Correlation functions of the integrable $SU(n)$ spin chain: analytic three-site evaluation, extension to $m\ge 4$ and $n>3$, and finite-temperature computation

The integrable $SU(n)$ spin chain with fundamental representation at each site is the prototypical higher-rank Bethe-ansatz-solvable system, yet for $n>2$ its correlation functions resisted exact evaluation for decades. Ribeiro and Klümper (2019) derived discrete functional equations, valid at any temperature, for a generalized density operator of the $SU(n)$ chain, and solved them at zero temperature for the $SU(3)$ chain up to three sites: two-site correlators were obtained in closed form in terms of values of the Hurwitz zeta function (a parameter-shifted generalization of the Riemann zeta function), while the three-site correlators were reduced to convolution integrals with integrands built from the digamma function (the logarithmic derivative of the Gamma function) that could only be evaluated numerically. Four extensions were explicitly left open and remain open: (i) an analytic evaluation of the three-site convolution integrals; (ii) the solution of the functional equations for four and more sites; (iii) the derivation and solution of the three-site equations for $SU(n)$ with $n>3$, starting with $SU(4)$; and (iv) the exact evaluation of three-point and higher correlators at finite temperature, which requires nonlinear integral equations for the generalized quantum transfer matrix and a scheme beyond the leading-eigenvalue derivative trick that yields only two-point functions. Independent reduced-qKZ computations, a nested version of the LeClair-Mussardo series (a form-factor expansion for thermal correlators) covering only one- and two-point functions, and separation-of-variables determinant formulae (obtained by a change of spectral variables that reduces the transfer-matrix eigenproblem to scalar form) for finite chains all leave this core untouched.

Background

The integrable $SU(n)$ spin chain with $n>2$ places the fundamental representation of $SU(n)$ at every site of a one-dimensional lattice, with interactions generated by the rational Yang-Baxter $R$-matrix (the fundamental solution of the Yang-Baxter consistency equation underlying exact solvability in one dimension); the spin-1/2 Heisenberg (XXX) chain is the $n=2$ member of this family. Its spectrum and thermodynamics have been understood since the 1970s through the nested Bethe ansatz (the Bethe-ansatz solution adapted to algebras of rank larger than one, in which several intertwined sets of rapidities are introduced), but its correlation functions were a longstanding open problem: for $n>2$ the rational $R$-matrix has no crossing symmetry, the property that underlies most exact correlation-function technology in the $SU(2)$ case.

Short-range correlations of a quantum spin chain in the thermodynamic limit are encoded in the reduced density matrix $D_m(\lambda_1,\dots,\lambda_m)$, the normalized expectation value of products of local operators on $m$ consecutive sites, generalized to arbitrary complex inhomogeneity parameters $\lambda_j$ (site-dependent shifts of the spectral parameter that preserve integrability). For the XXX chain, a discrete reduced quantum Knizhnik-Zamolodchikov (qKZ) equation — a difference equation in the inhomogeneities satisfied by the density matrix — was introduced by Boos, Jimbo, Miwa, Smirnov and Takeyama, leading to exact multiple-integral formulae; a factorization theorem eventually expressed all static correlators through a finite set of auxiliary generating functions (the Fermionic basis), giving closed forms in Riemann zeta values (Göhmann's lecture notes review this program). Finite temperature is handled with the quantum transfer matrix (QTM), the two-dimensional classical transfer matrix encoding the Gibbs ensemble through a Suzuki-Trotter decomposition (a slicing of the quantum partition function into a classical two-dimensional vertex model) with finite Trotter number $N$: thermal two-point functions follow from differentiating the leading QTM eigenvalue with respect to an inhomogeneity (Aufgebauer and Klümper, 2012).

