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Some Results on One-Dimensional Models with Broken and Deformed Symmetries

Francis N. C. Paraan

Published Sep 11, 2012• 0 citations
OpenAlex

Abstract

We present analytic results for the ground state properties and entanglement in the Lieb-Liniger model and q-deformed Affleck-Kennedy-Lieb-Tasaki (AKLT) models. The translational invariance of the Lieb-Liniger model is broken by an external harmonic potential in one case and by coarse-grained measurements of particle number in another. Meanwhile, anisotropy is introduced into the AKLT model by generalizing its SU(2) invariant hamiltonian into one that is SUq(2) invariant. The Lieb-Liniger model describes a one-dimensional gas of bosons that mutually interact by a zero-range interaction term. Under longitudinal harmonic confinement, the exact spectrum of the hamiltonian is known only for free and impenetrable bosons. We use a pseudopotential approach and perturbation theory to calculate the ground state energy of this gas near the limit of infinite repulsion. We further study entanglement in the periodic homogeneous Lieb-Liniger model. When the particle number in a spatial partition of the ground state is measured, entanglement in the resulting state arises only from interparticle interactions. We demonstrate that entanglement in these projected states increases monotonically with the strength of interactions. Finally, we discuss how anisotropy reduces entanglement in the valence-bond-solid ground state of the q-deformed AKLT model. We propose effective thermal models that describe block entanglement in terms of boundary degrees of freedom. The anisotropy parameter q enters these models as an effective temperature. | 116 pages

