Oscar Bolina, Pierluigi Contucci, Bruno Nachtergaele
Two discrete path integral formulations for the ground state of a spin-pinned quantum anisotropic XXZ Heisenberg chain are introduced. Their properties are discussed and two recursion relations are proved.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
继续探索
We derive a non-linear integral equation for the Bethe-ansatz solvable open XXZ spin chain of arbitrary length describing the lowest lying state with zero magnetization. For this case we show how to combine the integral representation with the known determinant formula of norms and scalar products.
We derive a novel multiple integral representation for a generating function of the $\s^z$-$\s^z$ correlation functions of the spin-$\2$ XXZ chain at finite temperature and finite, longitudinal magnetic field. Our work combines algebraic Bethe ansatz techniques for the calculation of matrix elements with the quantum transfer matrix approach to thermodynamics.
The correlation functions of the spin-1/2 XXZ chain in the ground state were expressed in the form of multiple integrals for -1<Δ\leq 1 and 1<Δ. In particular, adjacent four-point correlation functions were given as certain four-dimensional integrals. We show that these integrals can be reduced to polynomials with respect to specific one-dimensional integrals. The results give the polynomial representation of the third-neighbor correlation functions.
The correlation functions of the spin-1/2 XXZ spin chain in the ground state are expressed in the form of the multiple integrals. For -1< Delta <1, they were obtained by Jimbo and Miwa in 1996. Especially the next nearest-neighbour correlation functions are given as certain three-dimensional integrals. We shall show these integrals can be reduced to one-dimensional ones and thereby evaluate the values of the next nearest-neighbor correlation functions. We have also found that the remaining one-dimensinal integrals can be evaluated analytically, when nu = arccos(Delta)/pi is a rational number.
The half-infinite XXZ spin chain with a triangular boundary is considered in the massive regime. Two integral representations of correlation functions are proposed using bosonization. Sufficient conditions such that the expressions for triangular boundary conditions coincide with those for diagonal boundary conditions are identified. As an application, summation formulae of the boundary expectation values $\langle σ_1^a\rangle $ with $a=z,\pm$ are obtained. Exploiting the spin-reversal property, relations between $n$-fold integrals of elliptic theta functions are extracted.
Combining a lattice path integral formulation for thermodynamics with the solution of the quantum inverse scattering problem for local spin operators, we derive a multiple integral representation for the time-dependent longitudinal correlation function of the spin-1/2 Heisenberg XXZ chain at finite temperature and in an external magnetic field. Our formula reproduces the previous results in the following three limits: the static, the zero-temperature and the XY limits.
The calculation of the correlation functions of Bethe ansatz solvable models is very difficult problem. Among these solvable models spin 1/2 XXX chain has been investigated for a long time. Even for this model only the nearest neighbor and the second neighbor correlations were known. In 1990's multiple integral formula for the general correlations is derived. But the integration of this formula is also very difficult problem. Recently these integrals are decomposed to products of one dimensional integrals and at zero temperature, zero magnetic field and isotropic case, correlation functions ar…
We propose a new multiple integral representation for the correlation function <sigma_1^z sigma_{m+1}^z> of the XXZ spin-1/2 Heisenberg chain in the disordered regime. We show that for Delta=1/2 the integrals can be separated and computed exactly. As an example we give the explicit results up to the lattice distance m=8. It turns out that the answer is given as integer numbers divided by 2^[(m+1)^2].
Revising the derivation of the previous papers, for the integrable spin-$s$ XXZ chain we express any form factor in terms of a single sum over scalar products of the spin-1/2 XXZ chain. With the revised method we express the spin-$s$ XXZ correlation function of any given entry at zero temperature in terms of a single sum of multiple integrals.
