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arXiv:2208.13533 · Communications in Mathematical Physics

Nearest-Neighbour Correlation Functions for the Supersymmetric XYZ Spin Chain and Painlevé VI

Christian Hagendorf, Hjalmar Rosengren

Published Apr 1, 2024• 1 citation
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Abstract

Abstract We study nearest-neighbour correlation functions for the ground state of the supersymmetric XYZ spin chain with odd length and periodic boundary conditions. Under a technical assumption related to the Q-operator of the corresponding eight-vertex model, we show that they can be expressed exactly in terms of the Painlevé VI tau functions $$s_n$$ s n and $${{\bar{s}}}_n$$ s ¯ n introduced by Bazhanov and Mangazeev. Furthermore, we give an interpretation of the correlation functions in terms of the Painlevé VI Hamiltonian.

DOI
10.1007/s00220-024-04977-w

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Abstract We study nearest-neighbour correlation functions for the ground state of the supersymmetric XYZ spin chain with odd length and periodic boundary conditions. Under a technical assumption related to the Q-operator of the corresponding eight-vertex model, we show that they can be expressed exactly in terms of the Painlevé VI tau functions sns_nsn​ s n and sˉn{{\bar{s}}}_nsˉn​ s ¯ n introduced by Bazhanov and Mangazeev. Furthermore, we give an interpretation of the correlation functions in terms of the Painlevé VI Hamiltonian.
We analyze the ground state of a strongly interacting fermion chain with a supersymmetry. We conjecture a number of exact results, such as a hidden duality between weak and strong couplings. By exploiting a scale-free property of the perturbative expansions, we find exact expressions for the order parameters, yielding the critical exponents. We show that the ground state of this fermion chain and another model in the same universality class, the XYZ chain along a line of couplings, are both written in terms of the same polynomials. We demonstrate this explicitly for up to N = 24 sites and provide consistency checks for large N . These polynomials satisfy a recursion relation related to the Painlevé VI differential equation and, using a scale-free property of these polynomials, we derive a simple and exact formula for their limit.
We show that the XYZ spin chain along the special line of couplings J x J y +J x J z +J y J z =0 possesses a hidden mathcalN=(2,2)\\mathcal{N}=(2,2)mathcalN=(2,2) supersymmetry. This lattice supersymmetry is non-local and changes the number of sites. It extends to the full transfer matrix of the corresponding eight-vertex model. In particular, it is shown how to derive the supercharges from Baxter's Bethe ansatz. This analysis leads to new conjectures concerning the ground state for chains of odd length. We also discuss a correspondence between the spectrum of this XYZ chain and that of a manifestly supersymmetric staggered fermion chain
Abstract The square-lattice eight-vertex model with vertex weights a , b , c , d obeying the relation ( a 2 + a b ) ( b 2 + a b ) = ( c 2 + a b ) ( d 2 + a b ) and periodic boundary conditions is considered. It is shown that the transfer matrix of the model for L = 2 n + 1 vertical lines and periodic boundary conditions along the horizontal direction possesses the doubly degenerate eigenvalue Θ n = ( a + b ) 2 n + 1 . This proves a conjecture by Stroganov from 2001. The proof uses the supersymmetry of a related XYZ spin-chain Hamiltonian. The eigenstates of the transfer matrix corresponding to Θ n are shown to be the ground states of the spin-chain Hamiltonian. Moreover, for positive vertex weights Θ n is the largest eigenvalue of the transfer matrix.