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Physical Review Letters

Finite-size scaling in one-dimensional quantum liquid with long-range interaction

Norio Kawakami, Sung-Kil Yang

Published Oct 28, 1991• 117 citations
Publisher

Abstract

Finite-size scaling analysis is performed for one-dimensional Bose and Fermi systems with long-range interaction g/${\mathit{r}}^{2}$, 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.

DOI
10.1103/physrevlett.67.2493

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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.
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.
Asymptotic Bethe-ansatz solutions are obtained for one-dimensional quantum systems with a 1r2\frac{1}{{r}^{2}}r21​ long-range interaction by generalizing Sutherland's method to multicomponent quantum systems. We obtain the solutions to the supersymmetric $t\ensuremath{-}J$ model of Kuramoto and Yokoyama, the $\mathrm{SU}(\ensuremath{\nu})$ Haldane-Shastry model, and the multicomponent $t\ensuremath{-}J$ model. The excitation spectrum as well as bulk quantities are computed analytically. Conformal properties of low-energy excitations are discussed in connection with Luttinger liquid theory.
We introduce a general class of su(1|1) supersymmetric spin chains with long-range interactions which includes as particular cases the su(1|1) Inozemtsev (elliptic) and Haldane-Shastry chains, as well as the XX model. We show that this class of models can be fermionized with the help of the algebraic properties of the su(1|1) permutation operator and take advantage of this fact to analyze their quantum criticality when a chemical potential term is present in the Hamiltonian. We first study the low-energy excitations and the low-temperature behavior of the free energy, which coincides with that of a (1+1)-dimensional conformal field theory (CFT) with central charge c=1 when the chemical potential lies in the critical interval (0,E(π)), E(p) being the dispersion relation. We also analyze the von Neumann and Rényi ground state entanglement entropies, showing that they exhibit the logarithmic scaling with the size of the block of spins characteristic of a one-boson (1+1)-dimensional CFT. Our results thus show that the models under study are quantum critical when the chemical potential belongs to the critical interval, with central charge c=1. From the analysis of the fermion density at zero temperature, we also conclude that there is a quantum phase transition at both ends of the critical interval. This is further confirmed by the behavior of the fermion density at finite temperature, which is studied analytically (at low temperature), as well as numerically for the su(1|1) elliptic chain.
We attempt to construct U (1) conformal field theory (CFT) in the Tomonaga–Luttinger (TL) liquid with the 1/ r β long-range interaction (LRI). Treating the long-range forward scattering as a perturbation and applying CFT to it, we derive finite size scalings which depend on the power of the LRI. The obtained finite size scalings give nontrivial behaviours when β is odd and is close to 2. We find consistency between the analytical arguments and numerical results in the finite size scaling of energy.
Applying boundary conformal field theory we study the low-energy critical behavior of the Calogero-Sutherland model of BCNBC_NBCN​-type. The universality class of the model is found to be a {\it chiral} Tomonaga-Luttinger liquid. Various correlation exponents depending on the interaction strength are obtained.
The critical exponents for the charge, spin, electron, and superconducting correlation functions of the one-dimensional t-J model at t=J are obtained for arbitrary band filling. Our method is based on the Bethe-ansatz solution and the finite-size scaling analysis in conformal field theory. The long-distance behavior of the model belongs to the same universality class as the repulsive Hubbard model. Near half-filling the exponents take the values expected in the strong-correlation limit of the Hubbard model. In the low-density limit the exponents are those of the noninteracting system.