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arXiv:math-ph/0405009

Functional Integration with "Automorphic" Boundary Conditions and Correlators of Z-Components of Spins in the XY and XX Heisenberg Chains

Cyril Leonidovich Malyshev

Published May 4, 2004• 4 citations
arXivHTMLPDF

Abstract

Representations for the generating functionals of static correlators of $z$-components of spins in the XY and $XX$ Heisenberg spin chains are obtained in the form of sums of the fermionic functional integrals. The peculiarity of the functional integrals in question is because of the fact that the integration variables depend on the imaginary time ``automorphically''. In other words, the integration variables are multiplied with a certain complex number when the imaginary time is shifted by a period. Therefore, the corresponding boundary conditions at the ends of the imaginary time segment are not of the form corresponding to fermionic, or bosonic, variables taken in the Matsubara representation at nonzero temperature. In fact, one part of sites of the models corresponds to the integration variables which are subjected to the unusual boundary conditions, while the variables on the other sites depend on the imaginary time conventionally, i.e., as fermions (or bosons). Thus a situation, when an ``automorphic'' boundary condition is the same for all sites of a chain spin model, is generalized. The results of the functional integration are obtained in the form of determinants of the matrix operators which are regularized by means of the generalized zeta-function approach. The partition functions of the models and certain correlation functions at nonzero temperature are obtained explicitly thus demonstrating correctness of the functional integral representations proposed.

