No full-text available
To read the full-text of this research,
you can request a copy directly from the authors.
Emptiness formation probability and Painlevé V equation in the XY spin chain
... Whenever possible, comments will be made about the existence of universal terms and their comparison with field theory predictions. We will also obtain a universal formula in the scaling limit for the full counting statistics of the transverse magnetization and the domain walls by solving a Painlevé V equation [38,39]. The latter result applies to any system close to a quantum critical point within the Ising universality class. ...
... The function τ V is constructed in such a way that for x → 0, at short distances, log χ Mz (λ; x) coincides with Eq. (20), while for x → ∞, at large distances, log χ Mz (λ; x) is given by the Szegő theorem in Eq. (19). Following the ideas introduced in [38,39], to which we refer for any additional details, it is possible to write down an explicit power series expansion about x = 0 of the function τ V . The latter could be also extended to γ = 0, provided [39] one considers the scaling variable x = 2L| log |h||/γ; the first few terms of such an expansion are (34) where s 0 = −ψ(1 + β(λ)) − ψ(1 − β(λ)) + 3ψ(1) + 1, and ψ(z) is the Digamma function. ...
... Following the ideas introduced in [38,39], to which we refer for any additional details, it is possible to write down an explicit power series expansion about x = 0 of the function τ V . The latter could be also extended to γ = 0, provided [39] one considers the scaling variable x = 2L| log |h||/γ; the first few terms of such an expansion are (34) where s 0 = −ψ(1 + β(λ)) − ψ(1 − β(λ)) + 3ψ(1) + 1, and ψ(z) is the Digamma function. The expansion of τ V until order O(x 4 ) can be also straightforwardly obtained from [39]. ...
We calculate exactly cumulant generating functions (full counting statistics) for the transverse, staggered magnetization and the domain walls at zero temperature for a finite interval of the XY spin chain. In particular, we also derive a universal interpolation formula in the scaling limit for the full counting statistics of the transverse magnetization and the domain walls which is based on the solution of a Painlev\'e V equation. By further determining subleading corrections in a large interval asymptotics, we are able to test the applicability of conformal field theory predictions at criticality. As a byproduct, we also obtain exact results for the probability of formation of ferromagnetic and antiferromagnetic domains in both and basis in the ground state. The analysis hinges upon asympotic expansions of block Toeplitz determinants, for which we formulate and check numerically a new conjecture.
... For the 3 Author to whom any correspondence should be addressed. XY spin chain, exact expressions were obtained via free-fermion methods [2][3][4][5]. For the XXZ spin chain, techniques from quantum integrability allowed several authors to exactly compute the emptiness formation probability [6][7][8]. ...
... For |q| = 1 and φ ∈ R, they can thus be simultaneously diagonalised. 5 These values of q correspond to real values of the anisotropy parameter ∆ in the range [−1, 1]. ...
... To simplify further the determinant formula (6.15), we note the following identity for the Chebyshev polynomials, (6.17) which allows us to rewrite the matrix entries of M (5) in product form. In the resulting expressions, the factorials in the denominators coming from the right side of (6.17) depend only on the row label i and are factorised out of the denominator. ...
We define a new family of overlaps C N,m for the XXZ Hamiltonian on a periodic chain of length N. These are equal to the linear sums of the groundstate components, in the canonical basis, wherein m consecutive spins are fixed to the state ↑. We define the boundary emptiness formation probabilities as the ratios C N,m /C N,0 of these overlaps. In the associated six-vertex model, they correspond to correlation functions on a semi-infinite cylinder of perimeter N. At the combinatorial point Δ = − 1 2 , we obtain closed-form expressions in terms of simple products of ratios of integers.
... One can write the ground state in a configuration basis and look to the probabilities of different configurations. These probabilities are dubbed as formation probabilities and have been studied for subsystems of certain free fermions in depth [13,[16][17][18][19]. For results on the full system see [3,20]. ...
... In all of the considered models we first find the 2 L number of probabilities using Eq. (18). The largest size that we considered was L max = 42. ...
The Rényi (Shannon) entropy, i.e., Reα(Sh), of the ground state of quantum systems in local bases normally show a volume-law behavior. For a subsystem of quantum chains at a critical point there is an extra logarithmic subleading term with a coefficient which is universal. In this paper we study this coefficient for generic time-reversal translational invariant quadratic critical free fermions. These models can be parametrized by a complex function which has zeros on the unit circle. When the zeros on the unit circle do not have degeneracy and there is no zero outside of the unit circle we are able to classify the coefficient of the logarithm. In particular, we numerically calculate the Rényi (Shannon) entropy in a configuration basis for a wide variety of these models and show that there are two distinct classes. For systems with U(1) symmetry the coefficient is proportional to the central charge, i.e., one half of the number of points that one can linearize the dispersion relation of the system; for all the values of α with transition point at α=4. For systems without this symmetry, when α>1, this coefficient is again proportional to the central charge. However, the coefficient for α≤1 is a new universal number. Finally, by using the discrete version of the Bisognano-Wichmann modular Hamiltonian of the Ising chain we show that these coefficients are universal and dependent on the underlying CFT.
... One can write the ground state in configuration basis and look to the probabilities of different configurations. These probabilities are dubbed as formation probabilities and have been studied for subsystems of certain free fermions in depth [13,[16][17][18][19]. For results on the full system see [3,20]. ...
... In all of the considered models we first find the 2 L number of probabilities using the Eq. (18). The largest size that we considered was L max = 42. ...
