statistical physics. Finally, the review reports on an ongoing initiative to apply the method to the sampling problem in molecular simulation, that is, to real-world models of peptides, proteins, and polymers in aqueous solution.

... Both assumptions (i) and (ii) are satisfied for the Villain model, and the computation of the critical temperature from the BKT flow equations with J eff = J and μ eff v = μ v = Jπ 2 /2 perfectly reproduces high-precision MC results [49]. In the XY model case with μ = 0 one can efficiently compute the renormalization of the effective superfluid stiffness due to spin waves via functional RG, mean-field or low-temperature expansion, yielding estimates for the BKT critical temperature [32,50,51] within 5% of the numerically exact MC value [52][53][54]. Finally, the computation of the effective superfluid stiffness for 2D Fermi gases via Landau's quasiparticleexcitation formula produces a consistent picture for the dependence of the BKT temperature on the scattering length [30]. ...

... [55,56] to our MC data, we obtain a reliable estimate for the critical temperature T BKT (black points in Fig. 4), which TABLE I. Numerical coefficients for the effective vortex core energy using Eq. (22), based on the analysis of the MC data described in Fig. 3. (2) nicely reproduces the high-precision results [54,56] at μ = 0 (black star). The theoretical estimates for T BKT obtained by inserting the effective couplings ...

The Berezinskii-Kosterlitz-Thouless (BKT) mechanism describes universal vortex unbinding in many two-dimensional systems, including the paradigmatic XY model. However, most of these systems present a complex interplay between excitations at different length scales that complicates theoretical calculations of nonuniversal thermodynamic quantities. These difficulties may be overcome by suitably modifying the initial conditions of the BKT flow equations to account for noncritical fluctuations at small length scales. In this work, we perform a systematic study of the validity and limits of this two-step approach by constructing optimised initial conditions for the BKT flow. We find that the two-step approach can accurately reproduce the results of Monte Carlo simulations of the traditional XY model. To systematically study the interplay between vortices and spin-wave excitations, we introduce a modified XY model with increased vortex fugacity. We present large-scale Monte Carlo simulations of the spin stiffness and vortex density for this modified XY model and show that even at large vortex fugacity, vortex unbinding is accurately described by the nonperturbative functional renormalization group.

... III A 1 is not excluded. It is well known that corrections to BKT scaling in the two-dimensional XY model are very large [65]. In the eight-flavor system I have concentrated on fitting the leading terms only. ...

The existence of a strongly coupled ultraviolet fixed point in four-dimensional lattice models as they cross into the conformal window has long been hypothesized. The SU(3) gauge system with eight fundamental fermions is a good candidate to study this phenomenon as it is expected to be very close to the opening of the conformal window. I study the system using staggered lattice fermions in the chiral limit. My numerical simulations employ improved lattice actions that include heavy Pauli-Villars (PV) type bosons. This modification does not affect the infrared dynamics but greatly reduces the ultraviolet fluctuations, thus allowing the study of stronger renormalized couplings than previously possible. I consider two different PV actions and find that both show an apparent continuous phase transition in the eight-flavor system. I investigate the critical behavior using finite size scaling of the renormalized gradient flow coupling. The finite size scaling curve-collapse analysis predicts a first-order phase transition consistent with discontinuity exponent ν=1/4 in the system without PV bosons. The scaling analysis with the PV boson actions is not consistent with a first-order phase transition. The numerical data are well described by “walking scaling” corresponding to a renormalization group β function that just touches zero, β(g2)∼(g2−g⋆2)2, though second-order scaling cannot be excluded. Walking scaling could imply that the eight-flavor system is the opening of the conformal window, an exciting possibility that could be related to ’t Hooft anomaly cancellation of the system.

... The role of topological vortices in phase transitions in twodimensional (2D) superfluids was first explored by Kosterlitz and Thouless (KT) [1,2] and Berezhinskii [3]. A real-space renormalization group theory by Kosterlitz [4] was able to characterize many of the properties of the phase transition below the critical temperature T KT , and simulations of 2D XY spin lattices were able to verify many aspects of the theory [5,6]. However, the recursion relations of the theory blow up at temperatures above T KT , making it difficult to explore what happens in this region where the vortex density becomes large, approaching an average density of 1/3 per lattice site as T → ∞. ...

Vortex fluctuations above and below the critical Kosterlitz-Thouless (KT) transition temperature are characterized using simulations of the 2D XY model. The net winding number of vortices at a given temperature in a circle of radius $R$ is computed as a function of $R$. The average squared winding number is found to vary linearly with the perimeter of the circle at all temperatures above and below $T_{KT}$, and the slope with $R$ displays a sharp peak near the specific heat peak, decreasing then to a value at infinite temperature that is in agreement with an early theory by Dhar. We have also computed the vortex-vortex distribution functions, finding an asymptotic power-law variation in the vortex separation distance at all temperatures. In conjunction with a Coulomb-gas sum rule on the perimeter fluctuations, these can be used to successfully model the start of the perimeter-slope peak in the region below $T_{KT}$.

... In the case of the XY model, Fig. 1, for all values of the initial bond dimension D, we see a peak in C V around T ≈ 1 which is larger than the BKT transition temperature of T XY c ≈ 0.89 [43][44][45][46]. While the peak of C V does not match the transition temperature (as was previously known), it still happens to correlate well with the phase structure of the theory (which we can see from other observables below). ...

We study the effects of discretization on the U(1) symmetric XY model in two dimensions using the Higher Order Tensor Renormalization Group (HOTRG) approach. Regarding the $Z_N$ symmetric clock models as specific discretizations of the XY model, we compare those discretizations to ones from truncations of the tensor network formulation of the XY model based on a character expansion, and focus on the differences in their phase structure at low temperatures. We also divide the tensor network formulations into core and interaction tensors and show that the core tensor has the dominant influence on the phase structure. Lastly, we examine a perturbed form of the XY model that continuously interpolates between the XY and clock models. We examine the behavior of the additional phase transition caused by the perturbation as the magnitude of perturbation is taken to zero. We find that this additional transition has a non-zero critical temperature as the perturbation vanishes, suggesting that even small perturbations can have a significant effect on the phase structure of the theory.

... One of the major advantages of the stationary bootstrap is its ability to access the full sampling distribution and applicability to a large class of quantities, such as the specific heat, the Binder cumulant, normalized spatial correlations, and even the latent heat at a first-order transition. More complicated quantities involving multiple variables, such as the second-moment correlation length [22,[38][39][40] and the bulk and shear moduli [41], are within the scope of the method as well [19]. Another potential use of the method is to reliably estimate the peak locations of measured functions such as the structure factor. ...

In Markov-chain Monte Carlo simulations, estimating statistical errors or confidence intervals of numerically obtained values is an essential task. In this paper, we review several methods for error estimation, such as simple empirical estimation with multiple independent runs, the blocking method, and the stationary bootstrap method. We then study their performance when applied to an actual Monte-Carlo time series. We find that the stationary bootstrap method gives a reasonable and stable estimation for any quantity using only one single time series. In contrast, the simple estimation with few independent runs can be demonstratively erroneous. We further discuss the potential use of the stationary bootstrap method in numerical simulations.

