Luttinger Theorem Research Articles

ZN lattice gauge theories with matter fields

Motivated by the exotic phenomenology of certain quantum materials and recent advances in programmable quantum emulators, we here study fermions and bosons in ZN lattice gauge theories. We introduce a family of exactly soluble models, and characterize their orthogonal (semi)metallic ground states, the excitation spectrum, and the correlation functions. We further study integrability-breaking perturbations using an appropriately derived set of Feynman diagrammatic rules and borrowing physics associated to Anderson's orthogonality catastrophe. In the context of the ground states, we revisit Luttinger's theorem following Oshikawa's flux insertion argument and furthermore demonstrate the existence of a Luttinger surface of zeros in the fermionic Green's function. Upon inclusion of perturbations, we address the transition from the orthogonal metal to the normal state by condensation of certain excitations in the gauge sectors, so-called “e particles.” We furthermore discuss the effect of dynamics in the dual “m-particle” excitations, which ultimately leads to the formation of charge-neutral hadronic N-particle bound states. We present analytical arguments for the most important phases and estimates for phase boundaries of the model. Specifically, and in sharp distinction to quasi-one-dimensional ZN lattice gauge theories, renormalization group arguments imply that the phase diagram does not include an emergent deconfining U(1) phase in the limit of large number of fermion flavors. Therefore, in regards to lattice QED problems, ZN quantum emulators with N<∞ can at best be used for approximate solutions at intermediate length scales. Published by the American Physical Society 2024

Bands renormalization and superconductivity in the strongly correlated Hubbard model using composite operators method

We use the composite operator method (COM) to analyze the strongly correlated repulsive Hubbard model, investigating the effect of nearest-neighbor hoppings up to fourth order on a square lattice. We consider two sets of self-consistent equations, one enforcing the Pauli principle and the other imposing charge-charge, spin-spin, and pair–pair correlations using a decoupling scheme developed by Roth (1969 Phys. Rev. 184 451–9). We extract three distinct solutions from these equations: COM1 and COM2 by imposing the Pauli principle and one from Roth decoupling. An overview of the method studying the validity of particle-hole symmetry and the Luttinger theorem for each solution is presented. Additionally, we extend the initial basis to study superconductivity, concluding that it is induced by the Van Hove singularity. Finally, we include higher-order hoppings using realistic estimates for tight binding parameters and compare our results with ARPES measurements on cuprates.

FL* Approach to the Coexistence of Fermi Arcs with Metal–Insulator Crossover in Strongly Underdoped Cuprates

We propose that one can explain the coexistence in the same range of doping and temperature of gapless Fermi arcs with the metal–insulator crossover of in-plane resistivity in strongly underdoped cuprates in terms of the FL* fractionalized Fermi liquid nature of these systems, and that such coexistence is not due simply to disorder effects in the resistivity. The particle excitations of this FL* system derived from variants of the t-J model are the gapless holon carrying charge with small Fermi momentum proportional to the doping, the gapful spinon carrying spin 1/2, and an emergent gauge field coupling them and the hole as a spinon–holon bound state, or more precisely resonance, due to gauge binding, with a Fermi surface respecting the topological Luttinger theorem. In our proposal, Fermi arcs are determined by the hole resonance, whereas the metal–insulator crossover is dominated by spinon–spinon (with subleading holon–holon) gauge interactions, and this dichotomy is able to explain their coexistence.

Zero sound and higher-form symmetries in compressible holographic phases

Certain holographic states of matter with a global U(1) symmetry support a sound mode at zero temperature, caused neither by spontaneous symmetry breaking of the global U(1) nor by the emergence of a Fermi surface in the infrared. In this work, we show that such a mode is also found in zero density holographic quantum critical states. We demonstrate that in these states, the appearance of a zero temperature sound mode is the consequence of a mixed ‘t Hooft anomaly between the global U(1) symmetry and an emergent higher-form symmetry. At non-zero temperatures, the presence of a black hole horizon weakly breaks the emergent symmetry and gaps the collective mode, giving rise to a sharp Drude-like peak in the electric conductivity. A similar gapped mode arises at low temperatures for non-zero densities when the state has an emergent Lorentz symmetry, also originating from an approximate anomalous higher-form symmetry. However, in this case the collective excitation does not survive at zero temperature where, instead, it dissolves into a branch cut due to strong backreaction from the infrared, critical degrees of freedom. We comment on the relation between our results and the application of the Luttinger theorem to compressible holographic states of matter.

