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arXiv:1005.3202 · Physical Review E

Level density of spin chains of Haldane-Shastry type

Alberto Enciso B, Federico Finkel, Artemio González-López

Published Nov 15, 2010• 23 citations
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Abstract

We provide a rigorous proof of the fact that the level density of all known su(m) spin chains of Haldane-Shastry type associated with the A(N-1) root system approaches a Gaussian distribution as the number of spins N tends to infinity. Our approach is based on the study of the large-N limit of the characteristic function of the level density, using the description of the spectrum in terms of motifs and the asymptotic behavior of the transfer matrix.

DOI
10.1103/physreve.82.051117

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We provide a rigorous proof of the fact that the level density of all known su(m) spin chains of Haldane-Shastry type associated with the A(N-1) root system approaches a Gaussian distribution as the number of spins N tends to infinity. Our approach is based on the study of the large-N limit of the characteristic function of the level density, using the description of the spectrum in terms of motifs and the asymptotic behavior of the transfer matrix.
We show that the density of energy levels of a wide class of finite-dimensional quantum systems tends to a Gaussian distribution as the number of degrees of freedom increases. Our result is based on a variant of the central limit theorem which is especially suited to models whose partition function is explicitly known. In particular, we provide a theoretical explanation of the fact that the level density of several spin chains of Haldane-Shastry type is asymptotically Gaussian when the number of sites tends to infinity.
We analyze the thermodynamics and criticality properties of four families of su(m|n) supersymmetric spin chains of Haldane-Shastry (HS) type, related to both the A_{N-1} and the BC_{N} classical root systems. Using a known formula expressing the thermodynamic free energy per spin of these models in terms of the Perron (largest in modulus) eigenvalue of a suitable inhomogeneous transfer matrix, we prove a general result relating the su(kp|kq) free energy with arbitrary k=1,2,⋯, to the su(p|q) free energy. In this way we are able to evaluate the thermodynamic free energy per spin of several infinite families of supersymmetric HS-type chains, and study their thermodynamics. In particular, we show that in all cases the specific heat at constant volume features a single marked Schottky peak, which in some cases can be heuristically explained by approximating the model with a suitable multilevel system with equally spaced energies. We also study the critical behavior of the models under consideration, showing that the low-temperature behavior of their thermodynamic free energy per spin is the same as that of a (1+1)-dimensional conformal field theory with central charge c=m+n/2-1. However, using a motif-based description of the spectrum we prove that only the three families of su(1|n) chains of type A_{N-1} and the su(m|n) HS chain of BC_{N} type with m=1,2,3 (when the sign ɛ_{B} in the Hamiltonian takes the value -1 in the latter case) are truly critical.
The energy level density distributions of some Haldane-Shastry like spin chains associated with the AN − 1 root system have been computed recently by Enciso et al., exploiting the connection of these spin systems with inhomogeneous one-dimensional vertex models whose energy functions depend on the vertices through specific polynomials of first or second degree. Here we consider a much broader class of one-dimensional vertex models whose energy functions depend on the vertices through arbitrary polynomials of any possible degree. We estimate the order of mean and variance for such energy functions and show that the level density distribution of all vertex models belonging to this class asymptotically follow the Gaussian pattern for large number of vertices. We also present some numerical evidence in support of this analytical result.
We compute the partition function of the su(m)\text{su}(m)su(m) Polychronakos-Frahm spin chain of BCNB{C}_{N}BCN​ type by means of the freezing trick. We use this partition function to study several statistical properties of the spectrum, which turn out to be analogous to those of other spin chains of Haldane--Shastry type. In particular, we find that when the number of particles is sufficiently large the level density follows a Gaussian distribution with great accuracy. We also show that the distribution of (normalized) spacings between consecutive levels is of neither Poisson nor Wigner type but is qualitatively similar to that of the original Haldane--Shastry spin chain. This suggests that spin chains of Haldane--Shastry type are exceptional integrable models since they do not satisfy a well-known conjecture of Berry and Tabor, according to which the spacings distribution of a generic integrable system should be Poissonian. We derive a simple analytic expression for the cumulative spacings distribution of the BCNB{C}_{N}BCN​-type Polychronakos-Frahm chain using only a few essential properties of its spectrum such as the Gaussian character of the level density and the fact that the energy levels are equally spaced. This expression is shown to be in excellent agreement with the numerical data.
We derive an exact expression for the partition function of the su(m)\mathrm{su}(m)su(m) Haldane-Shastry spin chain, which we use to study the density of levels and the distribution of the spacing between consecutive levels. Our computations show that when the number of sites NNN is large enough, the level density is Gaussian to a very high degree of approximation. More surprisingly, we also find that the nearest-neighbor spacing distribution is not Poissonian, so that this model departs from the typical behavior for an integrable system. We show that the cumulative spacing distribution of the model can be well approximated by a simple functional law involving only three parameters.
Level density distributions for some Haldane-Shastry like spin chains associated with A N −1 root system have been computed recently by Enciso et al. , using the connection of these spin chains with inhomogeneous one-dimensional vertex models. The energy functions of such vertex models are given by specific polynomials of first or second degree. Here we consider a much broader class of one-dimensional vertex models whose energy functions are given by arbitrary polynomials of any possible degree and show that the level densities for this class of vertex models asymptotically follow the Gaussian distribution for large number of vertices.