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Straightforward solutions and degenerate properties of spin chains with reducible Hamiltonians

Shanhui Fan

Published Mar 31, 2018• 0 citations
OpenAlex

Abstract

The Jordan-Wigner transformation plays an important role in spin models. However, a common problem is that it introduces redundancy when dealing with the periodic boundary condition. In this paper, we theoretically reveal that redundant solutions are absent in a class of subtle spin chains when the Jordan-Wigner transformation is implemented. The holistic degeneracy defined for those systems coincides with the multiplicity (in the group-theory language) of the reduced Hamiltonian, and it is found to be even-fold only. For those systems, possible high degeneracy is found to appear in not only the ground state but also the excitation states. Besides, we take the one-dimensional compass model as the example to demonstrate our proposal. Our finding gets people rid of redundancy problems in some situations, and interesting degenerate properties are discovered in those systems.

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The Jordan-Wigner transformation plays an important role in spin models. However, a common problem is that it introduces redundancy when dealing with the periodic boundary condition. In this paper, we theoretically reveal that redundant solutions are absent in a class of subtle spin chains when the Jordan-Wigner transformation is implemented. The holistic degeneracy defined for those systems coincides with the multiplicity (in the group-theory language) of the reduced Hamiltonian, and it is found to be even-fold only. For those systems, possible high degeneracy is found to appear in not only the ground state but also the excitation states. Besides, we take the one-dimensional compass model as the example to demonstrate our proposal. Our finding gets people rid of redundancy problems in some situations, and interesting degenerate properties are discovered in those systems.
The Jordan–Wigner transformation plays an important role in spin models. However, the non-locality of the transformation implies that a periodic chain of N spins is not mapped to a periodic or an anti-periodic chain of lattice fermions. Since only the N − 1 bond is different, the effect is negligible for large systems, while it is significant for small systems. In this paper, it is interesting to find that a class of periodic spin chains can be exactly mapped to a periodic chain and an anti-periodic chain of lattice fermions without redundancy when the Jordan–Wigner transformation is implemented. For these systems, possible high degeneracy is found to appear in not only the ground state, but also the excitation states. Further, we take the one-dimensional compass model and a new XY-XY model ( σ x σ y − σ x σ y ) as examples to demonstrate our proposition. Except for the well-known one-dimensional compass model, we will see that in the XY-XY model, the degeneracy also grows exponentially with the number of sites.
We present an exact solution for a class of one-dimensional compass models which stand for interacting orbital degrees of freedom in a Mott insulator. By employing the Jordan-Wigner transformation we map these models on noninteracting fermions and discuss how spin correlations, high degeneracy of the ground state, and Z2{Z}_{2}Z2​ symmetry in the quantum compass model are visible in the fermionic language. Considering a zigzag chain of ions with singly occupied eg{e}_{g}eg​ orbitals (eg{e}_{g}eg​ orbital model) we demonstrate that the orbital excitations change qualitatively with increasing transverse field, and that the excitation gap closes at the quantum phase transition to a polarized state. This phase transition disappears in the quantum compass model with maximally frustrated orbital interactions which resembles the Kitaev model. Here we find that the finite transverse field destabilizes the orbital-liquid ground state with macroscopic degeneracy, and leads to peculiar behavior of the specific heat and orbital susceptibility at finite temperature. We show that the entropy and the cooling rate at finite temperature exhibit quite different behavior near the critical point for these two models.
When studying the quantum Ising chain in a transverse field, we usually transform the spin operators to fermion operators by applying the Jordan-Wigner transformation,i.e.we take it for granted that the spin model and corresponding fermionic model are equivalent naively. But deliberate treatment could reveal that the equivalence accompanies some delicate conditions.Redundant degrees of freedom should be projected out carefully, especially for finite length of chain. In this paper, we compare the two models in detail so as to find the conditions of equivalence of them,including the interaction, the boundary conditions, and odevity of the number of lattice sites.We also exemplify in detail how to project out the redundant degrees of freedom for the fermionic representation.
We consider the class of spin Hamiltonians on a 1D chain with periodic boundary conditions that are (i) translational invariant, (ii) commuting and (iii) scale invariant, where by the latter we mean that the ground state degeneracy is independent of the system size. We correspond a directed graph to a Hamiltonian of this form and show that the structure of its ground space can be read from the cycles of the graph. We show that the ground state degeneracy is the only parameter that distinguishes the phases of these Hamiltonians. Our main tool in this paper is the idea of Bravyi and Vyalyi (2005 Quantum Inform. Comput. 5 187–215) in using the representation theory of finite-dimensional C*-algebras to study commuting Hamiltonians.
Consider a generic quantum spin chain that can be mapped to free quadratic fermions via Jordan-Wigner (JW) transformation. In the presence of arbitrary boundary magnetic fields, this Hamiltonian is no longer a quadratic Hamiltonian after JW transformation. Using ancillary sites and enlarging the Hamiltonian we first introduce a bigger quadratic Hamiltonian. Then we diagonalize this enlarged Hamiltonian in its most generic form and show that all the states are degenerate because of the presence of a zero mode. The eigenstates of the original spin chain with boundary magnetic fields can be derived after appropriate projection. We study in depth the properties of the eigenstates of the enlarged Hamiltonian. In particular we find: 1) the eigenstates in configuration bases, 2) calculate all the correlation functions, 3) find the reduced density matrices, 4) calculate the entanglement entropy. We show that the generic eigenstate of the enlarged Hamiltonian (including the eigenstates of the original spin chain) breaks the parity number symmetry and consequently one needs to take care of some technicalities regarding the calculation of the reduced density matrix and entanglement entropy. Interestingly we show that the entanglement structure of these eigenstates is quite universal and independent of the Hamiltonian. We support our results by applying them to a couple of examples.