We show some symmetry relations among the correlation functions of the integrable higher-spin XXX and XXZ spin chains, where we explicitly evaluate the multiple integrals representing the one-point functions in the spin-1 case. We review the multiple-integral representations of correlation functions for the integrable higher-spin XXZ chains derived in a region of the massless regime including the anti-ferromagnetic point. Here we make use of the gauge transformations between the symmetric and asymmetric R-matrices, which correspond to the principal and homogeneous gradings, respectively, and w…
The half-infinite XXZ spin chain with a triangular boundary is considered in the massive regime. Two integral representations of form factors of local operators are proposed using bosonization. Sufficient conditions such that the expressions for triangular boundary conditions coincide with those for diagonal boundary conditions are identified. The expressions are compared with known results upon specializations.Using the spin-reversal property which relates the Hamiltonian with upper and lower triangular boundary conditions, new identities between multiple integrals of infinite products are ex…
We consider correlation functions of the form <vac|O|vac>', where |vac> is the vacuum eigenstate of an infinite antiferromagnetic XXZ chain, |vac>' is the vacuum eigenstate of an infinite XXZ chain which is split in two, and O is a local operator. The Hamiltonian of the split chain has no coupling between sites 1 and 0 and has a staggered magnetic field at these two sites; it arises from a tensor product of left and right boundary transfer matrices. We find a simple, exact expression for <vac|vac>' and an exact integral expression for general <vac|O|vac>' using the vertex operator approach. We…
For the integrable higher-spin XXX and XXZ spin chains we present multiple-integral representations for the correlation function of an arbitrary product of Hermitian elementary matrices in the massless ground state. We give a formula expressing it by a single term of multiple integrals. In particular, we explicitly derive the emptiness formation probability (EFP). We assume $2s$-strings for the ground-state solution of the Bethe ansatz equations for the spin-$s$ XXZ chain, and solve the integral equations for the spin-$s$ Gaudin matrix. In terms of the XXZ coupling $Δ$ we define $ζ$ by $Δ=\cos…
We derive compact multiple integral formulas for several physical spin correlation functions in the semi-infinite XXZ chain with a longitudinal boundary magnetic field. Our formulas follow from several effective re-summations of the multiple integral representation for the elementary blocks obtained in our previous article (I). In the free fermion point we compute the local magnetization as well as the density of energy profiles. These quantities, in addition to their bulk behavior, exhibit Friedel type oscillations induced by the boundary; their amplitudes depend on the boundary magnetic fiel…
Integral formulae for polynomial solutions of the quantum Knizhnik-Zamolodchikov equations associated with the R-matrix of the six-vertex model are considered. It is proved that when the deformation parameter q is equal to e^{+- 2 pi i/3} and the number of vertical lines of the lattice is odd, the solution under consideration is an eigenvector of the inhomogeneous transfer matrix of the six-vertex model. In the homogeneous limit it is a ground state eigenvector of the antiferromagnetic XXZ spin chain with the anisotropy parameter Delta equal to -1/2 and odd number of sites. The obtained integr…
We discuss some fundamental properties of the XXZ spin chain, which are important in the algebraic Bethe-ansatz derivation for the multiple-integral representations of the spin-s XXZ correlation function with an arbitrary product of elementary matrices. For instance, we construct Hermitian conjugate vectors in the massless regime and introduce the spin-s Hermitian elementary matrices.
We study the boundary free energy of the XXZ spin-$\tf{1}{2}$ chain subject to diagonal boundary fields. We first show that the representation for its finite Trotter number approximant obtained by Bortz, Frahm and Göhmann is related to the partition function of the six-vertex model with reflecting ends. Building on the Tsuchiya determinant representation for the latter quantity we are able to take the infinite Trotter number limit. This yields a representation for the surface free energy which involves the solution of the non-linear integral equation that governs the thermodynamics of the XXZ …
We address the problem of computing temperature correlation functions of the XXZ chain, within the approach developed in our previous works. In this paper we calculate the expected values of a fermionic basis of quasi-local operators, in the infinite volume limit while keeping the Matsubara (or Trotter) direction finite. The result is expressed in terms of two basic quantities: a ratio $ρ(\z)$ of transfer matrix eigenvalues, and a nearest neighbour correlator $ω(\z,ξ)$. We explain that the latter is interpreted as the canonical second kind differential in the theory of deformed Abelian integra…
We establish several properties of the solutions to the linear integral equations describing the infinite volume properties of the XXZ spin-1/2 chain in the disordered regime. In particular, we obtain lower and upper bounds for the dressed energy, dressed charge and density of Bethe roots. Furthermore, we establish that given a fixed external magnetic field (or a fixed magnetization) there exists a unique value of the boundary of the Fermi zone.