... As the temperature increases, the model undergoes a BKT transition driven by the unbinding of these vortex-antivortex pairs, so that at high temperatures lone (anti)vortices proliferate. The critical temperature is approximately T = 0.8929 [41] and the critical exponent of correlation length is ν = The average H1 persistence images in birthpersistence coordinates at different temperatures for the XY model with L = 30. ...

We use persistent homology and persistence images as an observable of three different variants of the two-dimensional XY model in order to identify and study their phase transitions. We examine models with the classical XY action, a topological lattice action, and an action with an additional nematic term. In particular, we introduce a new way of computing the persistent homology of lattice spin model configurations and, by considering the fluctuations in the output of logistic regression and k-nearest neighbours models trained on persistence images, we develop a methodology to extract estimates of the critical temperature and the critical exponent of the correlation length. We put particular emphasis on finite-size scaling behaviour and producing estimates with quantifiable error. For each model we successfully identify its phase transition(s) and are able to get an accurate determination of the critical temperatures and critical exponents of the correlation length.

... Intrinsic frequencies drive the XY model out-of-equilibrium as they account for a constant energy injection at the level of each spin. In equilibrium (σ = 0), the system exhibits a BKT transition at T KT ≈ 0.89 [2,3,50,51], characterized by a change in behavior of the correlation function C(r) = S i · S j |ri−rj |=r (the brackets denote the average over different realizations of noise and the intrinsic frequencies). Below T KT vortex-antivortex pairs bind, giving rise to critical correlations. ...

We consider a nonequilibrium extension of the 2D XY model, equivalent to the noisy Kuramoto model of synchronization with short-range coupling, where rotors sitting on a square lattice are self-driven by random intrinsic frequencies. We study the static and dynamic properties of topological defects (vortices) and establish how self-spinning affects the Berezenskii-Kosterlitz-Thouless phase transition scenario. The nonequilibrium drive breaks the quasi-long-range ordered phase of the 2D XY model into a mosaic of ordered domains of controllable size and results in self-propelled vortices that generically unbind at any temperature, featuring superdiffusion ⟨r2(t)⟩∼t3/2 with a Gaussian distribution of displacements. Our work provides a simple framework to investigate topological defects in nonequilibrium matter and sheds new light on the problem of synchronization of locally coupled oscillators.

... Theoretical results obtained by studying the x-y model, typically by computer simulations, are utilized both to ascertain whether a particular physical system experimentally investigated, believed to be in the same universality class, conforms with the KT paradigm, as well as to predict the behavior of systems yet unexplored [11][12][13][14]. Decades of computer simulation studies of the 2D x-y model, carried out on square lattices of size as large as L = 2 16 [15], have yielded very precise estimates of the superfluid transition temperature T c and of the critical exponents associated with the transition [15][16][17][18][19][20][21][22][23]. ...

We present results of large-scale Monte Carlo simulations of the 2D classical x-y model on the square lattice. We obtain high accuracy results for the superfluid fraction and for the specific heat as a function of temperature, for systems of linear size L up to 4096. Our estimate for the superfluid transition temperature is consistent with those furnished in all previous studies. The specific heat displays a well-defined peak, whose shape and position are independent of the size of the lattice for L > 256, within the statistical uncertainties of our calculations. The implications of these results on the interpretation of experiments on adsorbed thin films of He-4 are discussed.

... Theoretical results obtained by studying the x-y model, typically by computer simulations, are utilized both to ascertain whether a particular physical system experimentally investigated, believed to be in the same universality class, conforms with the KT paradigm, as well as to predict the behavior of systems yet unexplored [11][12][13][14]. Decades of computer simulation studies of the 2D x-y model, carried out on square lattices of size as large as L = 2 16 [15], have yielded very precise estimates of the superfluid transition temperature T c and of the critical exponents associated with the transition [15][16][17][18][19][20][21][22][23]. ...

We present results of large-scale Monte Carlo simulations of the 2D classical x-y model on the square lattice. We obtain high accuracy results for the superfluid fraction and for the specific heat as a function of temperature, for systems of size L×L with L up to 212. Our estimate for the superfluid transition temperature is consistent with those furnished in all previous studies. The specific heat displays a well-defined peak, whose shape and position are independent of the size of the lattice for L>28, within the statistical uncertainties of our calculations. The implications of these results on the interpretation of experiments on adsorbed thin films of 4He are discussed.

... Monte Carlo(1979) [35] 0.89 Monte Carlo(2005) [36] 0.8929 Monte Carlo(2012) [37] 0.89289 Monte Carlo(2013) [38] 0.8935 Series expansion(2009) [39] 0.89286 HOTRG(2014) [40] 0.8921 VUMPS(2019) [41] 0.8930 HOTRG(2020) [42] 0.89290 (5) Conclusion.-We analyzed the eigenspectrum of the renormalized tensors for the classical 2D XY model at fi-nite renormalization steps, in terms of finite-size scaling of CFT. The BKT transition is described in terms of two marginal couplings y K (spin wave stiffness) and y V (vortex fugacity), and the transition point can be identified with y K = y V where an SU(2) symmetry emerges. ...

Berezinskii-Kosterlitz-Thouless transition of the classical XY model is re-investigated, combining the Tensor Network Renormalization (TNR) and the Level Spectroscopy method based on the finite-size scaling of the Conformal Field Theory. By systematically analyzing the spectrum of the transfer matrix of the systems of various moderate sizes which can be accurately handled with a finite bond dimension, we determine the critical point removing the logarithmic corrections. This improves the accuracy by an order of magnitude over previous studies including those utilizing TNR. Our analysis also gives a visualization of the celebrated Kosterlitz Renormalization Group flow based on the numerical data.

... Since the inter-layer Josephson couplings are relevant at low temperatures, the relative phase becomes an important degree of freedom to alter the bulk properties of the uncoupled systems significantly, and even new phases may emerge in the low temperature phases [6][7][8][9][10]. Due to the lack of sharp thermodynamic signatures for the BKT transition, it remains a great challenge to separate the BKT transition from its adjacent phases, which obstructs the clarification of those novel phases and their phase transitions [11][12][13][14][15][16]. ...

We study a bilayer system of coupled two-dimensional XY models by using tensor-network methods, showing that the inter-layer coupling can significantly alter the uncoupled topological Berezinskii-Kosterlizt-Thouless (BKT) transition. In the tensor-network representation, the partition function is mapped into a product of one-dimensional quantum transfer operator, whose eigen-equation can be accurately solved by an algorithm of matrix product states. The entanglement entropy of this one-dimensional quantum correspondence exhibits singularity, which can be used to determine various phase transitions. In the low temperature phase, an inter-layer Ising long-range order is developed,accompanying with vortex-antivortex bindings in both intra-layers and inter-layers. For two identical coupled layers, the Ising transition coincides with the BKT transition at a multi-critical point. For two inequivalent coupled layers, however, there emerges an intermediate quasi-long-range ordered phase, where vortex-antivortex bindings occur only in the layer with the larger intra-layer coupling but half-vortex pairs associated with topological strings emerge in the other layer. These results are supported by the correlation functions of the XY spins and nematic spins, and can be used to understand the novel properties in bilayer systems of superfluids, superconductors and quantum Hall systems.

