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Cambridge University Press eBooks

The Hubbard Model

Hui Zhai

Published Jan 30, 2021• 0 citations
Publisher

Abstract

1. Introduce quantum phase transition in the Bose-Hubbard model. 2. Emphasize vanishing energy scales and universality at the quantum critical point. 3. Introduce microscopic theories to describe the quantum phase transition, and explicitly show the vanishing of energy scales and the critical exponents. 4. Illustrate the emergent Lorentz symmetry and the Higgs mode at the quantum critical point of the Bose-Hubbard model with the particle-hole symmetry. 5. Discuss an experimental probe of the superfluid to the Mott insulator phase transition. 6. Show that the repulsive and the attractive Fermi-Hubbard models are related by the particle-hole symmetry. 7. Discuss the origin of antiferromagnetic order in the repulsive Fermi-Hubbard model at half-filling. 8. Introduce the enlarged $SO(4)$ symmetry of the Fermi-Hubbard model at half-filling and zero spin imbalance and its physical concequence. 9. Introduce important unsolved challenge issues in the Fermi-Hubbard model. 10. Introduce the concept of eigenstate thermalization hypothesis and many-body localization as opposite to thermalization. 11. Introduce a few metrics to characterize the many-body localization. 12. Emphasize the role of entanglement entropy in characterizing quantum thermalization, and discuss how to measure entanglement entropy in ultracold atom systems.

