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arXiv:1906.03143 · Physical review. B./Physical review. B

Equilibrium dynamics of the XX chain

Frank Göhmann, Karol Kajetan Kozlowski, Jesko Sirker, Junji Suzuki

Published Oct 24, 2019• 33 citations
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Abstract

The equilibrium dynamics of the spin-$\frac{1}{2}$ XX chain is reexamined within a recently developed formalism based on the quantum transfer matrix and a thermal form-factor expansion. The transversal correlation function is evaluated in real time and space. The high-accuracy calculation reproduces several exact results in limiting cases as well as the well-known asymptotic formulas obtained by the matrix Riemann-Hilbert approach. Furthermore, comparisons to numerical data based on a direct evaluation of the Pfaffian as well as to asymptotic formulas obtained within nonlinear Luttinger liquid theory are presented.

DOI
10.1103/physrevb.100.155428

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The equilibrium dynamics of the spin-1/2 XX chain is re-examined within a recently developed formalism based on the quantum transfer matrix and a thermal form factor expansion. The transversal correlation function is evaluated in real time and space. The high-accuracy calculation reproduces several exact results in limiting cases as well as the well-known asymptotic formulas obtained by the matrix Riemann-Hilbert approach. Furthermore, comparisons to numerical data based on a direct evaluation of the Pfaffian as well as to asymptotic formulas obtained within non-linear Luttinger liquid theory are presented.
We analyze the transverse dynamical two-point correlation function of the XX chain by means of a thermal form factor series. The series is rewritten in terms of the resolvent and the Fredholm determinant of an integrable integral operator. This connects it with a matrix Riemann-Hilbert problem. We express the correlation function in terms of the solution of the matrix Riemann-Hilbert problem. The matrix Riemann-Hilbert problem is then solved asymptotically in the high-temperature limit. This allows us to obtain the leading high-temperature contribution to the two-point correlation function at any fixed space-time separation.
We present the new expressions for the formfactors of local operators for the XX- quantum spin chain as a Cauchy determinants. Using the known functional form of the correlator at large distances we propose the new expression for the constant for the asymptotics of the correlator as a Cauchy determinant. We calculate the momentum distribution for the general case of the XXZ- spin chain and point out that it is completely different from that for the Luttinger model (the system of fermions). For the XX chain we compare numerically the value of the lowest formfactor and the expectation value of momentum- zero operators which is determined by the functional form of the correlator. 1.Introduction. In the present paper we study the spin-spin equal-time correlator in the XX- spin chain
We derive expressions for the form factors of the quantum transfer matrix of the spin- X X Z chain which are suitable for taking the infinite Trotter number limit. These form factors determine the finitely many amplitudes in the leading asymptotics of the finite-temperature correlation functions of the model. We consider form factor expansions of the longitudinal and transversal two-point functions. Remarkably, the formulae for the amplitudes are in both cases of the same form. We also explain how to adapt our formulae to the description of ground-state correlation functions of the finite chain. The usefulness of our novel formulae is demonstrated by working out explicit results in the high- and low-temperature limits. We obtain, in particular, the large-distance asymptotics of the longitudinal two-point functions for small temperatures by summing up the asymptotically most relevant terms in the form factor expansion of a generating function of the longitudinal correlation functions. As expected, the leading term in the expansion of the corresponding two-point functions is in accordance with conformal field theory predictions. Here it is obtained for the first time by a direct calculation.
We study the out-of-equilibrium dynamics in the XX chain with step-like magnetic field, which maps to an inhomogeneous tight-binding chain after Jordan-Wigner transformation. We obtain exact analytic expressions for the fermionic two-point correlation function after a quantum quench from several initial product states, both homogeneous and inhomogeneous ones. This is achieved by using a combination of Fourier and Laplace transforms, which al low us to map the problem to a standard Riemann-Hilbert problem on the unit circle. For arbitrary positions and times the correlators are not expressed in terms of elementary functions. However, in the hydrodynamic limit x,y,t→∞x,y,t\to\inftyx,y,t→∞ with fixed ratios, we provide explicit formulas that depend only on the effective transmission coefficient across the origin. We benchmark our analytic predictions against exact numerical simulations and find excellent agreement in the hydrodynamic limit, apart from finite-time corrections.