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Progress of Theoretical Physics Supplement

Uq[SU(2)]-Invariant Spin Chains, Related Algebras and Finite-Size Corrections

Murray T. Batchelor

Published May 16, 2013• 1 citation
Publisher

Abstract

Exact results are derived, via the Temperley-Lieb-Jones algebra, for the ground state energy per site, surface energy, gap and central charge of the anisotropic generalization of the spin-1 biquadratic model.A recently observed simplification in the ground state energy of the U.[SU(2)]-invariant Zamolodchikov-Fateev model at q=e'x 14 is shown to be a direct consequence of a trivial representation of the related operator algebra.Similar points are conjectured to exist in the more general u.[SU(2)]-invariant spin-S chains at q=e 1 " 1 < 2 + 25 '.§ 1. Introduction

DOI
10.1143/ptp.102.39

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Uq[SU(2)]-Invariant Spin Chains, Related Algebras and Finite-Size Corrections

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Exact results are derived, via the Temperley-Lieb-Jones algebra, for the ground state energy per site, surface energy, gap and central charge of the anisotropic generalization of the spin-1 biquadratic model. A recently observed simplification in the ground state energy of the U.[SU(2)]-invariant Zamolodchikov-Fateev model at q=e'x14 is shown to be a direct consequence of a trivial representation of the related operator algebra. Similar points are conjectured to exist in the more general u.[SU(2)]-invariant spin-S chains at q=e 1 1<2+25'.

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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

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arXiv:cond-mat/9501035•Journal of Physics A Mathematical and General•Jul 7, 1995•3 citations
Exact results are derived, via the Temperley-Lieb-Jones algebra, for the ground state energy per site, surface energy, gap and central charge of the anisotropic generalization of the spin-1 biquadratic model.A recently observed simplification in the ground state energy of the U.[SU(2)]-invariant Zamolodchikov-Fateev model at q=e'x 14 is shown to be a direct consequence of a trivial representation of the related operator algebra.Similar points are conjectured to exist in the more general u.[SU(2)]-invariant spin-S chains at q=e 1 " 1 < 2 + 25 '.§ 1. Introduction
Exact results are derived, via the Temperley-Lieb-Jones algebra, for the ground state energy per site, surface energy, gap and central charge of the anisotropic generalization of the spin-1 biquadratic model. A recently observed simplification in the ground state energy of the U.[SU(2)]-invariant Zamolodchikov-Fateev model at q=e'x14 is shown to be a direct consequence of a trivial representation of the related operator algebra. Similar points are conjectured to exist in the more general u.[SU(2)]-invariant spin-S chains at q=e 1 1<2+25'.
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
We construct an open quantum spin chain from the “twisted” [Formula: see text]R matrix in the fundamental representation which has the quantum algebra symmetry Uq[ su (2)]. This anisotropic spin-1 chain is different from the Uq[ su (2)]-invariant chain constructed from the “untwisted” [Formula: see text] spin-1 R matrix (namely, the spin-1 XXZ chain of Fateev-Zamolodchikov with boundary terms) but, nevertheless, is also completely integrable. We discuss the general case of an R matrix of the type g(k), where k∈{1, 2, 3}, and g is any simple Lie algebra.
We consider the U q SU(2) invariant spin-1/2 XXZ quantum spin chain at the roots of unity q=exp(i pi /(m+1)), corresponding to different minimal models of conformal field theory. We conduct a numerical investigation of the correlation functions of U q SU(2) scalar two-point operators in order to find which operators in the minimal models they correspond to. Using graphical representations of the Temperley-Lieb algebra we are able to deal with chains of up to 28 sites. Depending on q, the correlation functions show different characteristics and finite-size behaviour. For m=2/3, which corresponds to the Lee-Yang edge singularity, we find the surface and bulk critical exponent -1/5. Together with the known result in the case m=3 (Ising model) this indicates that in the continuum limit the two-point operators involve conformal fields of spin-m-1/m+1. For other roots of unity q the chains are too short to determine the surface and bulk critical exponents.