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The landscape of the Hubbard model

Subir Sachdev

TL;DR

The article provides a comprehensive, gauge-theoretic survey of the Hubbard model across lattices and densities, focusing on Dirac-like low-energy physics on the honeycomb lattice and the emergence of symmetry-broken and topologically ordered phases. It develops and interconnects multiple theoretical frameworks—Gross-Neveu–Yukawa criticality, U(1) and SU(2) gauge theories, and rotor/ spinon decompositions—to describe semi-metals, Néel and VBS insulators, spin liquids, and fractionalized Fermi liquids, and extends these insights to nonzero temperature with AdS/CFT-inspired transport predictions. A central thrust is the unification of conventional and exotic phases via emergent gauge fields and deconfined criticality, including a deconfined QCD-like fixed point and a lattice-symmetry–driven VBS order. The work further links condensed-mmatter gauge theories to holographic metal constructions, illustrating how holography can capture universal quantum-critical transport and the short relaxation times characteristic of these strongly interacting 2+1D systems.

Abstract

I present a pedagogical survey of a variety of quantum phases of the Hubbard model. The honeycomb lattice model has a conformal field theory connecting the semi-metal to the insulator with Neel order. States with fractionalized excitations are linked to the deconfined phases of gauge theories. I also consider the confining phases of such gauge theories, and show how Berry phases of monopoles induce valence bond solid order. The triangular lattice model can display a metal-insulator transition from a Fermi liquid to a deconfined spin liquid, and I describe the theory of this transition. The bilayer triangular lattice is used to illustrate another compressible metallic phase, the `fractionalized Fermi liquid'. I make numerous connections of these phases and critical points to the AdS/CFT correspondence. In particular, I argue that two recent holographic constructions connect respectively to the Fermi liquid and fractionalized Fermi liquid phases.

The landscape of the Hubbard model

TL;DR

The article provides a comprehensive, gauge-theoretic survey of the Hubbard model across lattices and densities, focusing on Dirac-like low-energy physics on the honeycomb lattice and the emergence of symmetry-broken and topologically ordered phases. It develops and interconnects multiple theoretical frameworks—Gross-Neveu–Yukawa criticality, U(1) and SU(2) gauge theories, and rotor/ spinon decompositions—to describe semi-metals, Néel and VBS insulators, spin liquids, and fractionalized Fermi liquids, and extends these insights to nonzero temperature with AdS/CFT-inspired transport predictions. A central thrust is the unification of conventional and exotic phases via emergent gauge fields and deconfined criticality, including a deconfined QCD-like fixed point and a lattice-symmetry–driven VBS order. The work further links condensed-mmatter gauge theories to holographic metal constructions, illustrating how holography can capture universal quantum-critical transport and the short relaxation times characteristic of these strongly interacting 2+1D systems.

Abstract

I present a pedagogical survey of a variety of quantum phases of the Hubbard model. The honeycomb lattice model has a conformal field theory connecting the semi-metal to the insulator with Neel order. States with fractionalized excitations are linked to the deconfined phases of gauge theories. I also consider the confining phases of such gauge theories, and show how Berry phases of monopoles induce valence bond solid order. The triangular lattice model can display a metal-insulator transition from a Fermi liquid to a deconfined spin liquid, and I describe the theory of this transition. The bilayer triangular lattice is used to illustrate another compressible metallic phase, the `fractionalized Fermi liquid'. I make numerous connections of these phases and critical points to the AdS/CFT correspondence. In particular, I argue that two recent holographic constructions connect respectively to the Fermi liquid and fractionalized Fermi liquid phases.

Paper Structure

This paper contains 17 sections, 122 equations, 16 figures.

Table of Contents

  1. Introduction
  2. Semi-metal and antiferromagnetism on the honeycomb lattice
  3. Preliminaries
  4. Semi-metal
  5. Antiferromagnet
  6. Quantum phase transition
  7. Quantum impurities
  8. Electrical transport
  9. Non-zero temperatures and the AdS/CFT correspondence
  10. Quantum critical transport
  11. The AdS/CFT correspondence
  12. U(1) gauge theory and the valence bond solid on the honeycomb lattice
  13. SU(2) gauge theory and a phase diagram for the half-filled honeycomb lattice
  14. The Metal-insulator transition on the triangular lattice
  15. Fractionalized Fermi liquids
  16. ...and 2 more sections

Figures (16)

  • Figure 1: The honeycomb lattice with its A (green) and B (red) sublattices
  • Figure 2: The first Brillouin zone of the honeycomb lattice.
  • Figure 3: The lower band of the dispersion in Eq. (\ref{['spec']}) for $\mu=0$
  • Figure 4: The large $U$ state with antiferromagnetic (Néel) order.
  • Figure 5: The honeycomb lattice with a vacancy.
  • ...and 11 more figures