Some Significant Publications

Eduardo Fradkin and Leonard Susskind, Order and disorder in gauge systems and magnets , Physical Review D 17, 2637 (1978).

We show how phase transitions in Abelian two-dimensional spin and four-dimensional gauge systems can be understood in terms of condensation of topological objects. In the spin systems these objects are kinks and in the gauge systems either magnetic monopoles or fluxoids (quantized lines of magnetic flux). Four models are studied: two-dimensional Ising and \( XY \) models and four-dimensional \( \mathbb{Z}_2\) and \( U(1) \) gauge systems.

Eduardo Fradkin and Stephen H. Shenker, Phase diagrams of lattice gauge theories with Higgs fields, Physical Review D 19, 3682 (1979).

We study the phase diagram of lattice gauge theories coupled to fixed-length scalar (Higgs) fields. We consider several gauge groups: \(\mathbb{Z}_2\), \(U(1) \), and \(SU(N)\). We find that when the Higgs fields transform like the fundamental representation of the gauge group the Higgs and confining phases are smoothly connected, i.e., they are not separated by a phase boundary. When the Higgs fields transform like some representation other than the fundamental, a phase boundary may exist. This is the case for \(SU(N)\) with all the Higgs fields in the adjoint representation and for\( U(1)\) with all the Higgs fields in the charge-\(N\) \( (N>1) \) representation. We present an argument due to Wegner that indicates the stability of the pure gauge transition. Another phase, free charge or Coulomb, is generally present. In this regime, the spectrum of the theory contains massless gauge bosons (for continuous groups) and finite-energy states that represent free charge

Eduardo Fradkin, Mark Srednicki and Leonard Susskind, Fermion representations for the \( \mathbb{Z}_2\) lattice gauge theory in 2+1 dimensions, Physical Review D 21, 2885 (1980).

It is shown that the \( \mathbb{Z}_2 \) lattice gauge theory in three Euclidean dimensions is equivalent to a theory of locally interacting fermions. A Hamiltonian picture and a Lagrangian picture are derived and their relationship is discussed. The equivalent Fermi theory turns out to describe a theory of surfaces.

Eduardo Fradkin and Leo P. Kadanoff, Disorder variables and parafermions in two dimensional statistical mechanics, Nuclear Physics B 170, 1 (1980).

It is shown that "clock" type models in two-dimensional statistical mechanics possess order and disorder variables \( \varphi_n\; \) and \( \chi_m\; \) with \(n\) and \(m \) falling in the range \(1,2, \ldots, p\). These variables respectively describe abelian analogs to charged fields and the fields of 't Hooft monopoles with charges \(q = n/p \) and topological quantum number \(m \). They are related to one another by a dual symmetry. Products of these operators generate, via a short-distance expansion, para-fermion operators in which rotational symmetry and the internal symmetry group are tied together. The clock models in two dimensions are shown to be an ideal laboratory where these ideas have a very simple realization.

Eduardo Fradkin and Jorge E. Hirsch, Phase diagram of one-dimensional electron-phonon systems. I: The SSH model, Physical Review B 27, 1680 (1983).

We study the nature of the ground state of the Su-Schrieffer-Heeger model for electron-phonon interactions in one dimension in the half-filled-band case. We consider the cases of spinless electrons \( (n=1) \) and spin-1/2 electrons \( (n=2) \), and discuss the stability of the Peierls-dimerized ground state as a function of the ionic mass and electron-phonon coupling constant. We first consider the zero-mass limit of the theory and extend our results to finite mass using renormalization-group arguments. For spinless electrons, it is found that quantum fluctuations destroy the long-range dimerization order for the small electron-phonon coupling constant if the ionic mass is finite. For spin1/2 electrons, the system is dimerized for an arbitrary coupling constant and phonon frequency. Renormalization-group trajectories show that the low-energy behavior of the system is governed by the zero-mass limit of the theory, an n-component Gross-Neveu model. Monte Carlo simulations are performed for the model at finite phonon frequencies, and the results are compared with the static limit result. We study in particular the set of parameters appropriate for polyacetylene and find a 15 % reduction in the phonon order parameter due to fluctuations of the phonon field. A finite-size scaling analysis of the numerical data for the cases \( n=1\) and \(2\) is performed, which confirms the results obtained from the renormalization-group analysis.

Eduardo Fradkin, Roughening transition in quantum interfaces, Physical Review B 28, 5338(R) (1983).

The roughening transition in interfaces of quantum crystals and spin systems is studied. Quantum-mechanical solid-on-solid-like models are presented. I show that quantum interfaces of three-dimensional systems are generally smooth at zero temperature and display a classical roughening transition. In two dimensions, systems with discrete symmetry display a roughening transition at zero temperature.

Eduardo Fradkin, The \( N \)-color Ashkin-Teller Model in two dimensions: Solution in the large-\(N \) limit, Physical Review Letters 53, 1967 (1984).

The \(N\)-color Ashkin-Teller model is solved exactly in the large-\(N\) limit in two dimensions. The phase diagram is found. It is shown that for positive four-spin coupling the transition is first order while for negative four-spin coupling the transition is continuous and Ising type. The specific heat near the second-order transition is calculated and it is found to be finite at the transition because of large corrections to scaling. The latent heat and correlation length at the first-order transition are also calculated.

