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Journal of Physics A Mathematical and General

Conformal anomaly and surface energy for Potts and Ashkin-Teller quantum chains

Chris J. Hamer, G R W Quispel, Murray T. Batchelor

Published Nov 11, 1987• 209 citations
Publisher

Abstract

Exact equivalences between the critical quantum Potts and Ashkin-Teller chains and a modified XXZ Heisenberg chain have recently been derived by Alcaraz et al (1987). The leading finite-size corrections to the ground-state energies of these chains are derived using the methods of de Vega and Woynarovich (1985) and Eckle. Exact results are then obtained for the conformal anomaly of each model, and for the surface energy in the case of free boundaries.

DOI
10.1088/0305-4470/20/16/040

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New conjecture for the SUq(N) Perk-Schultz models

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In [1] (to which we refer to hereafter by I), based on numerical evidence, we present a series of conjectures about the eigenspectra of the SUq(N) invariant Perk-Schultz Hamiltonians, given by I.1 with p = 0. Subsequent extensive numerical work indicate that the conjecture (3) of I (I.85) is just a particular case of a more general one, relating the ground-state energy of diferent SUq(N) quantum chains. This new conjecture can be stated as follows: Conjecture: The difference of the ground-state energy of the SUq(N) and SUq ′(K) model in an open chain of lenght L is given by E0[SUq(N), q = e iπK/(N+K) ] − E0[SUq(N), q = e iπN/(N+K) ] = π(N − K) 2(1 − L) sin (), (1) 2(N + K) where N = K = 1, 2,... and E0[SUq(1), q)] = 0. The particular case N> 1 and K = 1 gives E0[SUq(N), q = e iπ/(N+1) π(N − 1)] = 2(1 − L) sin (), (2) 2(N + 1) recovering the earlier announced conjecture (I.85). The results (1) and (2) give exact finite-size corrections suporting earlier conjectures about the op-1 erator content of the Perk-Schultz models. As is well known [2] as a consequence of conformal invariance the finite-size corrections of the ground-state energy of critical chains with free boundary conditions are given by E0(L)/L = e ∞ + f∞/L − πc 24L 2 + o(L−2), (3) where e ∞ (f∞) is the ground-state energy per site (surface energy) at the bulk limit L → ∞, and c is the conformal anomaly of the effective underlying conformal field theory defined on a semi-infinte plane. The relations (3) and (2) imply that in the case of the SUq(N) model with q = exp(iπN/(N + 1)) the conformal anomaly has the value c = 0 for all N ≥ 2 and π(N − 1) e ∞ = −f ∞ = −2 sin (. (4) 2(N + 1) Moreover all the other finite-size corrections appearing in (3) are identically zero! This result can be understood from the conjectured operator content of the model. The conformal dimensions of the SUq(N) model are expected to be given by a generalized coulomb gas description x(n, m) = xp

•Jan 1, 2004•0 citations
Exact equivalences between the critical quantum Potts and Ashkin-Teller chains and a modified XXZ Heisenberg chain have recently been derived by Alcaraz et al (1987). The leading finite-size corrections to the ground-state energies of these chains are derived using the methods of de Vega and Woynarovich (1985) and Eckle. Exact results are then obtained for the conformal anomaly of each model, and for the surface energy in the case of free boundaries.
Eigenspectra of the critical quantum Ashkin-Teller and Potts chains with free boundaries can be obtained from that of the XXZ chain with free boundaries and a complex surface field. By deriving and solving numerically the Bethe ansatz equations for such boundaries the authors obtain eigenenergies of XXZ chains of up to 512 sites. The conformal anomaly and surface exponents of the quantum XXZ, Ashkin-Teller, and Potts chains are calculated by exploiting their relations with the mass gap amplitudes as predicted by conformal invariance.
Numerical solutions of the Bethe-Ansatz equations for the eigenenergies of XXZ Hamiltonian on very large chains are used to identify, via conformal invariance, the scaling dimensions of various two-dimensional models. With periodic boundary conditions, eight-vertex and Gaussian model operators are found. The scaling dimensions of the Ashkin-Teller and Potts models are obtained by the exact relating of eigenstates of their quantum Hamiltonians to those of the XXZ chain with modified boundary conditions. The irrelevant operators governing the dominant finite-size corrections are also identified.
The authors study the finite-size effects for the fourth-state one-dimensional quantum chain introduced by Kohmoto et al. (1981) using different boundary conditions. From the corrections to the ground-state energy for periodic boundary conditions they verify that the central charge of the Virasoro algebra is equal to one in the whole domain of criticality. Their numerical study suggests a new relation between the corrections to the ground-state energy with free boundary conditions and with periodic boundary conditions. Various critical exponents including the surface ones and those corresponding to the para-fermions are determined.
The method of de Vega and Woynarovitch (1985) is used to calculate finite-size corrections to the ground-state energy in different sectors for the XXZ Heisenberg chain. Finite-size scaling amplitudes and correction-to-scaling exponents in the critical region are derived. Using conformal invariance, a scaling dimension x=( pi - gamma )/2 pi is extracted corresponding to the electric field operator in the 8-vertex model: this confirms a conjecture of Baxter and Kelland (1974). Finite-size scaling properties near the Kosterlitz-Thouless critical point Delta =-1 are discussed.
In [1] (to which we refer to hereafter by I), based on numerical evidence, we present a series of conjectures about the eigenspectra of the SUq(N) invariant Perk-Schultz Hamiltonians, given by I.1 with p = 0. Subsequent extensive numerical work indicate that the conjecture (3) of I (I.85) is just a particular case of a more general one, relating the ground-state energy of diferent SUq(N) quantum chains. This new conjecture can be stated as follows: Conjecture: The difference of the ground-state energy of the SUq(N) and SUq ′(K) model in an open chain of lenght L is given by E0[SUq(N), q = e iπK/(N+K) ] − E0[SUq(N), q = e iπN/(N+K) ] = π(N − K) 2(1 − L) sin (), (1) 2(N + K) where N = K = 1, 2,... and E0[SUq(1), q)] = 0. The particular case N> 1 and K = 1 gives E0[SUq(N), q = e iπ/(N+1) π(N − 1)] = 2(1 − L) sin (), (2) 2(N + 1) recovering the earlier announced conjecture (I.85). The results (1) and (2) give exact finite-size corrections suporting earlier conjectures about the op-1 erator content of the Perk-Schultz models. As is well known [2] as a consequence of conformal invariance the finite-size corrections of the ground-state energy of critical chains with free boundary conditions are given by E0(L)/L = e ∞ + f∞/L − πc 24L 2 + o(L−2), (3) where e ∞ (f∞) is the ground-state energy per site (surface energy) at the bulk limit L → ∞, and c is the conformal anomaly of the effective underlying conformal field theory defined on a semi-infinte plane. The relations (3) and (2) imply that in the case of the SUq(N) model with q = exp(iπN/(N + 1)) the conformal anomaly has the value c = 0 for all N ≥ 2 and π(N − 1) e ∞ = −f ∞ = −2 sin (. (4) 2(N + 1) Moreover all the other finite-size corrections appearing in (3) are identically zero! This result can be understood from the conjectured operator content of the model. The conformal dimensions of the SUq(N) model are expected to be given by a generalized coulomb gas description x(n, m) = xp