We present a general study of a large family of exactly integrable quantum chains with multispin interactions. The exact integrability follows from the algebraic properties of the energy density operators defining the quantum chains. The Hamiltonians are characterized by a parameter $p=1,2,\ensuremath{\cdots}$ related to the number of interacting spins in the multispin interaction. In the general case, the quantum spins are of infinite dimension. In special cases, characterized by the parameter
N=2,3,..., the quantum chains describe the dynamics of
Z(N) quantum spin chains with open boundary conditions. The simplest case
p=1 corresponds to the free fermionic quantum Ising chain (
N=2) or the
Z(N) free parafermionic quantum chain. The eigenenergies of the quantum chains are given in terms of the roots of special polynomials, and for general values of
p the quantum chains are characterized by a free fermionic (
N=2) or free parafermionic (
N>2) eigenspectrum. The models have a special critical point when all coupling constants are equal. At this point, the ground-state energy is exactly calculated in the bulk limit, and our analytical and numerical analyses indicate that the models belong to universality classes of critical behavior with dynamical critical exponent
z=(p+1)/N and specific-heat exponent $\ensuremath{\alpha}=max{0,1\ensuremath{-}(p+1)/N}$.
We demonstrate using direct numerical diagonalization and extrapolation methods that boundary conditions have a profound effect on the bulk properties of a simple Z(N) model for N≥3 for which the model Hamiltonian is non-Hermitian. For N=2 the model reduces to the well-known quantum Ising model in a transverse field. For open boundary conditions, the Z(N) model is known to be solved exactly in terms of free parafermions. Once the ends of the open chain are connected by considering the model on a ring, the bulk properties, including the ground-state energy per site, are seen to differ dramatically with increasing N. Other properties, such as the leading finite-size corrections to the ground-state energy, the mass gap exponent, and the specific-heat exponent, are also seen to be dependent on the boundary conditions. We speculate that this anomalous bulk behavior is a topological effect.
We introduce a family of
Z(N) multispin quantum chains with a free fermionic (
N=2) or free parafermionic (
N>2) eigenspectrum. The models have
(p+1) interacting spins ($p=1,2,\ensuremath{\cdots}$), which are Hermitian in the
Z(2) (Ising) case and non-Hermitian for
N>2. We construct a set of mutually commuting charges that allows us to derive the eigenenergies in terms of the roots of polynomials generated by a recurrence relation of order
(p+1). In the critical limit we identify these polynomials with certain hypergeometric polynomials
p+1Fp. Also in the critical regime, we calculate the ground-state energy in the bulk limit and verify that they are given in terms of the Lauricella hypergeometric series. The models with special couplings are self-dual and at the self-dual point show a critical behavior with a dynamical critical exponent
zc=Np+1.
We calculated the spectral properties of two related families of non-Hermitian free-particle quantum chains with
N-multispin interactions (
N=2,3,…). The first family have a
Z(N) symmetry and are described by free parafermions. The second one has a
U(1) symmetry and are generalizations of
XX quantum chains described by free fermions. The eigenspectra of both free-particle families are formed by the combination of the same pseudo-energies. The models have a multicritical point with dynamical critical exponent
z=1. The finite-size behavior of their eigenspectra, as well as the entanglement properties of their ground state wave function, indicate the models are conformally invariant. The models with open and periodic boundary conditions show quite distinct physics due to their non-Hermiticity. The models defined with open boundaries have a single conformal invariant phase while the
XX multispin models show multiple phases with distinct conformal central charges in the periodic case. The critical exponents of the models are calculated for
N=3,4,5 and
6.
Abstract In this article we outline the historical development and key results obtained to date for free parafermionic spin chains. The concept of free parafermions provides a natural N-state generalization of free fermions, which have long underpinned the exact solution and application of widely studied quantum spin chains and their classical counterparts. In particular, we discuss the Baxter-Fendley free parafermionic Z(N) spin chain, which is a relatively simple non-Hermitian generalization of the Ising model.
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.