Journal of Statistical Mechanics Theory and Experiment

Published by IOP Publishing
Online ISSN: 1742-5468
Integrative approaches to the study of complex systems demand that one knows the manner in which the parts comprising the system are connected. The structure of the complex network defining the interactions provides insight into the function and evolution of the components of the system. Unfortunately, the large size and intricacy of these networks implies that such insight is usually difficult to extract. Here, we propose a method that allows one to systematically extract and display information contained in complex networks. Specifically, we demonstrate that one can (i) find modules in complex networks and (ii) classify nodes into universal roles according to their pattern of within- and between-module connections. The method thus yields a 'cartographic representation' of complex networks.
Difference between large bubble velocity v L and average foam velocity, rescaled by rv f , vs v f for different large-bubble sizes, on a semilog scale. The symbols on the graph correspond to different values of r.
The Cellular Potts Model (CPM) successfully simulates drainage and shear in foams. Here we use the CPM to investigate instabilities due to the flow of a single large bubble in a dry, monodisperse two-dimensional flowing foam. As in experiments in a Hele-Shaw cell, above a threshold velocity the large bubble moves faster than the mean flow. Our simulations reproduce analytical and experimental predictions for the velocity threshold and the relative velocity of the large bubble, demonstrating the utility of the CPM in foam rheology studies.
Cluster size histogram for the fly network. The cluster abundance for α = 1.5 and γ = 10 − 2 (red 
We suggest a fast method for finding possibly overlapping network communities of a desired size and link density. Our method is a natural generalization of the finite-T superparamagnetic Potts clustering introduced by Blatt et al (1996 Phys. Rev. Lett.76 3251) and the annealing of the Potts model with a global antiferromagnetic term recently suggested by Reichard and Bornholdt (2004 Phys. Rev. Lett.93 21870). Like in both cited works, the proposed generalization is based on ordering of the ferromagnetic Potts model; the novelty of the proposed approach lies in the adjustable dependence of the antiferromagnetic term on the population of each Potts state, which interpolates between the two previously considered cases. This adjustability allows one to empirically tune the algorithm to detect the maximum number of communities of the given size and link density. We illustrate the method by detecting protein complexes in high-throughput protein binding networks.
Illustration of a centipede, i.e. a spider with local constraint. All legs step to empty nearest neighbor sites at the same rate with adjacent legs staying within distance s from each other.
Illustration of a spider with global constraint. The legs can step independently to nearest neighbor empty sites within a distance S from each other.  
Molecular spiders are synthetic biomolecular systems which have 'legs' made of short single-stranded segments of DNA. Spiders move on a surface covered with single-stranded DNA segments complementary to legs. Different mappings are established between various models of spiders and simple exclusion processes. For spiders with simple gait and varying number of legs we compute the diffusion coefficient; when the hopping is biased we also compute their velocity.
Schematic illustration of the growth and catastrophic shrinkage of a microtubule as a function of time (adapted from [1]). 
Representative configuration of a microtubule, with a cap of length 4, three GTP + islands of lengths 1, 3, and 2, and three GDP − islands of lengths 3, 2, and a "tail" of length 5. The rest of the microtubule consists of GDP − . 
Cartoon of a microtubule. The GTP + regions (dark) get shorter further from the tip that advances as λt, while the GDP − regions (light) get longer. 
Schematic phase diagram for a microtubule in the λ-µ parameter space. 
We investigate an idealized model of microtubule dynamics that involves: (i) attachment of guanosine triphosphate (GTP) at rate λ, (ii) conversion of GTP to guanosine diphosphate (GDP) at rate 1, and (iii) detachment of GDP at rate μ. As a function of these rates, a microtubule can grow steadily or its length can fluctuate wildly. For μ = 0, we find the exact tubule and GTP cap length distributions, and power-law length distributions of GTP and GDP islands. For μ = ∞, we argue that the time between catastrophes, where the microtubule shrinks to zero length, scales as e(λ). We also discuss the nature of the phase boundary between a growing and shrinking microtubule.
Two recent theoretical models, Bai et al. (2004, 2007) and Tadigotla et al. (2006), formulated thermodynamic explanations of sequence-dependent transcription pausing by RNA polymerase (RNAP). The two models differ in some basic assumptions and therefore make different yet overlapping predictions for pause locations, and different predictions on pause kinetics and mechanisms. Here we present a comprehensive comparison of the two models. We show that while they have comparable predictive power of pause locations at low NTP concentrations, the Bai et al. model is more accurate than Tadigotla et al. at higher NTP concentrations. Pausing kinetics predicted by Bai et al. is also consistent with time-course transcription reactions, while Tadigotla et al. is unsuited for this type of kinetic prediction. More importantly, the two models in general predict different pausing mechanisms even for the same pausing sites, and the Bai et al. model provides an explanation more consistent with recent single molecule observations.
Scaling plot of the probability distribution for (a) node strengths for N = 2 12 , 2 14 , 2 16 for α = 0.40, (b) link weights for α = 0.40, for N = 2 16 , 2 18 , 2 20 .
(a) Variation of strength-degree correlation exponent β with α; open circles are data points and the solid line is the theoretical prediction of equation (5). (b) Plot of the link weight-degree correlation exponent θ with α for N = 2 16 ; the solid line is the plot of the fitting function θ = a + bα c .