In this work we study scalar products of Bethe vectors associated with the $XXZ$ spin chain with open boundary conditions. The scalar products are obtained as solutions of a system of functional equations. The description of scalar products through functional relations follows from a particular map having the reflection algebra as its domain and a function space as the codomain. Within this approach we find a multiple contour integral representation for the scalar products in which the homogeneous limit can be obtained trivially.
For the integrable spin-s XXZ chain we express explicitly any given spin-$s$ form factor in terms of a sum over the scalar products of the spin-1/2 operators. Here they are given by the operator-valued matrix elements of the monodromy matrix of the spin-1/2 XXZ spin chain. In the paper we call an arbitrary matrix element of a local operator between two Bethe eigenstates a form factor of the operator. We derive all important formulas of the fusion method in detail. We thus revise the derivation of the higher-spin XXZ form factors given in a previous paper. The revised method has several interes…
It was recently shown by Jimbo, Miwa and Smirnov that the correlation functions of a generalized XXZ chain associated with an inhomogeneous six-vertex model with disorder parameter $α$ and with arbitrary inhomogeneities on the horizontal lines factorize and can all be expressed in terms of only two functions $ρ$ and $ω$. Here we approach the description of the same correlation functions and, in particular, of the function $ω$ from a different direction. We start from a novel multiple integral representation for the density matrix of a finite chain segment of length $m$ in the presence of a dis…
We study the scalar products between Bethe states in the XXZ spin chain with anisotropy $|Δ|>1$ in the semi-classical limit where the length of the spin chain and the number of magnons tend to infinity with their ratio kept finite and fixed. Our method is a natural yet non-trivial generalization of similar methods developed for the XXX spin chain. The final result can be written in a compact form as a contour integral in terms of Faddeev's quantum dilogarithm function, which in the isotropic limit reduces to the classical dilogarithm function.
We consider a simple integrable model of a spin chain exhibiting the Magnetoelectric Effect (MEE). Starting from the periodic S=1/2 XXZ-chain with Dzyaloshinskii-Moriya terms, which we consider as a local electric polarization in the spirit of the Katsura-Nagaosa-Baladsky (KNB) mechanism, we perform the mapping onto the conventional XXZ-chain with twisted boundary conditions. Using the techniques of Quantum Transfer Matrix (QTM) and Non-Linear Integral Equations (NLIE) we obtain the magnetization, electric polarization and magnetoelectric tensor as functions of magnetic and electric field for …
We study the time evolution of the spin-1/2 XXZ chain initialized in a domain wall state, where all spins to the left of the origin are up, all spins to its right are down. The focus is on exact formulae, which hold for arbitrary finite (real or imaginary) time. In particular, we compute the amplitudes corresponding to the process where all but $k$ spins come back to their initial orientation, as a $k-$fold contour integral. These results are obtained using a correspondence with the six vertex model, and taking a somewhat complicated Hamiltonian/Trotter-type limit. Several simple applications …
The open XXZ spin chain with the anisotropy parameter $Δ=-\frac12$ and diagonal boundary magnetic fields that depend on a parameter $x$ is studied. For real $x>0$, the exact finite-size ground-state eigenvalue of the spin-chain Hamiltonian is explicitly computed. In a suitable normalisation, the ground-state components are characterised as polynomials in $x$ with integer coefficients. Linear sum rules and special components of this eigenvector are explicitly computed in terms of determinant formulas. These results follow from the construction of a contour-integral solution to the boundary quan…