Eduardo Fradkin, Critical behavior of disordered degenerate semiconductors, I: Models, symmetries and formalism, Physical Review B 33, 3257 (1986}).

A model that describes the qualitative properties of the electronic states of a disordered degenerate semiconductor with a finite number of degeneracy points is proposed. I introduce an effective Hamiltonian of the form of a Dirac operator coupled to randomly distributed fields. It is shown that there is a phase transition between the semimetal and metallic phases followed by a localization transition. The symmetry breaking associated with this transition is related to the non-symmorphic character of the space group. The density of states plays the role of the order parameter and the elastic mean free path is the correlation length. A path-integral representation is introduced and used to characterize the universality class of this transition. The lower critical dimension is 2. A mapping of the two-dimensional case to one-dimensional self-interacting Fermi systems is presented. Applications to zero-gap semiconductors and other systems are discussed.

Eduardo Fradkin, Critical behavior of disordered degenerate semiconductors, II: Spectrum and transport properties in mean-field-theory, Physical Review B 33, 3263 (1986).

The critical behavior of disordered degenerate semiconductors is studied within a mean-field theory valid when the number of degeneracy points is large. I show that above two dimensions there is a semimetal-metal transition at a critical impurity concentration. The mean free path and the one-particle density of states exhibit scaling behavior with universal exponents. The transition is smeared at nonzero temperature. An equation of state, relating temperature, disorder, and bare conductivity, is presented. In two dimensions, the semi-metallic phase is unstable. I show that a localization transition follows except in two dimensions where all states are localized. The bare conductivity appears to be a universal number in two dimensions. Applications to zero-gap semiconductors and other systems are discussed.

Eduardo Fradkin, Elbio Dagotto and Daniel Boyanovsky, Physical realization of the Parity Anomaly in Condensed Matter Physics, Physical Review Letters 57, 2967 (1986).

We show that a PbTe-type narrow-gap semiconductor with an antiphase boundary (or domain wall) has currents of abnormal parity and induced fractional charges. A model is introduced which reduces the problem to the physics of a Dirac equation with a soliton in background electric and magnetic fields. We show that this system is a physical realization of the parity anomaly.

Elbio Dagotto, Eduardo Fradkin and Adriana Moreo, \(SU(2) \) gauge invariance and order parameters in strongly coupled electronic systems, Physical Review B 38, 2926 (R) (1988).

We show that the spin-1/2 Heisenberg model (the strong-coupling limit of the Hubbard model) is also the strong-coupling limit of an \(SU(2)\) lattice gauge theory with fermions. The local \( SU(2)\) gauge symmetry is manifest. The role of this gauge invariance is investigated in both the Hamiltonian and path-integral formulations. Off half-filling, our results reveal the existence of a lattice \(SU(2)\) bosonic matrix field which is a natural candidate for a condensate order parameter.

Eduardo Fradkin and Michael Stone, Topological terms in one- and two-dimensional quantum Heisenberg antiferromagnets, Physical Review B 38, 2926 (R) (1988).

We show that the Wess-Zumino kinetic terms of individual spins combine to provide nontrivial topological terms for one-dimensional antiferromagnetic spin chains but do not produce a Hopf term for antiferromagnets on two-dimensional squares lattices. This result suggests that some other mechanism is needed to account for the neutral fermions which may be present in high-\(T_c\) materials.

Eduardo Fradkin, Jordan-Wigner Transformation for Quantum Spins Systems in Two Dimensions and Fractional Statistics, Physical Review Letters 63, 322 (1989).

I construct a Jordan-Wigner transformation for spin-one-half quantum systems on two-dimensional lattices. I show that the spin-one-half XY (i.e., a hard-core Bose system) is equivalent (on any two-dimensional Bravais lattice) to a system of spinless fermions and gauge fields satisfying the constraint that the gauge flux on a plaquette must be proportional to the spin (particle) density on site. The constraint is enforced by the addition of a Chern-Simons term of strength \( \theta\) to the Lagrangian of the theory. For the particular value \( \theta=1/2\pi \), the resulting particles are fermions. In general they are anyons. The implications of these results for quantum spin liquids are briefly discussed.

Eduardo Fradkin and Steven Kivelson, Short Range Resonating Valence Bond Theories and Superconductivity, Modern Physics Letters B 4, 2225 (1990).

The authors consider the nature of superconductivity near a spin-liquid state with a large spin-excitation-gap. They argue that the quantum-dimer-model with holes is a good approximation in this limit. The insulator is shown to be exactly equivalent to compact quantum electrodynamics, and has a massive spectrum. The doped system is a superconductor with a low density phase characterized by tightly bound pairs and a high density phase with two weakly coupled condensates.

Ana López and Eduardo fradkin The Fractional Quantum Hall Effect and Chern-Simons Gauge Theories, Physical Review B 44, 5246 (1991).