All crucial features of the recently observed real-world weighted networks are obtained in a model where the weight of a link is defined with a single non-linear parameter $\alpha$ as $w_{ij}=(s_is_j)^\alpha$, $s_i$ and $s_j$ are the strengths of two end nodes of the link and $\alpha$ is a continuously tunable positive parameter. In addition the definition of strength as $s_i= \Sigma_j w_{ij}$ results a self-organizing link weight dynamics leading to a self-consistent distribution of strengths and weights on the network. Using the Barab\'asi-Albert growth dynamics all exponents of the weighted networks which are continuously tunable with $\alpha$ are obtained. It is conjectured that the weight distribution should be similar in any scale-free network. Comment: 5 pages, 4 figures
We consider the physical combinatorics of critical lattice models and their associated conformal field theories arising in the continuum scaling limit. As examples, we consider A-type unitary minimal models and the level-1 sl(2) Wess-Zumino-Witten (WZW) model. The Hamiltonian of the WZW model is the $U_q(sl(2))$ invariant XXX spin chain. For simplicity, we consider these theories only in their vacuum sectors on the strip. Combinatorially, fermionic particles are introduced as certain features of RSOS paths. They are composites of dual-particles and exhibit the properties of quasiparticles. The particles and dual-particles are identified, through an energy preserving bijection, with patterns of zeros of the eigenvalues of the fused transfer matrices in their analyticity strips. The associated (m,n) systems arise as geometric packing constraints on the particles. The analyticity encoded in the patterns of zeros is the key to the analytic calculation of the excitation energies through the Thermodynamic Bethe Ansatz (TBA). As a by-product of our study, in the case of the WZW or XXX model, we find a relation between the location of the Bethe root strings and the location of the transfer matrix 2-strings. Comment: 57 pages, in version 2: typos corrected, some sentences clarified, one appendix removed
Through a series of exact mappings we reinterpret the Bernoulli model of sequence alignment in terms of the discrete-time totally asymmetric exclusion process with backward sequential update and step function initial condition. Using earlier results from the Bethe ansatz we obtain analytically the exact distribution of the length of the longest common subsequence of two sequences of finite lengths $X,Y$. Asymptotic analysis adapted from random matrix theory allows us to derive the thermodynamic limit directly from the finite-size result. Comment: 13 pages, 4 figures
A lattice model of critical spanning webs is considered for the finite cylinder geometry. Due to the presence of cycles, the model is a generalization of the known spanning tree model which belongs to the class of logarithmic theories with central charge c = −2. We show that in the scaling limit the universal part of the partition function for closed boundary conditions at both edges of the cylinder coincides with the character of symplectic fermions with periodic boundary conditions and for an open boundary at one edge and a closed one at the other coincides with the character of symplectic fermions with antiperiodic boundary conditions.
We report on large scale finite-temperature Monte Carlo simulations of the classical $120^\circ$ or $e_g$ orbital-only model on the simple cubic lattice in three dimensions with a focus towards its critical properties. This model displays a continuous phase transition to an orbitally ordered phase. While the correlation length exponent $\nu\approx0.665$ is close to the 3D XY value, the exponent $\eta \approx 0.15$ differs substantially from O(N) values. We also introduce a discrete variant of the $e_g$ model, called $e_g$-clock model, which is found to display the same set of exponents. Further, an emergent U(1) symmetry is found at the critical point $T_c$, which persists for $T<T_c$ below a crossover length scaling as $\Lambda \sim \xi^a$, with an unusually small $a\approx1.3$.
We study the domain wall partition function $Z_N$ for the $U_q(A_2^{(2)})$ (Izergin-Korepin) integrable $19$-vertex model on a square lattice of size $N$. $Z_N$ is a symmetric function of two sets of parameters: horizontal $\zeta_1,..,\zeta_N$ and vertical $z_1,..,z_N$ rapidities. For generic values of the parameter $q$ we derive the recurrence relation for the domain wall partition function relating $Z_{N+1}$ to $P_N Z_N$, where $P_N$ is the proportionality factor in the recurrence, which is a polynomial symmetric in two sets of variables $\zeta_1,..,\zeta_N$ and $z_1,..,z_N$. After setting $q^3=-1$ the recurrence relation simplifies and we solve it in terms of a Jacobi-Trudi-like determinant of polynomials generated by $P_N$.
By means of an algebraic Bethe ansatz approach we study the Zamolodchikov-Fateev and Izergin-Korepin vertex models with non-diagonal boundaries, characterized by reflection matrices with an upper triangular form. Generalized Bethe vectors are used to diagonalize the associated transfer matrix. The eigenvalues as well as the Bethe equations are presented.
Kenneth G. Wilson made deep and insightful contributions to statistical physics, particle physics, and related fields. He was also a helpful and thoughtful human being.
Lars Onsager and Bruria Kaufman calculated the partition function of the Ising model exactly in 1944 and 1949. Since then there have been many developments in the exact solution of similar, but usually more complicated, models. Here I shall mention a few, and show how some of the latest work seems to be returning once again to the properties observed by Onsager and Kaufman.
By the Bethe ansatz method we study the energy dispersion of a particle interacting by a local interaction with fermions (or hard core bosons) of equal mass in a one-dimensional lattice. We focus on the period of the Bloch oscillations, which turns out to be related to the Fermi wavevector of the Fermi sea and in particular on how this dispersion emerges as a collective effect in the thermodynamic limit. We also discuss the adiabatic coherent collective response of the system to an applied field.
We study the low energy behavior of the one dimensional Hubbard model across the Mott metal–insulator phase transition in an external magnetic field. In particular we calculate elements of the dressed charge matrix at the critical point of the Mott transition for arbitrary Hubbard repulsion and magnetization numerically and, in certain limiting cases, analytically. These results are combined with a non-perturbative effective field theory approach to reveal how the breaking of time reversal symmetry influences the Mott transition.
