Since the discovery of the pseudogap and Fermi arc states in underdoped cuprates, the understanding of such non-Fermi-liquid states and the associated violation of Luttinger's theorem have been the central theme in correlated electron systems. However, still lacking is a well-accepted theoretical framework to unambiguously explain these metallic states that are clearly beyond Landau's Fermi liquid and Luttinger's theorem of a Fermi surface and electron filling. Here, we design a lattice model of orthogonal metals with fermion and Ising matter fields coupled to topological order and, by solving the model via unbiased quantum Monte Carlo simulation at generic electron fillings, find that the system gives birth to phenomena of the Fermi arc and pseudogap in the single-particle spectrum that go beyond the Luttinger sum rule with broken Fermi surface but no symmetry breaking. The pseudogap and Fermi arcs coexist with a background of a deconfined ${Z}_{2}$ gauge field, and we further find that the confinement transition of the gauge field triggers a superconductivity instability and that the hopping of the gauge-neutral fermions brings the ``large'' Fermi surface back from the Fermi arc state. Our unbiased numerical results provide a concrete model realization and theoretical framework for the coupling between gauge field and fermions and, in the process, generate the rich phenomena of the pseudogap, the Fermi arc, and superconductivity in generic correlated electron systems.
Since the discovery of the pseudogap and Fermi arc states in underdoped cuprates, the understanding of such non-Fermi-liquid states and the associated violation of Luttinger's theorem have been the central theme in correlated electron systems. However, still lacking is a well-accepted theoretical framework to unambiguously explain these metallic states that are clearly beyond Landau's Fermi liquid and Luttinger's theorem of a Fermi surface and electron filling. Here, we design a lattice model of orthogonal metals with fermion and Ising matter fields coupled to topological order and, by solving the model via unbiased quantum Monte Carlo simulation at generic electron fillings, find that the system gives birth to phenomena of the Fermi arc and pseudogap in the single-particle spectrum that go beyond the Luttinger sum rule with broken Fermi surface but no symmetry breaking. The pseudogap and Fermi arcs coexist with a background of a deconfined Z2 gauge field, and we further find that the confinement transition of the gauge field triggers a superconductivity instability and that the hopping of the gauge-neutral fermions brings the ``large'' Fermi surface back from the Fermi arc state. Our unbiased numerical results provide a concrete model realization and theoretical framework for the coupling between gauge field and fermions and, in the process, generate the rich phenomena of the pseudogap, the Fermi arc, and superconductivity in generic correlated electron systems.
This article is based on a talk by S.S. at the Nambu Memorial Symposium at the University of Chicago. We review ideas on the nature of the metallic states of the hole-doped cuprate high temperature superconductors, with an emphasis on the connections between the Luttinger theorem for the size of the Fermi surface, topological quantum field theories (TQFTs), and critical theories involving changes in the size of the Fermi surface. We begin with the derivation of the Luttinger theorem for a Fermi liquid, using momentum balance during a process of flux-insertion in a lattice electronic model with toroidal boundary conditions. We then review the TQFT of the Z2 spin liquid, and demonstrate its compatibility with the toroidal momentum balance argument. This discussion leads naturally to a simple construction of `topological' Fermi liquid states: the fractionalized Fermi liquid (FL*) and the algebraic charge liquid (ACL). We present arguments for a description of the pseudogap metal of the cuprates using Z2-FL* or Z2-ACL states with Ising-nematic order. These pseudogap metal states are also described as Higgs phases of a SU(2) gauge theory. The Higgs field represents local antiferromagnetism, but the Higgs-condensed phase does not have long-range antiferromagnetic order: the magnitude of the Higgs field determines the pseudogap, the reconstruction of the Fermi surface, and the Ising-nematic order. Finally, we discuss the route to the large Fermi surface Fermi liquid via the critical point where the Higgs condensate and Ising nematic order vanish, and the application of Higgs criticality to the strange metal.
This review describes how topological order can reconstruct Fermi surfaces of metals, even in the absence of translational symmetry breaking. We begin with an introduction to topological order using Wegner's quantum Z2 gauge theory on the square lattice: the topological state is characterized by the expulsion of defects, carrying Z2 magnetic flux. The interplay between topological order and the breaking of global symmetry is described by the non-zero temperature statistical mechanics of classical XY models in dimension D=3; such models also describe the zero temperature quantum phases of bosons with short-range interactions on the square lattice at integer filling. The topological state is again characterized by the expulsion of certain defects, in a state with fluctuating symmetry-breaking order. The phase diagrams of the Z2 gauge theory and the XY models are obtained by embedding them in U(1) gauge theories, and by studying their Higgs and confining phases. These ideas are then applied to the single-band Hubbard model on the square lattice. A SU(2) gauge theory describes the fluctuations of spin-density-wave order, and its phase diagram is presented by analogy to the XY models. We obtain a class of zero temperature metallic states with fluctuating spin-density wave order, topological order associated with defect expulsion, reconstructed Fermi surfaces (with `chargon' or electron-like quasiparticles), but no broken symmetry. We conclude with the application of such metallic states to the pseudogap phase of the cuprates, and note the recent comparison with numerical studies of the Hubbard model. In a detour, we also discuss the influence of Berry phases, and how they can lead to deconfined quantum critical points: this applies to bosons on the square lattice at half-integer filling, and to quantum dimer models.
