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Physical review. B./Physical review. B

Generalizations of exactly solvable quantum spin models

Andrei A. Zvyagin

Published Mar 2, 2020• 6 citations
Publisher

Abstract

Several generalizations of one-dimensional and two-dimensional exactly solvable low-dimensional quantum spin-1/2 models, some of which contain off-diagonal terms of the exchange tensor, are proposed. It is shown that off-diagonal terms of the exchange (symmetric or nonsymmetric with respect to the exchange of spins) yield renormalization of the diagonal ones and phase shifts. The latter can be moved to eigenfunctions of the models and do not affect eigenvalues for open geometries of the lattices. For closed geometries of the lattices the phase shifts manifest themselves in the finite size corrections. The advantage of the proposed models is in their simplicity and in possible realizations of the models in a number of applications, e.g., for the description of the wide variety of correlated electron systems, ultracold atoms, and in the theory of topological quantum computation.

DOI
10.1103/physrevb.101.094403

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Several generalizations of one-dimensional and two-dimensional exactly solvable low-dimensional quantum spin-1/2 models, some of which contain off-diagonal terms of the exchange tensor, are proposed. It is shown that off-diagonal terms of the exchange (symmetric or nonsymmetric with respect to the exchange of spins) yield renormalization of the diagonal ones and phase shifts. The latter can be moved to eigenfunctions of the models and do not affect eigenvalues for open geometries of the lattices. For closed geometries of the lattices the phase shifts manifest themselves in the finite size corrections. The advantage of the proposed models is in their simplicity and in possible realizations of the models in a number of applications, e.g., for the description of the wide variety of correlated electron systems, ultracold atoms, and in the theory of topological quantum computation.
We present a detailed analysis of the spin models with near-neighbours interactions constructed in our previous paper (Enciso et al 2005 Phys. Lett. B 605 214) by a suitable generalization of the exchange operator formalism. We provide a complete description of a certain flag of finite-dimensional spaces of spin functions preserved by the Hamiltonian of each model. By explicitly diagonalizing the Hamiltonian in the latter spaces, we compute several infinite families of eigenfunctions of the above models in closed form in terms of generalized Laguerre and Jacobi polynomials.
I discuss many-body models for correlated fermions in two space dimensions which can be solved exactly using group theory. The simplest example is a model of a quantum Hall system: two-dimensional (2D) fermions in a constant magnetic field and a particular non-local four-point interaction. It is exactly solvable due to a dynamical symmetry corresponding to the Lie algebra gl ∞ ⊕ gl ∞ . There is an algorithm to construct all energy eigenvalues and eigenfunctions of this model. The latter are, in general, many-body states with spatial correlations. The model also has a non-trivial zero temperature phase diagram. I point out that this QH model can be obtained from a more realistic one using a truncation procedure generalizing a similar one leading to mean field theory. Applying this truncation procedure to other 2D fermion models I obtain various simplified models of increasing complexity which generalize mean field theory by taking into account non-trivial correlations but nevertheless are treatable by exact methods.
A method is proposed for constructing an exact ground-state wave function of a two-dimensional model with spin 1/2. The basis of the method is to represent the wave function by a product of fourth-rank spinors associated with the nodes of a lattice and the metric spinors corresponding to bonds between nearest neighbor nodes. The function so constructed is an exact wave function of a 14-parameter model. The special case of this model depending on one parameter is analyzed in detail. The ground state is always a nondegenerate singlet, and the spin correlation functions decay exponentially with distance. The method can be generalized for models with spin 1/2 to other types of lattices.
Quantum geometry has emerged as a guiding principle across atomic and condensed-matter physics, shaping the topological responses of Bloch bands and the stability of the correlated phases they host. Sublattice symmetry, though common among bipartite lattice models, has not yet been exploited to obtain closed-form quantum geometry in multiband systems. For this purpose, we derive a general expression for the QGT of sublattice-symmetric systems in terms of contributions from the individual sublattice sectors, and show that this symmetry renders the Bloch Hamiltonian of a paradigmatic four-band model, the quarter-flux Harper-Hofstadter model, anti-block-diagonal, analytically yielding the spectrum, eigenstates, and full quantum geometric tensor (QGT), including the Berry curvature and quantum metric, for all four bands. The model, describing charged particles on a two-dimensional square lattice subjected to a uniform magnetic field, has recently been realized experimentally with ultracold atoms, photons, and superconducting circuits. Finally, we evaluate fractional-Chern-insulator stability criteria analytically and quantify the lowest band of the quarter-flux Harper-Hofstadter model to be a nearly ideal Chern band. Our approach opens a route for studying also the quantum geometry of other sublattice-symmetric multiband systems.
We have introduced a type of universal transformation between spin variables si{\mathit{s}}_{\mathit{i}}si​ and ${\mathrm{\ensuremath{\sigma}}}_{\mathit{i}}$ to map the general spin-1 model onto the spin-1/2 Ising model and have employed the exact results of the latter to find the exactly solvable cases of the spin-1 model. Our transformations are not one-to-one correspondences between si{\mathit{s}}_{\mathit{i}}si​ and ${\mathrm{\ensuremath{\sigma}}}_{\mathit{i}}$ and they are applicable to all lattices, depending on the lattice structure only through the coordination number. On square, triangle, and honeycomb lattices the exact critical points are shown in certain subspaces of parameter space (K, J, L, H, and \ensuremath{\Delta}).
In recent years, topology has become increasingly prevalent in condensed matter physics. It has allowed us to understand, and even predict, a variety of striking and remarkable physical phenomena. The study of strongly interacting systems has similarly lavished us with a diverse range of exotic phases and unconventional transitions, many of which are still poorly understood. In this thesis we will explore the interplay between topology and interactions in an effort to uncover new and novel phases. First we study how interactions impact the quantum phase transition between a topologically non-trivial phase and a trivial phase. The combination of interactions and the low-energy degrees of freedom associated with the transition leads to the emergence of a dome of lattice-symmetry breaking nematic order. Such behaviour is reminiscent of a number of strongly correlated electronic systems. We move on to study the strongly interacting limit of one of the earliest and best-known non-interacting topological phases, Haldane's model of a Chern insulator. Recently realized with ultracold atoms in a shaken optical lattice, the model has a non-trivial topological invariant associated with its band structure. In the strongly interacting limit the spin degrees of freedom are all that survive and we find a rich phase diagram of magnetically ordered phases, using a combination of both classical and quantum techniques. Supplementing the model with an additional term we can `quantum-melt' one of these ordered states to produce a disordered, liquid state that we positively identify as a chiral spin liquid, a highly entangled state of matter with fractionalised excitations. We generalise this mechanism to other two dimensional lattices, uncovering a possible unifying framework with which to understand the emergence of chiral spin liquids in lattice spin models. Finally, motivated by groundbreaking experiments in the ultracold atoms community, we investigate a model of two-component bosons with an artificial spin-orbit coupling. The interplay between the lattice, interactions and spin-orbit coupling produces a variety of unusual superfluid phases. Using a novel Monte Carlo technique we reveal the finite temperature phase diagram that appears close to the Mott transition.