This paper gives a pedagogic derivation of the Bethe ansatz solution for 1D interacting anyons. This includes a demonstration of the subtle role of the anyonic phases in the Bethe ansatz arising from the anyonic commutation relations. The thermodynamic Bethe ansatz equations defining the temperature dependent properties of the model are also derived, from which some ground state properties are obtained.
We discuss in details a modified variational matrix-product-state algorithm for periodic boundary conditions, based on a recent work by P. Pippan, S.R. White and H.G. Everts, Phys. Rev. B 81, 081103(R) (2010), which enables one to study large systems on a ring (composed of N ~ 10^2 sites). In particular, we introduce a couple of improvements that allow to enhance the algorithm in terms of stability and reliability. We employ such method to compute the stiffness of one-dimensional strongly correlated quantum lattice systems. The accuracy of our calculations is tested in the exactly solvable spin-1/2 Heisenberg chain.
We study one-dimensional lattice systems with pair-wise interactions of in?nite range. We show projective convergence of Markov measures to the unique equilibrium state. For this purpose we impose a slightly stronger condition than summability of variations on the regularity of the interaction. With our condition we are able to explicitly obtain stretched exponential bounds for the rate of mixing of the equilibrium state. Finally we show convergence for the entropy of the Markov measures to that of the equilibrium state via the convergence of their topological pressure.
Exact numerical evaluation of the Rényi entanglement entropy for α = 1, 2, 5, ∞ as function of nn. For large nn these are perfectly described by the extensive asymptotic results reported as full (red) lines. Notice the presence of some non-analytic points for α = ∞. 
We analyze the entanglement properties of the asymptotic steady state after a quench from free to hard-core bosons in one dimension. The R\'enyi and von Neumann entanglement entropies are found to be extensive, and the latter coincides with the thermodynamic entropy of the Generalized Gibbs Ensemble (GGE). Computing the spectrum of the two-point function, we provide exact analytical results both for the leading extensive parts and the subleading terms for the entropies as well as for the cumulants of the particle number fluctuations. We also compare the extensive part of the entanglement entropy with the thermodynamic ones, showing that the GGE entropy equal the entanglement one and it is the double of the diagonal entropy.
The ground state values of the K-body local correlations for K ≤ 4 as a function of the dimensionless coupling γ (g 1 = 1 by definition). The exact values are represented by the solid lines, whereas the dashed lines show the empirical formula (7.14).
The quantity g 4 as a function of the dimensionless temperature τ for intermediate couplings. In the τ → 0 limit the three curves approach small (but non-zero) values which are shown in Figure 1.
We consider the K-body local correlations in the (repulsive) 1D Bose gas for general K, both at finite size and in the thermodynamic limit. Concerning the latter we develop a multiple integral formula which applies for arbitrary states of the system with a smooth distribution of Bethe roots, including the ground state and finite temperature Gibbs-states. In the cases K<=4 we perform the explicit factorization of the multiple integral. In the case of K=3 we obtain the recent result of Kormos, whereas our formula for K=4 is new. Numerical results are presented as well.
(Color online) Schematic diagram of the classical Heisenberg model in one dimension with two boundary heat baths at temperatures T l and T r , and local time varying mag- 
(Color online) Thermal current with frequency for different average energy of the bath E b = −0.1, −0.2 and −0.3 and same energy gradient ∆E = 0.015. Resonance enhances as the temperature decreases. (Inset) Log-log plot of the maximum thermal current Jm corresponding to ω = 1.5 for N = 500.
We study thermal transport in the one dimensional classical Heisenberg model driven by boundary heat baths in presence of a local time varying magnetic field that acts at one end of the system. The system is studied numerically using an energy conserving discrete-time odd even dynamics. We find that the steady state energy current shows thermal resonance as the frequency of the time- periodic forcing is varied. When the amplitude of the forcing field is increased the system exhibits multiple resonance peaks instead of a single peak. Both single and multiresonance survive in the thermodynamic limit and their magnitudes increase as the average temperature of the system is decreased. Finally we show that, although a reversed thermal current can be made to flow through the bulk for a certain range of the forcing frequency, the system fails to behave as a heat pump, thus revalidating the fact that thermal pumping is generically absent in such force-driven lattices.
The physics of the attractive one-dimensional Bose gas (Lieb–Liniger model) is investigated with techniques based on the integrability of the system. Combining a knowledge of particle quasi-momenta to exponential precision in the system size with determinant representations of matrix elements of local operators coming from the algebraic Bethe ansatz, we obtain rather general analytical results for the zero-temperature dynamical correlation functions of the density and field operators. Our results thus provide quantitative predictions for possible future experiments in atomic gases or optical waveguides.
We present a full identification of lattice model properties with their field theoretical counter parts in the continuum limit for a supersymmetric model for itinerant spinless fermions on a one dimensional chain. The continuum limit of this model is described by an $\mathcal{N}=(2,2)$ superconformal field theory (SCFT) with central charge c=1. We identify states and operators in the lattice model with fields in the SCFT and we relate boundary conditions on the lattice to sectors in the field theory. We use the dictionary we develop in this paper, to give a pedagogical explanation of a powerful tool to study supersymmetric models based on spectral flow. Finally, we employ the developed machinery to explain numerically observed properties of the particle density on the open chain presented in Beccaria et al. PRL 94:100401 (2005).
Fourier transform G(q, ω) of the correlation functions for ε = 1, N = 2000 and the three wavenumbers, q = π/100 (circles), 2π/100 (squares), and 3π/100 (diamonds). The solid line corresponds to the approximate analytic theory, i.e. it is obtained by Fourier transforming the function C(q, t) defined in (12,19). The same curves are plotted in the inset in log-log scales, where only positive frequencies are shown.
