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arXiv:hep-th/9409122

A Perturbative Approach to the Chiral Potts Model

A. Honecker

Published Sep 21, 1994• 4 citations
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Abstract

The massive high-temperature phase of the chiral Potts quantum chain is studied using perturbative methods. For the Z_3-chain we present high-temperature expansions for the groundstate energy and the dispersion relations of the two single-particle states as well as two-particle states at general values of the parameters. We also present a perturbative argument showing that a large class of massive Z_n-spin quantum chains have quasiparticle spectra with n-1 fundamental particles. It is known from earlier investigations that -at special values of the parameters- some of the fundamental particles exist only for limited ranges of the momentum. In these regimes our argument is not rigorous as one can conclude from a discussion of the radius of convergence of the perturbation series. We also derive correlation functions from a perturbative evaluation of the groundstate for the Z_3-chain. In addition to an exponential decay we observe an oscillating contribution. The oscillation length seems to be related to the asymmetry of the dispersion relations. We show that this relation is exact at special values of the parameters for general Z_n using a form factor expansion.

DOI
10.48550/arxiv.hep-th/9409122

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The massive high-temperature phase of the chiral Potts quantum chain is studied using perturbative methods. For the Z_3-chain we present high-temperature expansions for the groundstate energy and the dispersion relations of the two single-particle states as well as two-particle states at general values of the parameters. We also present a perturbative argument showing that a large class of massive Z_n-spin quantum chains have quasiparticle spectra with n-1 fundamental particles. It is known from earlier investigations that -at special values of the parameters- some of the fundamental particles exist only for limited ranges of the momentum. In these regimes our argument is not rigorous as one can conclude from a discussion of the radius of convergence of the perturbation series. We also derive correlation functions from a perturbative evaluation of the groundstate for the Z_3-chain. In addition to an exponential decay we observe an oscillating contribution. The oscillation length seems to be related to the asymmetry of the dispersion relations. We show that this relation is exact at special values of the parameters for general Z_n using a form factor expansion.
Using perturbative methods we derive new results for the spectrum and correlation functions of the general Z 3 -chiral Potts quantum chain in the massive low-temperature phase. Explicit calculations of the ground-state energy and the first excitations in the zero momentum sector give excellent approximations and confirm the general statement that the spectrum in the low-temperature phase of general Z n -spin quantum chains is identical to one in the high-temperature phase where the role of charge and boundary conditions are interchanged. Using a perturbative expansion of the ground state for the Z 3 model we are able to gain some insight into correlation functions. We argue that they might be oscillating and give estimates for the oscillation length as well as the correlation length.
First, an algebraic criterion for integrability is discussed -the so-called `superintegrability'- and some results on the classification of superintegrable quantum spin Hamiltonians based on sl(2) are obtained. Next, the massive phases of the Z_n-chiral Potts quantum spin chain (a model that violates parity) are studied in detail. It is shown that the excitation spectrum of the massive high-temperature phase can be explained in terms of n-1 fundamental quasiparticles. We compute correlation functions from a perturbative and numerical evaluation of the groundstate for the Z_3-chain. In addition to an exponential decay we observe an oscillating contribution. The oscillation length seems to be related to the asymmetry of the dispersion relations. We show that this relation is exact at special values of the parameters for general Z_n using a form factor expansion. Finally, we discuss several aspects of extended conformal algebras (W-algebras). We observe an analogy between boundary conditions for Z_n-spin chains and W-algebras and then turn to statements about the structure of W-algebras. In particular, we briefly summarize results on unifying structures present in the space of all quantum W-algebras.
Phase diagram and two-particle structure of the Z3Z_3Z3​-chiral Potts model
We calculate the low-lying part of the spectrum of the Z3Z_3Z3​-symmetrical chiral Potts quantum chain in its self-dual and integrable versions, using numerical diagonalisation of the hamiltonian for N≤12N \leq 12N≤12 sites and extrapolation $N \ra \infty$. From the sequences of levels crossing we show that the massive phases have oscillatory correlation functions. We calculate the wave vector scaling exponent. In the high-temperature massive phase the pattern of the low-lying levels can be explained assuming the existence of two particles, with Z3Z_3Z3​-charge Q ⁣= ⁣1Q\!=\!1Q=1 and Q ⁣= ⁣2Q\!=\!2Q=2, and their scattering states. In the superintegrable case the Q ⁣= ⁣2Q\!=\!2Q=2-particle has twice the mass of the Q ⁣= ⁣1Q\!=\!1Q=1-particle. Exponential convergence in NNN is observed for the single particle gaps, while power convergence is seen for the scattering levels. In the high temperature limit of the self-dual model the parity violation in the particle dispersion relation is equivalent to the presence of a macroscopic momentum $P_m = \pm \vph/3$, where $\vph$ is the chiral angle.
The authors calculate the lowest translationally invariant levels of the Z 3 - and Z 4 -symmetrical chiral Potts quantum chains, using numerical diagonalization of the Hamiltonian for N<or=12 and N<or=10 sites, respectively, and extrapolating n to infinity . In the high-temperature massive phase they find that the pattern of the low-lying zero momentum levels can be explained assuming the existence of n-1 particles carrying Z n charges Q=1, . . ., n-1 (mass m Q ), and their scattering states. In the superintegrable case the masses of the n-1 particles become proportional to their respective charges: m Q =Qm 1 . Exponential convergence in N is observed for the single-particle gaps, while power convergence is seen for the scattering levels. They also verify that qualitatively the same pattern appears for the self-dual and integrable cases. For general Z n they show that the energy-momentum relations of the particles show a parity nonconservation asymmetry which for very high temperatures is exclusive due to the presence of a macroscopic momentum P m =(1-2Q/n) phi , where phi is the chiral angle and Q is the Z n charge of the respective particle.
The McCoy-Roan integral representation for gaps of the integrable Z_n- symmetric Chiral Potts quantum chain is used to calculate the boundary of the incommensurable phase for various n. In the limit n -> \infty an analytic formula for this phase boundary is obtained. The McCoy-Roan formula gives the gaps in terms of a rapidity. For the lowest gap we conjecture the relation of this rapidity to the physical momentum in the high-temperature limit using symmetry properties and comparing the McCoy-Roan formula to high-temperature expansions and finite-size data.