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arXiv:2201.06588 · Physical review. E

Exactly solvable one-dimensional quantum models with gamma matrices

Yash Chugh, Kusum Dhochak, Uma Divakaran, Prithvi Narayan, Amit Kumar Pal

Published Aug 15, 2022• 2 citations
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Abstract

In this paper we write exactly solvable generalizations of one-dimensional quantum XY and Ising-like models by using 2^{d}-dimensional gamma matrices as the degrees of freedom on each site. We show that these models result in quadratic Fermionic Hamiltonians with Jordan-Wigner-like transformations. We illustrate the techniques using a specific case of four-dimensional gamma matrices and explore the quantum phase transitions present in the model.

DOI
10.1103/physreve.106.024114

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In this paper we write exactly solvable generalizations of one-dimensional quantum XY and Ising-like models by using 2^{d}-dimensional gamma matrices as the degrees of freedom on each site. We show that these models result in quadratic Fermionic Hamiltonians with Jordan-Wigner-like transformations. We illustrate the techniques using a specific case of four-dimensional gamma matrices and explore the quantum phase transitions present in the model.
It is shown that a solvable Hamiltonian can be obtained from a series of operators satisfying specific commutation relations. A transformation that diagonalize the Hamiltonian is obtained simultaneously. The two-dimensional Ising model with periodic interactions, the one-dimensional XY model with period 2, the transverse Ising chain, the one-dimensional Kitaev model and the cluster model, and other composite quantum spin chains are diagonalized following this procedure. The Jordan–Wigner transformation, the transformation from the Pauli spin operators to the Majorana fermion used by Shankar and Murthy, and the transformation introduced by Nambu, are special cases of this treatment.
We write down a class of two-dimensional quantum spin- 1 2 Hamiltonians whose eigenspectra are exactly solvable via the Jordan-Wigner transformation. The general structure corresponds to a suitable grid composed of XY or X-Ising spin chains and Z-Ising spin chains and is generalizable to higher dimensions. They can host stacks of one-dimensional spinless fermion liquids with gapless excitations and power-law correlations coexisting with ordered spin moments (localized spinless fermions). Bond-dependent couplings thus can be an alternate mechanism than geometric frustration of S U ( 2 ) -symmetric couplings to obtain spinless fermionic excitations. Put in a different way, bond-dependent couplings allow for an embedding of one-dimensional spinless fermion (Tomonoga-Luttinger) liquids and solids and also Majorana excitations in higher dimensions. They can accommodate a simpler set of site-local conserved quantities apart from the more intricate, interlocked set of plaquette-local or bond-local conserved quantities in Kitaev's honeycomb model with Majorana excitations. The proposed grid structure may provide an architecture for quantum engineering with controllable qubits.
The Jordan-Wigner transformation permits one to convert spin 1/2 operators into spinless fermion ones, or vice versa. In some cases, it transforms an interacting spin Hamiltonian into a noninteracting fermionic one, which is exactly solved at the mean-field level. Even when the resulting fermionic Hamiltonian is interacting, its mean-field solution can provide surprisingly accurate energies and correlation functions. Jordan-Wigner is, however, only one possible means of interconverting spin and fermionic degrees of freedom. Here, we apply several such techniques to the XXZ and J1-J2 Heisenberg models, as well as to the pairing or reduced Bardeen-Cooper-Schrieffer Hamiltonian, with the aim of discovering which of these mappings is most useful in applying fermionic mean-field theory to the study of spin Hamiltonians.
We introduce a new spin-fermion mapping, for arbitrary spin S generating the SU(2) group algebra, that constitutes a natural generalization of the Jordan-Wigner transformation for S = 1/2. The mapping, valid for regular lattices in any spatial dimension d, serves to unravel hidden symmetries. We illustrate the power of the transformation by finding exact solutions to lattice models previously unsolved by standard techniques. We also show the existence of the Haldane gap in S = 1 bilinear nearest-neighbor Heisenberg spin chains and discuss the relevance of the mapping to models of strongly correlated electrons. Moreover, we present a general spin-anyon mapping for the case d < or = 2.
Proposed is a generalization of Jordan-Wigner transform that allows to exactly fermionize a large family of quantum spin Hamiltonians in dimensions higher than one. The key steps are to enlarge the Hilbert space of the original model by adding to it a collection of stand-alone free spins and to use a combination of these auxiliary operators and the lattice spins to construct a proper fermion representation of the physical Hamiltonian. The transform is especially useful for lattice spin Hamiltonians, where two-spin interactions of XYXYXY type are either absent or exist only within one-dimensional chains and where the chains are coupled via two-spin interactions of Ising type, ring-exchange terms, or more general multispin interactions that involve an even number of spin operators from each chain. Using the proposed fermionization method we provide a simple argument suggesting that a spin Hamiltonian closely related to the ring-exchange model proposed by Paramekanti et al., [Phys. Rev. B 66, 054526 (2002)] indeed realizes a spin-liquid state.