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arXiv:1802.04453 · Physical review. E

Anomalous bulk behavior in the free parafermion Z(N) spin chain

Francisco Castilho Alcaraz, Murray T. Batchelor

Published Jun 8, 2018• 12 citations
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Abstract

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.

DOI
10.1103/physreve.97.062118

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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.
Abstract We consider the calculation of ground-state expectation values for the non-Hermitian spin chain described by free parafermions. For N = 2 the model reduces to the quantum Ising chain in a transverse field with open boundary conditions. Use is made of the Hellmann–Feynman theorem to obtain exact results for particular single site and nearest-neighbor ground-state expectation values for general N which are valid for sites deep inside the chain. These results are tested numerically for N = 3, including how they change as a function of distance from the boundary.
Abstract Results are given for the ground state energy and excitation spectrum of a simple N -state Z N spin chain described by free parafermions. The model is non-Hermitian for N ⩾ 3 with a real ground state energy and a complex excitation spectrum. Although having a simpler Hamiltonian than the superintegrable chiral Potts model, the model is seen to share some properties with it, e.g. the specific heat exponent α = 1 − 2 / N and the anisotropic correlation length exponents ν ∥ = 1 and ν ⊥ = 2 / N .
The influence of boundary conditions on bulk properties is generally expected to vanish in the thermodynamic limit. Here we show that this expectation can fail in an open transverse field Ising chain with modified boundary fields. By removing only the transverse fields acting on the two edge spins while leaving the bulk Hamiltonian unchanged, we uncover a boundary-induced switching of the topological spectral degeneracy. In contrast to the conventional open chain, where the topological degeneracy occurs in the ferromagnetic phase (∣g∣<1\left\vert g\right\vert <1∣g∣<1), the modified boundary condition shifts it to the paramagnetic phase (∣g∣>1\left\vert g\right\vert >1∣g∣>1). We demonstrate that the underlying mechanism originates from two exact local conserved quantities associated with the edge spins, which partition the Hilbert space into invariant sectors. Based on this structure, we establish an exact description in terms of domain-wall excitations, under which the Hamiltonian decomposes into two independent Kitaev chains. Complementarily, the Majorana representation reveals that the spectral switching originates from different unit-cell conventions and boundary terminations of the corresponding effective SSH lattice, leading to distinct boundary zero-mode structures without altering the bulk Hamiltonian. Furthermore, we show that the boundary-induced topological degeneracy remains observable at finite temperatures through non-Hermitian quench dynamics, providing a dynamical manifestation of the bulk-boundary correspondence. These results demonstrate that boundary engineering, through the introduction of local conserved boundary degrees of freedom, provides an effective route for controlling topological degeneracy in quantum spin chains without modifying bulk interactions.
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 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,...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 $\ensuremath{\alpha}=max{0,1\ensuremath{-}(p+1)/N}$.
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​.