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Journal of Physics Communications

Scaling corrections to the ground state energy of the spin-½ isotropic anti-ferromagnetic Heisenberg chain

Bernie G. Nickel

Published Nov 13, 2017• 2 citations
Publisher

Abstract

Solutions to the Bethe Ansatz equations for the ground state of spin-½ isotropic anti-ferromagnetic periodic Heisenberg chains to length L = 2 21 are obtained by combining Lagrange interpolation with Newton–Raphson iteration. The long chain lengths allow many powers of a renormalization group running coupling constant to be included in fits to the ground state energy and make possible the confirmation of the convergence of the leading logarithmic term. The amplitude of this term is consistent with that expected on the basis of conformal field theory and the connection of the discrete spin-½ system to the continuum Wess–Zumino–Witten model. This resolves a decades old discrepancy based on analysis of shorter chains. An analytical improvement to the Hulthén wave-vector distribution is also provided.

DOI
10.1088/2399-6528/aa9a15

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Solutions to the Bethe Ansatz equations for the ground state of spin-½ isotropic anti-ferromagnetic periodic Heisenberg chains to length L = 2 21 are obtained by combining Lagrange interpolation with Newton–Raphson iteration. The long chain lengths allow many powers of a renormalization group running coupling constant to be included in fits to the ground state energy and make possible the confirmation of the convergence of the leading logarithmic term. The amplitude of this term is consistent with that expected on the basis of conformal field theory and the connection of the discrete spin-½ system to the continuum Wess–Zumino–Witten model. This resolves a decades old discrepancy based on analysis of shorter chains. An analytical improvement to the Hulthén wave-vector distribution is also provided.
Evidence is presented that the solutions of the Bethe ansatz equations for spin-½ isotropic Heisenberg chains in fixed total spin and momentum sectors are the roots of single variable polynomials with integer (or integer based) coefficients. Such solutions are used as a starting point for investigation of long chain (critical region) properties. In the total spin S = 0 sector I conjecture explicit formulae for the Bethe string configuration labelling of all left and right tower excitations in the k = 1, SU(2) Wess-Zumino-Witten model.
A detailed account of the results of the numerical solution of the thermodynamic Bethe-ansatz equations for the isotropic ferromagnetic S=(1/2) Heisenberg chain is presented. The extrapolation procedure used in approximating the infinite set of coupled nonlinear integral equations is discussed. The data for the truncated integral equations are analyzed in terms of a finite-string-size scaling. Analytic expressions for the free energy and susceptibility for T\ensuremath{\rightarrow}0 are obtained. All the results are consistent with entropy S\ensuremath{\sim}(T/\ensuremath{\Vert}J\ensuremath{\Vert})1/2{)}^{1/2})1/2 and \ensuremath{\chi}\ensuremath{\sim} \ensuremath{\Vert}J\ensuremath{\Vert} / T2{T}^{2}T2 [${\mathrm{scrL}}^{\mathrm{\ensuremath{-}}1}$+(lnscrL)/scrL2{\mathrm{scrL}}^{2}scrL2+...]+O (${T}^{\mathrm{\ensuremath{-}}3/2}$) , where scrL=ln(\ensuremath{\Vert}J\ensuremath{\Vert}/T), suggesting the existence of a marginal variable. The logarithmic corrections reflect the analogy to the Kondo problem.
Leading and next-to-leading-order finite-size corrections to the ground and first excited states are calculated for the spin-1/2 anisotropic Heisenberg model in the critical region. The analytic results are compared to numerical data obtained for chains up to a length of N=1024. It is found that, near the isotropic point, the asymptotic region where the results obtained for N to infinity are applicable sets in at very large N values, and for obtaining good accuracy in fitting the numerical data one has to take into account several correction terms, even at large (N>100) chain lengths.
We consider the detailed structure of low-energy excitations in the periodic spin-1/2 XXZXXZXXZ Heisenberg chain. By performing a perturbative calculation of the nonlinear corrections to the Gaussian model obtained from bosonization, we determine the exact coefficients of asymptotic expansions in inverse powers of the system length NNN for a large number of low-lying excited energy levels. This allows us to calculate eigenenergies of the lattice model up to order $\mathcal{O}({N}^{\ensuremath{-}4})$, without having to solve the Bethe ansatz equations. At the same time, it is possible to express the exact eigenstates of the lattice model in terms of bosonic modes.
We develop an efficient method to perform density matrix renormalization group simulations of the SU(NNN) Heisenberg chain with open boundary conditions taking full advantage of the SU(NNN) symmetry of the problem. This method is an extension of the method previously developed for exact diagonalizations and relies on a systematic use of the basis of standard Young tableaux. Concentrating on the model with the fundamental representation at each site (i.e., one particle per site in the fermionic formulation), we have benchmarked our results for the ground-state energy up to N=8N=8N=8 and up to 420 sites by comparing them with Bethe ansatz results on open chains, for which we have derived and solved the Bethe ansatz equations. The agreement for the ground-state energy is excellent for SU(3) (12 digits). It decreases with NNN, but it is still satisfactory for N=8N=8N=8 (six digits). Central charges ccc are also extracted from the entanglement entropy using the Calabrese-Cardy formula and agree with the theoretical values expected from the SU(N)1\mathrm{SU}{(N)}_{1}SU(N)1​ Wess-Zumino-Witten conformal field theories.