We calculate 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 have 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.
A large family of multispin interacting one-dimensional quantum spin models with Z(N) symmetry and a free-particle eigenspectra are known in the literature. They are free-fermionic (N=2) and free-parafermionic (N≥2) quantum chains. The essential ingredient that implies the free-particle spectra is the fact that these Hamiltonians are expressed in terms of generators of a Z(N) exchange algebra. In all these known quantum chains, the number of spins in all the multispin interactions (range of interactions) is the same and, therefore, the models have a homogeneous interacting range. In this paper, we extend the Z(N) exchange algebra by introducing different models with a free-particle spectra, where the interaction ranges of the multispin interactions are no longer uniform and depend on the lattice sites (nonhomogeneous interacting range). We obtain the general conditions that the site-dependent ranges of the multispin interactions have to satisfy to ensure a free-particle spectra. Several simple examples are introduced. We study in detail the critical properties in the case where the range of interactions of the even (odd) sites is constant. The dynamical critical exponent is evaluated in several cases.
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 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.
Two new families of quantum spin chains with p multispin interactions were recently introduced in (1).One has Z(N) symmetry and describes free fermions (N = 2) and parafermions (N > 2) in the lattice.The other family is an extension of XY model with N multispin interactions, with U (1) symmetry and being solved by Jordan-Wigner transformation.Both families are non-Hermitian for N > 2 and with open boundary conditions, they share quasi-energies that are obtained from the roots of a given characteristic polynomial.In the present work, we show a general study of the conformal invariance properties and quantum information in a particular case of this new family of XY models that has three spins interactions (p = 2).Although this model is non-Hermitian, the entanglement entropy (as von Neumann and Rényi entropy) are studied by exploring the translation invariance and using the correlation matrix technique.The study of spectral properties of the model revealed that it has two distinct critical phases (with central charge c = 1 and c = 2, respectively) that depend on a given anisotropy parameter.With respect to entanglement entropy, our results follow the Cardy-Calabrese's conjecture for finite systems (2), obeying a logarithmic correction that is proportional to the central charge of the model.Furthermore, we observe that each Fermi point in the quantum chain contributes exactly 1/2 to the value of the central charge.In addition, oscillations were observed in the Rényi entropies for indices n = 2, 3, 4 and 6.The exponents governing the decay of these oscillations follow the conjecture of Xavier and Alcaraz (3), given by p n = 2x /n with x being the anomalous dimension of the energy operator of the model.
The relationship between the eigenspectrum of Ising and
XY quantum chains is well known. Although the Ising model has a
Z(2) symmetry and the
XY model a
U(1) symmetry, both models are described in terms of free-fermionic quasiparticles. The fermionic quasienergies are obtained by means of a Jordan-Wigner transformation. On the other hand, there exists in the literature a huge family of
Z(N) quantum chains whose eigenspectra, for
N>2, are given in terms of free parafermions, and they are not derived from the standard Jordan-Wigner transformation. The first members of this family are the
Z(N) free-parafermionic Baxter quantum chains. In this paper, we introduce a family of
XY models that, beyond two-body, also have
N-multispin interactions. Similar to the standard
XY model, they have a
U(1) symmetry and are also solved by the Jordan-Wigner transformation. We show that with appropriate choices of the
N-multispin couplings, the eigenspectra of these
XY models are given in terms of combinations of
Z(N) free-parafermionic quasienergies. In particular, all the eigenenergies of the
Z(N) free-parafermionic models are also present in the related free-fermionic
XY models. The correspondence is established via the identification of the characteristic polynomial, which fixes the eigenspectrum. In the
Z(N) free-parafermionic models, the quasienergies obey an exclusion circle principle that is not present in the related
N-multispin
XY models.