The R\'enyi (Shannon) entropy, i.e. , of the ground state of quantum systems in local bases normally show a volume-law behavior. For a subsystem of quantum chains at critical point there is an extra logarithmic subleading term with a coefficient which is universal. In this paper we study this coefficient for generic time-reversal translational invariant quadratic critical free fermions. These models can be parameterized by a complex function which has zeros on the unit circle. When the zeros on the unit circle do not have degeneracy and there is no zero outside of the unit circle we are able to classify the coefficient of the logarithm. In particular, we numerically calculate the R\'enyi (Shannon) entropy in configuration basis for wide variety of these models and show that there are two distinct classes. For systems with U(1) symmetry the coefficient is proportional to the central charge, i.e. one half of the number of points that one can linearize the dispersion relation of the system; for all the values of with transition point at . For systems without this symmetry, when this coefficient is again proportional to the central charge. However, the coefficient for is a new universal number. Finally, by using the discrete version of Bisognano-Wichmann modular Hamiltonian of the Ising chain we show that these coefficients are universal and dependent on the underlying CFT.
... The simplest instance is perhaps the probability to observe the totality of the spins on a finite interval of the chain pointing up or down, which is the so-called Emptiness Formation Probability. Such a quantity has been calculated analytically in a few integrable quantum chains, see [1,2,3,4,5,6,7,8]. In the scaling limit, next to a critical point, the Emptiness Formation Probability can be interpreted as statistical mechanics partition function on a cylinder or a strip with suitable boundary conditions [9]. ...
... , i 2r } which are in correspondence with the positions of the down spins in the state |σ . As a side remark, we observe that M in Eq. (11) has the same formal structure as the correlation matrix derived in [7] to evaluate the Emptiness Formation Probability, see also [8]. ...
We calculate exactly the probability to find the ground state of the XY chain in a given spin configuration in the transverse -basis. By determining finite-volume corrections to the probabilities for a wide variety of configurations, we obtain the universal Boundary Entropy at the critical point. The latter is a benchmark of the underlying Boundary Conformal Field Theory characterizing each quantum state. To determine the scaling of the probabilities, we prove a theorem that expresses, in a factorized form, the eigenvalues of a sub-matrix of a circulant matrix as functions of the eigenvalues of the original matrix. Finally, the Boundary Entropies are computed by exploiting a generalization of the Euler-MacLaurin formula to non-differentiable functions. It is shown that, in some cases, the spin configuration can flow to a linear superposition of Cardy states. Our methods and tools are rather generic and can be applied to all the periodic quantum chains which map to free-fermionic Hamiltonians.
... The probabilities of finding different states in this basis are called formation probabilities [40]. These probabilities, especially the emptiness formation probability, have been the subject of extensive research [28,34,40,[43][44][45][46][47][48][49][50][51][52][53][54][55]. The Shannon entropy, which is dependent on the choice of basis, is more experimentally accessible than the basis-independent entanglement entropy. ...
We investigate the Shannon entropy of the total system and its subsystems, as well as the subsystem Shannon mutual information, in quasiparticle excited states of free bosonic and fermionic chains and the ferromagnetic phase of the spin-1/2 XXX chain. For single-particle and double-particle states, we derive various analytical formulas for free bosonic and fermionic chains in the scaling limit. These formulas are also applicable to certain magnon excited states in the XXX chain in the scaling limit. We also calculate numerically the Shannon entropy and mutual information for triple-particle and quadruple-particle states in bosonic, fermionic, and XXX chains. We discover that Shannon entropy, unlike entanglement entropy, typically does not separate for quasiparticles with large momentum differences. Moreover, in the limit of large momentum difference, we obtain universal quantum bosonic and fermionic results that are generally distinct and cannot be explained by a semiclassical picture.
... The probabilities of finding different states in this basis are called formation probabilities [17]. These probabilities, especially the empty formation probability, have been the subject of extensive research [5,11,17,[20][21][22][23][24][25][26][27][28][29][30][31][32]. ...
In this paper, we investigate the Shannon entropy of the total system and its subsystems, as well as the subsystem Shannon mutual information, in quasiparticle excited states of free bosonic and fermionic chains and the ferromagnetic phase of the spin-1/2 XXX chain. Our focus is on single-particle and double-particle states, and we derive various analytical formulas for free bosonic and fermionic chains in the scaling limit. These formulas are also applicable to magnon excited states in the XXX chain under certain conditions. We discover that, unlike entanglement entropy, Shannon entropy does not separate when two quasiparticles have a large momentum difference. Moreover, in the large momentum difference limit, we obtain universal results for quantum spin chains that cannot be explained by a semiclassical picture of quasiparticles.
... We find it worth mentioning also that the sine kernel tau function (4.20) evaluated at z = 1 provides the emptiness formation probability [202][203][204] of the XX chain and in the double scaling limit [205][206][207][208]. More recently, the approach of [77] has been extended to compute the EE of two disjoint intervals on the XX chain separated by a single site [181]. ...
A bstract
We study the entanglement entropies of an interval on the infinite line in the free fermionic spinless Schrödinger field theory at finite density and zero temperature, which is a non-relativistic model with Lifshitz exponent z = 2. We prove that the entanglement entropies are finite functions of one dimensionless parameter proportional to the area of a rectangular region in the phase space determined by the Fermi momentum and the length of the interval. The entanglement entropy is a monotonically increasing function. By employing the properties of the prolate spheroidal wave functions of order zero or the asymptotic expansions of the tau function of the sine kernel, we find analytic expressions for the expansions of the entanglement entropies in the asymptotic regimes of small and large area of the rectangular region in the phase space. These expansions lead to prove that the analogue of the relativistic entropic C function is not monotonous. Extending our analyses to a class of free fermionic Lifshitz models labelled by their integer dynamical exponent z , we find that the parity of this exponent determines the properties of the bipartite entanglement for an interval on the line.