Scaling of the linewidth γ(q) of G(q, ω) with q for three different values of the coupling constant ε and N = 2000. The solid line corresponding to the expected power law q 3/2 is plotted for reference.
Fourier transform of the memory function Γ(q, ω) for the same parameter values as in figure 1 and for three wavenumbers, namely, π/100 (solid line), 2π/100 (dashed line), and 3π/100 (dotted line). The ω axis is suitably shifted in order to get rid of the oscillatory peak at frequency q, so that Γ(q, ω) reduces to M (q, ω). For comparison, the expected power-law ω −1/3 is also presented (thin solid line).
Quartic case: Fourier transform G(q, ω) of the correlation function for ε = 1, N = 2000 and the three wavenumbers q = π/100 (circles), 2π/100 (squares), and 3π/100 (diamonds). The solid line corresponds to the approximate analytical solution obtained by Fourier transforming the expression defined in (26,33).
Fourier transforms of the memory function for three values of the wave-vector for q = π/100 (solid line), 2π/100 (dashed line), and 3π/100 (dotted line). For small frequencies they tend to a constant as we expect by analytic solution.
We study the dynamics of long-wavelength fluctuations in one-dimensional (1D) many-particle systems as described by self-consistent mode-coupling theory. The corresponding non-linear integro-differential equations for the relevant correlators are solved analytically and checked numerically. In particular, we find that the memory functions exhibit a power-law decay accompanied by relatively fast oscillations. Furthermore, the scaling behaviour and, correspondingly, the universality class depend on the order of the leading non-linear term. In the cubic case, both viscosity and thermal conductivity diverge in the thermodynamic limit. In the quartic case, a faster decay of the memory functions leads to a finite viscosity, while the thermal conductivity exhibits an even faster divergence. Finally, our analysis puts on a firmer basis the previously conjectured connection between anomalous heat conductivity and anomalous diffusion.
It is known that exact traveling wave solutions exist for families of (n+1)-states stochastic one-dimensional non-equilibrium lattice models with open boundaries provided that some constraints on the reaction rates are fulfilled. These solutions describe the diffusive motion of a product shock or a domain wall with the dynamics of a simple biased random walker. The steady state of these systems can be written in terms of linear superposition of such shocks or domain walls. These steady states can also be expressed in a matrix product form. We show that in this case the associated quadratic algebra of the system has always a two-dimensional representation with a generic structure. A couple of examples for n=1 and n=2 cases are presented. Comment: 12 pages, 1 figure
We consider the Lieb-Liniger model for a gas of bosonic $\delta-$interacting particles. Using Algebraic Bethe Ansatz results we compute the thermodynamic limit of the form factors of the density operator between finite entropy eigenstates such as finite temperature states or generic non-equilibrium highly excited states. These form factors are crucial building blocks to obtain the thermodynamic exact dynamic correlation functions of such physically relevant states. As a proof of principle we compute an approximated dynamic structure factor by including only the simplest types of particle-hole excitations and show the agreement with known results.
Connectivity dependence of transition temperature 
The value of G for C = 4, β ′ = β and h ext = 0. The red curve represents the cumulant generating function calculated from the trivial solution. The blue curve represents that calculated from the nontrivial solution. The green curve represents the cumulant generating function calculated from the annealed average. 
We study $p$-spin glass models on regular random graphs. By analyzing the Franz-Parisi potential with a two-body cavity field approximation under the replica symmetric ansatz, we obtain a good approximation of the 1RSB transition temperature for $p=3$. Our calculation method is much easier than the 1RSB cavity method because the result is obtained by solving self-consistent equations with Newton's method.
Notations to define the non-linear transfer approach in d = 2 : we impose the boundary conditions m(x = 0, y) = 1 on the left boundary, and we study the surface magnetizations m(x = L, y) = 1 on the right boundary (see text for more details).
Critical point in d = 2 (here Jc = 0.335) : Evolution with L of the probability distribution PL(ln m surf L ) of the logarithm of surface magnetization. Inset : Corresponding fixed distribution of the rescaled variable v = (ln m surf L )/ ln m typ L .  
Critical point in d = 3 (here Jc = 0.1528) : Evolution with L of the probability distribution PL(ln m surf L ) of the logarithm of surface magnetization. Inset : Corresponding fixed distribution of the rescaled variable v = (ln m surf L )/ ln m typ L .
The 'Cavity-Mean-Field' approximation developed for the Random Transverse Field Ising Model on the Cayley tree [L. Ioffe and M. M\'ezard, PRL 105, 037001 (2010)] has been found to reproduce the known exact result for the surface magnetization in $d=1$ [O. Dimitrova and M. M\'ezard, J. Stat. Mech. (2011) P01020]. In the present paper, we propose to extend these ideas in finite dimensions $d>1$ via a non-linear transfer approach for the surface magnetization. In the disordered phase, the linearization of the transfer equations correspond to the transfer matrix for a Directed Polymer in a random medium of transverse dimension $D=d-1$, in agreement with the leading order perturbative scaling analysis [C. Monthus and T. Garel, arxiv:1110.3145]. We present numerical results of the non-linear transfer approach in dimensions $d=2$ and $d=3$. In both cases, we find that the critical point is governed by Infinite Disorder scaling. In particular exactly at criticality, the one-point surface magnetization scales as $\ln m_L^{surf} \simeq - L^{\omega_c} v$, where $\omega_c(d)$ coincides with the droplet exponent $\omega_{DP}(D=d-1)$ of the corresponding Directed Polymer model, with $\omega_c(d=2)=1/3$ and $\omega_c(d=3) \simeq 0.24$. The distribution $P(v)$ of the positive random variable $v$ of order O(1) presents a power-law singularity near the origin $P(v) \propto v^a$ with $a(d=2,3)>0$ so that all moments of the surface magnetization are governed by the same power-law decay $\bar{(m_L^{surf})^k} \propto L^{- x_S}$ with $x_S=\omega_c (1+a)$ independently of the order $k$.
