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arXiv:1407.6196 · Physical Review Letters

Exploring Unconventional Hubbard Models with Doubly Modulated Lattice Gases

Sebastian Greschner, Luis Santos, Dario Poletti

Published Oct 31, 2014• 51 citations
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Abstract

Recent experiments show that periodic modulations of cold atoms in optical lattices may be used to engineer and explore interesting models. We show that double modulation combining lattice shaking and modulated interactions allows for the engineering of a much broader class of lattice with correlated hopping, which we study for the particular case of one-dimensional systems. We show, in particular, that by using this double modulation it is possible to study Hubbard models with asymmetric hopping, which, contrary to the standard Hubbard model, present insulating phases with both parity and string order. Moreover, double modulation allows for the simulation of lattice models in unconventional parameter regimes, as we illustrate for the case of the spin-1/2 Fermi-Hubbard model with correlated hopping, a relevant model for cuprate superconductors.

DOI
10.1103/physrevlett.113.183002

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Recent experiments show that periodic modulations of cold atoms in optical lattices may be used to engineer and explore interesting models. We show that double modulation combining lattice shaking and modulated interactions allows for the engineering of a much broader class of lattice with correlated hopping, which we study for the particular case of one-dimensional systems. We show, in particular, that by using this double modulation it is possible to study Hubbard models with asymmetric hopping, which, contrary to the standard Hubbard model, present insulating phases with both parity and string order. Moreover, double modulation allows for the simulation of lattice models in unconventional parameter regimes, as we illustrate for the case of the spin-1/2 Fermi-Hubbard model with correlated hopping, a relevant model for cuprate superconductors.
By using a modulated magnetic field in a Feshbach resonance for ultracold fermionic atoms in optical lattices, we show that it is possible to engineer a class of models usually referred to as correlated-hopping models. These models differ from the Hubbard model in exhibiting additional density-dependent interaction terms that affect the hopping processes. In addition to the spin-SU(2) symmetry, they also possess a charge-SU(2) symmetry, which opens the possibility of investigating the $\ensuremath{\eta}$-pairing mechanism for superconductivity introduced by Yang for the Hubbard model. We discuss the known solution of the model in one dimension (where $\ensuremath{\eta}$ states have been found in the degenerate manifold of the ground state) and show that, away from the integrable point, quantum Monte Carlo simulations at half filling predict the emergence of a phase with coexisting incommensurate spin and charge order.
Fermionic atoms in a periodic optical lattice provide a realization of the single-band Hubbard model. Using quantum Monte Carlo simulations along with the maximum-entropy method, we evaluate the effect of a time-dependent perturbative modulation of the optical lattice amplitude on atomic correlations, revealed in the fraction of doubly occupied sites. We find that the effect of modulation depends strongly on the filling---the response of the double occupation is significantly different in the half-filled Mott insulator from that in the doped Fermi liquid region.
In this thesis, we discuss quantum phase transitions in low-dimensional optical lattices, namely one- and two-dimensional lattices. The dimensional confinement is realized in experiments by suppressing the hopping in the extra dimensions through a deep potential barrier that prevents the atoms to tunnel across dimensions. The dimensionality considerably affects the physical behavior of the system. For instance, in one dimension, the celebrated Fermi-liquid theory does not describe the many-body behavior of a fermionic system. There are no single particle excitations, but instead collective behavior, and spin and charge degrees of freedom are decoupled . In the first part of this thesis, we consider one-dimensional fermionic gases under the effect of time-dependent driving forces. First, we describe a proposal to realize non-standard Hubbard models that display correlated-hopping terms by modulating interactions through a time-dependent magnetic field. The extra terms describe hopping processes that depend on the particle-density difference in neighboring sites. The effective model is obtained in arbitrary spatial dimensions, but only in one-dimension it is known to be exactly solvable for specific values of the parameters. Moreover, an SU(2) symmetry in the charge sector allows to construct non-trivial superconducting eigenstates (eta-states), discovered by Yang as excited states of the Hubbard model. Then, we study a one-dimensional bipartite optical lattice with fermionic atoms at half-filling. The lattice potential barriers have heights alternating in magnitude. As a consequence, the hopping amplitudes also alternate in magnitude and the elementary unit cell has two sites. This system is known to display Peierls dimerization at half-filling: the fermions form dimers in neighboring sites that inhibit transport and open a gap in the spectrum. After applying shaking on this system we discover that one can