This review describes how topological order associated with the presence of emergent gauge fields can reconstruct Fermi surfaces of metals, even in the absence of translational symmetry breaking. We begin with an introduction to topological order using Wegner's quantum [Formula: see text] gauge theory on the square lattice: the topological state is characterized by the expulsion of defects, carrying [Formula: see text] magnetic flux. The interplay between topological order and the breaking of global symmetry is described by the non-zero temperature statistical mechanics of classical XY models in dimension D = 3; such models also describe the zero temperature quantum phases of bosons with short-range interactions on the square lattice at integer filling. The topological state is again characterized by the expulsion of certain defects, in a state with fluctuating symmetry-breaking order, along with the presence of emergent gauge fields. The phase diagrams of the [Formula: see text] gauge theory and the XY models are obtained by embedding them in U(1) gauge theories, and by studying their Higgs and confining phases. These ideas are then applied to the single-band Hubbard model on the square lattice. A SU(2) gauge theory describes the fluctuations of spin-density-wave order, and its phase diagram is presented by analogy to the XY models. We obtain a class of zero temperature metallic states with fluctuating spin-density wave order, topological order associated with defect expulsion, deconfined emergent gauge fields, reconstructed Fermi surfaces (with 'chargon' or electron-like quasiparticles), but no broken symmetry. We conclude with the application of such metallic states to the pseudogap phase of the cuprates, and note the recent comparison with numerical studies of the Hubbard model and photoemission observations of the electron-doped cuprates. In a detour, we also discuss the influence of Berry phases, and how they can lead to deconfined quantum critical points: this applies to bosons on the square lattice at half-integer filling, and to quantum dimer models.
Abstract The phenomenon of Mott insulation involves the localisation of itinerant electrons due to strong local repulsion. Upon doping, a pseudogap (PG) phase emerges—marked by selective gapping of the Fermi surface without conventional symmetry breaking in spin or charge channels. A key challenge is understanding how quasiparticle breakdown in the Fermi liquid (FL) gives rise to this enigmatic state, and how it connects to both the Mott insulating and superconducting phases. Here, we develop a renormalisation-based construction of strongly correlated lattice models that captures the emergence of the PG phase and its transition to a Mott insulator. Applying a many-body tiling scheme to the fixed-point impurity model uncovers a lattice model with electron interactions and Kondo physics. At half-filling, the interplay between Kondo screening and bath charge fluctuations in the impurity model leads to FL breakdown. This reveals a PG phase characterised by a non-FL (the Mott metal) residing on nodal arcs, gapped antinodal regions of the Fermi surface, and an anomalous scaling of the electronic scattering rate with frequency. The eventual confinement of holon–doublon excitations of this exotic metal obtains a continuous transition into the Mott insulator. Our results identify the PG as a distinct long-range entangled quantum phase, and offer a new route to Mott criticality beyond the paradigm of local quantum criticality.
Several years ago, there were rumors that metals might have an awkward cousin, whom people are sort of expecting but have never made acquaintance. As though he/she shares metallic features -- conducting -- but is distinctively different and even invisible if one approaches him/her in conventional manners -- acquiring no Fermi surface or quasiparticles. Of course, the real content of such metaphor ammounts to the fact that there exists new quantum metallic state that conducts electricity but is orthogonal to the established paradigm of Landau's Fermi-liquid (FL), and such state might hold the key of understanding the perplexing experimental observations of quantum metals that are beyond FL -- dubbed non-Fermi-liquid (nFL) -- ranging from the Cu- and Fe-based oxides, heavy fermion compounds to the recently discovered twisted graphene heterostructures. However, to realize such exotic state of matter, at least theoretically, one would need to construct a lattice model and solve it with unbiased quantum many-body machinery. Here, we achieve this goal by designing a 2D lattice model comprised of matter fields coupled with dynamic Z2 gauge fields, and studying it with sign-free quantum Monte Carlo simulations. We find as the bosonic matter fields become disordered, the system reacts with changing its nature from the conventional normal metal of FL with proper Fermi surface to an orthogonal metal of nFL without Fermi surface and quasiparticle and yet still responds to external perturbation with hidden fermionic degrees of freedom. Such a quantum phase transition from a normal metal to an orthogonal metal has never been seen before and certainly violated the Luttinger's theorem. These results are calling for the establishment of new paradigm of quantum metals and their transition with conventional ones. Rumor turns out to be true, as always, that metals indeed have an awkaward cousin.
Metallic quantum states coined strange metal (SM), with robust linear-T resistivity, have been widely observed in many quantum materials under strong electron correlations, ranging from high-Tc cuprate superconductors and organic superconductors to twisted multilayer graphene and MoTe2/WSe2 superlattices. Despite decades of intensive studies, the mystery of SM still defies any sensible theoretical explanation and has been a key puzzle in modern condensed matter physics. Here, we solve a doped Mott insulator model called an Ising-Kondo lattice, which includes static spin fluctuation. With Monte Carlo simulation, the Ising-Kondo lattice unambiguously exhibits SM phenomena accompanied with quantum critical scaling in observables, e.g., resistivity, susceptibility, and specific heat. A closer look at SM reveals the breakdown of Landau's Fermi liquid theory without any symmetry breaking, i.e., the violation of Luttinger theorem. Our paper reveals that the long-overlooked static fluctuations in literature may play an essential role in non-Fermi liquid behaviors in correlated electron systems.