We present a theory of the fractional quantum Hall effect (FQHE) based on a second-quantized fermion path-integral approach. We show that the problem of interacting electrons moving on a plane in the presence of an external magnetic field is equivalent to a family of systems of fermions bound to an even number of fluxes described by a Chern-Simons gauge field. The semiclassical approximation of this system has solutions that describe incompressible-liquid states, Wigner crystals, and soliton-like defects. The liquid states belong to the Laughlin sequence and to the first level of the hierarchy. We give a brief description of the FQHE for bosons and anyons in this picture. The semiclassical spectrum of collective modes of the FQHE states has a gap to all excitations. We derive an effective action for the Gaussian fluctuations and study the hydrodynamic regime. The dispersion curve for the magneto-plasmon is calculated in the low-momentum limit. We find a nonzero gap at ?c. The fractionally quantized Hall conductance is calculated and argued to be exact in this approximation. We also give an explicit derivation of the polarization tensor in the integer Hall regime and show that it is transverse.

Antonio H. Castro Neto and Eduardo Fradkin Bosonization of the Low Energy Excitations of Fermi Liquids, Physical Review Letters 72, 1393 (1994).

We bosonize the low energy excitations of Fermi liquids in any number of dimensions in the limit of long wavelengths. The bosons are a coherent superposition of electron-hole pairs and are related with the displacements of the Fermi surface in some arbitrary direction. A coherent-state path integral for the bosonized theory is derived and it is shown to represent histories of the shape of the Fermi surface. The Landau theory of Fermi liquids can be obtained from the formalism in the absence of nesting of the Fermi surface and singular interactions. We show that the Landau equation for sound waves is exact in the semiclassical approximation for the bosons.

Antonio H. Castro Neto and Eduardo Fradkin Exact Solution of the Landau Fixed Point via Bosonization, Physical Review B 51, 4084(1995).

We study, via bosonization, the Landau fixed point for the problem of interacting spinless fermions near the Fermi surface in dimensions higher than one. We rederive the bosonic representation of the Fermi operator and use it to find the general form of the fermion propagator for the Landau fixed point. Using a generalized Bogoliubov transformation we diagonalize exactly the bosonized Hamiltonian for the fixed point and calculate the fermion propagator (and the quasiparticle residue) for isotropic interactions (independently of their strength). We reexamine two well-known problems in this context: the screening of long-range potentials and the Landau damping of gauge fields. We also discuss the origin of the Luttinger fixed point in one dimension in contrast with the Landau fixed point in higher dimensions.

Eduardo Fradkin and Fidel A. Schaposnik The Fermion-Boson Mapping in Three Dimensional Quantum Field Theory, Physics Letters B 338, 253 (1994).

We discuss bosonization in three dimensions by establishing a connection between the massive Thirring model and the Maxwell-Chern-Simons theory. We show, to lowest order in inverse fermion mass, the identity between the corresponding partition functions; from this, a bosonization identity for the fermion current, valid for length scales long compared with the Compton wavelength of the fermion, is inferred. We present a non-local operator in the Thirring model which exhibits fractional statistics.

Eduardo Fradkin and Steven A. Kivelson, Modular Invariance, Self-Duality and The Phase Transition Between Quantum Hall Plateaus, Nuclear Physics B 474, 543 (1996).

We investigate the problem of the super-universality of the phase transition between different quantum Hall plateaus. We construct a set of models which give a qualitative description of this transition in a pure system of interacting charged particles. One of the models is manifestly invariant under both duality and periodic shifts of the statistical angle and, hence, it has a full modular invariance. We derive the transformation laws for the correlation functions under the modular group and use them to derive symmetry constraints for the conductances. These allow us to calculate exactly the conductivities at the modular fixed points. We show that, at least at the modular fixed points, the system is critical. Away from the fixed points, the behavior of the model is determined by extra symmetries such as time reversal. We speculate that if the natural connection between spin and statistics holds, the model may exhibit an effective analyticity at low energies. In this case, the conductance is completely determined by its behavior under modular transformations.

Nancy P.~Sandler, Claudio de C. Chamon and Eduardo Fradkin, Andreev reflection in the fractional quantum Hall effect, Physical Review B 57, 12324 (1998).

We study the reflection of electrons and quasiparticles on point-contact interfaces between fractional quantum Hall (FQH) states and normal metals (leads), as well as interfaces between two FQH states with mismatched filling fractions. We classify the processes taking place at the interface in the strong-coupling limit. In this regime a set of quasiparticles can decay into quasiholes on the FQH side and charge excitations on the other side of the junction. This process is analogous to an Andreev reflection in normal-metal/superconductor (N?S) interfaces.

Steven A.Kivelson, Eduardo Fradkin and Victor J.Emery, Electronic Liquid Crystal Phases of a Doped Mott Insulator. Nature 393, 550 (1998).

The character of the ground state of an antiferromagnetic insulator is fundamentally altered upon addition of even a small amount of charge. The added charge is concentrated into domain walls across which a \( \pi \) phase shift in the spin correlations of the host material is induced. In two dimensions, these domain walls are "stripes'' which are either insulating, or conducting, i. e. metallic rivers with their own low energy degrees of freedom. However, in arrays of one-dimensional metals, which occur in materials such as organic conductors, the interactions typically drive a transition to an insulating ordered charge density wave (CDW) state at low temperatures. Here it is shown that such a transition is eliminated if the zero-point energy of transverse stripe fluctuations is sufficiently large in comparison to the CDW coupling between stripes. As a consequence, there exist novel electronic quantum liquid crystal phases which constitute new states of matter, and which can be either high temperature superconductors or two-dimensional anisotropic "metallic'' non-Fermi liquids. Neutron scattering and other experiments in the cuprate superconductor, La\(_{1.6-x}\) Nd\(_{0.4}\) Sr\(_x\) Cu O\(_4\), already provide convincing evidence of the existence of these phases in at least one class of materials.