dynamically induce a phase transition to an unconventional metallic state with four Fermi-points. The second part of this thesis describes bosonic systems in two-dimensional optical lattices. First, we consider a square optical lattice with alternating deep-shallow wells arranged in a checkerboard pattern. This lattice is realized in the experimental laboratory of Prof. Hemmerich in Hamburg. In the experiment, the visibility is extracted from the time-of-flight imaging to quantify the phase coherence of the bosonic gas. We develop a mean-field theory and determine the phase diagram of this model. We interpret the experimental data by assuming that at large detuning, only the deeper wells are populated and form an imbalanced Mott insulator with empty shallow wells and integer density on the deeper wells. We support this hypothesis by large-scale Gutzwiller calculations. Then, we investigate a system of bosons in the Lieb lattice, which is the lattice geometry of the Cu-O plane in cuprates . We consider configurations of the optical potential where s and p orbitals sitting on neighboring wells become resonant and hybridize. The interest in this problem comes from the possibility to realize time-reversal broken phases, usually known as Varma phases for electronic systems, which have been proposed to explain the pseudo-gap regime of cuprate superconductors. We show the conditions under which the symmetry breaking in a four-band model can lead to states with loop currents. Moreover, the lowest branch of the excitation spectrum has a non-vanishing Chern number and therefore supports topological excitations.
We consider a strongly repulsive two-component Fermi gas in a one-dimensional optical lattice described in terms of a Hubbard Hamiltonian. We analyze the response of the system to a periodic modulation of the hopping amplitude in the presence of a large two-body interaction. By (essentially) the exact simulations of the time evolution, we find a nontrivial double occupancy frequency dependence. We show how the dependence relates to the spectral features of the system given by the Bethe ansatz. The discrete nature of the spectrum is clearly reflected in the double occupancy after a long enough modulation time. We also discuss the implications of the 1D results to experiments in higher dimensional systems.
Various topics at the interface between condensed matter physics and the physics of ultra-cold fermionic atoms in optical lattices are discussed. The lectures start with basic considerations on energy scales, and on the regimes in which a description by an effective Hubbard model is valid. Qualitative ideas about the Mott transition are then presented, both for bosons and fermions, as well as mean-field theories of this phenomenon. Antiferromagnetism of the fermionic Hubbard model at half-filling is briefly reviewed. The possibility that interaction effects facilitate adiabatic cooling is discussed, and the importance of using entropy as a thermometer is emphasized. Geometrical frustration of the lattice, by suppressing spin long-range order, helps revealing genuine Mott physics and exploring unconventional quantum magnetism. The importance of measurement techniques to probe quasiparticle excitations in cold fermionic systems is emphasized, and a recent proposal based on stimulated Raman scattering briefly reviewed. The unconventional nature of these excitations in cuprate superconductors is emphasized.
In the past fifteen years ultracold atoms confined to optical lattices, which are artificial crystals of light, have proven to be powerful tools in the study of strongly correlated many-body systems widely known from a condensed matter context. The detailed knowledge, high tunability and control of microscopic parameters has enabled the creation and control of highly non-trivial quantum systems for which a direct comparison of model calculations to experimental measurements is now possible. In order to gain precise information on the properties of ultracold gases in optical lattices, many powerful methods have been developed. In particular spectroscopic methods have enabled to probe different characteristic excitations. However, many fascinating properties of strongly interacting quantum many-body systems are not yet fully characterized. Therefore, this thesis explores further possibilities of lattice modulation spectroscopy which is a time-periodic modulation of the optical lattice amplitude. We theoretically investigate the response of different (mostly one-dimensional) cold atom systems to a lattice modulation by means of the time-dependent density matrix renormalization group method and different analytical approximations. In addition to the normal lattice modulation, we consider a superlattice modulation scheme where the lattice amplitude is modulated in a dimerized fashion, coupling to a certain class of excitations in the system. The lattice modulation spectroscopy is a versatile probe which we use to study different situations relevant for experimental applications. We show that a detection of signatures of the exotic bond order wave phase is feasible in a system of interacting fermions on a one-dimensional optical lattice with staggered energy offset. Moreover, we demonstrate that a temperature measurement of fermions confined to an optical lattice is possible as well as a determination of the interaction strength of bosonic atoms confined to a one-dimensional optical lattice. In addition, various excitations in all these systems are investigated in detail.