Figure A2. Workflow evolution provenance. In this example we have calculated the system size dependence of the spin gap of spin-1 and spin-1/2 Heisenberg chains, first calculating the gap by exact diagonalization and then performing fits to ∆( L ) = c/L and ∆( L ) = ∆ 0 + c exp( − L/ξ ). As expected from the Haldane conjecture the plots 
We present release 2.0 of the ALPS (Algorithms and Libraries for Physics Simulations) project, an open source software project to develop libraries and application programs for the simulation of strongly correlated quantum lattice models such as quantum magnets, lattice bosons, and strongly correlated fermion systems. The code development is centered on common XML and HDF5 data formats, libraries to simplify and speed up code development, common evaluation and plotting tools, and simulation programs. The programs enable non-experts to start carrying out serial or parallel numerical simulations by providing basic implementations of the important algorithms for quantum lattice models: classical and quantum Monte Carlo (QMC) using non-local updates, extended ensemble simulations, exact and full diagonalization (ED), the density matrix renormalization group (DMRG) both in a static version and a dynamic time-evolving block decimation (TEBD) code, and quantum Monte Carlo solvers for dynamical mean field theory (DMFT). The ALPS libraries provide a powerful framework for programers to develop their own applications, which, for instance, greatly simplify the steps of porting a serial code onto a parallel, distributed memory machine. Major changes in release 2.0 include the use of HDF5 for binary data, evaluation tools in Python, support for the Windows operating system, the use of CMake as build system and binary installation packages for Mac OS X and Windows, and integration with the VisTrails workflow provenance tool. The software is available from our web server at
The geometry of the D3Q19 lattice with lattice vectors c i as defined in equation (3).  
Velocity difference fields for different approaches: inflow and outflow velocity constraints in the direction perpendicular to the boundary plane (a), inflow and outflow velocity tilted and of parabolic shape as in the analytic solution, but with N z x and N z y set equal to zero (b), and the boundary conditions derived in section 4 with the correct choice for N z x and N z y (c). The velocity difference vectors are scaled for the drawing by a factor of 1500 and the absolute value of the velocity difference is color coded from blue (small) to red (large).
Poiseuille flow between two parallel walls driven by a body force. The simulation data are averaged in each lattice plane parallel to the walls and agree up to floating point precision with the calculated Poiseuille profile.  
(a) Velocity profile in a square channel averaged along the yz-planes for the no-slip boundary condition (squares), for the node based bounce back rule (triangles), mid-link bounce back (top-down triangles), and the analytical solution. The no-slip boundary condition collapses with the analytical solution, whereas the on-site bounce back boundary condition shows a kink in the profile close to the boundary nodes. The numerical results are obtained on a 32 3 lattice and for the analytical solution the sum in equation (38) is truncated after 50 terms. A 2D profile of the z-velocity in an xy cross section is displayed as an inset. (b) The relative error for system sizes between 8 and 64 nodes: in the corners next to the boundary the largest relative deviations occur. Note that on the boundary nodes, where the truncation error of equation (38) is largest, the deviation between the simulation and approximated analytical solution is negligible. Therefore, the deviation can be taken as a measure for the quality of the numerical result. (c) The averaged error for different lattice resolutions versus the number of lattice nodes in each dimension confirming the second-order accuracy of the no-slip boundary condition.  
On-site boundary conditions are often desired for lattice Boltzmann simulations of fluid flow in complex geometries such as porous media or microfluidic devices. The possibility to specify the exact position of the boundary, independent of other simulation parameters, simplifies the analysis of the system. For practical applications it should allow to freely specify the direction of the flux, and it should be straight forward to implement in three dimensions. Furthermore, especially for parallelized solvers it is of great advantage if the boundary condition can be applied locally, involving only information available on the current lattice site. We meet this need by describing in detail how to transfer the approach suggested by Zou and He to a D3Q19 lattice. The boundary condition acts locally, is independent of the details of the relaxation process during collision and contains no artificial slip. In particular, the case of an on-site no-slip boundary condition is naturally included. We test the boundary condition in several setups and confirm that it is capable to accurately model the velocity field up to second order and does not contain any numerical slip.
We study the O(n) loop model on a dynamically triangulated disk, with a new type of boundary conditions, discovered recently by Jacobsen and Saleur. The partition function of the model is that of a gas of self and mutually avoiding loops covering the disk. The Jacobsen-Saleur (JS) boundary condition prescribes that the loops that do not touch the boundary have fugacity n in [-2,2], while the loops touching at least once the boundary are given different fugacity y. The class of JS boundary conditions, labeled by the real number y, contains the Neumann (y=n) and Dirichlet (y=1) boundary conditions as particular cases. Here we consider the dense phase of the loop gas, where we compute the two-point boundary correlators of the L-leg operators with mixed Neumann-JS boundary condition. The result coincides with the boundary two-point function in Liouville theory, derived by Fateev, Zamolodchikov and Zamolodchikov. The Liouville charge of the boundary operators match, by the KPZ correspondence, with the L-leg boundary exponents conjectured by JS.