... The blocks areD mm = ∆ m−m andÑ mm = N m−m . The EFP has been extensively studied in the Ising/XY chains at equilibrium [92][93][94][95], in which case the block Toeplitz matrics simplifies to a simple Toeplitz matrix (after rotation to the Majorana fermionic operators) due toD being real. The latter is not the case in the state after the linear quench that we consider here and, in general, one has to deal with a full block Toeplitz matrix. ...
Linear quench of the transverse field drives the quantum Ising chain across a quantum critical point from the paramagnetic to the ferromagnetic phase. We focus on normal and anomalous quadratic correlators between fermionic kink creation and annihilation operators. They depend not only on the Kibble-Zurek (KZ) correlation length but also on a dephasing length scale, which differs from the KZ length by a logarithmic correction. Additional slowing down of the ramp in the ferromagnetic phase further increases the dephasing length and suppresses the anomalous correlator. The quadratic correlators enter Pfaffians that yield experimentally relevant kink correlation functions, the probability distribution of ferromagnetic domain sizes, and, closely related, emptiness formation probability. The latter takes the form of a Pfaffian of a block Toeplitz matrix that allows for some analytic asymptotes. Finally, we obtain further insight into the structure of the state at the end of the ramp by interpreting it as a paired state of fermionic kinks characterized by its pair wave function. All those quantities are sensitive to quantum coherence between eigenstates with different numbers of kinks, thus making them a convenient probe of the quantumness of a quantum simulator platform.
... In this subsection we study non-local quantities called formation probabilities. The most known of this kind of observables is the emptiness formation probability (EFP) P ± which has a long history, see [65][66][67][68][69][70][71][72][73][74][75][76][77]. P + (P − ) is the probability of having all the spins in a block of length pointing up (down) in the σ z j basis. ...
We carry out a comprehensive comparison between the exact modular Hamiltonian
and the lattice version of the Bisognano-Wichmann (BW) one in one-dimensional
critical quantum spin chains. As a warm-up, we first illustrate how the trace
distance provides a more informative mean of comparison between reduced density
matrices when compared to any other Schatten n-distance, normalized or not.
In particular, as noticed in earlier works, it provides a way to bound other
correlation functions in a precise manner, i.e., providing both lower and upper
bounds. Additionally, we show that two close reduced density matrices, i.e.
with zero trace distance for large sizes, can have very different modular
Hamiltonians. This means that, in terms of describing how two states are close
to each other, it is more informative to compare their reduced density matrices
rather than the corresponding modular Hamiltonians. After setting this
framework, we consider the ground states for infinite and periodic XX spin
chain and critical Ising chain. We provide robust numerical evidence that the
trace distance between the lattice BW reduced density matrix and the exact one
goes to zero as for large length of the interval . This
provides strong constraints on the difference between the corresponding
entanglement entropies and correlation functions. Our results indicate that
discretized BW reduced density matrices reproduce exact entanglement entropies
and correlation functions of local operators in the limit of large subsystem
sizes. Finally, we show that the BW reduced density matrices fall short of
reproducing the exact behavior of the logarithmic emptiness formation
probability in the ground state of the XX spin chain.
... In this subsection we study non-local quantities called formation probabilities. The most known of this kind of observables is the emptiness formation probability (EFP) P ± which has a long history, see [65][66][67][68][69][70][71][72][73][74][75][76][77]. P + (P − ) is the probability of having all the spins in a block of length pointing up (down) ...
We carry out a comprehensive comparison between the exact modular Hamiltonian and the lattice version of the Bisognano-Wichmann (BW) one in one-dimensional critical quantum spin chains. As a warm-up, we first illustrate how the trace distance provides a more informative mean of comparison between reduced density matrices when compared to any other Schatten n-distance, normalized or not. In particular, as noticed in earlier works, it provides a way to bound other correlation functions in a precise manner, i.e., providing both lower and upper bounds. Additionally, we show that two close reduced density matrices, i.e. with zero trace distance for large sizes, can have very different modular Hamiltonians. This means that, in terms of describing how two states are close to each other, it is more informative to compare their reduced density matrices rather than the corresponding modular Hamiltonians. After setting this framework, we consider the ground states for infinite and periodic XX spin chain and critical Ising chain. We provide robust numerical evidence that the trace distance between the lattice BW reduced density matrix and the exact one goes to zero as for large length of the interval . This provides strong constraints on the difference between the corresponding entanglement entropies and correlation functions. Our results indicate that discretized BW reduced density matrices reproduce exact entanglement entropies and correlation functions of local operators in the limit of large subsystem sizes. Finally, we show that the BW reduced density matrices fall short of reproducing the exact behavior of the logarithmic emptiness formation probability in the ground state of the XX spin chain.
Linear quench of the transverse field drives the quantum Ising chain across a quantum critical point from the paramagnetic to the ferromagnetic phase. We focus on normal and anomalous quadratic correlators between fermionic kink creation and annihilation operators. They depend not only on the Kibble-Zurek (KZ) correlation length but also on a dephasing length scale, which differs from the KZ length by a logarithmic correction. Additional slowing down of the ramp in the ferromagnetic phase further increases the dephasing length and suppresses the anomalous correlator. The quadratic correlators enter Pfaffians that yield experimentally relevant kink correlation functions, the probability distribution of ferromagnetic domain sizes, and, closely related, emptiness formation probability. The latter takes the form of a Pfaffian of a block Toeplitz matrix that allows for some analytic asymptotes. Finally, we obtain further insight into the structure of the state at the end of the ramp by interpreting it as a paired state of fermionic kinks characterized by its pair wave function. All those quantities are sensitive to quantum coherence between eigenstates with different numbers of kinks, thus making them a convenient probe of the quantumness of a quantum simulator platform.
