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arXiv:2010.01116 · Physical review. B./Physical review. B

Integrable quantum spin chains with free fermionic and parafermionic spectrum

Francisco Castilho Alcaraz, Rodrigo A. Pimenta

Published Dec 31, 2020• 36 citations
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Abstract

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 $\ensuremath{\alpha}=max{0,1\ensuremath{-}(p+1)/N}$.

DOI
10.1103/physrevb.102.235170

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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.

Physical review. E•Nov 19, 2021•16 citations
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⋯p=1,2,\ensuremath{\cdots}p=1,2,\ensuremath⋯ 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,...N=2,3,...N=2,3,..., the quantum chains describe the dynamics of Z(N)Z(N)Z(N) quantum spin chains with open boundary conditions. The simplest case p=1p=1p=1 corresponds to the free fermionic quantum Ising chain (N=2N=2N=2) or the Z(N)Z(N)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 ppp the quantum chains are characterized by a free fermionic (N=2N=2N=2) or free parafermionic (N>2N>2N>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)/Nz=(p+1)/Nz=(p+1)/N and specific-heat exponent \ensuremathα=max0,1\ensuremath−(p+1)/N\ensuremath{\alpha}=max{0,1\ensuremath{-}(p+1)/N}\ensuremathα=max0,1\ensuremath−(p+1)/N.
We introduce a family of Z(N)Z(N)Z(N) multispin quantum chains with a free fermionic (N=2N=2N=2) or free parafermionic (N>2N>2N>2) eigenspectrum. The models have (p+1)(p+1)(p+1) interacting spins ($p=1,2,\ensuremath{\cdots}$), which are Hermitian in the Z(2)Z(2)Z(2) (Ising) case and non-Hermitian for N>2N>2N>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)(p+1)(p+1). In the critical limit we identify these polynomials with certain hypergeometric polynomials p+1Fp{}_{p+1}{F}_{p}p+1​Fp​. 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=p+1N{z}_{c}=\frac{p+1}{N}zc​=Np+1​.
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 calculated the spectral properties of two related families of non-Hermitian free-particle quantum chains with NNN-multispin interactions (N=2,3,…N=2,3,\ldotsN=2,3,…). The first family have a Z(N)Z(N)Z(N) symmetry and are described by free parafermions. The second one has a U(1)U(1)U(1) symmetry and are generalizations of XXXXXX 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=1z=1z=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 XXXXXX multispin models show multiple phases with distinct conformal central charges in the periodic case. The critical exponents of the models are calculated for N=3,4,5N=3,4,5N=3,4,5 and 666.
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.
Free-parafermionic Z(N)Z(N)Z(N) and free-fermionic XYXYXY quantum chains
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.
The relationship between the eigenspectrum of Ising and XYXYXY quantum chains is well known. Although the Ising model has a Z(2)Z(2)Z(2) symmetry and the XYXYXY model a U(1)U(1)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)Z(N)Z(N) quantum chains whose eigenspectra, for N>2N>2N>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)Z(N)Z(N) free-parafermionic Baxter quantum chains. In this paper, we introduce a family of XYXYXY models that, beyond two-body, also have NNN-multispin interactions. Similar to the standard XYXYXY model, they have a U(1)U(1)U(1) symmetry and are also solved by the Jordan-Wigner transformation. We show that with appropriate choices of the NNN-multispin couplings, the eigenspectra of these XYXYXY models are given in terms of combinations of Z(N)Z(N)Z(N) free-parafermionic quasienergies. In particular, all the eigenenergies of the Z(N)Z(N)Z(N) free-parafermionic models are also present in the related free-fermionic XYXYXY models. The correspondence is established via the identification of the characteristic polynomial, which fixes the eigenspectrum. In the Z(N)Z(N)Z(N) free-parafermionic models, the quasienergies obey an exclusion circle principle that is not present in the related NNN-multispin XYXYXY models.