The effects of locally random magnetic fields are considered in a nonequilibrium Ising model defined on a square lattice with nearest-neighbors interactions. In order to generate the random magnetic fields, we have considered random variables $\{h\}$ that change randomly with time according to a double-gaussian probability distribution, which consists of two single gaussian distributions, centered at $+h_{o}$ and $-h_{o}$, with the same width $\sigma$. This distribution is very general, and can recover in appropriate limits the bimodal distribution ($\sigma\to 0$) and the single gaussian one ($ho=0$). We performed Monte Carlo simulations in lattices with linear sizes in the range $L=32 - 512$. The system exhibits ferromagnetic and paramagnetic steady states. Our results suggest the occurence of first-order phase transitions between the above-mentioned phases at low temperatures and large random-field intensities $h_{o}$, for some small values of the width $\sigma$. By means of finite size scaling, we estimate the critical exponents in the low-field region, where we have continuous phase transitions. In addition, we show a sketch of the phase diagram of the model for some values of $\sigma$. Comment: 13 pages, 9 figures, accepted for publication in JSTAT
Spin-spin correlation as a function of r for constant (solid line) and variable (pointed line) defect strength. We set β = 10. The dashed line indicates the defect-free universal behavior r −1/4 .
We consider the critical spin-spin correlation function of the 2D Ising model with a line defect which strength is an arbitrary function of position. By using path-integral techniques in the continuum description of this model in terms of fermion fields, we obtain an analytical expression for the correlator as functional of the position dependent coupling. Thus, our result provides one of the few analytical examples that allows to illustrate the transit of a magnetic system from scaling to non-scaling behavior in a critical regime. We also show that the non-scaling behavior obtained for the spin correlator along a non-uniformly altered line of an Ising model remains unchanged in the Ashkin-Teller model.
Potential U (φ) of a quantum field theory with kink excitations. 
Residue equation for the matrix element on the kink states. 
We study the spectrum of Landau-Ginzburg theories in 1+1 dimensions using the truncated conformal space approach employing a compactified boson. We study these theories both in their broken and unbroken phases. We first demonstrate that we can reproduce the expected spectrum of a $\Phi^2$ theory (i.e. a free massive boson) in this framework. We then turn to $\Phi^4$ in its unbroken phase and compare our numerical results with the predictions of two-loop perturbation theory, finding excellent agreement. We then analyze the broken phase of $\Phi^4$ where kink excitations together with their bound states are present. We confirm the semiclassical predictions for this model on the number of stable kink-antikink bound states. We also test the semiclassics in the double well phase of $\Phi^6$ Landau-Ginzburg theory, again finding agreement.
Helicity modulus Υ 2 of the 5-state clock model at the inverse temperature β = 1/k B T = 1.11 versus e −m/70 where m is the number of states of the truncated Hilbert spaces of the left and right blocks during the DMRG process. The different curves correspond to different lattice sizes L and the straight lines to linear fits.
Scaling function F Υ (up to a factor) of the 6-state clock model with respect to the scaling argument (β BKT − β) (ln L/L 0 ) 2. The different symbols correspond to different lattice sizes. The left figure corresponds to the high-temperature BKT transition while the right one corresponds to the lowtemperature transition.
The two Berezinskii-Kosterlitz-Thouless phase transitions of the two-dimensional 5-state clock model are studied on infinite strips using the DMRG algorithm. Because of the open boundary conditions, the helicity modulus $\Upsilon_2$ is computed by imposing twisted magnetic fields at the two boundaries. Its scaling behavior is in good agreement with the existence of essential singularities with $\sigma=1/2$ at the two transitions. The predicted universal values of $\Upsilon_2$ are shown to be reached in the thermodynamic limit. The fourth-order helicity modulus is observed to display a dip at the low-temperature BKT transition, like the XY model. Finally, the scaling behavior of magnetization at the low temperature transition is compatible with $\eta=1/4$.
The 2D Euler equations is the basic example of fluid models for which a microcanical measure can be constructed from first principles. This measure is defined through finite-dimensional approximations and a limiting procedure. Creutz's algorithm is a microcanonical generalization of the Metropolis-Hasting algorithm (to sample Gibbs measures, in the canonical ensemble). We prove that Creutz's algorithm can sample finite-dimensional approximations of the 2D Euler microcanonical measures (incorporating fixed energy and other invariants). This is essential as microcanonical and canonical measures are known to be inequivalent at some values of energy and vorticity distribution. Creutz's algorithm is used to check predictions from the mean-field statistical mechanics theory of the 2D Euler equations (the Robert-Sommeria-Miller theory). We found full agreement with theory. Three different ways to compute the temperature give consistent results. Using Creutz's algorithm, a first-order phase transition never observed previously, and a situation of statistical ensemble inequivalence are found and studied. Strikingly, and contrasting usual statistical mechanics interpretations, this phase transition appears from a disordered phase to an ordered phase (with less symmetries) when energy is increased. We explain this paradox.
We examine the scaling behavior of the entanglement entropy for the 2D quantum dimer model (QDM) at criticality and derive the universal finite sub-leading correction $\gamma_{QCP}$. We compute the value of $\gamma_{QCP}$ without approximation working directly with the wave function of a generalized 2D QDM at the Rokhsar-Kivelson QCP in the continuum limit. Using the replica approach, we construct the conformal boundary state corresponding to the cyclic identification of $n$-copies along the boundary of the observed region. We find that the universal finite term is $\gamma_{QCP}=\ln R$ where $R$ is the compactification radius of the bose field theory quantum Lifshitz model, the effective field theory of the 2D QDM at quantum criticality. It is shown that, at least in the cylinder geometry, no other universal contributions to the entanglement entropy are possible. We also demonstrated that the entanglement spectrum of the critical wave function on a large but finite region is described by the characters of the underlying conformal field theory. A geometric interpretation is given to the R{\'e}nyi entropy for this system at criticality, and it is shown that this is formally related to the problems of quantum Brownian motion on $n$-dimensional lattices or equivalently a system of strings interacting with a brane containing a background electromagnetic field and can be written as an expectation value of a vertex operator.