T-W relation and free energy of the antiperiodic XXZ chain with η=iπ/3 at a finite temperature
本文通过t-W方法研究各向异性参数η=iπ/3的反周期XXZ链热力学，利用零点参数化转移矩阵本征值，构建非线性积分方程并计算出有限温度下的自由能。
We present exact results on the thermal conductivity of the one-dimensional spin-1/2 XXZ model in the massive antiferromagnetic and ferromagnetic regimes. The thermal Drude weight is calculated by a lattice path integral formulation. Numerical results for wide ranges of temperature and anisotropy as well as analytical results in the low and high temperature limits are presented. At finite temperature, the thermal Drude weight is finite and hence there is non-dissipative thermal transport even in the massive regime. At low temperature, the thermal Drude weight behaves as $D(T)\sim \exp(-δ/T)/\s…
We develop a geometric representation for the ground state of the spin-1/2 quantum XXZ ferromagnetic chain in terms of suitably weighted random walks in a two-dimensional lattice. The path integral model so obtained admits a genuine classical statistical mechanics interpretation with a translation invariant Hamiltonian. This new representation is used to study the interface ground states of the XXZ model. We prove that the probability of having a number of down spins in the up phase decays exponentially with the sum of their distances to the interface plus the square of the number of down spin…
We consider the spectrum of correlation lengths of the spin-$\frac{1}{2}$ XXZ chain in the antiferromagnetic massive regime. These are given as ratios of eigenvalues of the quantum transfer matrix of the model. The eigenvalues are determined by integrals over certain auxiliary functions and by their zeros. The auxiliary functions satisfy nonlinear integral equations. We analyse these nonlinear integral equations in the low-temperature limit. In this limit we can determine the auxiliary functions and the expressions for the eigenvalues as functions of a finite number of parameters which satisfy…
Using the algebraic Bethe ansatz method, and the solution of the quantum inverse scattering problem for local spins, we obtain multiple integral representations of the $n$-point correlation functions of the XXZ Heisenberg spin-$1 \over 2$ chain in a constant magnetic field. For zero magnetic field, this result agrees, in both the massless and massive (anti-ferromagnetic) regimes, with the one obtained from the q-deformed KZ equations (massless regime) and the representation theory of the quantum affine algebra ${\cal U}_q (\hat{sl}_2)$ together with the corner transfer matrix approach (massive…
Using algebraic Bethe ansatz and the solution of the quantum inverse scattering problem, we compute compact representations of the spin-spin correlation functions of the XXZ-1/2 Heisenberg chain in a magnetic field. At lattice distance m, they are typically given as the sum of m terms. Each term n of this sum, n = 1,...,m is represented in the thermodynamic limit as a multiple integral of order 2n+1; the integrand depends on the distance as the power m of some simple function. The root of these results is the derivation of a compact formula for the multiple action on a general quantum state of…
We identify the Taylor coefficients of the transfer matrices corresponding to quantum toroidal algebras with the elliptic local and non-local integrals of motion introduced by Kojima, Shiraishi, Watanabe, and one of the authors. That allows us to prove the Litvinov conjectures on the Intermediate Long Wave model. We also discuss the (gl(m),gl(n)) duality of XXZ models in quantum toroidal setting and the implications for the quantum KdV model. In particular, we conjecture that the spectrum of non-local integrals of motion of Bazhanov, Lukyanov, and Zamolodchikov is described by Gaudin Bethe ans…