Eduardo Fradkin, Chetan Nayak, Alexei Tsvelik and Frank Wilczek. A Chern-Simons Effective Field Theory for the Pfaffian Quantum Hall State. Nuclear Physics B 516, 704 (1998).

We present a low-energy effective field theory describing the universality class of the Pfaffian quantum Hall state. To arrive at this theory, we observe that the edge theory of the Pfaffian state of bosons at \(\nu=1\) is an SU(2)\(_2\) Kac-Moody algebra. It follows that the corresponding bulk effective field theory is an \( SU(2) \) Chern-Simons theory with coupling constant \( k=2 \). The effective field theories for other Pfaffian states, such as the fermionic one at \(\nu=1/2\) are obtained by a flux-attachment procedure. We discuss the non-Abelian statistics of quasiparticles in the context of this effective field theory.

Eduardo Fradkin and Steven Kivelson. Liquid Crystal Phases of Quantum Hall Systems. Physical Review B 59, 8065 (1999).

Mean-field calculations for the two dimensional electron gas (2DEG) in a large magnetic field with a partially filled Landau level with index \(N>1\) consistently yield "stripe-ordered'' charge-density wave ground-states, for much the same reason that frustrated phase separation leads to stripe ordered states in doped Mott insulators. We have studied the effects of quantum and thermal fluctuations about such a state and show that they can lead to a set of electronic liquid crystalline states, particularly a stripe-nematic phase which is stable at \(T>0\). Recent measurements of the longitudinal resistivity of a set of quantum Hall devices have revealed that these systems spontaneously develop, at low temperatures, a very large anisotropy. We interpret these experiments as evidence for a stripe nematic phase, and propose a general phase diagram for this system.

Eduardo Fradkin, Chetan Nayak, and Kareljan Schoutens. Landau-Ginzburg Theories for Non-Abelian Quantum Hall States. Nuclear Physics B 546, 711 (1999).

We construct Landau-Ginzburg effective field theories for fractional quantum Hall states -- such as the Pfaffian state -- which exhibit non-Abelian statistics. These theories rely on a Meissner construction which increases the level of a non-Abelian Chern-Simons theory while simultaneously projecting out the unwanted degrees of freedom of a concomitant enveloping Abelian theory. We describe this construction in the context of a system of bosons at Landau level filling factor \(\nu=1\), where the non-Abelian symmetry is a dynamically-generated SU(2) continuous extension of the discrete particle-hole symmetry of the lowest Landau level. We show how the physics of quasiparticles and their non-Abelian statistics arises in this Landau-Ginzburg theory. We describe its relation to edge theories -- where a coset construction plays the role of the Meissner projection -- and discuss extensions to other states.

Victor J. Emery, Eduardo Fradkin, Steven A. Kivelson and Tom C. Lubensky, Quantum Theory of the Smectic Metal State in Stripe Phases, Physical Review Letters 85, 2160 (2000).

We present a theory of the electron smectic fixed point of the stripe phases of doped layered Mott insulators. We show that in the presence of a spin gap three phases generally arise: (a) a smectic superconductor, (b) an insulating stripe crystal and (c) a smectic metal. The latter phase is a stable two-dimensional anisotropic non-Fermi liquid. In the absence of a spin gap there is also a more conventional Fermi-liquid-like phase. The smectic superconductor and smectic metal phases (or glassy versions thereof) may have already been seen in Nd-doped LSCO.

Vadim Oganesyan, Steven A. Kivelson, and Eduardo Fradkin, Quantum Theory of a Nematic Fermi Fluid , Physical Review B 64, 195109 (2001).

We develop a microscopic theory of the electronic nematic phase proximate to an isotropic Fermi liquid in both two and three dimensions. Explicit expressions are obtained for the small amplitude collective excitations in the ordered state; remarkably, the nematic Goldstone mode (the director wave) is overdamped except along special directions dictated by symmetry. At the quantum critical point we find a dynamical exponent of \( z=3 \), implying stability of the Gaussian fixed point. The leading perturbative effect of the overdamped Goldstone modes leads to a breakdown of Fermi-liquid theory in the nematic phase and to strongly angle-dependent electronic self energies around the Fermi surface. Other metallic liquid-crystal phases, e.g., a quantum hexatic, behave analogously.

R. Moessner, S. L. Sondhi and Eduardo Fradkin, Short-ranged RVB physics, quantum dimer models and Ising gauge theories, Physical Review B 65, 024504 (2001).

