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arXiv:cond-mat/9701188 · International Journal of Modern Physics B

Exact Solution to the Moment Problem for the XY Chain

Nicholas S. Witte

Published May 20, 1997• 7 citations
arXivHTMLPDF

Abstract

We present the exact solution to the moment problem for the spin-1/2 isotropic antiferromagnetic XY chain with explicit forms for the moments with respect to the Néel state, the cumulant generating function, and the Resolvent Operator. We verify the correctness of the Horn–Weinstein Theorems, but the analytic structure of the generating function <e-tH> in the complex t-plane is quite different from that assumed by the t-Expansion and the Connected Moments Expansion due to the vanishing gap. This function has a finite radius of convergence about t=0, and for large t has a leading descending algebraic series E(t)-E0~ At-2. The Resolvent has a branch cut and essential singularity near the ground state energy of the form G(s)/s~ B|s+1|-3/4 exp (C|s+1|1/2). Consequently extrapolation strategies based on these assumptions are flawed and in practise we find that the CMX methods are pathological, and cannot be applied, while numerical evidence for two of the t-expansion methods indicates a clear asymptotic convergence behaviour with truncation order.

DOI
10.1142/s0217979297000769

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We present the exact solution to the moment problem for the spin-1/2 isotropic antiferromagnetic XY chain with explicit forms for the moments with respect to the Néel state, the cumulant generating function, and the Resolvent Operator. We verify the correctness of the Horn–Weinstein Theorems, but the analytic structure of the generating function <e-tH> in the complex t-plane is quite different from that assumed by the t-Expansion and the Connected Moments Expansion due to the vanishing gap. This function has a finite radius of convergence about t=0, and for large t has a leading descending algebraic series E(t)-E0~ At-2. The Resolvent has a branch cut and essential singularity near the ground state energy of the form G(s)/s~ B|s+1|-3/4 exp (C|s+1|1/2). Consequently extrapolation strategies based on these assumptions are flawed and in practise we find that the CMX methods are pathological, and cannot be applied, while numerical evidence for two of the t-expansion methods indicates a clear asymptotic convergence behaviour with truncation order.
Abstract We obtain the ground state magnetization of the Heisenberg and XXZ spin chains in a magnetic field h as a series in , where h c is the smallest field for which the ground state is fully polarized. All the coefficients of the series can be computed in closed form through a recurrence formula that involves only algebraic manipulations. For some values of the anisotropy parameter the expansion is numerically observed to be convergent in the full range . To that end we express the free energy at mean magnetization per site as a series in whose coefficients can be similarly recursively computed in closed form. This series converges for all . The recurrence is nothing but the Bethe equations when their roots are written as a double series in their corresponding Bethe number and in . It can also be used to derive the corrections in finite size, that correspond to the spectrum of a free compactified boson whose Luttinger parameter can be expanded as a similar series. The method presumably applies to a large class of models: it also successfully applies to a case where the Bethe roots lie on a curve in the complex plane.
The ground-state energy of the XY antiferromagnetic chains with spin-1/2 and spin-1 are obtained by a real space renormalization group method. For S=1/2 a large spin cluster is used in the mapping to improve the accuracy and a boundary theory is applied to extrapolate to the result which is in excellent agreement with the exact Bethe Ansatz result. For S=1 the ground-state energy of the model is obtained by using the renormalization group method and a finite chain calculation for lattices up to 16 sites. The two results are in very good agreement.
We calculate exactly cumulant generating functions (full counting statistics) for the transverse, staggered magnetization, and the domain walls at zero temperature for a finite interval of the XY spin chain. In particular, we also derive a universal interpolation formula in the scaling limit for the full counting statistics of the transverse magnetization and the domain walls which is based on the solution of a Painlevé V equation. By further determining subleading corrections in a large interval asymptotics, we are able to test the applicability of conformal field theory predictions at criticality. As a by-product, we also obtain exact results for the probability of formation of ferromagnetic and antiferromagnetic domains in both the σ^{z} and σ^{x} basis in the ground state. The analysis hinges upon asymptotic expansions of block Toeplitz determinants, for which we formulate and check numerically a different conjecture.
A zero-temperature renormalization group (RG) approach is applied to the one-dimensional, spin-1/2 antiferromagnetic Heisenberg dimerized (alternating) chain. Specifically, the ground state energy and lowest-lying spectral excitations are examined. The calculation indicates the existence of a gap in the spectrum of the dimerized chain which vanishes only in the limit of a uniform spin chain, in contrast to a recent Green’s function approach. The RG results are in reasonable agreement with numerical extrapolations on the exact eigenvalue spectrum of finite chains of up to 12 spins. Both methods are compared with several other approximate treatments of the Heisenberg system, and tested by comparison with exact results for the spin-1/2 XY dimerized chain.
We consider the non-Hermitian XY spin chain with open boundary conditions when the anisotropy parameter is extended to complex values. By analyzing the quasi-Hamiltonian matrix, we demonstrate that the free-fermion structure of the quasi-energy spectrum coincides with that of the Hermitian model and construct the corresponding biorthogonal fermionic basis away from exceptional points (EPs). We make use of an explicit Chebyshev-polynomial representation of the open-boundary eigenvectors in which the quasi-energy ε\varepsilonε is the natural spectral variable. This quasi-energy polynomial form is particularly useful at EPs, because EPs correspond to repeated roots of the same boundary polynomial, making the construction of generalized eigenvectors by ε\varepsilonε-differentiation transparent. At EPs, where the quasi-Hamiltonian becomes defective, we derive the Jordan normal form and construct the associated generalized eigenvectors, which yields the correct counting of independent many-body eigenstates. We further show that EPs act as branch points in the complex anisotropy plane, leading to the characteristic permutation of eigenenergies and eigenstates upon encirclement. The branch-cut structure of the biorthogonal eigenstates provides direct evidence for the exchange of eigenstates when an EP is encircled. These results provide an analytically controlled many-body platform for studying EP physics and non-Hermitian topology beyond momentum-space descriptions.
We investigate the ground state of the alternating spin-1/2 XY chain with the anisotropy H = J ∑ i [1+(-1) i δ][(1+γ) S i x S i +1 x +(1-γ) S i y S i +1 y ], J >0, where 0≤γ≤1 and 0≤δ≤1. From the exact results of the ground state energy calculation, the ground state is found to be either one of the following three states: the spin fluid state (γ=δ=0), the effective singlet pair state (ES state; δ>γ) and the Néel state (γ>δ). We also exactly calculate the asymptotic form of the longitudinal spin correlation < S i z S j z > in the long distance limit. Both its functional form and the parameter dependence of the correlation length are quite consistent with our picture for the ES state and the Néel state.