... The leading behavior of g(L) is 1/ [2 ln(L) + C] from Weber and Minnhagen [68]. We can also include higher order corrections and take g(L) = 1/ [2 ln(L) + C + ln(C/2 + ln(L))] + A/ ln 2 (L) [69][70][71][72][73] to further decrease the error. The goal of this method is to find the best data collapse of the rescaled energy gap ∆E s = L∆E [1 + g (L)] near the phase transition point in the parameter space (Y c , b, C, A). ...

The O(2) model in Euclidean space-time is the zero-gauge-coupling limit of the compact scalar quantum electrodynamics. We obtain a dual representation of it called the charge representation. We study the quantum phase transition in the charge representation with a truncation to "spin $S$", where the quantum numbers have an absolute value less or equal to $S$. The charge representation preserves the gapless-to-gapped phase transition even for the smallest spin truncation $S = 1$. The phase transition for $S = 1$ is an infinite-order Gaussian transition with the same critical exponents $\delta$ and $\eta$ as the Berezinskii-Kosterlitz-Thouless (BKT) transition, while there are true BKT transitions for $S \ge 2$. The essential singularity in the correlation length for $S = 1$ is different from that for $S \ge 2$. The exponential convergence of the phase transition point is studied in both Lagrangian and Hamiltonian formulations. We discuss the effects of replacing the truncated $\hat{U}^{\pm} = \exp(\pm i \hat{\theta})$ operators by the spin ladder operators $\hat{S}^{\pm}$ in the Hamiltonian. The marginal operators vanish at the Gaussian transition point for $S = 1$, which allows us to extract the $\eta$ exponent with high accuracy.

... As the temperature T changes from low to high, phase transitions will take place for the 3D classical O(3) and the 5-state ferromagnetic Potts models. The critical points T c of the 3D classical O(3) and the 5-state ferromagnetic Potts models as well as the g c of the 3D plaquette model described above have been calculated with high accuracy in the literature [83][84][85][86][87]. The construction of the configurations used for the NN testing -As pointed out in the main text, bulk quantities satisfying certain conditions are suitable for employing to construct the configurations used for the NN testing. ...

A universal supervised neural network (NN) relevant to compute the associated criticalities of real experiments studying phase transitions is constructed. The validity of the built NN is examined by applying it to calculate the criticalities of several three-dimensional (3D) models on the cubic lattice, including the classical $O(3)$ model, the 5-state ferromagnetic Potts model, and a dimerized quantum antiferromagnetic Heisenberg model. Particularly, although the considered NN is only trained one time on a one-dimensional (1D) lattice with 120 sites, yet it has successfully determined the related critical points of the studied 3D systems. Moreover, real configurations of states are not used in the testing stage. Instead, the employed configurations for the prediction are constructed on a 1D lattice of 120 sites and are based on the bulk quantities or the microscopic states of the considered models. As a result, our calculations are ultimately efficient in computation and the applications of the built NN is extremely broaden. Considering the fact that the investigated systems vary dramatically from each other, it is amazing that the combination of these two strategies in the training and the testing stages lead to a highly universal supervised neural network for learning phases and criticalities of 3D models. Based on the outcomes presented in this study, it is favorably probable that much simpler but yet elegant machine learning techniques can be constructed for fields of many-body systems other than the critical phenomena.

... However, in certain cases the behavior is more complex, due to the appearance of logarithmic terms [40]. They may be induced by the presence of marginal RG perturbations, as happening in BKT transitions in U(1)-symmetric systems [50,51,206,311,312], or by resonances between the RG eigenvalues, as it occurs in transitions belonging to the 2d Ising universality class [40, 119] 20 or to the 3d O(N )-vector universality class in the large-N limit [310,313,314]. We also mention that quantum particle systems at fixed chemical potential may show further peculiar features when an infinite number of level crossings occurs as the system size increases. ...

The many-body physics at quantum phase transitions is dominated by a subtle interplay between quantum and thermal fluctuations, emerging in the low-temperature limit. In this review, we first give a pedagogical introduction to the equilibrium behavior of systems in that context, whose scaling framework is essentially developed by exploiting the quantum-to-classical mapping and the renormalization-group theory of critical phenomena at continuous phase transitions. Then we specialize to protocols entailing the out-of-equilibrium quantum dynamics, such as instantaneous quenches and slow passages across quantum transitions. These are mostly discussed within dynamic scaling frameworks, obtained by appropriately extending the equilibrium scaling laws. We review phenomena at first-order quantum transitions as well, whose peculiar scaling behaviors are characterized by an extreme sensitivity to the boundary conditions, giving rise to exponentials or power laws for the same bulk system. In the last part, we cover aspects related to the effects of dissipative interactions with an environment, through suitable generalizations of the dynamic scaling at quantum transitions. The presentation is limited to issues related to, and controlled by, the quantum transition developed by closed many-body systems, treating the dissipation as a perturbation of the critical regimes, as for the temperature at the zero-temperature quantum transition. We focus on the physical conditions giving rise to a nontrivial interplay between critical modes and various dissipative mechanisms, generally realized when the involved mechanism excites only the low-energy modes of the quantum transitions.

... By means of Monte Carlo canonical simulations the critical temperature of the transition is fitted through the divergence of the spatial correlation length ξ ∼ exp {b[(T − T c )/T c ] −1/2 } in the spin-spin correlation function s i · s j ∼ exp (−a|i − j|/ξ), where a and b are constants. Even though there has been some controversy about the extent of the critical region, as was discussed in a thorough analysis in Ref. [33] confirmed in Ref. [34], the transition temperature is fixed at k B T c /J = 0.8929, in excellent agreement with early findings of k B T c /J 0.894 reported in [35][36][37]. While we estimate a higher transition temperature with respect to this value, it is worth mentioning that results worked out by means of nonperturbative Renormalization Group in [38] gave 0.9 < k B T c /J < 1 for the critical temperature, and by means of functional renormalization in [39] the authors found 0.91 < k B T c /J < 1.02. ...

Phase transitions do not necessarily correspond to a symmetry breaking phenomenon. This is the case of the Kosterlitz–Thouless (KT) phase transition in a two-dimensional classical XY model, a typical example of a transition stemming from a deeper phenomenon than a symmetry-breaking. Actually, the KT transition is a paradigmatic example of the successful application of topological concepts to the study of phase transition phenomena in the absence of an order parameter. Topology conceptually enters through the meaning of defects in real space. In the present work, the same kind of KT phase transition in a two-dimensional classical XY model is tackled by resorting again to a topological viewpoint, however focussed on the energy level sets in phase space rather than on topological defects in real space. Also from this point of view, the origin of the KT transition can be attributed to a topological phenomenon. In fact, the transition is detected through peculiar geometrical changes of the energy level sets which, after a theorem in differential topology, are direct probes of topological changes of these level sets.

