Analytic long-time and long-distance asymptotics of finite-temperature order-parameter correlations in the anisotropic quantum XY chain

The spin-1/2 XY chain in a transverse field is exactly solvable by a Jordan–Wigner mapping to free fermions, and Iorgov (2011) derived closed factorized form factors of the spin operators on the finite periodic chain. Yet the long-time and long-distance asymptotics of the order-parameter ($\sigma^x$) two-point function at finite temperature, for which Krones and Stolze (2011) provided numerical evidence of exponential decay in both space and time, have never been derived analytically for the anisotropic chain ($0<\gamma<1$). Only the isotropic limit ($\gamma=0$; Its–Izergin–Korepin–Slavnov 1993 and modern rederivations) and, very recently, the decay rates of the Ising limit ($\gamma=1$) are understood. The problem is to obtain the complete finite-temperature asymptotic behavior — exponential decay rates (proving or correcting the Krones–Stolze conjecture), prefactors including the time-like power-law factor, and the space-like/time-like and high-temperature crossovers — for the anisotropic chain, with the broader goal of controlled thermal form-factor expansions of multipoint dynamical correlators. The problem remains open.

Background

The spin-$1/2$ XY chain in a transverse field is the one-dimensional Hamiltonian

$H=-\sum_{j}\Big[\tfrac{J(1+\gamma)}{2}\,\sigma^x_j\sigma^x_{j+1}+\tfrac{J(1-\gamma)}{2}\,\sigma^y_j\sigma^y_{j+1}+h\,\sigma^z_j\Big],$

where $\sigma^{x,y,z}_j$ are Pauli matrices, $h$ is a magnetic field perpendicular to the ordering plane, and the anisotropy $\gamma$ interpolates between the isotropic XX chain ($\gamma=0$) and the transverse-field Ising chain ($\gamma=1$). A Jordan–Wigner transformation — the standard mapping that replaces spins by lattice fermions, attaching to each fermion operator a string of $(-1)$ factors from the preceding sites — brings $H$ to a quadratic (free-fermion) form, which is why the model counts as exactly solved. Correlations of $\sigma^z$ then reduce to free-fermion density correlations and are elementary. The order-parameter correlator $\langle\sigma^x_i(t)\sigma^x_j(0)\rangle$ is of a different nature: $\sigma^x$ carries the full Jordan–Wigner string, so its two-point function is a Pfaffian (a signed square root of a determinant) whose size grows with distance, and its exact decomposition over eigenstates is a form-factor expansion — a spectral sum over matrix elements $\langle {\rm vac}|\sigma^x|n\rangle$ (form factors) weighted by Boltzmann factors $e^{-\beta E_n}$ at inverse temperature $\beta=1/T$.

Iorgov (2011) derived closed factorized formulas for the $\sigma^x$ and $\sigma^y$ form factors of the finite periodic chain, using separation of variables in the $N=2$ Baxter–Bazhanov–Stroganov $\tau^{(2)}$ model (a solvable two-dimensional lattice model whose Ising-sector eigenstates encode the XY-chain spectrum). With them he re-derived the known zero-temperature large-distance asymptotics of $\langle\sigma^x_0\sigma^x_d\rangle$ in the disordered phase without the usual Toeplitz-determinant machinery, and posed as the open target the finite-temperature dynamics: it is, in his words, interesting to derive analytically the long-time and long-distance asymptotic behavior of the $\sigma^x$ spin correlations at finite temperature, which had just been observed numerically by Krones and Stolze.