We study scaling of the ground-state fidelity in neighborhoods of quantum critical points in a model of interacting spinfull fermions - a BCS-like model. Due to the exact diagonalizability of the model, in one and higher dimensions, scaling of the ground-state fidelity can be analyzed numerically with great accuracy, not only for small systems but also for macroscopic ones, together with the crossover region between them. Additionally, in one-dimensional case we have been able to derive a number of analytical formulae for fidelity and show that they accurately fit our numerical results; these results are reported in the article. Besides regular critical points and their neighborhoods, where well-known scaling laws are obeyed, there is the multi-critical point and critical points in its proximity where anomalous scaling behavior is found. We consider also scaling of fidelity in neighborhoods of critical points where fidelity oscillates strongly as the system size or the chemical potential is varied. Our results for a one-dimensional version of a BCS-like model are compared with those obtained by Rams and Damski in similar studies of a quantum spin chain - an anisotropic XY model in transverse magnetic field.
The distribution of internal shear stresses in a 2D dislocation system is investigated for when external shear stress is applied. This problem serves as a natural continuation of the previous work of Csikor and Groma (2004 Phys. Rev. B 58 2969), where an analytical result was given for the stress distribution function at zero applied stress. First, the internal stress distribution generated by a set of randomly positioned ideal dislocation dipoles is studied. Analytical calculations are carried out for this case. The theoretical predictions are checked by numerical simulations, showing perfect agreement. It is found that for real relaxed dislocation configurations the role of dislocation multipoles cannot be neglected, but the theory presented can still be applied.
Size dependence of the maxima of the specific heat for the pure (filled squares; data taken from reference [108]) and the random bond (filled and open circles for r = 0.818 and open triangles for r = 0.6) square SAF model. The inset shows a power law fit for the case r = 0.818 for L ≥ 80 giving a negative value for the exponent α/ν of the order of −0.12(6).
We investigate and contrast, via entropic sampling based on the Wang-Landau algorithm, the effects of quenched bond randomness on the critical behavior of two Ising spin models in 2D. The random bond version of the superantiferromagnetic (SAF) square model with nearest- and next-nearest-neighbor competing interactions and the corresponding version of the simple Ising model are studied and their general universality aspects are inspected by a detailed finite-size scaling (FSS) analysis. We find that, the random bond SAF model obeys weak universality, hyperscaling, and exhibits a strong saturating behavior of the specific heat due to the competing nature of interactions. On the other hand, for the random Ising model we encounter some difficulties for a definite discrimination between the two well-known scenarios of the logarithmic corrections versus the weak universality. Yet, a careful FSS analysis of our data favors the field-theoretically predicted logarithmic corrections.
Cross sectional slices through the 3D volumes in (Q h , Q k , ћω) space rendered from the MAPS data. Colour shading is the intensity in neutron scattering with red stronger and blue weaker. Slices are shown for temperatures a) 21.3 K which is well below T N , in the three-dimensionally ordered phase. Well defined transverse spin waves are observed across the Brillouin zone. b) 46.9 K, above T N but with significant short range order. Spin waves are observed although with increased lifetime broadening. The spin waves are over-damped for wave lengths longer than the correlation length. c) 100.7 K, above the Curie Weiss temperature. Over-damped behaviour alone is seen in the paramagnetic phase. Note that these slices are taken from the measured data. No background has been subtracted and in particular incoherent scattering is seen at zero energy transfer.
Crossovers in temperature and wave-vector of the dynamical behaviour in Rb 2 MnF 4. Above the Curie-Weiss transition temperature, 5 . 87 = Θ CW K [24], the material is paramagnetic and the dynamics is over-damped. Also, for wave-vectors smaller than the inverse correlation length, taken from Lee et al [3], our data and computations also show over-damped behaviour. At small wave vectors, ZB q q << , dynamical scaling is proposed and has been observed at k=0 for Rb 2 MnF 4 in a field by Christianson et al. [12]. The field counters the small anisotropy and promotes the scaling behaviour. Spin-waves are observed over the rest of the wave-vector and temperature region i.e. ( ) κ << T q and
We report comprehensive inelastic neutron scattering measurements of the magnetic excitations in the 2D spin-5/2 Heisenberg antiferromagnet Rb2MnF4 as a function of temperature from deep in the Neel ordered phase up to paramagnetic, 0.13 < kBT/4JS < 1.4. Well defined spin-waves are found for wave-vectors larger than the inverse correlation length $\eta^{-1}$ for temperatures up to near the Curie-Weiss temperature, $\Theta_{CW}$. For wave-vectors smaller than $\eta^{-1}$, relaxational dynamics occurs. The observed renormalization of spin-wave energies, and evolution of excitation line-shapes, with increasing temperature are quantitatively compared with finite-temperature spin-wave theory, and computer simulations for classical spins. Random phase approximation calculations provide a good description of the low-temperature renormalisation of spin-waves. In contrast, lifetime broadening calculated using the first Born approximation shows, at best, modest agreement around the zone boundary at low temperatures. Classical dynamics simulations using an appropriate quantum-classical correspondence were found to provide a good description of the intermediate- and high-temperature regimes over all wave-vector and energy scales, and the crossover from quantum to classical dynamics observed around $\Theta_{CW}/S$, where the spin S=5/2. A characterisation of the data over the whole wave-vector/energy/temperature parameter space is given. In this, $T^2$ behaviour is found to dominate the wave-vector and temperature dependence of the line widths over a large parameter range, and no evidence of hydrodynamic behaviour or dynamical scaling behaviour found within the accuracy of the data sets.