... At each site of the square lattice spins take values in S 1 and are governed by H XY = − i,j cos (θ i − θ j ). There is a well-known Kosterlitz-Thouless phase transition at T XY ≈ 0.892 (see [22,23], among others). This is an infinite-order phase transition where at low temperatures there are bound vortex-antivortex pairs while at high temperatures free vortices proliferate and spins are randomly oriented. ...

We apply persistent homology to the task of discovering and characterizing phase transitions, using lattice spin models from statistical physics for working examples. Persistence images provide a useful representation of the homological data for conducting statistical tasks. To identify the phase transitions, a simple logistic regression on these images is sufficient for the models we consider, and interpretable order parameters are then read from the weights of the regression. Magnetization, frustration and vortex-antivortex structure are identified as relevant features for characterizing phase transitions.

... There is a well-known KT phase transition at T XY ≈ 0.892 (see [38,39], among others). This is an infinite-order phase transition where at low temperatures there are bound vortex-antivortex pairs while at high temperatures free vortices proliferate and spins are randomly oriented. ...

We apply modern methods in computational topology to the task of discovering and characterizing phase transitions. As illustrations, we apply our method to four two-dimensional lattice spin models: the Ising, square ice, XY, and fully-frustrated XY models. In particular, we use persistent homology, which computes the births and deaths of individual topological features as a coarse-graining scale or sublevel threshold is increased, to summarize multiscale and high-point correlations in a spin configuration. We employ vector representations of this information called persistence images to formulate and perform the statistical task of distinguishing phases. For the models we consider, a simple logistic regression on these images is sufficient to identify the phase transition. Interpretable order parameters are then read from the weights of the regression. This method suffices to identify magnetization, frustration, and vortex-antivortex structure as relevant features for phase transitions in our models. We also define "persistence" critical exponents and study how they are related to those critical exponents usually considered.

Recently, a colossal magnetoresistance (CMR) was observed in EuCd2P2, a compound that does not fit the conventional mixed-valence paradigm. Instead, experimental evidence points to a resistance driven by strong magnetic fluctuations within the two-dimensional ferromagnetic planes of the layered antiferromagnetic structure. While the experimental results have not yet been fully understood, a recent theory relates the CMR to a topological vortex-antivortex unbinding, i.e., Berezinskii-Kosterlitz-Thouless (BKT), phase transition. Motivated by these observations, in this work we explore the magnetic phases hosted by a microscopic classical magnetic model for EuCd2P2, which easily generalizes to other Eu A-type antiferromagnetic compounds. Using Monte Carlo techniques to probe the specific heat and the helicity modulus, we show that our model can exhibit a vortex-antivortex unbinding phase transition. We find that this phase transition displays the same sensitivity to in-plane magnetization, interlayer coupling, and easy-plane anisotropy that is observed experimentally in the CMR signal, providing qualitative numerical evidence that the effect is related to a magnetic BKT transition.

We have studied the critical properties of the three-dimensional random anisotropy Heisenberg model by means of numerical simulations using the Parallel Tempering method. We have simulated the model with two different disorder distributions, cubic and isotropic ones, with two different anisotropy strengths for each disorder class. For the case of the anisotropic disorder, we have found evidence of universality by finding critical exponents and universal dimensionless ratios independent of the strength of the disorder. In the case of isotropic disorder distribution the situation is very involved: we have found two phase transitions in the magnetization channel which are merging for larger lattices remaining a zero magnetization low-temperature phase. Studying this region using a spin-glass order parameter we have found evidence for a spin-glass phase transition. We have estimated effective critical exponents for the spin-glass phase transition for the different values of the strength of the isotropic disorder, discussing the crossover regime.

We use persistent homology and persistence images as an observable of three variants of the two-dimensional XY model to identify and study their phase transitions. We examine models with the classical XY action, a topological lattice action, and an action with an additional nematic term. In particular, we introduce a way of computing the persistent homology of lattice spin model configurations and, by considering the fluctuations in the output of logistic regression and k-nearest neighbor models trained on persistence images, we develop a methodology to extract estimates of the critical temperature and the critical exponent of the correlation length. We put particular emphasis on finite-size scaling behavior and producing estimates with quantifiable error. For each model we successfully identify its phase transition(s) and are able to get an accurate determination of the critical temperatures and critical exponents of the correlation length.

Using a supervised neural network (NN) trained once on a one-dimensional lattice of 200 sites, we calculate the Berezinskii–Kosterlitz–Thouless phase transitions of the two-dimensional (2D) classical XY and the 2D generalized classical XY models. In particular, both the bulk quantities Binder ratios and the spin states of the studied systems are employed to construct the needed configurations for the NN prediction. By applying semiempirical finite-size scaling to the relevant data, the critical points obtained by the NN approach agree well with the known results established in the literature. This implies that for each of the considered models, the determination of its various phases requires only a little information. The outcomes presented here demonstrate convincingly that the employed universal NN is not only valid for the symmetry breaking related phase transitions, but also works for calculating the critical points of the phase transitions associated with topology. The efficiency of the used NN in the computation is examined by carrying out several detailed benchmark calculations.

A universal supervised neural network (NN) relevant to compute the associated critical points of real experiments studying phase transitions is constructed. The validity of the built NN is examined by applying it to calculate the critical points of several three-dimensional (3D) and two-dimensional (2D) models, including the 3D classical O(3) model, the 3D 5-state ferromagnetic Potts model, a 3D dimerized quantum antiferromagnetic Heisenberg model, and the 2D XY model. Particularly, although the considered NN is only trained once on a one-dimensional (1D) lattice with 120 sites, it has successfully determined the related critical points of the studied systems. Moreover, real configurations of states are not used in the testing stage. Instead, the employed configurations for the NN prediction are constructed on a 1D lattice of 120 sites and are based on the bulk quantities or the spin states of the considered models. As a result, our calculations are ultimately efficient in computation, and the application of the built NN in studying the phase transitions of physical systems is extremely broad. Considering the fact that the investigated systems vary dramatically from each other, it is amazing that the combination of these two strategies in the training and the testing stages leads to a highly universal supervised neural network for learning phases of 3D and 2D models. Based on the outcomes presented in this study, it is favorably probable that much simpler but elegant machine learning techniques can be constructed for fields of many-body systems other than the critical phenomena.

Berezinskii-Kosterlitz-Thouless transition of the classical XY model is reinvestigated, combining the tensor network renormalization (TNR) and the level spectroscopy method based on the finite-size scaling of the conformal field theory. By systematically analyzing the spectrum of the transfer matrix of the systems of various moderate sizes, which can be accurately handled with a finite bond dimension, we determine the critical point removing the logarithmic corrections. This improves the accuracy by an order of magnitude over previous studies including those utilizing TNR. Our analysis also gives a visualization of the celebrated Kosterlitz renormalization group flow based on the numerical data.