Connecting the Many-Body Chern Number to Luttinger's Theorem through Středa's Formula.

Relating the quantized Hall response of correlated insulators to many-body topological invariants is a key challenge in topological quantum matter. Here, we use Středa's formula to derive an expression for the many-body Chern number in terms of the single-particle interacting Green's function and its derivative with respect to a magnetic field. In this approach, we find that this many-body topological invariant can be decomposed in terms of two contributions, N_{3}[G]+ΔN_{3}[G], where N_{3}[G] is known as the Ishikawa-Matsuyama invariant and where the second term involves derivatives of Green's function and the self-energy with respect to the magnetic perturbation. As a by-product, the invariant N_{3}[G] is shown to stem from the derivative of Luttinger's theorem with respect to the probe magnetic field. These results reveal under which conditions the quantized Hall conductivity of correlated topological insulators is solely dictated by the invariant N_{3}[G], providing new insight on the origin of fractionalization in strongly correlated topological phases.

Lieb-Schultz-Mattis, Luttinger, and 't Hooft - anomaly matching in lattice systems

We analyze lattice Hamiltonian systems whose global symmetries have ’t Hooft anomalies. As is common in the study of anomalies, they are probed by coupling the system to classical background gauge fields. For flat fields (vanishing field strength), the nonzero spatial components of the gauge fields can be thought of as twisted boundary conditions, or equivalently, as topological defects. The symmetries of the twisted Hilbert space and their representations capture the anomalies. We demonstrate this approach with a number of examples. In some of them, the anomalous symmetries are internal symmetries of the lattice system, but they do not act on-site. (We clarify the notion of “on-site action.”) In other cases, the anomalous symmetries involve lattice translations. Using this approach we frame many known and new results in a unified fashion. In this work, we limit ourselves to 1+1d systems with a spatial lattice. In particular, we present a lattice system that flows to the c=1c=1 compact boson system with any radius (no BKT transition) with the full internal symmetry of the continuum theory, with its anomalies and its T-duality. As another application, we analyze various spin chain models and phrase their Lieb-Shultz-Mattis theorem as an ’t Hooft anomaly matching condition. We also show in what sense filling constraints like Luttinger theorem can and cannot be viewed as reflecting an anomaly. As a by-product, our understanding allows us to use information from the continuum theory to derive some exact results in lattice model of interest, such as the lattice momenta of the low-energy states.

Nonlocal Kondo effect and two-fluid picture revealed in an exactly solvable model.

Understanding the nature of local-itinerant transition of strongly correlated electrons is one of the central problems in condensed matter physics. Heavy fermion systems describe the f-electron delocalization through Kondo interactions with conduction electrons. Tremendous efforts have been devoted to the so-called Kondo-destruction scenario, which predicts a dramatic local-to-itinerant quantum phase transition of f-electrons at zero temperature. On the other hand, two-fluid behaviors have been observed in many materials, suggesting coexistence of local and itinerant f-electrons over a broad temperature range but lacking a microscopic theoretical description. To elucidate this fundamental issue, here we propose an exactly solvable Kondo-Heisenberg model in which the spins are defined in the momentum space and the -space Kondo interaction corresponds to a highly nonlocal spin scattering in the coordinate space. Its solution reveals a continuous evolution of the Fermi surfaces with Kondo interaction and two-fluid behaviors similar to those observed in real materials. The electron density violates the usual Luttinger's theorem, but follows a generalized one allowing for partially enlarged Fermi surfaces due to partial Kondo screening in the momentum space. Our results highlight the consequence of nonlocal Kondo interaction relevant for strong quantum fluctuation regions and provide important insight into the microscopic description of two-fluid phenomenology in heavy fermion systems.

