This chapter finally moves beyond the quantum rotor models which have been the complete focus of our attention so far in Part II. Our motivation is two-fold: to introduce the coherent state path integral, which plays an important role in developing the field theory for many interesting quantum phase transitions; and to provide a deeper and more complete explanation of our claimed connection between the N = 2 rotor model and the experiments on ultracold bosonic atoms in an optical lattice which was claimed in Sections 1.3 and 1.4.3. We do this by studying the boson Hubbard model, which has a direct connection to the microscopic Hamiltonian of the ultracold atoms. The Hubbard model was originally introduced as a description of the motion of electrons in transition metals, with the motivation of understanding their magnetic properties. This original model remains a very active subject of research today, and important progress has been made in recent years by examining its properties in the limit of large spatial dimensionality [160, 165]. In this chapter, we examine only the much simpler “boson Hubbard model,” following the analysis in an important paper by Fisher et al. [148]. As the name implies, the elementary degrees of freedom in this model are spinless bosons, which take the place of the spin-1/2 fermionic electrons in the original Hubbard model. These bosons could represent Cooper pairs of electrons undergoing Josephson tunneling between superconducting islands, helium atoms moving on a substrate, or ultracold atoms in an optical lattice.This chapter finally moves beyond the quantum rotor models which have been the complete focus of our attention so far in Part II. Our motivation is two-fold: to introduce the coherent state path integral, which plays an important role in developing the field theory for many interesting quantum phase transitions; and to provide a deeper and more complete explanation of our claimed connection between the N = 2 rotor model and the experiments on ultracold bosonic atoms in an optical lattice which was claimed in Sections 1.3 and 1.4.3. We do this by studying the boson Hubbard model, which has a direct connection to the microscopic Hamiltonian of the ultracold atoms. The Hubbard model was originally introduced as a description of the motion of electrons in transition metals, with the motivation of understanding their magnetic properties. This original model remains a very active subject of research today, and important progress has been made in recent years by examining its properties in the limit of large spatial dimensionality [160, 165]. In this chapter, we examine only the much simpler “boson Hubbard model,” following the analysis in an important paper by Fisher et al. [148]. As the name implies, the elementary degrees of freedom in this model are spinless bosons, which take the place of the spin-1/2 fermionic electrons in the original Hubbard model. These bosons could represent Cooper pairs of electrons undergoing Josephson tunneling between superconducting islands, helium atoms moving on a substrate, or ultracold atoms in an optical lattice.

Cambridge University Press eBooksApr 7, 20110 citations

Ultracold atoms in optical lattices

Immanuel Bloch

Oxford University Press eBooksNov 15, 20120 citations

Ultracold atoms in novel optical lattices

Nathan E. Lundblad

Bulletin of the American Physical SocietyNov 9, 20120 citations

Feshbach modulation spectroscopy of the Fermi-Hubbard model

Andreas Dirks, Karlis Mikelsons, H. R. Krishnamurthy +1 more

In the vicinity of a Feshbach resonance, a system of ultracold atoms in an optical lattice undergoes rich physical transformations which involve molecule formation and hopping of molecules on the lattice and thus goes beyond a single-band Hubbard model description. We explore theoretically the response of this system to a harmonic modulation of the magnetic field, and thus of the scattering length, across the Feshbach resonance. In the regime in which the single-band Hubbard model is still valid, we provide results for the doublon production as a function of the various parameters, such as frequency, amplitude, etc., that characterize the field modulation, as well as the lattice depth. The method may uncover a route towards the efficient creation of ultracold molecules and also provide an alternative to conventional lattice-depth-modulation spectroscopy.In the vicinity of a Feshbach resonance, a system of ultracold atoms in an optical lattice undergoes rich physical transformations which involve molecule formation and hopping of molecules on the lattice and thus goes beyond a single-band Hubbard model description. We explore theoretically the response of this system to a harmonic modulation of the magnetic field, and thus of the scattering length, across the Feshbach resonance. In the regime in which the single-band Hubbard model is still valid, we provide results for the doublon production as a function of the various parameters, such as frequency, amplitude, etc., that characterize the field modulation, as well as the lattice depth. The method may uncover a route towards the efficient creation of ultracold molecules and also provide an alternative to conventional lattice-depth-modulation spectroscopy.

arXiv:1408.0206Physical Review ANov 16, 20154 citations

Site-resolved observables in the doped spin-imbalanced triangular Hubbard model

Davis Garwood, Jirayu Mongkolkiattichai, Liyu Liu +2 more

The suppression of antiferromagnetic ordering in geometrically frustrated Hubbard models leads to a variety of exotic quantum phases including quantum spin liquids and chiral states. Here, we focus on the Hubbard model on one of the simplest frustrated lattice geometries, a triangular lattice. Motivated by the recent realization of ultracold fermionic atoms in triangular optical lattices, we study the properties of the triangular-lattice Hubbard model through a numerical linked-cluster expansion algorithm. We investigate the Mott insulator transition finding a critical interaction ${U}_{c}/t=7.0(2)$ and use spatial two- and three-point correlation functions to explore doped and imbalanced systems. Our results demonstrate that many interesting features occur at temperatures previously obtained for ultracold fermions in optical lattices and are accessible by upcoming experiments. Our calculations will be helpful for thermometry in ultracold atom quantum simulators and can guide experimental searches for exotic quantum phases in atomic triangular Hubbard quantum simulators.The suppression of antiferromagnetic ordering in geometrically frustrated Hubbard models leads to a variety of exotic quantum phases including quantum spin liquids and chiral states. Here, we focus on the Hubbard model on one of the simplest frustrated lattice geometries, a triangular lattice. Motivated by the recent realization of ultracold fermionic atoms in triangular optical lattices, we study the properties of the triangular-lattice Hubbard model through a numerical linked-cluster expansion algorithm. We investigate the Mott insulator transition finding a critical interaction ${U}_{c}/t=7.0(2)$ and use spatial two- and three-point correlation functions to explore doped and imbalanced systems. Our results demonstrate that many interesting features occur at temperatures previously obtained for ultracold fermions in optical lattices and are accessible by upcoming experiments. Our calculations will be helpful for thermometry in ultracold atom quantum simulators and can guide experimental searches for exotic quantum phases in atomic triangular Hubbard quantum simulators.

arXiv:2205.12233Physical Review AJul 11, 202216 citations

{U}_{c}/t=7.0(2)

and use spatial two- and three-point correlation functions to explore doped and imbalanced systems. Our results demonstrate that many interesting features occur at temperatures previously obtained for ultracold fermions in optical lattices and are accessible by upcoming experiments. Our calculations will be helpful for thermometry in ultracold atom quantum simulators and can guide experimental searches for exotic quantum phases in atomic triangular Hubbard quantum simulators.