We calculate exactly cumulant generating functions (full counting statistics) for the transverse, staggered magnetization, and the domain walls at zero temperature for a finite interval of the XY spin chain. In particular, we also derive a universal interpolation formula in the scaling limit for the full counting statistics of the transverse magnetization and the domain walls which is based on the solution of a Painlevé V equation. By further determining subleading corrections in a large interval asymptotics, we are able to test the applicability of conformal field theory predictions at criticality. As a by-product, we also obtain exact results for the probability of formation of ferromagnetic and antiferromagnetic domains in both the σz and σx basis in the ground state. The analysis hinges upon asymptotic expansions of block Toeplitz determinants, for which we formulate and check numerically a different conjecture.
We show that the computation of formation probabilities (FP's) in the configuration basis and the full counting statistics of observables in quadratic fermionic Hamiltonians are equivalent to the calculation of emptiness formation probability (EFP) in the Hamiltonian with a defect. In particular, we first show that the FP of finding a particular configuration in the ground state is equivalent to the EFP of the ground state of the quadratic Hamiltonian with a defect. Then, we show that the probability of finding a particular value for any quadratic observable is equivalent to a FP problem and ultimately leads to the calculation of EFP in the ground state of a Hamiltonian with a defect. We provide exact determinant formulas for the FP in generic quadratic fermionic Hamiltonians. For applications of our formalism we study the statistics of the number of particles and kinks. Our conclusions can be extended also to quantum spin chains, which can be mapped to free fermions via Jordan-Wigner transformation. In particular, we provide an exact solution to the problem of the transverse field XY chain with a staggered line defect. We also study the distribution of magnetization and kinks in the transverse field XY chain numerically and show how the dual nature of these quantities manifests itself in the distributions.
We consider the symmetry resolved Rényi entropies in the one dimensional tight binding model, equivalent to the spin-1/2 XX chain in a magnetic field. We exploit the generalised Fisher–Hartwig conjecture to obtain the asymptotic behaviour of the entanglement entropies with a flux charge insertion at leading and subleading orders. The contributions are found to exhibit a rich structure of oscillatory behaviour. We then use these results to extract the symmetry resolved entanglement, determining exactly all the non-universal constants and logarithmic corrections to the scaling that are not accessible to the field theory approach. We also discuss how our results are generalised to a one-dimensional free fermi gas.
A bstract
We show in details that the all order genus expansion of the two-cut Hermitian cubic matrix model reproduces the perturbative expansion of the H 1 Argyres-Douglas theory coupled to the Ω background. In the self-dual limit we use the Painlevé/gauge correspondence and we show that, after summing over all instanton sectors, the two-cut cubic matrix model computes the tau function of Painlevé II without taking any double scaling limit or adding any external fields. We decode such solution within the context of transseries. Finally in the Nekrasov-Shatashvili limit we connect the H 1 and the H 0 Argyres-Douglas theories to the quantum mechanical models with cubic and double well potentials.
We develop the theory of irregular conformal blocks of the Virasoro algebra. In previous studies, expansions of irregular conformal blocks at regular singular points were obtained as degeneration limits of regular conformal blocks; however, such expansions at irregular singular points were not clearly understood. This is because precise definitions of irregular vertex operators had not been provided previously. In this paper, we present precise definitions of irregular vertex operators of two types and we prove that one of our vertex operators exists uniquely. Then, we define irregular conformal blocks with at most two irregular singular points as expectation values of given irregular vertex operators. Our definitions provide an understanding of expansions of irregular conformal blocks and enable us to obtain expansions at irregular singular points. As an application, we propose conjectural formulas of series expansions of the tau functions of the fifth and fourth Painlevé equations, using expansions of irregular conformal blocks at an irregular singular point.
We find a connection between logarithmic formation probabilities of two
disjoint intervals of quantum critical spin chains and the Casimir energy of
two aligned needles in two dimensional classical critical systems. Using this
connection we provide a formula for the logarithmic formation probability of
two disjoint intervals in generic 1+1 dimensional critical systems. The
quantity is depenedent on the full structure of the underlying conformal field
theory and so useful to find the universality class of the critical system. The
connection that we find also provides a very efficient numerical method to
calculate the Casimir energy between needles using quantum critical chains. The
agreement between numerical results performed on critical transverse field
Ising model and XX chain with our exact results is very good. We also comment
on the mutual R\'enyi information of two disjoint intervals.
We elaborate on a recently conjectured relation of Painlevé transcendents and 2D conformal field theory. General solutions of Painlevé VI, V and III are expressed in terms of c = 1 conformal blocks and their irregular limits, AGT related to instanton partition functions in supersymmetric gauge theories with Nf = 0, 1, 2, 3, 4. The resulting combinatorial series representations of Painlevé functions provide an efficient tool for their numerical computation at finite values of the argument. The series involves sums over bipartitions which, in the simplest cases, coincide with Gessel expansions of certain Toeplitz determinants. Considered applications include Fredholm determinants of classical integrable kernels, scaled gap probability in the bulk of the Gaussian Unitary Ensemble, and all-order conformal perturbation theory expansions of correlation functions in the sine–Gordon field theory at the free-fermion point.
Using its multiple integral representation, we compute the large distance asymptotic behaviour of the emptiness formation probability of the XXZ spin-1/2 Heisenberg chain in the massless regime.
In this paper, we compute the emptiness formation probability of a (twisted-) periodic XXZ spin chain of finite length at , thus proving the formulas conjectured by Razumov and Stroganov (2001 J. Phys. A: Math. Gen.34 3185 and 5335–40). The result is obtained by exploiting the fact that the ground state of the inhomogeneous XXZ spin chain at satisfies a set of qKZ equations associated with .