We apply modern methods in computational topology to the task of discovering and characterizing phase transitions. As illustrations, we apply our method to four two-dimensional lattice spin models: the Ising, square ice, XY, and fully frustrated XY models. In particular, we use persistent homology, which computes the births and deaths of individual topological features as a coarse-graining scale or sublevel threshold is increased, to summarize multiscale and high-point correlations in a spin configuration. We employ vector representations of this information called persistence images to formulate and perform the statistical task of distinguishing phases. For the models we consider, a simple logistic regression on these images is sufficient to identify the phase transition. Interpretable order parameters are then read from the weights of the regression. This method suffices to identify magnetization, frustration, and vortex-antivortex structure as relevant features for phase transitions in our models. We also define “persistence” critical exponents and study how they are related to those critical exponents usually considered.

The many-body physics at quantum phase transitions shows a subtle interplay between quantum and thermal fluctuations, emerging in the low-temperature limit. In this review, we first give a pedagogical introduction to the equilibrium behavior of systems in that context, whose scaling framework is essentially developed by exploiting the quantum-to-classical mapping and the renormalization-group theory of critical phenomena at continuous phase transitions. Then we specialize to protocols entailing the out-of-equilibrium quantum dynamics, such as instantaneous quenches and slow passages across quantum transitions. These are mostly discussed within dynamic scaling frameworks, obtained by appropriately extending the equilibrium scaling laws. We review phenomena at first-order quantum transitions as well, whose peculiar scaling behaviors are characterized by an extreme sensitivity to the boundary conditions, giving rise to exponentials or power laws for the same bulk system. In the last part, we cover aspects related to the effects of dissipative interactions with an environment, through suitable generalizations of the dynamic scaling at quantum transitions. The presentation is limited to issues related to, and controlled by, the quantum transition developed by closed many-body systems, treating the dissipation as a perturbation of the critical regimes, as for the temperature at the zero-temperature quantum transition. We focus on the physical conditions giving rise to a nontrivial interplay between critical modes and various dissipative mechanisms, generally realized when the involved mechanism excites only the low-energy modes of the quantum transitions.

The O(2) model in Euclidean space-time is the zero-gauge-coupling limit of the compact scalar quantum electrodynamics. We obtain a dual representation of it called the charge representation. We study the quantum phase transition in the charge representation with a truncation to “spin S,” where the quantum numbers have an absolute value less than or equal to S. The charge representation preserves the gapless-to-gapped phase transition even for the smallest spin truncation S=1. The phase transition for S=1 is an infinite-order Gaussian transition with the same critical exponents δ and η as the Berezinskii-Kosterlitz-Thouless (BKT) transition, while there are true BKT transitions for S≥2. The essential singularity in the correlation length for S=1 is different from that for S≥2. The exponential convergence of the phase-transition point is studied in both Lagrangian and Hamiltonian formulations. We discuss the effects of replacing the truncated Û±=exp(±iθ̂) operators by the spin ladder operators Ŝ± in the Hamiltonian. The marginal operators vanish at the Gaussian transition point for S=1, which allows us to extract the η exponent with high accuracy.

We simulate the two-dimensional XY model in the flow representation by a worm-type algorithm, up to linear system size L=4096, and study the geometric properties of the flow configurations. As the coupling strength K increases, we observe that the system undergoes a percolation transition Kperc from a disordered phase consisting of small clusters into an ordered phase containing a giant percolating cluster. Namely, in the low-temperature phase, there exhibits a long-ranged order regarding the flow connectivity, in contrast to the quasi-long-range order associated with spin properties. Near Kperc, the scaling behavior of geometric observables is well described by the standard finite-size scaling ansatz for a second-order phase transition. The estimated percolation threshold Kperc=1.1053(4) is close to but obviously smaller than the Berezinskii-Kosterlitz-Thouless (BKT) transition point KBKT=1.1193(10), which is determined from the magnetic susceptibility and the superfluid density. Various interesting questions arise from these unconventional observations, and their solutions would shed light on a variety of classical and quantum systems of BKT phase transitions.

We theoretically and numerically investigate a two-dimensional O(2) model where an order parameter is convected by shear flow. We show that a long-range phase order emerges in two dimensions as a result of anomalous suppression of phase fluctuations by the shear flow. Furthermore, we use the finite-size scaling theory to demonstrate that a phase transition to the long-range ordered state from the disordered state is second order. At a transition point far from equilibrium, the critical exponents turn out to be close to the mean-field value for equilibrium systems.

We derive the self-consistent harmonic approximation for the 2D XY model with non-local interactions. The resulting equation for the variational couplings holds for any form of the spin-spin coupling as well as for any dimension. Our analysis is then specialized to power-law couplings decaying with the distance r as in order to investigate the robustness, at finite σ, of the Berezinskii-Kosterlitz-Thouless (BKT) transition, which occurs in the short-range limit σ → ∞. We propose an ansatz for the functional form of the variational couplings and show that for any σ > 2 the BKT mechanism occurs. The present investigation provides an upper bound σ ∗ = 2 for the critical threshold σ ∗ above which the traditional BKT transition persists in spite of the non-local nature of the couplings.

A system of hard spheres exhibits physics that is controlled only by their density. This comes about because the interaction energy is either infinite or zero, so all allowed configurations have exactly the same energy. The low-density phase is liquid, while the high-density phase is crystalline, an example of “order by disorder” as it is driven purely by entropic considerations. Here we study a family of hard spin models, which we call hard-core spin models, where we replace the translational degrees of freedom of hard spheres with the orientational degrees of freedom of lattice spins. Their hard-core interaction serves analogously to divide configurations of the many spin system into allowed and disallowed sectors. We present detailed results on the square lattice in d=2 for a set of models with Zn symmetry, which generalize Potts models, and their U(1) limits, for ferromagnetic and antiferromagnetic senses of the interaction, which we refer to as exclusion and inclusion models. As the exclusion and inclusion angles are varied, we find a Kosterlitz-Thouless phase transition between a disordered phase and an ordered phase with quasi-long-ranged order, which is the form order by disorder takes in these systems. These results follow from a set of height representations, an ergodic cluster algorithm, and transfer matrix calculations.

We study the phase transitions of three-dimensional (3D) classical O(3) model and two-dimensional (2D) classical XY model, as well as both the quantum phase transitions of 2D and 3D dimerized spin-1/2 antiferromagnets, using the technique of supervised neural network (NN). Moreover, unlike the conventional approaches commonly used in the literature, the training sets employed in our investigation are neither the theoretical nor the real configurations of the considered systems. Remarkably, with such an unconventional set up of the training stage in conjunction with some semiexperimental finite-size scaling formulas, the associated critical points determined by the NN method agree well with the established results in the literature. The outcomes obtained here imply that certain unconventional training strategies, like the one used in this study, are not only cost-effective in computation but are also applicable for a wide range of physical systems.