The half-infinite XXZ open spin chain with general integrable boundary conditions is considered within the recently developed `Onsager's approach'. Inspired by the finite size case, for any type of integrable boundary conditions it is shown that the transfer matrix is simply expressed in terms of the elements of a new type of current algebra recently introduced. In the massive regime $-1<q<0$, level one infinite dimensional representation ($q-$vertex operators) of the new current algebra are constructed in order to diagonalize the transfer matrix. For diagonal boundary conditions, known result…
Dressed energy of the XXZ chain in the complex plane
本文研究有限磁场、0<Δ<1 无质量反铁磁 XXZ 链的修饰能量，通过其复平面扩展刻画低温量子转移矩阵 Bethe 根聚集的 Re ε(λ)=0 曲线，并给出正区域下界。
This work constructs a well-defined and operational form factor expansion in a model having a massless spectrum of excitations. More precisely, the dynamic two-point functions in the massless regime of the XXZ spin-1/2 chain are expressed in terms of properly regularised series of multiple integrals. These series are obtained by taking, in an appropriate way, the thermodynamic limit of the finite volume form factor expansions. The series are structured in way allowing one to identify directly the contributions to the correlator stemming from the conformal-type excitations on the Fermi surface …
We consider the computation of the Loschmidt echo after quantum quenches in the interacting $XXZ$ Heisenberg spin chain both for real and imaginary times. We study two-site product initial states, focusing in particular on the Néel and tilted Néel states. We apply the Quantum Transfer Matrix (QTM) approach to derive generalized TBA equations, which follow from the fusion hierarchy of the appropriate QTM's. Our formulas are valid for arbitrary imaginary time and for real times at least up to a time $t_0$, after which the integral equations have to be modified. In some regimes, $t_0$ is seen to …
Spin Drude weight for the integrable XXZ chain with arbitrary spin
本文利用广义流体动力学精确计算了任意自旋可积 XXZ 链在零磁场下的有限温度自旋 Drude 权重，通过融合层级和 TBA 方程揭示了其分形结构，并实现了高温极限的系统计算。
It was previously established that the critical staggered XXZ spin chain provides a lattice regularization of the black hole CFT. We reconsider the continuum limit of this spin chain with the exact method of non-linear integral equations (NLIEs), paying particular attention to the effects of a singular integration kernel. With the help of the NLIEs, we rederive the continuous black hole spectrum, but also numerically match the density of states of the spin chain with that of the CFT, which is a new result. Finally, we briefly discuss the integrable structure of the black hole CFT and the ident…
Correlation functions for open XXZ spin 1/2 quantum chains with unparallel boundary magnetic fields
针对非平行边界磁场下的开XXZ自旋1/2量子链，基于量子变量分离框架推导了局域算符的关联函数，给出了有限链结果和半无限链的多重积分表达式。
On correlation functions for the open XXZ chain with non-longitudinal boundary fields : the case with a constraint
本文将带特殊非平行边界场的XXZ自旋1/2开链局域算符矩阵元的研究，推广到场满足一个约束的一般非纵向边界情形，在SoV框架下通过齐次TQ方程描述部分能谱，得到半无限链极限下矩阵元的多重积分表示。
Two-parameter families of matrix product operator integrals of motion in Heisenberg spin chains
本文研究了Heisenberg自旋链中不同各向异性情况下双参数矩阵积算符守恒量族的构造方法，并探讨了其在Floquet协议中的应用。
The ground states of the spin-$ S $ antiferromagnetic chain $H_\textrm{AF}$ with a projection-based interaction and the spin-$ 1/2$ XXZ-chain $ H_\textrm{XXZ} $ at anisotropy parameter $Δ=\cosh(λ) $ share a common loop representation in terms of a two-dimensional functional integral which is similar to the classical planar $Q$-state Potts model at $ \sqrt Q= 2S+1 =2\cosh(λ)$. The multifaceted relation is used here to directly relate the distinct forms of translation symmetry breaking which are manifested in the ground states of these two models: dimerization for $H_\textrm{AF}$ at all $S> 1/2$…
One of the key steps in recent work on the correlation functions of the XXZ chain was to regularize the underlying six-vertex model by a disorder parameter $α$. For the regularized model it was shown that all static correlation functions are polynomials in only two functions. It was further shown that these two functions can be written as contour integrals involving the solutions of a certain type of linear and non-linear integral equations. The linear integral equations depend parametrically on $α$ and generalize linear integral equations known from the study of the bulk thermodynamic propert…