We propose to describe correlations in classical and quantum systems in terms of full counting statistics of a suitably chosen discrete observable. The method is illustrated with two exactly solvable examples: the classical one-dimensional Ising model and the quantum spin-1/2 XY chain. For the one-dimensional Ising model, our method results in a phase diagram with two phases distinguishable by the long-distance behavior of the Jordan-Wigner strings. For the anisotropic spin-1/2 XY chain in a transverse magnetic field, we compute the full counting statistics of the magnetization and use it to classify quantum phases of the chain. The method, in this case, reproduces the previously known phase diagram. We also discuss the relation between our approach and the Lee-Yang theory of zeros of the partition function.
A bstract
Generic Painlevé VI tau function τ ( t ) can be interpreted as four-point correlator of primary fields of arbitrary dimensions in 2D CFT with c = 1. Using AGT combinatorial representation of conformal blocks and determining the corresponding structure constants, we obtain full and completely explicit expansion of τ ( t ) near the singular points. After a check of this expansion, we discuss examples of conformal blocks arising from Riccati, Picard, Chazy and algebraic solutions of Painlevé VI.
The problem of the form of the ‘arctic’ curve of the six-vertex model with domain wall boundary conditions in its disordered
regime is addressed. It is well-known that in the scaling limit the model exhibits phase-separation, with regions of order
and disorder sharply separated by a smooth curve, called the arctic curve. To find this curve, we study a multiple integral
representation for the emptiness formation probability, a correlation function devised to detect spatial transition from order
to disorder. We conjecture that the arctic curve, for arbitrary choice of the vertex weights, can be characterized by the
condition of condensation of almost all roots of the corresponding saddle-point equations at the same, known, value. In explicit
calculations we restrict to the disordered regime for which we have been able to compute the scaling limit of certain generating
function entering the saddle-point equations. The arctic curve is obtained in parametric form and appears to be a non-algebraic
curve in general; it turns into an algebraic one in the so-called root-of-unity cases. The arctic curve is also discussed
in application to the limit shape of q-enumerated (with 0<q
≤4) large alternating sign matrices. In particular, as q→0 the limit shape tends to a nontrivial limiting curve, given by a relatively simple equation.
Six-vertex model-Domain wall boundary conditions-Correlation functions-Emptiness formation probability-Random matrix models-Spatial phase separation-Condensation hypothesis-Alternating-sign matrices-Limit shapes
Two genuinely quantum mechanical models for an antiferromagnetic linear chain with nearest neighbor interactions are constructed and solved exactly, in the sense that the ground state, all the elementary excitations and the free energy are found. A general formalism for calculating the instantaneous correlation between any two spins is developed and applied to the investigation of short- and long-range order. Both models show nonvanishing long-range order in the ground state for a range of values of a certain parameter λ which is analogous to an anisotropy parameter in the Heisenberg model. A detailed comparison with the Heisenberg model suggests that the latter has no long-range order in the isotropic case but finite long-range order for any finite amount of anisotropy. The unreliability of variational methods for determining long-range order is emphasized. It is also shown that for spin systems having rather general isotropic Heisenberg interactions favoring an antiferromagnetic ordering, the ground state is nondegenerate and there is no energy gap above the ground state in the energy spectrum of the total system.
We demonstrate that for the sine-Gordon theory at the free fermion point, the 2-point correlation functions of the fields eiαΦ for 0 < α < 1 can be parameterized in terms of a solution to the sinh-Gordon equation. This result is derived by summing over intermediate multiparticle states and using the form factors to express this as a Fredholm determinant. The proof of the differential equations relies on a 2 graded multiplication law satisfied by the integral operators of the Fredholm determinant. Using this methodology, we give a new proof of the differential equations which govern the spin and disorder field correlations in the Ising model.
We review recent progress in limit laws for the one-dimensional asymmetric
simple exclusion process (ASEP) on the integer lattice. The limit laws are
expressed in terms of a certain Painlev\'e II function. Furthermore, we take
this opportunity to give a brief survey of the appearance of Painlev\'e
functions in statistical physics.
The short-distance assymptotics of the τ-function associated to the 2-point function of the two-dimensional Ising model is computed as a function of the integration constant defined from the long-distance behavior of the τ-function. The result is expressible in terms of the Barnes double gamma function (equivalently, the BarnesG-function).
In this paper we show how an infinite system of coupled Toda-type nonlinear
differential equations derived by one of us can be used efficiently to
calculate the time-dependent pair-correlations in the Ising chain in a
transverse field. The results are seen to match extremely well long large-time
asymptotic expansions newly derived here. For our initial conditions we use new
long asymptotic expansions for the equal-time pair correlation functions of the
transverse Ising chain, extending an old result of T.T. Wu for the 2d Ising
model. Using this one can also study the equal-time wavevector-dependent
correlation function of the quantum chain, a.k.a. the q-dependent diagonal
susceptibility in the 2d Ising model, in great detail with very little
computational effort.
