The past decade has seen a substantial rejuvenation of interest in the study of quantum phase transitions, driven by experiments on cuprate superconductors, heavy fermion materials, organic conductors, and related compounds. Although quantum phase transitions in simple spin systems, like the Ising model in a transverse field, were studied in the early 1970s, much of the subsequent theoretical work examined a particular example: the metal–insulator transition. While this is a subject of considerable experimental importance, the greatest theoretical progress was made for the case of the Anderson transition of non-interacting electrons, which is driven by localization of the electronic states in the presence of a random potential. The critical properties of this transition of noninteracting electrons constituted the primary basis upon which most condensed matter physicists have formed their intuition on the behavior of the systems near a quantum phase transition. However, it is clear that strong electronic interactions play a crucial role in the systems of current interest noted earlier, and simple paradigms for the behavior of such systems near quantum critical points are not widely known. It is the purpose of this book to move interactions to center stage by describing and classifying the physical properties of the simplest interacting systems undergoing a quantum phase transition. The effects of disorder will be neglected for the most part but will be considered in the concluding chapters. Our focus will be on the dynamical properties of such systems at nonzero temperature, and it will become apparent that these differ substantially from the noninteracting case.
Two of the iconic phases of the hole-doped cuprate materials are the intermediate temperature pseudogap metal and the lower temperature
d-wave superconductor. Following the suggestion of P. W. Anderson, there were early theories of these phases as doped quantum spin liquids. However, these theories have had difficulties with two prominent observations: (i) angle-dependent magnetoresistance measurements (ADMR) in the pseudogap metal, including observation of the Yamaji effect, present convincing evidence of small hole pockets which can tunnel coherently between square lattice layers (Fang et al., Nature Physics 18, 558 (2022); Chan et al., Nature Physics 21, 1753 (2025)) and (ii) the velocities of the nodal Bogoliubov quasiparticles in the
d-wave superconductor are highly anisotropic, with
vF≫vΔ (Chiao et al., Phys. Rev. B 62, 3554 (2000)). These notes review how the fractionalized Fermi Liquid (FL*) state, which dopes quantum spin liquids with gauge-neutral electron-like quasiparticles, resolves both difficulties. Theories of insulating quantum spin liquids employing fractionalization of the electron spin into bosonic or fermionic partons are discussed. Doping the bosonic parton theory leads to a holon metal theory: while not appropriate for the cuprate pseudogap, this theory is argued to apply to ultracold atom experiments on the Lieb lattice. Doping the fermionic parton theory leads to a
d-wave superconductor with nearly isotropic quasiparticle velocities. The construction of the FL* state in a single band Hubbard-type model is described using a quantum dimer model, followed by a more realistic description using the Ancilla Layer Model (ALM), which is then used to obtain the theory of the pseudogap and the
d-wave superconductor. The ALM also leads to a wavefunction for the FL* state of the Hubbard model, which agrees with ultracold atom observations.
Spin and charge are two fundamental degrees of freedom in strongly correlated electron systems. In this thesis we study low-energy effective theory and phase transitions of spin and charge in a variety of strongly correlated systems. First, we study effective theory of magnetic phase transition in Sp(4) spin system and iron-based superconductors. Quantum phase transitions beyond Landau paradigm has been proposed in SU(2) spill model. We generalize this paradigm to Sp(4) model, which can be realized using spin-[special characters omitted] cold atoms on optical lattice. In iron-based superconductors, we study low-energy effective theory for magnetic and structural phase transitions, and present a global phase diagram of the two phase transitions. Second, we study spin and charge fluctuations on triangular lattice, and compare our results with experiments on organic Mott insulator κ-(ET) 2(CN)2. We show that the spin liquid state observed in experiments can be explained by Z2 spin liquid theory which contains spin-½ spinon and Z2 vison excitations. We also argue that near the Mott-insulator-to-metal transition there are fractionalized spinless charge excitations, which form exciton bound states near the transition. At last, we study spin and charge excitations in high-T c cuprates. We describe fluctuating two-dimensional metallic antiferromagnets by a gauge-theoretic description of an 'algebraic charge liquid' involving spinless fermions and a spin S = 1/2 complex scalar. We propose a phenomenological effective lattice Hamiltonian which describes the binding of these particles into gauge-neutral, electron-like excitations, and describe its implications for the electron spectral function across the entire Brillouin zone. We discuss connections of our results to photoemission experiments in the pseudogap regime of the cuprate superconductors.
This modern text describes the remarkable developments in quantum condensed matter physics following the experimental discoveries of quantum Hall effects and high temperature superconductivity in the 1980s. After a review of the phases of matter amenable to an independent particle description, entangled phases of matter are described in an accessible and unified manner. The concepts of fractionalization and emergent gauge fields are introduced using the simplest resonating valence bond insulator with an energy gap, the Z2 spin liquid. Concepts in band topology and the parton method are then combined to obtain a large variety of experimentally relevant gapped states. Correlated metallic states are described, beginning with a discussion of the Kondo effect on magnetic impurities in metals. Metals without quasiparticle excitations are introduced using the Sachdev-Ye-Kitaev model, followed by a discussion of critical Fermi surfaces and strange metals. Numerous end-of-chapter problems expand readers' comprehension and reinforce key concepts.