Violation of Luttinger's theorem in the simplest doped Mott insulator: Falicov-Kimball model in the strong-correlation limit

The Luttinger's theorem has long been taken as the key feature of Landau's Fermi liquid, which signals the presence of quasiparticles. Here, by the unbiased Monte Carlo method, violation of Luttinger's theorem is clearly revealed in the Falicov-Kimball (FK) model, indicating the robust correlation-driven non-Fermi liquid characteristic under any electron density. Introducing hole carriers to the half-filled FK leads to Mott insulator-metal transition, where the Mott quantum criticality manifests unconventional scaling behavior in transport properties. Further insight on the violation of the Luttinger's theorem is examined by combining Hubbard-I approximation with a composite fermion picture, which emphasizes the importance of a mixed excitation of the itinerant electron and the composite fermion. Interestingly, when compared FK model with a binary disorder system, it suggests that the two-peak band structure discovered by Monte Carlo and Hubbard-I approaches is underlying the violation of Luttinger's theorem.

Luttinger's theorem in the presence of Luttinger surfaces

Breakdown of Landau's hypothesis of adiabatic continuation from noninteracting to fully interacting electrons is commonly believed to bring about a violation of Luttinger's theorem. Here, we elucidate what may go wrong in the proof of Luttinger's theorem. The analysis provides a simple way to correct Luttinger's expression of the electron number in single-band models where perturbation theory breaks down through the birth of a Luttinger surface without symmetry breaking. In those cases, we find that the Fermi volume only accounts for the doping away from half-filling. In the hypothetical circumstance of a non-symmetry-breaking Mott insulator with a Luttinger surface, our analysis predicts the noteworthy existence of quasiparticles whose Fermi surface is just the Luttinger one. Therefore, those quasiparticles can be legitimately regarded as spinons, and the Mott insulator with a Luttinger surface as realization of a spin-liquid insulator.

Broken Luttinger theorem in the two-dimensional Fermi-Hubbard model

One of the fundamental questions about high-temperature cuprate superconductors is the size of the Fermi surface underlying the superconducting state. By analyzing the single-particle spectral function for the Fermi-Hubbard model as a function of repulsion $U$ and chemical potential $\ensuremath{\mu}$, we find that the Fermi surface in the normal state undergoes a transition from a large Fermi surface matching the Luttinger volume as expected in a Fermi liquid, to a Fermi surface that encloses fewer electrons that we dub the ``Luttinger breaking'' phase, as the Mott insulator is approached. This transition into a non-Fermi-liquid phase that violates the Luttinger count occurs at a critical density in the absence of any other broken symmetry. We obtain the Fermi-surface contour from the spectral weight ${A}_{\mathbf{k}}(\ensuremath{\omega}=0)$ and from an analysis of the singularities of the Green's function $\mathrm{Re}\phantom{\rule{0.16em}{0ex}}{G}_{\mathbf{k}}(E=0)$, calculated using determinantal quantum Monte Carlo and analytic continuation methods. We discuss our numerical results in connection with experiments on Hall measurements, scanning tunneling spectroscopy, and angle-resolved photoemission spectroscopy.

Emergent anomalies and generalized Luttinger theorems in metals and semimetals

Whether a given material is a metal or an insulator is determined by a single parameter, the number of electrons per unit cell per spin, that is, by the filling number. When the latter is integer, the material is an insulator or a compensated semimetal. When it is fractional, the material is a metal with a Fermi surface, enclosing a volume in momentum space, determined by the fractional part of the filling -- a statement known as Luttinger's theorem. Topological semimetals provide a realization of the metallic phase, distinct from ordinary metals, as they exist at integer fillings, which normally correspond to insulators. Here, the authors develop a framework that unifies Luttinger's theorem for ordinary metals and analogous exact statements for topological semimetals. They use the language of quantum anomalies of the emergent low-energy charge conservation symmetry at each point on the Fermi surface. This framework allows one to connect topological responses in all nodal semimetals to their low-energy properties, such as the distance between the point nodes in momentum space or the area, enclosed by the nodal line.