We study an asymptotic behavior of a special correlator known as the Emptiness Formation Probability (EFP) for the one-dimensional anisotropic XY spin-1/2 chain in a transverse magnetic field. This correlator is essentially the probability of formation of a ferromagnetic string of length n in the antiferromagnetic ground state of the chain and plays an important role in the theory of integrable models. For the XY Spin Chain, the correlator can be expressed as the determinant of a Toeplitz matrix and its asymptotical behaviors for throughout the phase diagram are obtained using known theorems and conjectures on Toeplitz determinants. We find that the decay is exponential everywhere in the phase diagram of the XY model except on the critical lines, i.e. where the spectrum is gapless. In these cases, a power-law prefactor with a universal exponent arises in addition to an exponential or Gaussian decay. The latter Gaussian behavior holds on the critical line corresponding to the isotropic XY model, while at the critical value of the magnetic field the EFP decays exponentially. At small anisotropy one has a crossover from the Gaussian to the exponential behavior. We study this crossover using the bosonization approach. Comment: 40 pages, 9 figures, 1 table. The poor quality of some figures is due to arxiv space limitations. If You would like to see the pdf with good quality figures, please contact Fabio Franchini at "fabio.franchini@stonybrook.edu"
We study a correlation function for the one-dimensional isotropic model ( model), which is called the Emptiness Formation Probability (EFP). It is the probability of the formation of a ferromagnetic string in the anti-ferromagnetic ground state. Using the expression of the EFP as a Toeplitz determinant, we discuss its asymptotic behaviors. We also compare the analytical results with numerical calculations as the density-matrix renormalization group and the quantum Monte-Carlo method. Comment: 20 pages, 7 figures, to be published in J. Phys. Soc. Jpn
The quantum inverse scattering method is a means of finding exact solutions of two-dimensional models in quantum field theory and statistical physics (such as the sine-Gordon equation or the quantum non-linear Schrödinger equation). These models are the subject of much attention amongst physicists and mathematicians. The present work is an introduction to this important and exciting area. It consists of four parts. The first deals with the Bethe ansatz and calculation of physical quantities. The authors then tackle the theory of the quantum inverse scattering method before applying it in the second half of the book to the calculation of correlation functions. This is one of the most important applications of the method and the authors have made significant contributions to the area. Here they describe some of the most recent and general approaches and include some new results. The book will be essential reading for all mathematical physicists working in field theory and statistical physics.
We prove a Fredholm determinant and short-distance series representation of the Painlevé V tau function τt associated with generic monodromy data. Using a relation of τt to two different types of irregular c = 1 Virasoro conformal blocks and the confluence from Painlevé VI equation, connection formulas between the parameters of asymptotic expansions at 0 and i∞ are conjectured. Explicit evaluations of the connection constants relating the tau function asymptotics as t → 0, +∞, i∞ are obtained. We also show that irregular conformal blocks of rank 1, for arbitrary central charge, are obtained as confluent limits of the regular conformal blocks.
Similarly to the system Hamiltonian, a subsystem's reduced density matrix is composed of blocks characterized by symmetry quantum numbers (charge sectors). We present a geometric approach for extracting the contribution of individual charge sectors to the subsystem's entanglement measures within the replica trick method, via threading appropriate conjugate Aharonov-Bohm fluxes through a multi-sheet Riemann surface. Specializing to the case of 1+1D conformal field theory, we obtain general exact results for the entanglement entropies and spectrum, and apply them to a variety of systems, ranging from free and interacting fermions to spin and parafermion chains, and verify them numerically. We find that the total entanglement entropy, which scales as , is composed of contributions of individual subsystem charge sectors for interacting fermion chains, or even contributions when total spin conservation is also accounted for. We also explain how measurements of the contribution to the entanglement from separate charge sectors can be performed experimentally with existing techniques.
Entanglement entropy may display a striking new symmetry under Möbius transformations. This symmetry was analysed in our previous work for the case of a non-critical (gapped) free homogeneous fermionic chain invariant under parity and charge conjugation. In the present work we extend and analyse this new symmetry in several directions. First, we show that the above mentioned symmetry also holds when parity and charge conjugation invariance are broken. Second we extend this new symmetry to the case of critical (gapless) theories. Our results are further supported by numerical analysis. For some particular cases, analytical demonstrations show the validity of the extended symmetry. We finally discuss the intriguing parallelism of this new symmetry and space-time conformal transformations.
We first provide a useful formula to calculate the probability of occurrence
of different configurations (formation probabilities) in a generic free fermion
system. We then study the scaling of these probabilities with respect to the
size in the case of critical transverse-field XY-chain in the bases.
In the case of the transverse field Ising model, we show that all the "crystal"
configurations follow the formulas expected from conformal field theory (CFT).
In the case of critical XX chain, we show that the only configurations that
follow the formulas of the CFT are the ones which respect the filling factor of
the system. By repeating all the calculations in the presence of open and
periodic boundary conditions we find further support to our classification of
different configurations. Using the developed technique, we also study Shannon
information of a subregion in our system. Finally, we study the evolution of
formation probabilities, Shannon information and Shannon mutual information
after a quantum quench in free fermion system.
There is an intimate relation between entanglement entropy and Riemann
surfaces. This fact is explicitly noticed for the case of quadratic fermionic
Hamiltonians with finite range couplings. After recollecting this fact, we make
a comprehensive analysis of the action of the M\"obius transformations on the
Riemann surface. We are then able to uncover the origin of some symmetries and
dualities of the entanglement entropy already noticed recently in the
literature. These results give further support for the use of entanglement
entropy to analyse phase transition.
Random matrix theory, both as an application and as a theory, has evolved rapidly over the past fifteen years.Log-Gases and Random Matricesgives a comprehensive account of these developments, emphasizing log-gases as a physical picture and heuristic, as well as covering topics such as beta ensembles and Jack polynomials.Peter Forrester presents an encyclopedic development of log-gases and random matrices viewed as examples of integrable or exactly solvable systems. Forrester develops not only the application and theory of Gaussian and circular ensembles of classical random matrix theory, but also of the Laguerre and Jacobi ensembles, and their beta extensions. Prominence is given to the computation of a multitude of Jacobians; determinantal point processes and orthogonal polynomials of one variab the Selberg integral, Jack polynomials, and generalized hypergeometric functions; Painlev transcendents; macroscopic electrostatistics and asymptotic formulas; nonintersecting paths and models in statistical mechanics; and applications of random matrix theory. This is the first textbook development of both nonsymmetric and symmetric Jack polynomial theory, as well as the connection between Selberg integral theory and beta ensembles. The author provides hundreds of guided exercises and linked topics, makingLog-Gases and Random Matricesan indispensable reference work, as well as a learning resource for all students and researchers in the field.