Quantum dimer models are believed to capture the essential physics of antiferromagnetic phases dominated by short-ranged valence bond configurations. We show that these models arise as particular limits of Ising ( \(\mathbb{Z}_2\) ) gauge theories, but that in these limits the system develops a larger local \( U(1) \) invariance that has different consequences on different lattices. Conversely, we note that the standard \( \mathbb{Z}_2\) gauge theory is a generalized quantum dimer model, in which the particular relaxation of the hardcore constraint for the dimers breaks the \( U(1) \) down to \( \mathbb{Z}_2\). These mappings indicate that at least one realization of the Senthil-Fisher proposal for fractionalization is exactly the short ranged resonating valence bond (RVB) scenario of Anderson and of Kivelson, Rokhsar and Sethna. They also suggest that other realizations will require the identification of a local low energy, Ising link variable and a natural constraint. We also discuss the notion of topological order in \( \mathbb{Z}_2\) gauge theories and its connection to earlier ideas in RVB theory. We note that this notion is not central to the experiment proposed by Senthil and Fisher to detect vortices in the conjectured \( \mathbb{Z}_2\) gauge field.

S.A. Kivelson, E.Fradkin, V.Oganesyan, I.P.Bindloss, J.M.Tranquada, A.Kapitulnik and C.Howald, How to detect fluctuating order in the high-temperature superconductors, Reviews of Modern Physics 75, 1201 (2003).

We discuss fluctuating order in a quantum disordered phase proximate to a quantum critical point, with particular emphasis on fluctuating stripe order. Optimal strategies for extracting information concerning such local order from experiments are derived with emphasis on neutron scattering and scanning tunneling microscopy. These ideas are tested by application to two model systems - the exactly solvable one dimensional electron gas with an impurity, and a weakly-interacting 2D electron gas. We extensively review experiments on the cuprate high-temperature superconductors which can be analyzed using these strategies. We adduce evidence that stripe correlations are widespread in the cuprates. Finally, we compare and contrast the advantages of two limiting perspectives on the high-temperature superconductor: weak coupling, in which correlation effects are treated as a perturbation on an underlying metallic (although renormalized) Fermi liquid state, and strong coupling, in which the magnetism is associated with well defined localized spins, and stripes are viewed as a form of micro-phase separation. We present quantitative indicators that the latter view better accounts for the observed stripe phenomena in the cuprates.

Eddy Ardonne, Paul Fendley and Eduardo Fradkin, Topological Order and Conformal Quantum Critical Points, Annals Phys. 310, 493 (2004).

We discuss a certain class of two-dimensional quantum systems which exhibit conventional order and topological order, as well as two-dimensional quantum critical points separating these phases. All of the ground-state equal-time correlators of these theories are equal to correlation functions of a local two-dimensional classical model. The critical points therefore exhibit a time-independent form of conformal invariance. These theories characterize the universality classes of two-dimensional quantum dimer models and of quantum generalizations of the eight-vertex model, as well as \(\mathbb{Z}_2\) and non-abelian gauge theories. The conformal quantum critical points are relatives of the Lifshitz points of three-dimensional anisotropic classical systems such as smectic liquid crystals. In particular, the ground-state wave functional of these quantum Lifshitz points is just the statistical (Gibbs) weight of the ordinary 2D free boson, the 2D Gaussian model. The full phase diagram for the quantum eight-vertex model exhibits quantum critical lines with continuously-varying critical exponents separating phases with long-range order from a \(\mathbb{Z}_2\) deconfined topologically-ordered liquid phase. We show how similar ideas also apply to a well-known field theory with non-abelian symmetry, the strong-coupling limit of 2+1-dimensional Yang-Mills gauge theory with a Chern-Simons term. The ground state of this theory is relevant for recent theories of topological quantum computation.

Eduardo Fradkin, David Huse, Roderich Moessner, Vadim Oganesyan and Shivaji L. Sondhi, On bipartite Rokhsar-Kivelson points and Cantor deconfinement, Physical Review B 69, 224415 (2004).

Quantum dimer models on bipartite lattices exhibit Rokhsar-Kivelson (RK) points with exactly known critical ground states and deconfined spinons. We examine generic, weak, perturbations around these points. In \(d=2+1\) we find a first order transition between a "plaquette'' valence bond crystal and a region with a devil's staircase of commensurate and incommensurate valence bond crystals. In the part of the phase diagram where the staircase is incomplete, the incommensurate states exhibit a gapless photon and deconfined spinons on a set of finite measure, almost but not quite a deconfined phase in a compact \( U(1)\) gauge theory in \(d=2+1\)! In \(d=3+1\) we find a continuous transition between the \(U(1)\) resonating valence bond (RVB) phase and a deconfined staggered valence bond crystal. In an appendix we comment on analogous phenomena in quantum vertex models, most notably the existence of a continuous transition on the triangular lattice in \(d=2+1\).

Paul Fendley and Eduardo Fradkin, Realizing non-Abelian fractional statistics in time-reversal invariant systems, Physical Review B 72, 024412 (2005).

We construct a series of 2+1-dimensional models whose quasiparticles obey non-Abelian statistics. The adiabatic transport of quasiparticles is described by using a correspondence between the braid matrix of the particles and the scattering matrix of 1+1-dimensional field theories. We discuss in depth lattice and continuum models whose braiding is that of SO(3) Chern-Simons gauge theory, including the simplest type of non-Abelian statistics, involving just one type of quasiparticle. The ground-state wave function of an \( SO(3) \) model is related to a loop description of the classical two-dimensional Potts model. We discuss the transition from a topological phase to a conventionally-ordered phase, showing in some cases there is a quantum critical point.