Characterizing the scaling with the total particle number (N) of the largest eigenvalue of the one-body density matrix (λ0) provides information on the occurrence of the off-diagonal long-range order (ODLRO) according to the Penrose-Onsager criterion. Setting λ0∼NC0, then C0=1 corresponds in ODLRO. The intermediate case, 0<C0<1, corresponds in translational invariant systems to the power-law decaying of (nonconnected) correlation functions and it can be seen as identifying quasi-long-range order. The goal of the present paper is to characterize the ODLRO properties encoded in C0 (and in the corresponding quantities Ck≠0 for excited natural orbitals) exhibited by homogeneous interacting bosonic systems at finite temperature for different dimensions in presence of short-range repulsive potentials. We show that Ck≠0=0 in the thermodynamic limit. In one dimension it is C0=0 for nonvanishing temperature, while in three dimensions it is C0=1 (C0=0) for temperatures smaller (larger) than the Bose-Einstein critical temperature. We then focus our attention to D=2, studying the XY and the Villain models, and the weakly interacting Bose gas. The universal value of C0 near the Berezinskii-Kosterlitz-Thouless temperature TBKT is 7/8. The dependence of C0 on temperatures between T=0 (at which C0=1) and TBKT is studied in the different models. An estimate for the (nonperturbative) parameter ξ entering the equation of state of the two-dimensional Bose gases is obtained using low-temperature expansions and compared with the Monte Carlo result. We finally discuss a “double jump” behavior for C0, and correspondingly of the anomalous dimension η, right below TBKT in the limit of vanishing interactions.

The helicity modulus of the quasi-one-dimensional classical XY model is examined in connection with liquid He confined in nanopores. The helicity modulus Υp is suppressed with phase slippage, and its suppression is found to be well described with the scaling form obtained from the Tomonaga-Luttinger liquid theory in the one-dimensional limit. For phase slippage to occur, the system has to overcome an energy barrier, and the helicity modulus Υp suppressed by phase slippage is not necessarily observed in experiments. The energy barrier for phase slippage is explicitly calculated. Using the result, the helicity modulus Υs protected by the energy barrier is found to survive up to a higher temperature than Υp. The relation of the present results to experiments on liquid He in nanopores is also discussed.

This paper presents a pedagogical review of duality (in the sense of Kramers and Wannier) and its application to a wide range of field theories and statistical systems. Most of the article discusses systems in arbitrary dimensions with discrete or continuous Abelian symmetry. Globally and locally symmetric interactions are treated on an equal footing. For convenience, most of the theories are formulated on a d-dimensional (Euclidean) lattice, although duality transformations in the continuum are briefly described. Among the familiar theories considered are the Ising model, the x-y model, the vector Potts model, and the Wilson lattice gauge theory with a ZN or U(1) symmetry, all in various dimensions. These theories are all members of a more general heirarchy of theories with interactions which are distinguished by their geometrical character. For all these Abelian theories it is shown that the duality transformation maps the high-temperature (or, for a field theory, large coupling constant) region of the theory into the low-temperature (small coupling constant) region of the dual theory, and vice versa. The interpretation of the dual variables as disorder parameters is discussed. The formulation of the theories in terms of their topological excitations is presented, and the role of these excitations in determining the phase structure of the theories is explained. Among the other topics discussed are duality for the Abelian Higgs model and related models, duality transformations applied to random systems (such as theories of a spin glass), duality transformations in the "lattice Hamiltonian" formalism, and a description of attempts to construct duality transformations for theories with a non-Abelian symmetry, both on the lattice and in the continuum.

The massive continuum limit of the (1+1)-dimensional O(2) nonlinear σ-model (XY-model) is studied using its equivalence to the sine-Gordon model at its asymptotically free point. It is shown that leading lattice artifacts are universal but they vanish only as inverse powers of the logarithm of the correlation length. Such leading artifacts are calculated for the case of the scattering phase shifts and the correlation function of the Noether current using the bootstrap S-matrix and perturbation theory respectively.

A new definition of order called topological order is proposed for two-dimensional systems in which no long-range order of the conventional type exists. The possibility of a phase transition characterized by a change in the response of the system to an external perturbation is discussed in the context of a mean field type of approximation. The critical behaviour found in this model displays very weak singularities. The application of these ideas to the xy model of magnetism, the solid-liquid transition, and the neutral superfluid are discussed. This type of phase transition cannot occur in a superconductor nor in a Heisenberg ferromagnet.

We study the fully frustrated XY model on a square lattice with the
use of Monte Carlo simulations. We find a phase transition at a finite
temperature TI with the specific-heat data being consistent with
a logarithmic divergence. The helicity modulus Ï jumps to zero with
a value Ï/kBTâ³2/Ï at a Tâ²TI. The application of frustrated XY models
to the behavior of coupled Josephson junction arrays is discussed.

A new approach to the 2-D classical planar magnet is proposed. The Hamiltonian is replaced by an approximate one, which, in contrast with previous approximations, preserves the correct symmetry of the problem; the existence of a phase transition without long range order at some temperature Tc is confirmed; quantitative evaluations of the spin pair correlation function below T c, and of the transition temperature Tc are given; the results are in good agreement at low T with the self consistent harmonic approximation (S.C.H.A.). A transition is predicted at a temperature KB Tc = 1.7 Js2, in very good agreement with predictions based on series expansions, whereas the critical exponents are those predicted by Kosterlitz, in particular γ = ∞. On donne une nouvelle méthode d'étude des systèmes planaires magnétiques à 2 dimensions : le Hamiltonien est remplacé par une approximation qui, contrairement aux approximations antérieures, préserve la symétrie correcte du problème. On confirme l'existence d'une transition à Tc et on donne une évaluation quantitative de Tc et de la fonction de corrélation de paire au-dessous de Tc. Les résultats sont en accord à basse température avec ceux de l'approximation de Hartree. On prévoit une transition à une température KB Tc = 1,7 Js 2, en bon accord avec les prédictions des développements en séries à haute température, alors que les exposants critiques sont ceux prévus par Kosterlitz, notamment γ = ∞.

An exactly solvable model of the crystal-vacuum interface is constructed which exhibits a roughening transition. The model is obtained as a special limit of a ferromagnetic Ising model and it is isomorphic to the symmetric six-vertex model. Some of the thermodynamic properties of the system are discussed.

Recent exact predictions for the massive scaling limit of the two dimensional XY-model are based on the equivalence with the sine-Gordon theory and include detailed results on the finite size behavior. The so-called step-scaling function of the mass gap is simulated with very high precision and found consistent with analytic results in the continuum limit. To come to this conclusion, an also predicted form of a logarithmic decay of lattice artifacts was essential to use for the extrapolation. Comment: 16 pages, 3 figures

We present a new analysis of the rare decay KL→
π0e+e- taking into account important
experimental progress that has recently been achieved in measuring
KL→ π0γγ and
KS→ π0e+e-. This
includes a brief review of the direct CP-violating component, a
calculation of the indirect CP-violating contribution, which is now
possible after the measurement of KS→
π0e+e-, and a reanalysis of the
CP-conserving part. The latter is shown to be negligible, based on
experimental input from KL→
π0γγ, a more general treatment of the form
factor entering the dispersive contribution, and on a comparison with
the CP-violating rate, which can now be estimated reliably. We predict
B( KL→ π0e+e-)=(3.2
+1.2-0.8)×10 -11 in the Standard
Model, dominated by CP violation with a sizable contribution
(˜40%) from the direct effect, largely through interference with
the indirect one. Methods to deal with the severe backgrounds for
KL→ π0e+e- using
Dalitz-plot analysis and time-dependent KL- KS
interference are also briefly discussed.

