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.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.

Cambridge University Press eBooksJan 30, 20210 citations

Quantum phase transitions in a novel Fermi-Bose Hubbard model

Lincoln D. Carr, Daniel E. Sheehy, Murray Holland

Bulletin of the American Physical SocietyMar 23, 20050 citations

Bose-Fermi Mapping in Hubbard Models at Imaginary Chemical Potential and Phase-Induced Fermionization

Evangelos Georgiou Filothodoros

We find a mapping between the attractive Fermi-Hubbard model and the repulsive Bose-Hubbard model at finite temperature and at imaginary chemical potential $μ=iθ$ . We show, by using a large $N$ -expansion, that the partition functions of the two models are related by a simple shift $θ\to θ+ π$ . This condition maps the BCS--BEC crossover of attractive fermions to a Bose--Fermi crossover (fermion-like occupation) of repulsive bosons. Central feature of this correspondence plays the thermal kernel $g(βE,ϕ),$ whose analytic continuation $g_B(βE,ϕ) = g_F(βE,ϕ+π)$ governs the bosonic and fermionic sectors. Interestingly, we are able to find that the special angles $ϕ= 2π/3,4π/3$ for fermions correspond to $ϕ= π/3,5π/3$ for bosons, marking the boundaries of a universal thermal window. We further argue that the present mechanism shows that fermionization can occur at finite interaction strength through a thermodynamic effect induced by the imaginary chemical potential. This suggests that it is a new way of fermionization (not a change in statistics but a fermion-like behaviour) unlike the Tonks--Girardeau limit, where fermionization arises from an infinite repulsive interaction and anyonic or Floquet-engineered systems where transmutation emerges from modified statistics or dynamics. Essentially, the phase $ϕ$ is a statistical parameter; by twisting the thermal phase, it generates fermion-like behaviour without hard-core constraints or infinite repulsion but only by using thermodynamics. We derive the gap equation and number equation for the bosonic model, highlighting the role of the imaginary chemical potential as a statistical regulator. Our results provide a unified framework for understanding crossovers in interacting lattice systems.We find a mapping between the attractive Fermi-Hubbard model and the repulsive Bose-Hubbard model at finite temperature and at imaginary chemical potential $μ=iθ$. We show, by using a large $N$-expansion, that the partition functions of the two models are related by a simple shift $θ\to θ+ π$. This condition maps the BCS--BEC crossover of attractive fermions to a Bose--Fermi crossover (fermion-like occupation) of repulsive bosons. Central feature of this correspondence plays the thermal kernel $g(βE,ϕ),$ whose analytic continuation $g_B(βE,ϕ) = g_F(βE,ϕ+π)$ governs the bosonic and fermionic sectors. Interestingly, we are able to find that the special angles $ϕ= 2π/3,4π/3$ for fermions correspond to $ϕ= π/3,5π/3$ for bosons, marking the boundaries of a universal thermal window. We further argue that the present mechanism shows that fermionization can occur at finite interaction strength through a thermodynamic effect induced by the imaginary chemical potential. This suggests that it is a new way of fermionization (not a change in statistics but a fermion-like behaviour) unlike the Tonks--Girardeau limit, where fermionization arises from an infinite repulsive interaction and anyonic or Floquet-engineered systems where transmutation emerges from modified statistics or dynamics. Essentially, the phase $ϕ$ is a statistical parameter; by twisting the thermal phase, it generates fermion-like behaviour without hard-core constraints or infinite repulsion but only by using thermodynamics. We derive the gap equation and number equation for the bosonic model, highlighting the role of the imaginary chemical potential as a statistical regulator. Our results provide a unified framework for understanding crossovers in interacting lattice systems.

arXiv:2605.10418May 11, 20260 citations

Phase diagrams of the Bose-Fermi-Hubbard-Model: Analytical and numerical studies

Alexander Mering, Michael Fleischhauer

Bulletin of the American Physical SocietyJun 7, 20070 citations

Repulsive Fermi and Bose Polarons in Quantum Gases

Francesco Scazza, Matteo Zaccanti, Pietro Massignan +2 more

Polaron quasiparticles are formed when a mobile impurity is coupled to the elementary excitations of a many-particle background. In the field of ultracold atoms, the study of the associated impurity problem has attracted a growing interest over the last fifteen years. Polaron quasiparticle properties are essential to our understanding of a variety of paradigmatic quantum many-body systems realized in ultracold atomic gases and in the solid state, from imbalanced Bose–Fermi and Fermi–Fermi mixtures to fermionic Hubbard models. In this topical review, we focus on the so-called repulsive polaron branch, which emerges as an excited many-body state in systems with underlying attractive interactions such as ultracold atomic mixtures, and is characterized by an effective repulsion between the impurity and the surrounding medium. We give a brief account of the current theoretical and experimental understanding of repulsive polaron properties, for impurities embedded in both fermionic and bosonic media, and we highlight open issues deserving future investigations.Polaron quasiparticles are formed when a mobile impurity is coupled to the elementary excitations of a many-particle background. In the field of ultracold atoms, the study of the associated impurity problem has attracted a growing interest over the last fifteen years. Polaron quasiparticle properties are essential to our understanding of a variety of paradigmatic quantum many-body systems realized in ultracold atomic gases and in the solid state, from imbalanced Bose–Fermi and Fermi–Fermi mixtures to fermionic Hubbard models. In this topical review, we focus on the so-called repulsive polaron branch, which emerges as an excited many-body state in systems with underlying attractive interactions such as ultracold atomic mixtures, and is characterized by an effective repulsion between the impurity and the surrounding medium. We give a brief account of the current theoretical and experimental understanding of repulsive polaron properties, for impurities embedded in both fermionic and bosonic media, and we highlight open issues deserving future investigations.

arXiv:2204.06984AtomsMay 27, 202287 citations

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 find a mapping between the attractive Fermi-Hubbard model and the repulsive Bose-Hubbard model at finite temperature and at imaginary chemical potential

μ=iθ

. We show, by using a large

N

-expansion, that the partition functions of the two models are related by a simple shift

θ\to θ+ π

. This condition maps the BCS--BEC crossover of attractive fermions to a Bose--Fermi crossover (fermion-like occupation) of repulsive bosons. Central feature of this correspondence plays the thermal kernel

g(βE,ϕ),

whose analytic continuation

g_B(βE,ϕ) = g_F(βE,ϕ+π)

governs the bosonic and fermionic sectors. Interestingly, we are able to find that the special angles

ϕ= 2π/3,4π/3

for fermions correspond to

ϕ= π/3,5π/3

for bosons, marking the boundaries of a universal thermal window. We further argue that the present mechanism shows that fermionization can occur at finite interaction strength through a thermodynamic effect induced by the imaginary chemical potential. This suggests that it is a new way of fermionization (not a change in statistics but a fermion-like behaviour) unlike the Tonks--Girardeau limit, where fermionization arises from an infinite repulsive interaction and anyonic or Floquet-engineered systems where transmutation emerges from modified statistics or dynamics. Essentially, the phase

ϕ

is a statistical parameter; by twisting the thermal phase, it generates fermion-like behaviour without hard-core constraints or infinite repulsion but only by using thermodynamics. We derive the gap equation and number equation for the bosonic model, highlighting the role of the imaginary chemical potential as a statistical regulator. Our results provide a unified framework for understanding crossovers in interacting lattice systems.