Low-temperature spectrum of the quantum transfer matrix of the XXZ chain in the massless regime
本文通过XXZ链在无质量态和低温极限下的研究，严格证明了量子转移矩阵谱中与Bethe Ansatz相关的部分，并揭示了其与共形场论的关联。
We consider the longitudinal dynamical two-point function of the XXZ quantum spin chain in the antiferromagnetic massive regime. It has a series representation based on the form factors of the quantum transfer matrix of the model. The $n$th summand of the series is a multiple integral accounting for all $n$-particle $n$-hole excitations of the quantum transfer matrix. In previous works the expressions for the form factor amplitudes appearing under the integrals were either again represented as multiple integrals or in terms of Fredholm determinants. Here we obtain a representation which reduce…
On correlation functions of the open XYZ spin 1/2 chain with boundary fields related by a constraint
本文研究带约束边界场的开式XYZ自旋1/2链，用量子变量分离法等推导关联函数基本块的精确表示，发现三类开式自旋链的关联函数构建块结构相似且无需非局域尾算符。
We find novel site-dependent Lax operators in terms of which we demonstrate exact solvability of a dissipatively driven XYZ spin-1/2 chain in the Zeno limit of strong dissipation, with jump operators polarizing the boundary spins in arbitrary directions. We write the corresponding nonequilibrium steady state using an inhomogeneous MPA, where the constituent matrices satisfy a simple set of linear recurrence relations. Although these matrices can be embedded into an infinite-dimensional auxiliary space, we have verified that they cannot be simultaneously put into a tridiagonal form, not even in…
This script is based on the notes the author prepared to give a set of six lectures at the Les Houches School "Integrability in Atomic and Condensed Matter Physics" in the summer of 2018. The school had its focus on the application of integrability based methods to problems in non-equilibrium statistical mechanics. The lectures were meant to complement this subject with background material on the equilibrium statistical mechanics of quantum spin chains from a vertex model perspective. The author was asked to provide a minimal introduction to quantum spin systems including notions like the redu…
Thermodynamics of the spin-$1/2$ Heisenberg-Ising chain at high temperatures: a rigorous approach
本文在量子逆散射方法下严格证明高温XXZ自旋链积分表示所依赖的极限交换、最大本征值及辅助方程解等猜想，并导出次主导本征值与关联长度的大$T$渐近展开。
Finite temperature properties of an integrable zigzag ladder chain
本研究通过面内六顶点模型推导出具有三自旋相互作用的可积量子哈密顿量，并利用量子转移矩阵方法分析其有限温度性质，发现其具有不同的有序性和临界指数，且在临界区域表现出破缺宇称对称性的量子自旋液体行为。
Evaluating a lattice path integral in terms of spectral data and matrix elements pertaining to a suitably defined quantum transfer matrix, we derive form-factor series expansions for the dynamical two-point functions of arbitrary local operators in fundamental Yang-Baxter integrable lattice models at finite temperature. The summands in the series are parameterised by solutions of the Bethe Ansatz equations associated with the eigenvalue problem of the quantum transfer matrix. We elaborate on the example of the XXZ chain for which the solutions of the Bethe Ansatz equations are sufficiently wel…
We explore the connections between the theories of stochastic analysis and discrete quantum mechanical systems. Naturally these connections include the Feynman-Kac formula, and the Cameron-Martin-Girsanov theorem. More precisely, the notion of the quantum canonical transformation is employed for computing the time propagator, in the case of generic dynamical diffusion coefficients. Explicit computation of the path integral leads to a universal expression for the associated measure regardless of the form of the diffusion coefficient and the drift. This computation also reveals that the drift pl…
扫码添加微信好友，提出您的宝贵建议 👇
💡 备注请填写：网站反馈
发表机构由 AI 从论文文本抽取，如发现错误，欢迎反馈修正建议。