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 $\ensuremath{\alpha}=max{0,1\ensuremath{-}(p+1)/N}$.
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.
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.
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.
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 study the effects of quenched disorder in a class of quantum chains with (
p+1)-multispin interactions exhibiting a free fermionic spectrum, paying special attention to the case
p=2. Depending if disorder couples to (i) all the couplings or just to (ii) some of them, we have two distinct physical scenarios. In case (i), we find that the transitions of the model are governed by a universal infinite-randomness critical point surrounded by quantum Griffiths phases similarly as happens to the random transverse-field Ising chain. In case (ii), we find that quenched disorder becomes an irrelevant perturbation: the clean critical behavior is stable and Griffiths phases are absent. Beyond the perturbative regime, disorder stabilizes a line of finite-randomness critical points (with nonuniversal critical exponents), that ends in a multicritical point of infinite-randomness type. In that case, quantum Griffiths phases also appear surrounding the finite-disorder transition point. We have characterized the correlation functions and the low-temperature thermodynamics of these chains. Our results are derived from a strong-disorder renormalization-group technique and from finite-size scaling analysis of the spectral gap computed exactly (up to sizes $\ensuremath{\sim}{10}^{7}$) via an efficient new numerical method recently introduced in the literature [Phys. Rev. B 104, 174206 (2021)].