What is known around this question: at zero temperature in the gapped phases the decay is exponential with distance, with controlled asymptotics. At any $T>0$ all spin correlations of the chain decay exponentially with distance, with temperature-dependent correlation lengths — a classical result for the XY model, whose determinantal representations of the spin correlations go back to Barouch and McCoy (1971). For the isotropic chain $\gamma=0$ the full finite-temperature space-time asymptotics are exact: Its, Izergin, Korepin and Slavnov (1993) showed that $\langle\sigma^x_i(t)\sigma^x_j(0)\rangle$ decays exponentially in both space and time with explicitly computable rates, by representing it as a temperature-dependent Fredholm determinant (the determinant of an integral operator) and analyzing the associated Riemann–Hilbert problem (a boundary-value problem for piecewise analytic functions) by the nonlinear steepest-descent method; the derivation rests on a mapping of the correlator to a classical integrable nonlinear equation that has no known anisotropic counterpart. Krones and Stolze (2011) computed the anisotropic correlators numerically: exponential decay in space and time at $T>0$; a space-like regime ($n>v_0 t$, with $v_0$ the maximal quasiparticle group velocity and $n$, $t$ the space and time separations) in which the equal-time exponential persists; a time-like regime ($n<v_0 t$) with a temperature-dependent rate and a power-law prefactor; and, in the autocorrelation $n=0$, a crossover at high temperature from the exponential decay to the rigorous infinite-temperature Gaussian of Brandt–Jacoby and Capel–Perk (the $n\neq 0$ correlator vanishes at $T=\infty$). They conjectured a natural generalization of the 1993 rate formula,

$f(n,t)=\frac{1}{2\pi}\int_{-\pi}^{\pi}dk\,\Big|n-t\,\frac{d\varepsilon_k}{dk}\Big|\,\ln\Big|\tanh\frac{\beta\varepsilon_k}{2}\Big|,$

with $\varepsilon_k$ the single-particle fermion energy — the signed dispersion of Krones and Stolze's Eq. (6), $\varepsilon_k=-\mathrm{sign}(h+J\cos k)\sqrt{(h+J\cos k)^2+\gamma^2J^2\sin^2 k}$ in their convention, where the spin operators are $S^\alpha=\sigma^\alpha/2$, so their $J$ and $h$ are twice the couplings in the Hamiltonian above — together with the time-like prefactor $\propto t^{4\nu^2}$ where $\nu=(1/2\pi)\ln|\tanh(\beta\varepsilon_{k_0}/2)|$ is evaluated at the momentum $k_0$ satisfying $d\varepsilon_{k_0}/dk=n/t$ (their Eqs. (8)–(11)) — but no analytic derivation, no prefactor constants, and no controlled crossover description exists for $\gamma\neq 0$. Beyond the two-point function, Iorgov's motivation was the hope that the factorized form factors would yield controlled spectral expansions for multipoint dynamical correlators at finite temperature; the two-point asymptotics above is the concrete benchmark step of that program.

Problem Statement

Derive analytically the long-time and long-distance asymptotic behavior of the finite-temperature ($T>0$) order-parameter two-point function $\langle\sigma^x_i(t)\,\sigma^x_{i+n}(0)\rangle_T$ of the spin-$1/2$ XY chain in a transverse field at anisotropy $0<\gamma<1$ (and general fields, at least the subcritical regime $|h|<J$ including $h=0$), where $n$ and $t$ are the spatial and temporal separations. The regime $0<\gamma<1$ is the genuinely unsolved interior: the isotropic limit $\gamma=0$ is solved exactly and the Ising limit $\gamma=1$ has recently seen exact decay rates, but neither boundary result extends to the anisotropic chain. A complete answer must jointly determine, as one coherent asymptotic description: (i) the exponential decay in space and in time, with explicit decay rates as functions of $(\gamma,h,T,n,t)$ — in particular establishing, or correcting, the Krones–Stolze conjectured generalization of the Its–Izergin–Korepin–Slavnov rate formula; (ii) the prefactors, including the power-law factor in the time-like regime; and (iii) the crossover structure — the transition between the space-like regime $n>v_0 t$ and the time-like regime $n<v_0 t$ at the maximal quasiparticle group velocity $v_0$, and the high-temperature crossover of the autocorrelation ($n=0$) between the finite-temperature exponential decay and the rigorous infinite-temperature Gaussian — as observed numerically. The derivation should proceed from an exact representation of the correlator, such as Iorgov's factorized finite-chain form factors or the equivalent Pfaffian/Fredholm-determinant representation, and any candidate formula must reproduce the known Its–Izergin–Korepin–Slavnov result at $\gamma=0$ and be consistent with the known transverse-field-Ising decay rates at $\gamma=1$. The motivating extension, explicitly left open by the source, is the use of the same form factors to build controlled, convergent, asymptotically tractable spectral expansions for multipoint dynamical correlators at finite temperature; the two-point asymptotics above constitutes the well-defined benchmark and completion boundary of that program.

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