Michael Lawler, Daniel Barci, Victoria Fernandez, Eduardo Fradkin, and Luis Oxman, Non-perturbative behavior of the quantum phase transition to a nematic Fermi fluid, Physical Review B 73, 085101 (2006).

We discuss shape (Pomeranchuk) instabilities of the Fermi surface of a two-dimensional Fermi system using bosonization. We consider in detail the quantum critical behavior of the transition of a two dimensional Fermi fluid to a nematic state which breaks spontaneously the rotational invariance of the Fermi liquid. We show that higher dimensional bosonization reproduces the quantum critical behavior expected from the Hertz-Millis analysis, and verify that this theory has dynamic critical exponent \(z=3\). Going beyond this framework, we study the behavior of the fermion degrees of freedom directly, and show that at quantum criticality as well as in the the quantum nematic phase (except along a set of measure zero of symmetry-dictated directions) the quasi-particles of the normal Fermi liquid are generally wiped out. Instead, they exhibit short ranged spatial correlations that decay faster than any power-law, with the law \(|x|^{-1}\; \exp(-\textrm{const.} \;|x|^{1/3}\; )\) and we verify explicitly the vanishing of the fermion residue utilizing this expression. In contrast, the fermion auto-correlation function has the behavior \(|t|^{-1}\; \exp(-{\rm const}.\; |t|^{-2/3}\; )\). In this regime we also find that, at low frequency, the single-particle fermion density-of-states behaves as \(N^*(\omega)=N^*(0) \;+ B \: \omega^{2/3}\; \log\omega +...\), where \(N^*(0)\) is larger than the free Fermi value, \(N(0)\), and \(B\) is a constant. These results confirm the non-Fermi liquid nature of both the quantum critical theory and of the nematic phase.

Michael J. Lawler and Eduardo Fradkin, Local Quantum Criticality in the Nematic Quantum Phase Transition of a Fermi Fluid, Physical Review B 75, 033304 (2007)

We discuss the finite temperature properties of the fermion correlation function near the fixed point theory of the nematic quantum critical point (QCP) of a metallic Fermi system. We show that though the fixed point theory is above its upper critical dimension, the equal time fermion correlation function takes on a universal scaling form in the vicinity of the QCP. We find that in the quantum critical regime, this equal-time correlation function has an ultra local behavior in space, while the low-frequency behavior of the equal-position auto correlation function is that of a Fermi liquid up to subdominant terms. This behavior should also apply to other quantum phase transitions of metallic Fermi systems.

Eduardo Fradkin and Joel Moore, Entanglement entropy of 2D conformal quantum critical points: hearing the shape of a quantum drum, Physical Review Letters 97, 050404 (2006).

The entanglement entropy of a pure quantum state of a bipartite system \(A \cup B\) is defined as the von Neumann entropy of the reduced density matrix obtained by tracing over one of the two parts. Critical ground states of local Hamiltonians in one dimension have entanglement that diverges logarithmically in the subsystem size, with a universal coefficient that for conformally invariant critical points is related to the central charge of the conformal field theory. We find the entanglement entropy for a standard class of \(z=2\) quantum critical points in two spatial dimensions with scale invariant ground state wave functions: in addition to a non-universal "area law'' contribution proportional to the size of the \(A B\) boundary, there is generically a universal logarithmically divergent correction. This logarithmic term is completely determined by the geometry of the partition into subsystems and the central charge of the field theory that describes the equal-time correlations of the critical wavefunction.

Erez Berg, Eduardo Fradkin, Eun-Ah Kim, Steven Kivelson, Vadim Oganesyan, John M. Tranquada, and Shoucheng Zhang, Dynamical layer decoupling in a stripe-ordered high \(T_c\) superconductor, Physical Review Letters 99, 127003 (2007).

In the stripe-ordered state of a strongly correlated two-dimensional electronic system, under a set of special circumstances, the superconducting condensate, like the magnetic order, can occur at a nonzero wave vector corresponding to a spatial period double that of the charge order. In this case, the Josephson coupling between near neighbor planes, especially in a crystal with the special structure of La\(_{2-x}\) Ba\(_x \) Cu O\(_4\) , vanishes identically. We propose that this is the underlying cause of the dynamical decoupling of the layers recently observed in transport measurements at x=1/8.

Shying Dong, Eduardo Fradkin, Robert G. Leigh, and Sean Nowling, Topological Entanglement Entropy in Chern-Simons Theories and Quantum Hall Fluids , Journal of High Energy Physics JHEP 2008, 016 (2008).

We compute directly the entanglement entropy of spatial regions in Chern-Simons gauge theories in 2+1 dimensions using surgery. We consider the possible dependence of the entanglement entropy on the topology of the spatial manifold and on the vacuum state on that manifold. The entanglement entropy of puncture insertions (quasiparticles) is discussed in detail for a few cases of interest. We show that quite generally the topological entanglement entropy is determined by the modular \( \mathcal{S}\)-matrix of the associated rational conformal field theory as well as by the fusion rules and fusion coefficients. We use these results to determine the universal topological piece of the entanglement entropy for Abelian and non-Abelian quantum Hall fluids. As a byproduct we present the calculation of the modular \( \mathcal{S} \)-matrix of two coset RCFTs of interest.