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We present analytic results for the ground state properties and entanglement in the Lieb-Liniger model and q-deformed Affleck-Kennedy-Lieb-Tasaki (AKLT) models. The translational invariance of the Lieb-Liniger model is broken by an external harmonic potential in one case and by coarse-grained measurements of particle number in another. Meanwhile, anisotropy is introduced into the AKLT model by generalizing its SU(2) invariant hamiltonian into one that is SUq(2) invariant. The Lieb-Liniger model describes a one-dimensional gas of bosons that mutually interact by a zero-range interaction term. Under longitudinal harmonic confinement, the exact spectrum of the hamiltonian is known only for free and impenetrable bosons. We use a pseudopotential approach and perturbation theory to calculate the ground state energy of this gas near the limit of infinite repulsion. We further study entanglement in the periodic homogeneous Lieb-Liniger model. When the particle number in a spatial partition of the ground state is measured, entanglement in the resulting state arises only from interparticle interactions. We demonstrate that entanglement in these projected states increases monotonically with the strength of interactions. Finally, we discuss how anisotropy reduces entanglement in the valence-bond-solid ground state of the q-deformed AKLT model. We propose effective thermal models that describe block entanglement in terms of boundary degrees of freedom. The anisotropy parameter q enters these models as an effective temperature. | 116 pages
In the ground state of gapped systems, the entanglement entropy of a subsystem A scales with the length of the boundary of A. This observation suggests that the entanglement properties of the sub- system can be described in terms of degrees of freedom living in the boundary of A. We will discuss the the connection between en- tanglement properties and effective boundary descriptions in spin systems in one and two dimensions. In one dimension we present analytic results for the spin S = 1, Affleck-Kennedy-Lieb-Tasaki (AKLT) ground state entanglement, characterized by negativity and entanglement spectrum. We also discuss a generalization of the AKLT model, based on the quantum group Uq (sl(2)) for gen- eral integer spin S. In two dimensions, we study two spin systems whose ground state can be written in terms of tensor product states of bond dimension two, the AKLT model in the hexagonal lattice and the Ising projected entangled pair state (Ising PEPS) in the square lattice. We show how the reduced density matrix of a parti- tion is associated with a thermal state of a one dimensional model along the boundary of that partition. We also present arguments supporting this correspondence for arbitrary gapped systems. Fi- nally we discuss the behavior of this boundary theory when the original two dimensional model is tuned through a quantum phase transition,126 pages
We exactly calculate the reduced density matrix of matrix product states (MPS). Our compact result enables one to perform analytic studies of entanglement in MPS. In particular, we consider the MPS ground states of two anisotropic spin chains. One is a q -deformed Affleck-Kennedy-Lieb-Tasaki (AKLT) model and the other is a general spin-1 quantum antiferromagnet with nearest-neighbor interactions. Our analysis shows how anisotropy affects entanglement on different continuous parameter manifolds. We also construct an effective boundary spin model that describes a block of spins in the ground state of the q -deformed AKLT Hamiltonian. The temperature of this effective model is given in terms of the deformation parameter q .
We study the ground-state properties and excitation spectrum of the Lieb-Liniger model, i.e. the one-dimensional Bose gas with repulsive contact interactions. We solve the Bethe-Ansatz equations in the thermodynamic limit by using an analytic method based on a series expansion on orthogonal polynomials developed in and push the expansion to an unprecedented order. By a careful analysis of the mathematical structure of the series expansion, we make a conjecture for the analytic exact result at zero temperature and show that the partially resummed expressions thereby obtained compete with accurate numerical calculations. This allows us to evaluate the density of quasi-momenta, the ground-state energy, the local two-body correlation function and Tan’s contact. Then, we study the two branches of the excitation spectrum. Using a general analysis of their properties and symmetries, we obtain novel analytical expressions at arbitrary interaction strength which are found to be extremely accurate in a wide range of intermediate to strong interactions.
Abstract In 1963, Lieb and Liniger solved exactly a one-dimensional model of bosons interacting by a repulsive δ -potential and calculated the ground state in the thermodynamic limit. In the present work, we extend this model to a potential of three δ -functions—one of them is repulsive and the other two are attractive—modeling some aspects of the interaction between atoms, and we present an approximate solution for a dilute gas. In this limit, for low energy states, the results are found to be reduced to the ones of an effective Lieb–Liniger model with an effective δ -function of strength c eff and the regime of stability is identified. This may shed light on some aspects of interacting bosons.
Affleck, Kennedy, Lieb, and Taski (AKLT) constructed an exemplary spin-3/2 valence-bond solid (VBS) state on the hexagonal lattice, which is the ground state of an isotropic quantum antiferromagnet and possesses no spontaneous magnetization but finite correlation length. This is distinct from the N\'eel ordered state of the spin-3/2 Heisenberg model on the same lattice. Niggemann, Kl\"umper, and Zittartz then generalized the AKLT Hamiltonian to one family invariant under spin rotation about the zzz axis. The ground states of this family can be parameterized by a single parameter that deforms the AKLT state, and this system exhibits a quantum phase transition between the VBS and N\'eel phases, as the parameter increases from the AKLT point to large anisotropy. We investigate the opposite regime when the parameter decreases from the AKLT point and find that there appears to be a Berezinskii-Kosterlitz-Thouless-like transition from the VBS phase to an XY phase. Such a transition also occurs in the deformation of other types of AKLT states with triplet-bond constructions on the same lattice. However, we do not find such an XY-like phase in the deformed AKLT models on other trivalent lattices, such as square-octagon, cross, and star lattices. On the star lattice, the deformed family of AKLT states remain in the same phase as the isotropic AKLT state throughout the whole region of the parameter. However, for two triplet-bond generalizations, the triplet VBS phase is sandwiched between two ferromagnetic phases (for large and small deformation parameters, respectively), which are characterized by spontaneous magnetizations along different axes. Along the way, we also discuss how various deformed AKLT states can be used for the purpose of universal quantum computation.
Two-dimensional (spin-2) Affleck-Kennedy-Lieb-Tasaki (AKLT) type valence bond solids on a square lattice are known to be symmetry-protected topological (SPT) gapped spin liquids [S. Takayoshi, P. Pujol, and A. Tanaka Phys. Rev. B 94, 235159 (2016)]. Using the projected entangled pair state framework, we extend the construction of the AKLT state to the case of SU(3)\text{SU}(3)SU(3), relevant for cold atom systems. The entanglement spectrum is shown to be described by an alternating SU(3)\text{SU}(3)SU(3) chain of ``quarks'' and ``antiquarks'', subject to exponentially decaying (with distance) Heisenberg interactions, in close similarity with its SU(2)\text{SU}(2)SU(2) analog. We discuss the SPT feature of the state.