DOI
10.48550/arxiv.math-ph/0405009

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Representations for the generating functionals of static correlators of zzz-components of spins in the XY and XXXXXX Heisenberg spin chains are obtained in the form of sums of the fermionic functional integrals. The peculiarity of the functional integrals in question is because of the fact that the integration variables depend on the imaginary time automorphically''. In other words, the integration variables are multiplied with a certain complex number when the imaginary time is shifted by a period. Therefore, the corresponding boundary conditions at the ends of the imaginary time segment are not of the form corresponding to fermionic, or bosonic, variables taken in the Matsubara representation at nonzero temperature. In fact, one part of sites of the models corresponds to the integration variables which are subjected to the unusual boundary conditions, while the variables on the other sites depend on the imaginary time conventionally, i.e., as fermions (or bosons). Thus a situation, when an automorphic'' boundary condition is the same for all sites of a chain spin model, is generalized. The results of the functional integration are obtained in the form of determinants of the matrix operators which are regularized by means of the generalized zeta-function approach. The partition functions of the models and certain correlation functions at nonzero temperature are obtained explicitly thus demonstrating correctness of the functional integral representations proposed.
A number of problems in statistical physics can be reformulated in terms of a two-state system evolving in a random field. The corresponding evolution operator can be written in the form of time-ordered operator exponential. Functional formalism allows us to rewrite the latter as a product of usual matrix exponentials using a nonlinear change of functional integration variables. In this review I present this formalism applied to two physical systems the quantum Heisenberg magnet and one-dimensional quantum mechanics in a spatially random potential. First, I derive a representation of the partition function of a quantum Heisenberg ferromagnet as a functional integral over number valued fields (a real one and a complex one) free of constraints. The fields of integration as functions of time obey initial conditions instead of the usual periodic boundary conditions. This is a manifestation of the finite-dimensionality of the space of spin states. In the subsequent sections I study the one-dimensional localization problem. The change of functional integration variables gives simultaneously explicit expressions for the averaging weight and for the Green function of the stationary Schrödinger equation. It allows to compute density correlators of arbitrary orders. The generalization to the case of different energy correlators (Berezinskii-Gor’kov equations) is considered too. In the present review such technical points as regularizations of functional integrals and transformations are discussed in more details than in the original papers.
We explain how to compute correlation functions at zero temperature within the framework of the quantum version of the Separation of Variables (SoV) in the case of a simple model: the XXX Heisenberg chain of spin 1/2 with twisted (quasi-periodic) boundary conditions. We first detail all steps of our method in the case of anti-periodic boundary conditions. The model can be solved in the SoV framework by introducing inhomogeneity parameters. The action of local operators on the eigenstates are then naturally expressed in terms of multiple sums over these inhomogeneity parameters. We explain how to transform these sums over inhomogeneity parameters into multiple contour integrals. Evaluating these multiple integrals by the residues of the poles outside the integration contours, we rewrite this action as a sum involving the roots of the Baxter polynomial plus a contribution of the poles at infinity. We show that the contribution of the poles at infinity vanishes in the thermodynamic limit, and that we recover in this limit for the zero-temperature correlation functions the multiple integral representation that had been previously obtained through the study of the periodic case by Bethe Ansatz or through the study of the infinite volume model by the q-vertex operator approach. We finally show that the method can easily be generalized to the case of a more general non-diagonal twist: the corresponding weights of the different terms for the correlation functions in finite volume are then modified, but we recover in the thermodynamic limit the same multiple integral representation than in the periodic or anti-periodic case, hence proving the independence of the thermodynamic limit of the correlation functions with respect to the particular form of the boundary twist.
This paper is a continuation of [J. Phys. A: Math. Theor. 55, 405203 (2022)], in which a set of matrix elements of local operators was computed for the XXZ spin-1/2 open chain with a particular case of unparallel boundary fields. Here, we extend these results to the more general case in which both fields are non-longitudinal and related by one constraint, allowing for a partial description of the spectrum by usual Bethe equations. More precisely, the complete spectrum and eigenstates can be characterized within the Separation of Variables (SoV) framework. One uses here the fact that, under the constraint, a part of this SoV spectrum can be described via solutions of a usual, homogeneous, TQ-equation, with corresponding transfer matrix eigenstates coinciding with generalized Bethe states. We explain how to generically compute the action of a basis of local operators on such kind of states, and this under the most general boundary condition on the last site of the chain. As a result, we can compute the matrix elements of some of these basis elements in any eigenstate described by the homogenous TQ-equation. Assuming, following a conjecture of Nepomechie and Ravanini, that the ground state itself can be described in this framework, we obtain multiple integral representations for these matrix elements in the half-infinite chain limit, generalizing those previously obtained in the case of longitudinal boundary fields and in the case of the special boundary conditions considered in [J. Phys. A: Math. Theor. 55, 405203 (2022)].
Abstract Functional integral representations are constructed for Fermions with spin 1/2, in which the fields satisfy directly by construction the constraints (e.g., exclusion of double occupancy of a site) appearing in recent models in the theory of high‐temperature superconductivity. Thus, the enforcement of the constraints by delta functions in the integration measure is avoided. Perelomov's concept of generalized coherent states is used. However, in constructing such representations, exponential functions of linear combinations of operators (which are difficult to disentangle) are avoided, as is the construction and reduction of the invariant measure. Instead, an ansatz is used for the resolution of the unity operator. This approach also provides more freedom in choosing the appropriate fields. Several new and simple representations with only few elementary fields are given. The representation already used by Wiegmann is recovered. In this case and in any other cases the integration measure is explicitly given. In all these representations, the original Fermi operators are substituted by the product of a spin independent Graßmann field and a spin dependent bosonic (complex) field in accordance with the physical idea of separation of charge and spin degrees of freedom. It is further shown how a change in the integration measure eliminates also zero occupancy (the case of the Heisenberg antiferromagnet). The absence of an explicit delta function constraint in the functional integral is reflected in a special form of the kinetic part of the action. The considered representations are compared with that of the slave boson method.
Using the algebraic Bethe Ansatz, we derive a matrix product representation of the exact Bethe-Ansatz states of the six-vertex Heisenberg chain (either XXX or XXZ and spin-1/2) with open boundary conditions. In this representation, the components of the Bethe eigenstates are expressed as traces of products of matrices that act on a tensor product of auxiliary spaces. As compared to the matrix product states of the same Heisenberg chain but with periodic boundary conditions, the dimension of the exact auxiliary matrices is enlarged as if the conserved number of spin-flips considered would have been doubled. This result is generic for any non-nested integrable model, as is clear from our derivation, and we further show this by providing an additional example of the same matrix product state construction for a well-known model of a gas of interacting bosons. Counterintuitively, the matrices do not depend on the spatial coordinate despite the open boundaries, and thus they suggest generic ways of exploiting (emergent) translational invariance both for finite size and in the thermodynamic limit.