Erez Berg, Eduardo Fradkin, Steven A. Kivelson, and John M. Tranquada, Striped Superconducting Order: How the cuprates intertwine spin, charge and superconducting orders, New Journal of Physics 11, 115004 (2009).

Recent transport experiments in the original cuprate high temperature superconductor, La\(_{2-x}\) Ba\(_x\) Cu O\(_4\), have revealed a remarkable sequence of transitions and crossovers that give rise to a form of dynamical dimensional reduction, in which a bulk crystal becomes essentially superconducting in two directions while it remains poorly metallic in the third. We identify these phenomena as arising from a distinct new superconducting state, the 'striped superconductor', in which the superconducting order is spatially modulated, so that its volume average value is zero. Here, in addition to outlining the salient experimental findings, we sketch the order parameter theory of the state, stressing some of the ways in which a striped superconductor differs fundamentally from an ordinary (uniform) superconductor, especially concerning its response to quenched randomness. We also present the results of density matrix renormalization group calculations on a model of interacting electrons in which sign oscillations of the superconducting order are established. Finally, we speculate concerning the relevance of this state to experiments in other cuprates, including recent optical studies of La\(_{2-x}\) Sr\(_x\) Cu O\(_4\) in a magnetic field, neutron scattering experiments in underdoped Y Ba\(_2\) Cu\(_3\) O\(_{6+x}\) and a host of anomalies seen in STM and ARPES studies of Bi\(_2\) Sr\(_2\) Ca Cu\(_2\) O\(_{8+\delta}\) .

Erez Berg, Eduardo Fradkin, and Steven A. Kivelson, Striped Superconducting Order: Charge \(4e\) superconductivity from pair density wave order in certain high temperature superconductors, Nature Physics 5, 830 (2009).

A number of spectacular experimental anomalies have been discovered recently in certain cuprates, notably La\(_{2-x}\) Sr\(_x\) Cu O\(_4\) and La\(_{1.6-x}\) Nd\(_{0.4} \) Sr\(_x\) CuO\(_4\), which show unidirectional spin and charge order (known as stripe order). We have recently proposed to interpret these observations as evidence for a new `striped' superconducting state, in which the superconducting order parameter is modulated in space, such that its average is precisely zero. Here, we show that thermal melting of the striped superconducting state can lead to a number of unusual phases, of which the most novel is a charge-\(4e\) superconducting state, with a corresponding fractional flux quantum \(h c/4e\). These are never-before-observed states of matter, which, moreover, cannot arise from the conventional Bardeen-Cooper-Schrieffer mechanism. Thus, direct confirmation of their existence, even in a small subset of the cuprates, could have much broader implications for our understanding of high-temperature superconductivity. We propose experiments to observe fractional flux quantization, which could confirm the existence of these states.

Kai Sun, Hong Yao, Eduardo Fradkin, and Steven A. Kivelson, Topological Insulators and Nematic Phases from Spontaneous Symmetry Breaking in 2D Fermi Systems with a Quadratic Band Crossing, Physical Review Letters 103, 056811 (2009).

We investigate the stability of a quadratic band-crossing point (QBCP) in 2D fermionic systems. At the noninteracting level, we show that a QBCP exists and is topologically stable for a Berry flux \(\pm 2\pi \) if the point symmetry group has either fourfold or sixfold rotational symmetries. This putative topologically stable free-fermion QBCP is marginally unstable to arbitrarily weak short-range repulsive interactions. We consider both spinless and spin-1/2 fermions. Four possible ordered states result: a quantum anomalous Hall phase, a quantum spin Hall phase, a nematic phase, and a nematic-spin-nematic phase.

Eduardo Fradkin, Steven A. Kivelson, Michael J. Lawler, James P. Eisenstein, and Andrew Mackenzie, Nematic Fermi Fluids in Condensed Matter Physics, Annual Review of Condensed Matter Physics 1, 153 (2010).

Correlated electron fluids can exhibit a startling array of complex phases, among which one of the more surprising is the electron nematic, a translationally invariant metallic phase with a spontaneously generated spatial anisotropy. Classical nematics generally occur in liquids of rod-like molecules; given that electrons are point like, the initial theoretical motivation for contemplating electron nematics came from thinking of the electron fluid as a quantum melted electron crystal, rather than a strongly interacting descendent of a Fermi gas. Dramatic transport experiments in ultra-clean quantum Hall systems in 1999 and in Sr\(_3\) Ru\(_2\) O\(_7\) in a strong magnetic field in 2007 established that such phases exist in nature. In this article, we briefly review the theoretical considerations governing nematic order, summarize the quantum Hall and Sr\(_3\) Ru\(_2\) O\(_7\) experiments that unambiguously establish the existence of this phase, and survey some of the current evidence for such a phase in the cuprate and Fe-based high temperature superconductors.

Erez Berg, Eduardo Fradkin and Steven A. Kivelson, Pair Density Wave correlations in the Kondo-Heisenberg Model, Physical Review Letters 105, 146403 (2010).

