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arXiv:1301.6311

Absence of finite size correction at the combinatorial point of the integrable higher spin XXZ chain

Kohei Motegi

Published Jan 27, 2013• 0 citations
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Abstract

We investigate the integrable higher spin XXZ chain at the Razumov-Stroganov point. We present a method to evaluate the exact value of the eigenvalue which is conjectured to correspond to the groundstate of the Hamiltonian for finite size chain from the Baxter Q operator. This allows us to examine the exact total energy difference between different number of total sites, from which we find strong evidence for the absence of finite size correction to the groundstate energy.

DOI
10.48550/arxiv.1301.6311

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We investigate the integrable higher spin XXZ chain at the Razumov-Stroganov point. We present a method to evaluate the exact value of the eigenvalue which is conjectured to correspond to the groundstate of the Hamiltonian for finite size chain from the Baxter Q operator. This allows us to examine the exact total energy difference between different number of total sites, from which we find strong evidence for the absence of finite size correction to the groundstate energy.
We study the one-dimensional S=12S=\frac{1}{2}S=21​ XXZXXZXXZ model on a finite lattice at zero temperature, varying the exchange anisotropy $\ensuremath{\Delta}$ and the number of sites NNN of the lattice. Special emphasis is given to the model with $\ensuremath{\Delta}=\frac{1}{2}$ and NNN odd, whose ground state, the so-called Razumov-Stroganov state, has a peculiar structure and no finite-size corrections to the energy per site. We find that such model corresponds to a special point on the $\ensuremath{\Delta}$ axis which separates the region where adding spin-pairs increases the energy per site from that where the longer the chain the lower the energy. Entanglement properties do not hold surprises for $\ensuremath{\Delta}=\frac{1}{2}$ and NNN odd. Finite-size corrections to the energy per site non trivially vanish also in the ferromagnetic $\ensuremath{\Delta}\ensuremath{\rightarrow}\ensuremath{-}{1}^{+}$ isotropic limit, which is consequently addressed; in this case, peculiar features of some entanglement properties, due to the finite length of the chain and related with the change in the symmetry of the Hamiltonian, are evidenced and discussed. In both the above models the absence of finite-size corrections to the energy per site is related to a peculiar structure of the ground state, which has permitted us to provide new exact analytic expressions for some correlation functions.
The method of de Vega and Woynarovitch (1985) is used to calculate finite-size corrections to the ground-state energy in different sectors for the XXZ Heisenberg chain. Finite-size scaling amplitudes and correction-to-scaling exponents in the critical region are derived. Using conformal invariance, a scaling dimension x=( pi - gamma )/2 pi is extracted corresponding to the electric field operator in the 8-vertex model: this confirms a conjecture of Baxter and Kelland (1974). Finite-size scaling properties near the Kosterlitz-Thouless critical point Delta =-1 are discussed.
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The methods of de Vega and Woynarovitch (1985) are used to calculate finite-size corrections to the ground state energy in different sectors for the XXZ Heisenberg chain, in the critical region -1< Delta <1. The finite-size scaling amplitude for the mass gap between the lowest lying sectors is derived. Using conformal invariance, the scaling dimension is extracted for an associated operator, corresponding to the electric field operator in the 8-vertex model. The conjecture of Baxter and Kelland (1974) for the electric field exponent is confirmed.
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