We present a thorough study of the static properties of 2D models of spin-ice type on the square lattice or, in other words, the sixteen-vertex model. We use extensive Monte Carlo simulations to determine the phase diagram and critical properties of the finite dimensional system. We put forward a suitable mean-field approximation, by defining the model on carefully chosen trees. We employ the cavity (Bethe-Peierls) method to derive self-consistent equations, the fixed points of which yield the equilibrium properties of the model on the tree-like graph. We compare mean-field and finite dimensional results. We discuss our findings in the context of experiments in artificial two dimensional spin ice.
We investigate the out of equilibrium dynamics of the two-dimensional XY model when cooled across the Berezinskii-Kosterlitz-Thouless (BKT) phase transition using different protocols. We focus on the evolution of the growing correlation length and the density of topological defects (vortices). By using Monte Carlo simulations we first determine the time and temperature dependence of the growing correlation length after an infinitely rapid quench from above the transition temperature to the quasi-long range order region. The functional form is consistent with a logarithmic correction to the diffusive law and it serves to validate dynamic scaling in this problem. This analysis clarifies the different dynamic roles played by bound and free vortices. We then revisit the Kibble-Zurek mechanism in thermal phase transitions in which the disordered state is plagued with topological defects. We provide a theory of quenching rate dependence in systems with the BKT-type transition that goes beyond the equilibrium scaling arguments. Finally, we discuss the implications of our results to a host of physical systems with vortex excitations including planar ferromagnets and liquid crystals as well as the Ginzburg-Landau approach to bidimensional freely decaying turbulence.
Strength of resonant activation as measured by the relative depth of: mean first passage time τ and inter-quantile width q0.9 − q0.1 for 2D (top panel) and 3D (bottom panel) setups.  
Resonant activation is one of classical effects demonstrating constructive role of noise. In resonant activation cooperative action of barrier modulation process and noise lead to the optimal escape kinetics as measured by the mean first passage time. Resonant activation has been observed in versatilities of systems for various types of barrier modulation processes and noise types. Here, we show that resonant activation is also observed in 2D and 3D systems driven by bi-variate and tri-variate $\alpha$-stable noises. Strength of resonant activation is sensitive to the exact value of the noise parameters. In particular, the decrease in the stability index $\alpha$ results in the disappearance of the resonant activation.
We discuss invariant measures of partial differential equations such as the 2D Euler or Vlasov equations. For the 2D Euler equations, starting from the Liouville theorem, valid for N-dimensional approximations of the dynamics, we define the microcanonical measure as a limit measure where N goes to infinity. When only the energy and enstrophy invariants are taken into account, we give an explicit computation to prove the following result: the microcanonical measure is actually a Young measure corresponding to the maximization of a mean-field entropy. We explain why this result remains true for more general microcanonical measures, when all the dynamical invariants are taken into account. We give an explicit proof that these microcanonical measures are invariant measures for the dynamics of the 2D Euler equations. We describe a more general set of invariant measures and discuss briefly their stability and their consequence for the ergodicity of the 2D Euler equations. The extension of these results to the Vlasov equations is also discussed, together with a proof of the uniqueness of statistical equilibria, for Vlasov equations with repulsive convex potentials. Even if we consider, in this paper, invariant measures only for Hamiltonian equations, with no fluxes of conserved quantities, we think this work is an important step towards the description of non-equilibrium invariant measures with fluxes.
We study the behaviour of the 2d Ising model in the symmetric high temperature phase in presence of a small magnetic perturbation. We successfully compare the quantum field theory predictions for the shift in the mass spectrum of the theory with a set of high precision transfer matrix results. Our results rule out a prediction for the same quantity obtained some years ago with strong coupling methods.
We study the fate of the 2d kinetic q-state Potts model after a sudden quench to zero temperature. Both ground states and complicated static states are reached with non-zero probabilities. These outcomes resemble those found in the quench of the 2d Ising model; however, the variety of static states in the q-state Potts model (with q>=3) is much richer than in the Ising model, where static states are either ground or stripe states. Another possibility is that the system gets trapped on a set of equal-energy blinker states where a subset of spins can flip ad infinitum; these states are similar to those found in the quench of the 3d Ising model. The evolution towards the final energy is also unusual---at long times, sudden and massive energy drops may occur that are accompanied by macroscopic reordering of the domain structure. This indeterminacy in the zero-temperature quench of the kinetic Potts model is at odds with basic predictions from the theory of phase-ordering kinetics. We also propose a continuum description of coarsening with more than two equivalent ground states. The resulting time-dependent Ginzburg-Landau equations reproduce the complex cluster patterns that arise in the quench of the kinetic Potts model.
In this article, we show that the double scaling limit correlation functions of a random matrix model when two cuts merge with degeneracy $2m$ (i.e. when $y\sim x^{2m}$ for arbitrary values of the integer $m$) are the same as the determinantal formulae defined by conformal $(2m,1)$ models. Our approach follows the one developed by Berg\`{e}re and Eynard in \cite{BergereEynard} and uses a Lax pair representation of the conformal $(2m,1)$ models (giving Painlev\'e II integrable hierarchy) as suggested by Bleher and Eynard in \cite{BleherEynard}. In particular we define Baker-Akhiezer functions associated to the Lax pair to construct a kernel which is then used to compute determinantal formulae giving the correlation functions of the double scaling limit of a matrix model near the merging of two cuts.
Top-cited authors
Vincent D. Blondel
  • Université Catholique de Louvain - UCLouvain
Renaud Lambiotte
  • University of Oxford
Jean-Loup Guillaume
  • La Rochelle Université
Pasquale Calabrese
  • University of Basel
Albert Diaz-Guilera
  • University of Barcelona