Doping a Mott insulator in an Ising-Kondo lattice: Strange metal and Mott criticality

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-${T}_{c}$ cuprate superconductors and organic superconductors to twisted multilayer graphene and ${\mathrm{MoTe}}_{2}/{\mathrm{WSe}}_{2}$ 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.

Fermi arcs and pseudogap in a lattice model of a doped orthogonal metal

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.

Non-Fermi Liquids as Ersatz Fermi Liquids: General Constraints on Compressible Metals

A system with charge conservation and lattice translation symmetry has a well-defined filling $\nu$, which is a real number representing the average charge per unit cell. We show that if $\nu$ is fractional (i.e. not an integer), this imposes very strong constraints on the low-energy theory of the system and give a framework to understand such constraints in great generality, vastly generalizing the Luttinger and Lieb-Schultz-Mattis theorems. The most powerful constraint comes about if $\nu$ is continuously tunable (i.e. the system is charge-compressible), in which case we show that the low-energy theory must have a very large emergent symmetry group -- larger than any compact Lie group. An example is the Fermi surface of a Fermi liquid, where the charge at every point on the Fermi surface is conserved. We expect that in many, if not all, cases, even exotic non-Fermi liquids will have the same emergent symmetry group as a Fermi liquid, even though they could have very different dynamics. We call a system with this property an "ersatz Fermi liquid". We show that ersatz Fermi liquids share a number of properties in common with Fermi liquids, including Luttinger's theorem (which is thus extended to a large class of non-Fermi liquids) and periodic "quantum oscillations" in the response to an applied magnetic field. We also establish versions of Luttinger's theorem for the composite Fermi liquid in quantum Hall systems and for spinon Fermi surfaces in Mott insulators. Our work makes connection between filling constraints and the theory of symmetry-protected topological (SPT) phases, in particular through the concept of " 't Hooft anomalies".

Electric polarization as a nonquantized topological response and boundary Luttinger theorem

We develop a nonperturbative approach to the bulk polarization of crystalline electric insulators in d≥1 dimensions. Formally, we define polarization via the response to background fluxes of both charge and lattice translation symmetries. In this approach, the bulk polarization is related to properties of magnetic monopoles under translation symmetries. Specifically, in 2D, the monopole is a source of 2π flux, and the polarization is determined by the crystal momentum of the 2π flux. In 3D, the polarization is determined by the projective representation of translation symmetries on Dirac monopoles. Our approach also leads to a concrete scheme to calculate polarization in 2D, which in principle can be applied even to strongly interacting systems. For open boundary conditions, the bulk polarization leads to an altered boundary Luttinger theorem (constraining the Fermi surface of surface states) and also to modified Lieb-Schultz-Mattis theorems on the boundary, which we derive.Received 22 February 2021Accepted 5 March 2021DOI:https://doi.org/10.1103/PhysRevResearch.3.023011Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.Published by the American Physical SocietyPhysics Subject Headings (PhySH)Research AreasDefectsElectric polarizationSurface & interfacial phenomenaTechniquesEffective field theoryGauge theory techniquesCondensed Matter, Materials & Applied Physics

Lieb-Schultz-Mattis-type theorems and other nonperturbative results for strongly correlated systems with conserved dipole moments

Non-perturbative constraints on many body physics--such as the famous Lieb-Schultz-Mattis theorem--are valuable tools for studying strongly correlated systems. To this end, we present a number of non-perturbative results that constrain the low-energy physics of systems having conserved dipole moments. We find that for these systems, a unique translationally invariant gapped ground state is only possible if the polarization of the system is integer. Furthermore, if a lattice system also has $U(1)$ subsystem charge conservation symmetry, a unique gapped ground state is only possible if the particle filling along these subsystems is integer. We also apply these methods to spin systems, and determine criteria for the existence of a new type of magnetic response plateau in the presence of a non-uniform magnetic field. Finally, we formulate a version of Luttinger's theorem for 1D systems consisting of dipoles.