Ribeiro and Klümper (2019) circumvented the absence of crossing symmetry for $n>2$ by enlarging the density matrix to a generalized density operator $\mathbb{D}_m$ whose auxiliary lines carry arbitrary mixtures of fundamental and anti-fundamental representations. Its matrix elements, normalized by leading QTM eigenvalues, satisfy a closed set of discrete functional equations that hold at any finite Trotter number, hence in principle at any temperature. At zero temperature they solved these equations for two sites and general $n$ (the $SU(3)$ specialization appearing in closed form through the Hurwitz zeta function $\zeta(s,a)=\sum_{k\ge 0}(k+a)^{-s}$, e.g. the two-site coefficient $\alpha_{33}(0,0)=\frac{1}{24}(2-\pi/\sqrt{3}-3\ln 3)$), and for three sites in the $SU(3)$ case: after reducing the problem through the 42-dimensional singlet sector of the generalized density operator (the invariant part of the operator space under the global $SU(3)$ symmetry) to 11 independent equations for the generalized mixed density operator, and finally to three decoupled functional equations, the solution is expressed through convolution integrals whose integrands are products of rational functions and the digamma function $\psi(z)=\Gamma'(z)/\Gamma(z)$, evaluated only numerically, e.g. $\langle P_{12}P_{23}\rangle=0.191368820116674$. The correlators were shown not to factorize into two-point functions, in contrast to the $SU(2)$ chain.

The conclusions of that work delimit precisely what remains of the program: the analytic evaluation of the three-site convolution integrals (the required Fourier transform of a rational function times a digamma function was left open); the solution of the functional equations for $m\ge 4$, called completely open; the three-site functional equations and their solution for $SU(n)$ with $n>3$, at least for $SU(4)$; and the exact finite-temperature evaluation of correlations, which requires nonlinear integral equations (NLIEs, integral equations for auxiliary functions characterizing the spectrum of the transfer matrix) for the generalized QTM (the quantum transfer matrix enlarged by $m\cdot n$ auxiliary spaces carrying the inhomogeneous lines of the density operator) and, more challengingly, a method for three-point functions at finite temperature, since the eigenvalue-derivative trick gives only two-point functions.

Problem Statement

The problem is to complete the exact evaluation of short-range correlation functions of the integrable $SU(n)$ spin chain with $n>2$ beyond the frontier reached by Ribeiro and Klümper (2019) — zero temperature, $SU(3)$, distances $m\le 3$ — along the four directions they identified. A full solution delivers the following, each stated so that it can be checked against known limiting data.

(1) Closed form. An analytic evaluation of the $m=3$, $SU(3)$ correlation functions currently known only through the convolution integrals that solve the three-site functional equations, whose integrands are built from rational functions times digamma functions: these convolution integrals (equivalently, the Fourier transform of a product of a rational function with a digamma function) must be computed analytically, going beyond the already-published convolution-integral representations, functional equations, and numerical evaluations. The result must reproduce the published numerical values, e.g. $\langle P_{12}P_{23}\rangle=0.191368820116674$ in the thermodynamic limit.

(2) Higher distance. The derivation and solution of the discrete functional equations for $m\ge 4$: at least the explicit four-site case for $SU(3)$, and in full generality the solution for arbitrary $m$, i.e. exact four-site (and beyond) density-matrix elements in the thermodynamic limit with correct normalization, single-site trace reduction properties, and homogeneous limits.

(3) Higher rank. The discrete functional equations for $m=3$ in $SU(n)$ with $n>3$ — at least for $SU(4)$ — together with their solution, so that three-site correlators are available beyond rank two.

(4) Finite temperature. The exact evaluation of $m\ge 3$ correlation functions at temperature $T>0$, at least for the $SU(3)$ chain: a characterization of the generalized quantum transfer matrix sufficient for computation (the anticipated route being nonlinear integral equations for its leading eigenvalue and eigenstates, which also yields the two-point thermal correlators), plus a computational scheme for three-point and higher thermal correlators, for which differentiating the leading eigenvalue with respect to an inhomogeneity does not work. Results must reduce, in the $T\to 0$ limit, to the known zero-temperature values.

These four deliverables jointly complete one scientific objective — the exact correlation functions of the integrable $SU(n)$ chain at arbitrary short distance, rank, and temperature — and they are posed exactly as the source work delimits them; progress on individual components is meaningful, but the problem is solved only when all four are settled. Within components (2)–(4), the named concrete case (four sites for $SU(3)$; $SU(4)$; the $SU(3)$ chain) is a first milestone rather than the completion boundary: each of these components is settled only when its full-generality statement is attained — arbitrary $m$ for (2), general $SU(n)$ with $n>3$ for (3), and arbitrary temperature for (4).

References

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