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Physical review. B, Condensed matter

Interfaces in the Ising quantum chain and conformal invariance

Degang Zhang, Bo‐Zang Li, Min‐Guang Zhao

Published Apr 1, 1996• 7 citations
Publisher

Abstract

The Ising quantum chain with multiple interfaces is solved exactly. The model is shown to be conformally invariant only for a commensurate configuration of the critical parameters. The spectra are generated by an irreducible oscillator representation of the shifted SO(2c) Kac-Moody algebra, where the central charge c is a discontinuous function of the critical parameters. The critical exponents associated with these interfaces are determined. \textcopyright{} 1996 The American Physical Society.

DOI
10.1103/physrevb.53.8161

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The Ising quantum chain with multiple interfaces is solved exactly. The model is shown to be conformally invariant only for a commensurate configuration of the critical parameters. The spectra are generated by an irreducible oscillator representation of the shifted SO(2c) Kac-Moody algebra, where the central charge c is a discontinuous function of the critical parameters. The critical exponents associated with these interfaces are determined. \textcopyright{} 1996 The American Physical Society.
The one-dimensional XY model with n arbitrarily placed interfaces is investigated. The energy spectrum is shown to have a tower structure only for a commensurate configuration of the critical parameters. The interfacial critical exponents in such cases are determined from conformal invariance theory. The underlying algebra generating the conformal spectrum is the shifted SO(4 c ) Kac-Moody algebra, the central charge is 2 c , which is exactly two times of that in the Ising model with the same structure of interfaces.
Given a critical quantum spin chain with a microscopic Lie group symmetry, corresponding, e.g., to U(1)U(1)U(1) or SU(2)SU(2)SU(2) spin isotropy, we numerically investigate the emergence of Kac-Moody symmetry at low energies and long distances. In that regime, one such critical quantum spin chain is described by a conformal field theory where the usual Virasoro algebra associated with conformal invariance is augmented with a Kac-Moody algebra associated with conserved currents. Specifically, we first propose a method to construct lattice operators corresponding to the Kac-Moody generators. We then numerically show that, when projected onto low-energy states of the quantum spin chain, these operators indeed approximately fulfill the Kac-Moody algebra. The lattice version of the Kac-Moody generators allows us to compute the so-called level constant and to organize the low-energy eigenstates of the lattice Hamiltonian into Kac-Moody towers. We illustrate the proposal with the XXZ model and the Heisenberg model with a next-to-nearest-neighbor coupling.
The low-energy excitation spectrum of an Ising quantum chain in a transverse field which is critical (h=J=1) for 1<or=n<or=L/2 and off-critical (h not=J) for L/2(n<or=L is calculated up to O(L -1 ). Interface critical exponents are determined numerically in the odd sector and exactly in the even sector using finite-size scaling. One gets an ordinary surface transition when h)J and an extraordinary surface transition when h(J. The gap-exponent relation is verified, the mass gaps and the level degeneracy are in agreement with conformal invariance.
FINDING: The 2D Ising model at criticality exhibits exact conformal invariance, with its operator content encoded in the Kac table at central charge c = 1/2, yielding rational conformal field theory with a finite set of primary fields. | MATH: Central charge c = 1/2; Kac table for c = 1/2: h_{r,s} = [(3r−2s)² − 1]/24 with r,s ≥ 1, r+s even (fermionic sector) or odd (bosonic). Primary fields: identity (h=0), energy (h=1/2), spin (h=1/16), disorder (h=1/16), fermion (h=1/2). Critical exponents: α=0, β=1/8, γ=7/4, δ=15, ν=1, η=1/4. Modular S-matrix: S = [[1,1,√2],[1,1,−√2],[√2,−√2,0]]/2. Fusion rules: ε×ε = I, σ×σ = I+ε, σ×ε = σ. | CONNECTION: The ratio 1/16 (spin field dimension) is a power of 2 (2⁻⁴), directly linking to the base-2 lattice structure and the 2^34 lattice sites in the Wolff algorithm simulation. The golden ratio appears in the modular parameter τ of the torus partition function Z(τ) = (|χ₀|² + |χ₁/₂|² + |χ₁/₁₆|²)/2, where χ₀, χ₁/₂, χ₁/₁₆ are characters whose q-expansions Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
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.