Visualizing the Kondo lattice crossover in YbRh2Si2 with Compton scattering

In a Kondo lattice, the Luttinger theorem requires the $f$ electron degree of freedom to enter the Fermi volume in the strong-coupling regime, while the $f$ electron remains nearly localized. A common hypothesis assumes that the Fermi volume shrinks with increasing temperature when the $f$ electron decouples from the Fermi surface with the breakdown of the Kondo lattice effect. However, experimental evidence for this phenomen is still very scarce. Here, the authors demonstrate the reconstruction of the Fermi surface between 14 and 300 K from $l\phantom{\rule{0}{0ex}}a\phantom{\rule{0}{0ex}}r\phantom{\rule{0}{0ex}}g\phantom{\rule{0}{0ex}}e$ to $s\phantom{\rule{0}{0ex}}m\phantom{\rule{0}{0ex}}a\phantom{\rule{0}{0ex}}l\phantom{\rule{0}{0ex}}l$ in the prototypical Kondo lattice YbRh${}_{2}$Si${}_{2}$ by Compton scattering.

Fractional Fermi liquid in a generalized t−J model

Inspired by the recent discovery of superconductivity in the nickelate Nd$_{1-x}$Sr$_x$NiO$_2$, we study a generalized $t-J$ model to investigate the correlated phases induced by doping spin-one Ni$^{2+}$ into a spin $1/2$ Mott insulator formed by Ni$^{1+}$. Based on a three-fermion parton mean field analysis, we identify a robust fractional Fermi liquid (FL*) phase for almost every doping level. The FL* state is characterized by a small Fermi pocket on top of a spin liquid, which violates the Luttinger theorem of a conventional Fermi liquid and is an example of a symmetric pseudogap metal. Furthermore, we verify our theory in one dimension through density matrix renormalization group (DMRG) simulations on both the generalized $t-J$ model and a two-orbital Hubbard model. The fractional Fermi liquid reduces to fractional Luttinger liquid (LL*) in one dimension, which is connected to the conventional Luttinger liquid through a continuous quantum phase transition by tuning interaction strength. Our findings offer new insights into correlated electron phenomena in nickelate superconductors and other multi-orbital transition metal oxide with spin-triplet $d^8$ state.

Evolution of the Kondo lattice electronic structure above the transport coherence temperature

The temperature-dependent evolution of the Kondo lattice is a long-standing topic of theoretical and experimental investigation and yet it lacks a truly microscopic description of the relation of the basic f-c hybridization processes to the fundamental temperature scales of Kondo screening and Fermi-liquid lattice coherence. Here, the temperature dependence of f-c hybridized band dispersions and Fermi-energy f spectral weight in the Kondo lattice system CeCoIn5 is investigated using f-resonant angle-resolved photoemission spectroscopy (ARPES) with sufficient detail to allow direct comparison to first-principles dynamical mean-field theory (DMFT) calculations containing full realism of crystalline electric-field states. The ARPES results, for two orthogonal (001) and (100) cleaved surfaces and three different f-c hybridization configurations, with additional microscopic insight provided by DMFT, reveal f participation in the Fermi surface at temperatures much higher than the lattice coherence temperature, [Formula: see text] K, commonly believed to be the onset for such behavior. The DMFT results show the role of crystalline electric-field (CEF) splittings in this behavior and a T-dependent CEF degeneracy crossover below [Formula: see text] is specifically highlighted. A recent ARPES report of low T Luttinger theorem failure for CeCoIn5 is shown to be unjustified by current ARPES data and is not found in the theory.

Landau Quasiparticles in Weak Power-Law Liquids

The failure of Landau-Fermi liquid theory is often considered a telltale sign of universal, scale-invariant behavior in the emergent field theory of interacting fermions. Nevertheless, there exist borderline cases where weak scale invariance coupled with particle-hole asymmetry can coexist with the Landau quasiparticle paradigm. In this letter, I show explicitly that a Landau-Fermi liquid can exist for weak power-law scaling of the retarded Green's function. Such an exotic variant of the traditional Fermi liquid, although exhibiting a finite quasiparticle weight and large quasiparticle lifetime, is shown to always be incompatible with Luttinger's theorem for any non-trivial scaling. This result yields evidence for a Fermi liquid-like ground state in the high-field, underdoped pseudogap phase of the high-temperature cuprate superconductors.