We study the behavior of 〈σ0x(t)σnx(0)〉 and 〈σ0y(t)σny(0)〉 for the transverse Ising chain at the critical magnetic field at T = 0. Explicit results are obtained for the three distinct regions where t → ∞ and n → ∞with 0 ≤ n t<1, 1 < n t, or t = n + n1 3 ( z 2) where z is fixed of order one. In this latter region the general Painlevé V solution is shown to reduce to a Painlevé II function. We use our results to discuss the general problem of long-time behavior of Toda equations with slowly decaying initial values.
Using a multiple integral representation for the correlation functions, we compute the emptiness formation probability of the XXZ spin-½ Heisenberg chain at anisotropy Δ = ½. We prove that it is expressed in terms of the number of alternating sign matrices.
We consider a special correlation function in the isotropic spin-\half
Heisenberg antiferromagnet. It is the probability of finding a ferromagnetic
string of (adjacent) spins in the antiferromagnetic ground state. We give two
different representations for this correlation function. Both of them are exact
at any distance, but one becomes more effective for the description of long
distance behaviour, the other for the description of short distance behaviour.
We compute exactly the spin-spin correlation functions 〈σ0,0σM,N〉 for the two-dimensional Ising model on a square lattice in zero magnetic field for T>Tc and T<Tc. We then analyze the correlation functions in the scaling limit T→Tc,M2+N2→∞ such that (T-Tc) is fixed. In this scaling limit 〈σ0,0σM,N〉=R-1/4F±(t)+R-5/4F1±(t)+o(R-5/4), where t is the scaling variable R/ξ and F±(t) and F1±(t) are the scaling functions (ξ is the correlation length). We derive exact expressions for these scaling functions, in terms of a Painlevé function of the third kind and analyze both the small- and large-t behavior. A table of values for F±(t) (good to ten significant digits) is also given. As an application we computer the coefficients C0± and C1± in the expansion kBTχ(T)=C0±|1-Tc/T|-7/4+C1±|1-Tc/T|-3/4+O(1) of the zero-field susceptibility χ(T) as T→Tc±.
Introduction and ExamplesPrevious ResultsExtensions of the Classical TheoremSpecial Results and ConjecturesSummary
We study the correlation function 〈σ0x(t)σnx(0)〉 of the transverse Ising model in a critical field whose hamiltonian is . At an arbitrary temperature T we relate the autocorrelation to a Fredholm determinant. Moreover at T = 0 the correlations are given by a Painlevé V function for all n. The long-time asymptotic behavior of this function is found and the connection problem is studied. This result contains oscillatory terms which are related to the density of states at the Brillouin zone boundary.
We review the asymptotic behavior of a class of Toeplitz (as well as related Hankel and Toeplitz + Hankel) determinants which
arise in integrable models and other contexts. We discuss Szego, Fisher-Hartwig asymptotics, and how a transition between
them is related to the Painlevé V equation. Certain Toeplitz and Hankel determinants reduce, in certain double-scaling limits,
to Fredholm determinants which appear in the theory of group representations, in random matrices, random permutations and
partitions. The connection to Toeplitz determinants helps to evaluate the asymptotics of related Fredholm determinants in
situations of interest, and we review the corresponding results.
KeywordsToeplitz matrices-random matrices
We compute the entropy of entanglement in the ground states of a general class of quantum spin-chain Hamiltonians — those that are related to quadratic forms of Fermi operators — between the first N spins and the rest of the system in the limit of infinite total chain length. We show that the entropy can be expressed in terms of averages over the classical compact groups and establish an explicit correspondence between the symmetries of a given Hamiltonian and those characterizing the Haar measure of the associated group. These averages are either Toeplitz determinants or determinants of combinations of Toeplitz and Hankel matrices. Recent generalizations of the Fisher-Hartwig conjecture are used to compute the leading order asymptotics of the entropy as N. This is shown to grow logarithmically with N. The constant of proportionality is determined explicitly, as is the next (constant) term in the asymptotic expansion. The logarithmic growth of the entropy was previously predicted on the basis of numerical computations and conformal-field-theoretic calculations. In these calculations the constant of proportionality was determined in terms of the central charge of the Virasoro algebra. Our results therefore lead to an explicit formula for this charge. We also show that the entropy is related to solutions of ordinary differential equations of Painlev type. In some cases these solutions can be evaluated to all orders using recurrence relations.
The emptiness formation probability in the six-vertex model with domain wall boundary conditions is considered. This correlation function allows one to address the problem of limit shapes in the model. We apply the quantum inverse scattering method to calculate the emptiness formation probability for the inhomogeneous model. For the homogeneous model, the result is given both in terms of certain determinant and as a multiple integral representation.
A result from Palmer, Beatty and Tracy suggests that the two-point function of certain spinless scaling fields in a free Dirac theory on the Poincaré disk can be described in terms of Painlevé VI transcendents. We complete and verify this description by fixing the integration constants in the Painlevé VI transcendent describing the two-point function, and by calculating directly in a Dirac theory on the Poincaré disk the long distance expansion of this two-point function and the relative normalization of its long and short distance asymptotics. The long distance expansion is obtained by developing the curved-space analogue of a form factor expansion, and the relative normalization is obtained by calculating the one-point function of the scaling fields in question. The long distance expansion in fact provides part of the solution to the connection problem associated with the Painlevé VI equation involved. Calculations are done using the formalism of angular quantization.
