XXZ Spin Chain with the Asymmetry Parameter Δ = −1/2: Evaluation of the Simplest Correlators

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We consider a finite XXZ spin chain with periodic boundary conditions and an odd number of sites. It appears that for the special value of the asymmetry parameter Δ = −1/2, the ground state of this system described by the Hamiltonian \(H_{xxz} = - \sum\nolimits_{j = 1}^N {\left\{ {{{\sigma _j^x \sigma _{j + 1}^x + \sigma _j^y \sigma _{j + 1}^y - \sigma _j^z \sigma _{j + 1}^z } \mathord{\left/ {\vphantom {{\sigma _j^x \sigma _{j + 1}^x + \sigma _j^y \sigma _{j + 1}^y - \sigma _j^z \sigma _{j + 1}^z } 2}} \right. \kern-\nulldelimiterspace} 2}} \right\}} \) has the energy E0 = −3N/2. Although the ground state is antiferromagnetic, we can find the corresponding solution of the Bethe equations. Specifically, we can explicitly construct a trigonometric polynomial Q(u) of degree n = (N−1)/2, whose zeros are the parameters of the Bethe wave function for the ground state of the system. As is known, this polynomial satisfies the Baxter T–Q equation. This equation also has a second independent solution corresponding to the same eigenvalue of the transfer matrix T. We use this solution to find the derivative of the ground-state energy of the XXZ chain with respect to the crossing parameter η. This derivative is directly related to one of the spin–spin correlators, which appears to be \(\left\langle {\sigma _j^z \sigma _{j + 1}^z } \right\rangle = {{ - 1} \mathord{\left/ {\vphantom {{ - 1} 2}} \right. \kern-\nulldelimiterspace} 2} + {3 \mathord{\left/ {\vphantom {3 2}} \right. \kern-\nulldelimiterspace} 2}N^2 \). In turn, this correlator gives the average number of spin strings for the ground state of the chain, \(\left\langle {N_{string} } \right\rangle = {{\left( {{3 \mathord{\left/ {\vphantom {3 8}} \right. \kern-\nulldelimiterspace} 8}} \right)\left( {N - 1} \right)} \mathord{\left/ {\vphantom {{\left( {{3 \mathord{\left/ {\vphantom {3 8}} \right. \kern-\nulldelimiterspace} 8}} \right)\left( {N - 1} \right)} N}} \right. \kern-\nulldelimiterspace} N}\). All these simple formulas fail if the number N of chain sites is even.