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

C. Malyshev|ArXiv.org|May 4, 2004

Quantum many-body systems24 references3 citations

TL;DR

This paper develops a functional integral approach for computing static correlators of z-components of spins in XY and XX Heisenberg chains at finite temperature, using a novel class of 'automorphic' boundary conditions in imaginary time. The method generalizes standard fermionic/bosonic boundary conditions by allowing integration variables to transform non-trivially under imaginary time translations, leading to determinant representations regularized via zeta functions and yielding exact results consistent with known solutions.

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.

Motivation & Objective

• To develop a functional integral representation for static correlators of z-component spins in XY and XX Heisenberg chains at finite temperature.
• To generalize standard fermionic or bosonic boundary conditions in imaginary time by introducing 'automorphic' boundary conditions, where fields transform under time translations by multiplication with a complex phase.
• To demonstrate the consistency and correctness of the proposed functional integral formalism by deriving explicit expressions for partition functions and correlation functions.
• To extend the method of [23] to the XY model, which is equivalent to quasi-free fermions via a Bogoliubov transformation.
• To show that the resulting determinant representations, regularized via the zeta-function approach, reproduce known results from other methods such as Bethe ansatz and multiple integral representations.

Proposed method

• The generating functional for spin z-component correlators is expressed as a Gaussian functional integral over Grassmann variables with 'automorphic' boundary conditions in imaginary time.
• The boundary conditions are inhomogeneous: only a subset of lattice sites (first m sites) are subject to automorphic behavior, while the remaining sites obey standard periodic or anti-periodic conditions.
• The functional integral is transformed from a multiple integral over Grassmann coherent states into a continuum path integral with a quadratic action in the exponent.
• The resulting determinant representations are regularized using the generalized zeta-function method to handle divergences and ensure convergence.
• The method allows computation of correlation functions via Fredholm determinants of matrix operators, with explicit expressions derived for the thermodynamic limit.
• The formalism is validated by recovering known results such as the magnetization and two-point correlation functions in the XX limit.

Experimental results

Research questions

• RQ1 Can functional integral representations with non-standard 'automorphic' boundary conditions be used to compute finite-temperature correlators in integrable spin chains?
• RQ2 How does the introduction of spatially inhomogeneous automorphic boundary conditions—valid only on a subset of lattice sites—affect the structure of the generating functional?
• RQ3 Do the determinant representations derived via zeta-regularization yield results consistent with established solutions in the XX and XY models?
• RQ4 Can the method be extended to models with translationally inhomogeneous boundary conditions or non-uniform couplings?
• RQ5 Is the functional integral formalism self-consistent, as evidenced by agreement with results from Bethe ansatz and multiple integral representations?

Key findings

• The generating functional for z-component spin correlators in the XY and XX chains is expressed as a sum of fermionic functional integrals with 'automorphic' boundary conditions in imaginary time.
• The functional integral formalism yields exact expressions for the partition function and correlation functions, including the magnetization and two-point correlation functions.
• In the thermodynamic limit, the spin z-component expectation value is given by $\sigma^z = 1 - \frac{1}{\pi} \int_{-\pi}^{\pi} \frac{dq}{1 + e^{\beta \varepsilon_q}} $, matching known results.
• The two-point correlation function is derived as $\langle\sigma_{m+1}^z \sigma_1^z\rangle = (\sigma^z)^2 - \frac{1}{\pi^2} \left| \int_{-\pi}^{\pi} \frac{e^{imq}}{1 + e^{\beta \varepsilon_q}} dq \right|^2 $, consistent with literature.
• The zeta-regularization procedure ensures convergence and correctness of the determinant representations, with final results agreeing with those from Bethe ansatz and multiple integral methods.
• The method generalizes previous approaches by allowing non-uniform boundary conditions in imaginary time, applicable to models with spatially varying couplings or boundary conditions.