We study the emptiness formation probability (EFP) for the spin 1/2 XXZ spin chain. EFP P(n) detects a formation of ferromagnetic string of the length n in the ground state. It is expected that EFP decays in a Gaussian way for large strings P(n)∼n−γC−n2. Here, we propose the explicit expressions for the rate of Gaussian decay C as well as for the exponent γ. In order to confirm the validity of our formulas, we employed an ab initio simulation technique of the density-matrix renormalization group to simulate XXZ spin chain of sufficient length. Furthermore, we performed Monte Carlo integration of the Jimbo–Miwa multiple integral for P(n). Those numerical results for P(n) support our formulas fairly definitely.
We study an asymptotic behavior of the probability of formation of a ferromagnetic string (referred to as EFP) of length n in the ground state of the one-dimensional anisotropic XY model in a transversal magnetic field as n→∞. We find that it is exponential everywhere in the phase diagram of the XY model except at the critical lines where the spectrum is gapless. One of those lines corresponds to the isotropic XY model where EFP decays in a Gaussian way, as was shown in [J. Phys. Soc. Jpn. 70 (2001) 3535]. The other lines are at the critical value of the magnetic field. There, we show that EFP is still exponential but acquires a non-trivial power-law prefactor with a universal exponent.
We study the probability of formation of ferromagnetic string in the antiferromagnetic spin-1/2 XXZ chain. We show that in the limit of long strings with weak magnetization per site the bosonization technique can be used to address the problem. At zero temperature the obtained probability is Gaussian as a function of the length of the string. At finite but low temperature there is a crossover from the Gaussian behavior at intermediate lengths of strings to the exponential decay for very long strings. Although the weak magnetization per site is a necessary small parameter justifying our results, the extrapolation of obtained results to the case of maximally ferromagnetic strings is in qualitative agreement with known numerics and exact results. The effect of an external magnetic field on the probability of formation of ferromagnetic strings is also studied.
The conjecture of Fisher and Hartwig, published in 1968, describes the asymptotic expansion of Toeplitz determinants with singular generating functions. For more than twenty years progress was made in extending the validity of the conjecture, but recent computer experiments led to counterexamples that show the limits of the original conjecture and pointed the way to a revised conjecture. This paper describes the history of the problem, several numerical examples, and the revised conjecture.
We discuss the status of the Fisher-Hartwig conjecture concerning the asymptotic expansion of a class of Toeplitz determinants with singular generating functions. A counterexample is given for a nonrational generating function; and we formulate a generalized Fisher-Hartwig conjecture.
We present an investigation of the massless, two-dimentional, interacting field theories. Their basic property is their invariance under an infinite-dimensional group of conformal (analytic) transformations. It is shown that the local fields forming the operator algebra can be classified according to the irreducible representations of Virasoro algebra, and that the correlation functions are built up of the “conformal blocks” which are completely determined by the conformal invariance. Exactly solvable conformal theories associated with the degenerate representations are analyzed. In these theories the anomalous dimensions are known exactly and the correlation functions satisfy the systems of linear differential equations.
Boundary operators is conformal field theory are considered as arising from the juxtaposition of different types of boundary conditions. From this point of view, the operator content of the theory in an annulus may be related to the fusion rules. By considering the partition function in such a geometry, we give a simple derivation of the Verlinde formula.
The quantal system of Bose particles described by the non-linear Schrödinger equation i∂φ/∂t = -∂2φ/∂x2 + cφ∗φ2, with c= cxf∞ and via the ground state with finite particle density, is the 1- dimensional gas of impenetrable bosons studied by M. Girardeau, T.D. Schultz, A. Lenard, H.G. Vaidya and C.A. Tracy. We show that the 2-point (resp. 2n-point) function, or the 1-particle (resp. n-particle) reduced density matrix, of this system satisfies a non-linear differential equation (resp. a system of non-linear partial differential equations) of Painlevé type. Derivation of these equations is based on the link between field operators in a Clifford group and monodromy preserving deformation theory, which was previously established and applied to the 2-dimensional Ising model and other problems. Several related topics are also discussed.
We obtain asymptotic expansions for Toeplitz determinants corresponding to a
family of symbols depending on a parameter t. For t positive, the symbols
are regular so that the determinants obey Szeg\H{o}'s strong limit theorem. If
t=0, the symbol possesses a Fisher-Hartwig singularity. Letting we
analyze the emergence of a Fisher-Hartwig singularity and a transition between
the two different types of asymptotic behavior for Toeplitz determinants. This
transition is described by a special Painlev\'e V transcendent. A particular
case of our result complements the classical description of Wu, McCoy, Tracy,
and Barouch of the behavior of a 2-spin correlation function for a large
distance between spins in the two-dimensional Ising model as the phase
transition occurs.
The Gel'fand-Levitan and Marchenko formalisms for solving the inverse scattering problem are applied together to a single set of scattering phase-shifts. The result is an identity relating two different types of Fredholm determinant. As an application of the method, an asymptotic formula of high accuracy is derived for a particular Fredholm determinant that determines the level-spacing distribution-function in the theory of random matrices.
We consider correlation functions of the spin-\half XXX and XXZ Heisenberg chains in a magnetic field. Starting from the algebraic Bethe Ansatz we derive representations for various correlation functions in terms of determinants of Fredholm integral operators. Comment: 23 pages, TeX, BONN-TH-94-14, revised version: typos corrected
Using exact expressions for the Ising form factors, we give a new very simple proof that the spin-spin and disorder-disorder correlation functions are governed by the Painlev\'e III non linear differential equation. We also show that the generating function of the correlation functions of the descendents of the spin and disorder operators is a N-soliton, , -function of the sinh-Gordon hierarchy. We discuss a relation of our approach to isomonodromy deformation problems, as well as further possible generalizations.