DOI
10.1017/9781108595216.009

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•May 17, 2019•0 citations
  1. Introduce quantum phase transition in the Bose-Hubbard model. 2. Emphasize vanishing energy scales and universality at the quantum critical point. 3. Introduce microscopic theories to describe the quantum phase transition, and explicitly show the vanishing of energy scales and the critical exponents. 4. Illustrate the emergent Lorentz symmetry and the Higgs mode at the quantum critical point of the Bose-Hubbard model with the particle-hole symmetry. 5. Discuss an experimental probe of the superfluid to the Mott insulator phase transition. 6. Show that the repulsive and the attractive Fermi-Hubbard models are related by the particle-hole symmetry. 7. Discuss the origin of antiferromagnetic order in the repulsive Fermi-Hubbard model at half-filling. 8. Introduce the enlarged SO(4)SO(4)SO(4) symmetry of the Fermi-Hubbard model at half-filling and zero spin imbalance and its physical concequence. 9. Introduce important unsolved challenge issues in the Fermi-Hubbard model. 10. Introduce the concept of eigenstate thermalization hypothesis and many-body localization as opposite to thermalization. 11. Introduce a few metrics to characterize the many-body localization. 12. Emphasize the role of entanglement entropy in characterizing quantum thermalization, and discuss how to measure entanglement entropy in ultracold atom systems.
We report results of quantum Monte Carlo simulations of the Bose-Hubbard model in three dimensions. Critical parameters for the superfluid-to-Mott-insulator transition are determined with significantly higher accuracy than has been done in the past. In particular, the position of the critical point at filling factor n=1n=1n=1 is found to be at (U∕t)c=29.34(2){(U∕t)}_{\mathrm{c}}=29.34(2)(U∕t)c​=29.34(2), and the insulating gap $\ensuremath{\Delta}$ is measured with accuracy of a few percent of the hopping amplitude ttt. We obtain the effective mass of particle and hole excitations in the insulating state---with explicit demonstration of the emerging particle-hole symmetry and relativistic dispersion law at the transition tip---along with the sound velocity in the strongly correlated superfluid phase. These parameters are the necessary ingredients to perform analytic estimates of the low temperature $(T⪡\ensuremath{\Delta})$ thermodynamics in macroscopic samples. We present accurate thermodynamic curves, including these for specific heat and entropy, for typical insulating (U∕t=40)(U∕t=40)(U∕t=40) and superfluid (t∕U=0.0385)(t∕U=0.0385)(t∕U=0.0385) phases. Our data can serve as a basis for accurate experimental thermometry, and a guide for appropriate initial conditions if one attempts to use interacting bosons in quantum information processing.
Ultracold atomic gases may be the ultimate quantum simulator.These isolated systems have the lowest temperatures in the observable universe, and their properties and interactions can be precisely and accurately tuned across a full spectrum of behaviors, from few-body physics to highly-correlated many-body effects.The ability to impose potentials on and tune interactions within ultracold gases to mimic complex systems mean they could become a theorist's playground.One of their great strengths, however, is also one of the largest obstacles to this dream: isolation.This thesis touches on both of these themes.First, methods to characterize phases and quantum critical points, and to construct finite temperature phase diagrams using experimentally accessible observables in the Bose Hubbard model are discussed.Then, the transition from a weakly to a strongly interacting Bose-Fermi mixture in the continuum is analyzed using zero temperature numerical techniques.Real materials can be emulated by ultracold atomic gases loaded into optical lattice potentials.We discuss the characteristics of a single boson species trapped in an optical lattice (described by the Bose Hubbard model) and the hallmarks of the quantum critical region that separates the superfluid and the Mott insulator ground states.We propose a method to map the quantum critical region using the single, experimentally accessible, local quantity R, the ratio of compressibility to local number fluctuations.The procedure to map a phase diagram with R is easily generalized to inhomogeneous systems and generic many-body Hamiltonians.We illustrate it here using quantum Monte Carlo simulations of the 2D Bose Hubbard model.Secondly, we investigate the transition from a degenerate Fermi gas weakly coupled List of Tables Table Page 2.1 Techniques to Probe Atoms in Optical Lattices . . . . . .
Introduction. Solvable Models, Rigorous Results, and Advances in Formalism: The Hubbard Model: Some Rigorous Results and Open Problems E.H. Lieb. On the Bethe Ansatz Soluble Degenerate Hubbard Model H. Frahm, A. Schadschneider. Infinite in all Dimensions: Large Coupling, High Dimensions, Many Components: The Mott Transition in Infinite Dimensions: Old Ideas and Some Surprises G. KIotliar, M.J. Rozenberg. The Hubbard Model with Infinite Interaction: Magnetic Properties V.Ya. Krivnov, et al. Fermi Liquid versus Luttinger Liquid: Two Particle Scattering and Orthogonality Catastrophe in the Hubbard Model W. Metzner. Conservation Laws in Normal Metals: Luttinger Liquid vs. Fermi Liquid C. Di Castro, et al. Perturbative, Mean Field, Variational, and Numerical Studies: Hartree-Fock and RPA Studies of the Hubbard Model G. Guinea, et al. The Wavefunction Renormalization Constant for the One- and Two-band Hubbard Hamiltonians in Two Dimensions E. Louis, et al. Experiments and Physical Applications: Electron Spectroscopy and Hubbard: Issues and Opportunities J.W. Allen, et al. On Electrical Properties of Chalcogenide Glassy Semiconductors in the Framework of Hubbard Model with Negative Correlation Energy S.D. Savransky. 34 additional articles. Index.
Many-body localization (MBL) of a disordered interacting boson system in one dimension is studied numerically at the filling faction one-half. The von Neumann entanglement entropy SvN is commonly used to detect the MBL phase transition but remains challenging to be directly measured. Based on the U(1) symmetry from the particle number conservation, SvN can be decomposed into the particle number entropy SN and the configuration entropy SC. In light of the tendency that the eigenstate’s SC nears zero in the localized phase, we introduce a quantity describing the deviation of SN from the ideal thermalization distribution; finite-size scaling analysis illustrates that it shares the same phase transition point with SvN but displays the better critical exponents. This observation hints that the phase transition to MBL might largely be determined by SN and its fluctuations. Notably, the recent experiments [A. Lukin, et al., Science 364, 256 (2019); J. Léonard, et al., Nat. Phys. 19, 481 (2023)] demonstrated that this deviation can potentially be measured through the SN measurement. Furthermore, our investigations reveal that the thermalized states primarily occupy the low-energy section of the spectrum, as indicated by measures of localization length, gap ratio, and energy density distribution. This low-energy spectrum of the Bose model closely resembles the entire spectrum of the Fermi (or spin XXZ) model, accommodating a transition from the thermalized to the localized states. While, owing to the bosonic statistics, the high-energy spectrum of the model allows the formation of distinct clusters of bosons in the random potential background. We analyze the resulting eigenstate properties and briefly summarize the associated dynamics. To distinguish between the phase regions at the low and high energies, a probing quantity based on the structure of SvN is also devised. Our work highlights the importance of symmetry combined with entanglement in the study of MBL. In this regard, for the disordered Heisenberg XXZ chain, the recent pure eigenvalue analyses in [J. Suntajs, et al., Phys. Rev. E 102, 062144 (2020)] would appear inadequate, while methods used in [A. Morningstar, et al., Phys. Rev. B 105, 174205 (2022)] that spoil the U(1) symmetry could be misleading.
One of the distinguishing achievements of physics has been the ability to describe the Universe based upon underlying microscopic principles. When there are many participating objects being governed by these microscopic principles, the resultant behavior often harbors emergent collective phenomena that may not be obviously related to the microscopic description. Phase transitions, quantum or classical, are a canonical example of such many-body phenomena that are understood in equilibrium through a framework of thermodynamic constraints, statistical ensembles, and a growth of system-wide correlations at the transition. However, a new class of quantum phase transitions has recently emerged for non-equilibrium systems with behavior that is not captured by this traditional framework. In the case of isolated quantum systems, these transitions occur among excited eigenstates and requires knowledge of the structure of these individual eigenstates in the quantum system. Understanding what drives such a phase transition requires a deeper microscopic knowledge of the role of correlations at the transition. \nThe use of ultracold atoms in optical lattices has paved the way for faithfully realizing quantum phase transitions. In particular, the advent of the quantum gas microscope realized an experimental platform that is highly controllable and provides microscopic access to the many-bodied wave functions. All experiments presented in this thesis use such a quantum gas microscope architecture where a quantum degenerate gas of Rb is used to probe the behavior of non-equilibrium quantum systems in one spatial dimension.\nThis thesis briefly discusses some studies of entanglement in equilibrium quantum phase transitions near the ground state of the system. In the first experiment, we directly measure the second-order R\\'enyi entanglement entropy that develops as a Bose-Hubbard system transitions from a Mott insulator to a superfluid. This is followed by a study of entanglement growth at the transition in the transverse Ising model. \nThese studies progress to the investigation of two particular non-equilibrium quantum phases and their transition into one another: quantum thermalization and many-body localization. We investigate the development of entanglement entropy that mimics thermodynamic entropy for an isolated quantum system which locally thermalizes under its own unitary dynamics. We then investigate the only known robust exception to this thermalizing behavior by studying the onset of many-body localization due to disorder. We observe the breakdown of thermalization in this phase and the logarithmically slow growth of entanglement throughout the system, which identifies the localized system to be an interacting one. Lastly, we investigate the role of correlations in the critical dynamics that emerge at high-orders where the system transitions from thermalizing behavior to many-body localization.