At low temperatures, ice has a residual entropy, presumably caused by an indeterminacy in the positions of the hydrogen atoms. While the oxygen atoms are in a regular lattice, each O-H-O bond permits two possible positions for the hydrogen atom, subject to certain constraints called the “ice condition.” The statement of the problem in two dimensions is to find the number of ways of drawing arrows on the bonds of a square planar net so that precisely two arrows point into each vertex. If N is the number of molecules and (for large N) W
N
is the number of arrangements, then S = Nk lnW. Our exact result is W = (4/3)3/2.

Using two sets of high-precision Monte Carlo data for the two-dimensional XY model in the Villain formulation on square L×L lattices, the scaling behavior of the susceptibility χ and correlation length ξ at the Kosterlitz-Thouless phase transition is analyzed with emphasis on multiplicative logarithmic corrections (ln L)-2r in the finite-size scaling region and (ln ξ)-2r in the high-temperature phase near criticality, respectively. By analyzing the susceptibility at criticality on lattices of size up to 5122 we obtain r=-0.0270(10), in agreement with recent work of Kenna and Irving on the finite-size scaling of Lee-Yang zeros in the cosine formulation of the XY model. By studying susceptibilities and correlation lengths up to ξ≈140 in the high-temperature phase, however, we arrive at quite a different estimate of r=0.0560(17), which is in good agreement with recent analyses of thermodynamic Monte Carlo data and high-temperature series expansions of the cosine formulation.

A systematic renormalisation group technique for studying the 2D sine-Gordon theory (Coulomb gas, XY model) near its phase transition is presented. The new results are (a) higher order terms in the flow equations beyond those of Kosterlitz (1974) give rise to a new universal quantity; (b) this in turn gives the universal form as well as the relative coefficient of the next-to-leading term in the correlation function of the XY model; (c) the free energy (1PI vacuum sum) is calculated after the singularity at beta 2=4 pi is treated; (d) vortices with multiple charges are shown to be irrelevant; (e) symmetry breaking fields are analysed systematically. The main ideas that the sine-Gordon theory can be defined as a double expansion in alpha (fugacity) and delta = beta 2/8 pi -1 (distance from the critical temperature at alpha =0). Wave-function and coupling constant ( alpha ) renormalisations are necessary and sufficient, around beta 2=8 pi where cos phi acquires dimension 2, for functions with elementary SG fields. This gives rise to renormalisation of beta . The renormalisability is proved to the order calculated in the context of the SG theory, and in general, by using the equivalence to the Thirring-Schwinger model. The renormalised beta 2 plays a role analogous to the dimension in a phi 4 theory-8 pi being the critical dimension. beta 2>8 pi gives an infrared asymptotically free theory which leads to the well-known fixed line. The infrared properties are understood by analogy with the non-linear sigma model.

We propose to compute the running coupling in lattice gauge theories and non-linear σ-models through a finite-size scaling analysis. The idea is shown to work well in the O(3) σ-model, where we succeed to determine (in, say, a momentum subtraction scheme) at energies up to about 30 times the mass of the particles in the theory.

The quantum field theory describing the massive O(2) non-linear sigma-model is investigated through two non-perturbative constructions: the form factor bootstrap based on integrability and the lattice formulation as the XY model. The S-matrix, the spin and current two-point functions, as well as the 4-point coupling are computed and critically compared in both constructions. On the bootstrap side a new parafermionic super selection sector is found; in the lattice theory a recent prediction for the (logarithmic) decay of lattice artifacts is probed.

We describe an efficient position space technique to calculate lattice Feynman integrals in infinite volume. The method applies to diagrams with massless propagators. For illustration a set of two-loop integrals is worked out explicitly. An important ingredient is an observation of Vohwinkel that the free lattice propagator can be evaluated recursively and is expressible as a linear function of its values near the origin.

We examine the Kosterlitz-Thouless universality class and show that essential scaling at this type of phase transition is not self-consistent unless multiplicative logarithmic corrections are included. In the case of specific heat these logarithmic corrections are identified analytically. To identify those corresponding to the susceptibility we set up a numerical method involving the finite-size scaling of Lee-Yang zeroes. We also study the density of zeroes and introduce a new concept called index scaling. We apply the method to the XY model and the c]osely related step model in two dimensions. The critical parameters (including logarithmic corrections) of the step model are compatible with those of the XY model indicating that both models belong to the same universality class. This result then raises questions over how a vortex binding scenario can be the driving mechanism for the phase transition. Furthermore, the logarithmic corrections identified numerically by our methods of fitting are not in agreement with the renormalization group predictions of Kosterlitz and Thouless.

We confirm the Kosterlitz-Thouless scenario of the roughening transition for three different Solid-On-Solid models: the Discrete Gaussian model, the Absolute-Value-Solid-On-Solid model and the dual transform of the XY-model with standard (cosine) action. The method is based on a matching of the renormalization group flow of the candidate models with the flow of a bona fide KT model, the exactly solvable BCSOS model. The Monte Carlo simulations are performed using efficient cluster algorithms. We obtain high precision estimates for the critical couplings and other non-universal quantities. For the XY-model with cosine action our critical coupling estimate is βXYR = 1.1197 (5). For the roughening coupling of the Discrete Gaussian and the Absolute-Value-Solid-On-Solid model we find KDGR = 0.6645 (6) and KASOSR = 0.8061 (3), respectively.

By expressing thermodynamic functions in terms of the edge and density of Lee-Yang zeroes, we relate the scaling behaviour of the specific heat to that of the zero field magnetic susceptibility in the thermodynamic limit of the XY-model in two dimensions. Assuming that finite-size scaling holds, we show that the conventional Kosterlitz-Thouless scaling predictions for these thermodynamic functions are not mutually compatible unless they are modified by multiplicative logarithmic corrections. We identify these logarithmic corrections analytically in the case of the specific heat and numerically in the case of the susceptibility. The techniques presented here are general and can be used to check the compatibility of scaling behaviour of odd and even thermodynamic functions in other models too.

The critical properties of the xy model with nearest-neighbour interactions
on a two-dimensional square lattice are studied by a renormalization
group technique. The mean magnetization is zero for all temperatures,
and the transition is from a state of finite to one of infinite susceptibility.
The correlation length is found to diverge faster than any power
of the deviation from the critical temperature. Analogues of the
strong scaling laws are derived and the critical exponents, eta ,
and delta , are the same as for the two-dimensional Ising model.

