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arXiv:cond-mat/9212021 · Physical review. B, Condensed matter

Test of conformal invariance in a one-dimensional quantum liquid with long-range interactions

Rudolf A. Römer, Bill Sutherland

Published Sep 1, 1993• 12 citations
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Abstract

We study the momentum distribution of one-dimensional Bose and Fermi systems with long-range interaction g/${\mathit{r}}^{2}$ for the values g=-1/2, 0, 4, singled out by random-matrix theory. By making explicit use of the predictions of conformal invariance in periodic quantum chains of finite length, we can show that the critical exponents of the momentum distribution are independent of density and in excellent agreement with estimates obtained from c=1 conformal finite-size scaling analysis. Furthermore we show how the complete functional form of the one-particle reduced density matrix in a finite system considerably simplifies the numerical extrapolation of critical exponents for conformally invariant systems.

DOI
10.1103/physrevb.48.6058

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We study the momentum distribution of one-dimensional Bose and Fermi systems with long-range interaction g/r2{\mathit{r}}^{2}r2 for the values g=-1/2, 0, 4, singled out by random-matrix theory. By making explicit use of the predictions of conformal invariance in periodic quantum chains of finite length, we can show that the critical exponents of the momentum distribution are independent of density and in excellent agreement with estimates obtained from c=1 conformal finite-size scaling analysis. Furthermore we show how the complete functional form of the one-particle reduced density matrix in a finite system considerably simplifies the numerical extrapolation of critical exponents for conformally invariant systems.
Finite-size scaling analysis is performed for one-dimensional Bose and Fermi systems with long-range interaction g/r2{\mathit{r}}^{2}r2, based on the exact solution of Sutherland. The low-energy behavior of this model is shown to be described by the c=1 conformal field theory. Exact formulas for various correlation exponents are obtained. At a special point g=4, the model belongs to the same universality class as the Haldane-Shastry model, the effective theory of which is the level-1 SU(2) Kac-Moody theory.
The ground-state properties of a quantum spin chain with three-spin couplings are investigated using finite-size calculations. The results are compared with the predictions of conformal invariance for finite systems at the critical point. The critical point. The critical exponents eta E and eta S of energy density and spin correlation functions are calculated and estimated to be eta E approximately=1.30 and eta S approximately=0.25. The results suggest that this model is conformally invariant with a central charge very close to or equal to one.
We present the finite-size scaling theory of one-dimensional quantum critical systems in the presence of boundaries. While the finite-size spectrum in the conformal limit, namely of a conformal field theory with conformally invariant boundary conditions, is related to the dimensions of boundary operators by Cardy, the actual spectra of lattice models are affected by both bulk and boundary perturbations and contain non-universal boundary energies. We obtain a general expression of the finite-size energy levels in the presence of bulk and boundary perturbations. In particular, a generic boundary perturbation related to the energy-momentum tensor gives rise to a renormalization of the effective system size. We verify our field-theory formulation by comparing the results with the exact solution of the critical transverse-field Ising chain and with accurate numerical results on the critical three-state Potts chain obtained by Density-Matrix Renormalization Group.
In this work, we investigate the critical properties of the Bose-Einstein condensation transition in a model of non-interacting bosons whose one-particle hamiltonian in the tight-binding approximation includes long-range couplings in a linear chain with periodic boundary conditions. The couplings along the chain are described by a kinetic hopping term which is responsible by the quantum mobility of the particles and whose amplitude decays with the distance between the sites as a power-law with a characteristic exponent a. We show that the density of states in the vicinity of the lower band edge of the energy spectrum strongly depends on a. In the re- gime of 1 < α < 2, the density of states vanishes as a power-law, similar to the behavior found for bosons in hypercubic lattices with dimension d = 2/(α -1). We determine the critical properties of the Bose-Einstein condensation transition that takes place in this regime using a hypothesis of a single characteristic length scale and a finite-size scaling analysis. In the regime of 3/2 < α < 2, the critical exponents are quite similar to those that characterize the transition in the corresponding hypercubic lattice. On the other hand, in the regime of 1 < α < 3/2, the exponents obtained from the single length scale hypothesis are quite distinct form those on the hypercubic lattice. This result indicates that the single length scale hypothesis can not correctly capture the critical exponents in the regime where the effective dimensionality of the model is above the upper critical dimension above which the transition has a mean-field critical behavior.
The effects of a long-range electronic potential on a one-dimensional chain of spinless fermions are investigated by numerical techniques (exact diagonalization of rings with up to 30 sites complemented by finite size analysis) and analytic calculations. Due to a competition between the 2kF{2k}_{F}2kF​ oscillations in the density and the (very slow) log divergence of the long wavelength part of the Coulomb potential, the metallic character of the system is enhanced at intermediate (and up to quite large) length scales. Despite some similarities, we found that this quasimetallic regime is not of the Luttinger liquid type, as evidenced by strong deviations from predictions of conformal field theory and in agreement with the picture of a (very weakly pinned) Wigner crystal. When the strength of the Coulomb interaction is substantially increased, we observe a smooth crossover to a strongly localized charge density wave.