The classical planar Heisenberg model is studied at low temperatures
by means of renormalization theory and a series of exact transformations.
A numerical study of the Migdal recursion relation suggests that
models with short-range isotropic interactions rapidly become equivalent
to a simplified model system proposed by Villain. A series of exact
transformations then allows us to treat the Villain model analytically
at low temperatures. To lowest order in a parameter which becomes
exponentially small with decreasing temperature, we reproduce results
obtained previously by Kosterlitz. We also examine the effect of
symmetry-breaking crystalline fields on the isotropic planar model.
A numerical study of the Migdal recursion scheme suggests that these
fields (which must occur in real quasi-two-dimensional crystals)
are strongly relevant variables, leading to critical behavior distinct
from that found for the planar model. However, a more exact low-temperature
treatment of the Villain model shows that hexagonal crystalline fields
eventually become irrelevant at temperatures below the Tc of the
isotropic model. Isotropic planar critical behavior should be experimentally
accessible in this case. Nonuniversal behavior may result if cubic
crystalline fields dominate the symmetry breaking. Interesting duality
transformations, which aid in the analysis of symmetry-breaking fields
are also discussed.

The ordered state of a d-dimensional isotropic system with an n-vector
(nâ¥2) order parameter is considered. By the imposition of suitable
boundary conditions it is shown how to define explicitly a helicity
modulus Ï(T) which measures the free-energy increment associated
with "twisting" the direction of the order parameter. For a Bose
system the superfluid density is seen to be Ïs(T)=(m / â)2Ï(T). A
critical exponent v is defined by Ï(T)â¼|T-Tc|v as TâTc; for an ideal
Bose gas and spherical model (nââ), v=1 is an exact result for all
d>2. The difficulties of defining a correlation length in the ordered
phase are discussed. A full scaling theory of the correlations avoids
these problems and may be linked to a phenomenological hydrodynamic
approach, to clarify and rederive Josephson's relation v=2Î²-Î·Î½=2-Î±-2Î½.
This reduces to v=(d-2)Î½ (used by some authors with d=3), only if
one accepts d-dependent, "hyperscaling" relations such as d Î½=2-Î±;
however, both these latter relations fail for the ideal Bose gas
when d>4. An alternative derivation of the formula v=2-Î±-2Î½ is based
on the scaling theory for systems with a large but finite dimension.

The temperature scale of relevance for the vortex excitations in the two-dimensional XY model is identified. The size dependence of the helicity modulus at constant vortex temperature is then obtained by Monte Carlo simulations. The Monte Carlo data are then compared with the results for the length-dependent dielectric function from the Kosterlitz renormalization-group equations. The agreement is excellent for temperatures up to and slightly above Tc. The critical temperature is determined to be Tc=0.892 13(10). The temperature dependence of the characteristic length right above Tc is also obtained and found to be in agreement with recent results from high-temperature series expansions.

We use finite-size scaling of Lee-Yang partition function zeros to study the critical behavior of the two-dimensional step or sgn O(2) model. We present evidence that, like the closely related XY model, this has a phase transition from a disordered high-temperature phase to a low-temperature massless phase where the model remains critical. The critical parameters (including logarithmic corrections) are compatible with those of the XY model indicating that both models belong to the same universality class.

The three-dimensional classical XY model on a cubic lattice has been studied using Monte Carlo simulations. The finite-size scaling method of the phenomenological renormalization group has been used to calculate the critical exponents ν, v, and β of the correlation length, helicity modulus, and order parameter. Good agreement with series-expansion results is obtained.

We report two sets of high-precision Monte Carlo simulations of the two-dimensional XY model in Villain's formulation on large square lattices, employing the single-cluster update algorithm. In one set of simulations we use improved estimators to study the correlation length xi and susceptibility chi in the high-temperature phase. On a 1200×1200 lattice this allows measurements up to xi~=140 with statistical errors less than 0.25%. We judge quantitatively the advantage of using improved estimators, estimate autocorrelation times, and compare the numerical efficiency of various definitions of the correlation length. From least-square fits to these data we find clear support for the exponential divergence predicted by Kosterlitz and Thouless. The other set of simulations is performed in the vicinity of the transition point. Here we apply finite-size-scaling theory to obtain an estimate for the exponent eta at criticality.

A Monte Carlo algorithm is presented that updates large clusters of spins simultaneously in systems at and near criticality. We demonstrate its efficiency in the two-dimensional O(n) σ models for n=1 (Ising) and n=2 (x-y) at their critical temperatures, and for n=3 (Heisenberg) with correlation lengths around 10 and 20. On lattices up to 1282 no sign of critical slowing down is visible with autocorrelation times of 1-2 steps per spin for estimators of long-range quantities.

We recalculate the two-loop beta-functions in the
two-dimensional sine-Gordon model in a two-parameter expansion
around the asymptotically free point. Our results agree with
those of Amit et al (Amit D J, Goldschmidt Y Y and Grinstein G
1980 J. Phys. A:
Math. Gen. 13 585).

I present a new improved estimator for the correlation function of 2D nonlinear sigma models. Numerical tests for the 2D XY model and the 2D O(3)-invariant vector model were performed. For small physical volume, i.e. a lattice size small compared to the to the bulk correlation length, a reduction of the statistical error of the finite system correlation length by a factor of up to 30 compared to the cluster-improved estimator was observed. This improvement allows for a very accurate determination of the running coupling proposed by M. L"uscher et al. for 2D O(N)-invariant vector models. Comment: 20 pages, LaTeX + 2 ps figures, CERN-TH.7375/94

We study the roughening transition of the dual of the 2D XY model, of the Discrete Gaussian model, of the Absolute Value Solid-On-Solid model and of the interface in an Ising model on a 3D simple cubic lattice. The investigation relies on a renormalization group finite size scaling method that was proposed and successfully tested a few years ago. The basic idea is to match the renormalization group flow of the interface observables with that of the exactly solvable BCSOS model. Our estimates for the critical couplings are $\beta_R^{XY}=1.1199(1)$, $K_R^{DG}=0.6653(2)$ and $K_R^{ASOS}=0.80608(2)$ for the XY-model, the Discrete Gaussian model and the Absolute Value Solid-On-Solid model, respectively. For the inverse roughening temperature of the Ising interface we find $K_R^{Ising}= 0.40758(1)$. To the best of our knowledge, these are the most precise estimates for these parameters published so far.

Using the $x-y$ model and a non-local updating scheme called cluster Monte Carlo, we calculate the superfluid density of a two dimensional superfluid on large-size square lattices $L \times L$ up to $400\times 400$. This technique allows us to approach temperatures close to the critical point, and by studying a wide range of $L$ values and applying finite-size scaling theory we are able to extract the critical properties of the system. We calculate the superfluid density and from that we extract the renormalization group beta function. We derive finite-size scaling expressions using the Kosterlitz-Thouless-Nelson Renormalization Group equations and show that they are in very good agreement with our numerical results. This allows us to extrapolate our results to the infinite-size limit. We also find that the universal discontinuity of the superfluid density at the critical temperature is in very good agreement with the Kosterlitz-Thouless-Nelson calculation and experiments. Comment: 13 pages, postscript file

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