We show, using density-matrix renormalization-group calculations complemented by field-theoretic arguments, that the spin-gapped phase of the one dimensional Kondo-Heisenberg model exhibits quasi-long-range superconducting correlations only at a nonzero momentum. The local correlations in this phase resemble those of the pair-density-wave state which was recently proposed to describe the phenomenology of the striped ordered high-temperature superconductor La\(_{2-x}\) Ba\(_x\) Cu O\(_4\) , in which the spin, charge, and superconducting orders are strongly intertwined.

Taylor L. Hughes, Robert G. Leigh and Eduardo Fradkin, Torsional Response and Dissipationless Viscosity in Topological Insulators, Physical Review Letters 102, 075502 (2011).

We consider the viscoelastic response of the electronic degrees of freedom in 2D and 3D topological insulators (TI's). Our primary focus is on the 2D Chern insulator which exhibits a bulk dissipationless viscosity analogous to the quantum Hall viscosity predicted in integer and fractional quantum Hall states. We show that the dissipationless viscosity is the response of a TI to torsional deformations of the underlying lattice geometry. The viscoelastic response also indicates that crystal dislocations in Chern insulators will carry momentum density. We briefly discuss generalizations to 3D which imply that time-reversal invariant TI's will exhibit a quantum Hall viscosity on their surfaces.

AtMa Chan, Taylor L. Hughes, Shinsei Ryu, and Eduardo Fradkin, Effective field theories for topological insulators by functional bosonization, Physical Review B 87, 085132 (2013).

Effective field theories that describe the dynamics of a conserved \( U(1) \) current in terms of "hydrodynamic" degrees of freedom of topological phases in condensed matter are discussed in general dimension \( D=d+1 \) using the functional bosonization technique. For noninteracting topological insulators (superconductors) with a conserved \( U(1) \) charge and characterized by an integer topological invariant [more specifically, they are topological insulators in the complex symmetry classes (class A and AIII), and in the "primary series" of topological insulators, in the eight real symmetry classes], we derive the BF-type topological field theories supplemented with the Chern-Simons (when \( D \) is odd) or the \( \theta \) (when \( D \) is even) terms. For topological insulators characterized by a \( \mathbb{Z}_2\) topological invariant (the first and second descendants of the primary series), their topological field theories are obtained by dimensional reduction. Building on this effective field theory description for noninteracting topological phases, we also discuss, following the spirit of the parton construction of the fractional quantum Hall effect by Block and Wen, the putative "fractional" topological insulators and their possible effective field theories, and use them to determine the physical properties of these nontrivial quantum phases.

Yizhi You, and Eduardo Fradkin, Field Theory of Nematicity in the Spontaneous Quantum Anomalous Hall Effect, Physical Review B 88, 235124 (2013).

We derive from a microscopic model the effective theory of nematic order in a system with a spontaneous quantum anomalous Hall effect in two dimensions. Starting with a model of two-component fermions (a spinor field) with a quadratic band crossing and short-range four-fermion marginally relevant interactions we use \( 1/N \) expansion and bosonization methods to derive the effective field theory for the hydrodynamic modes associated with the conserved currents and with the local fluctuations of the nematic order parameter. We focus on the vicinity of the quantum phase transition from the isotropic Mott Chern insulating phase to a phase in which time-reversal symmetry breaking coexists with nematic order, the nematic Chern insulator. The topological sector of the effective field theory is a background field (BF)/Chern-Simons gauge theory. We show that the nematic order parameter field couples with the Maxwell-type terms of the gauge fields as the space components of a locally fluctuating metric tensor. The nematic field has \( z= 2 \) dynamic scaling exponent. The low-energy dynamics of the nematic order parameter is found to be governed by a Berry phase term. By means of a detailed analysis of the coupling of the spinor field of the fermions to the changes of their local frames originating from long-wavelength lattice deformations, we calculate the Hall viscosity of this system and show that in this system it is not the same as the Berry phase term in the effective action of the nematic field, but both are related to the concept of torque Hall viscosity, which we introduce here.

Gil Young Cho, Yizhi You, and Eduardo Fradkin, Geometry of Fractional Quantum Hall Fluids, Physical Review B 90, 115139 (2014).

We use the field theory description of the fractional quantum Hall states to derive the universal response of these topological fluids to shear deformations and curvature of their background geometry, i.e., the Hall viscosity, and the Wen-Zee term. To account for the coupling to the background geometry, we show that the concept of flux attachment needs to be modified and use it to derive the geometric responses from Chern-Simons theories. We show that the resulting composite particles minimally couple to the spin connection of the geometry. We derive a consistent theory of geometric responses from the Chern-Simons effective field theories and from parton constructions, and apply it to both Abelian and non-Abelian states.

Krishna Kumar, Kai Sun and Eduardo Fradkin, Chern-Simons theory for magnetization plateaus of the spin-1/2 XXZ quantum Heisenberg model on the kagome lattice, Physical Review B 90, 174409 (2014).

Frustrated spin systems on kagome lattices have long been considered to be a promising candidate for realizing exotic spin-liquid phases. Recently, there has been a lot of renewed interest in these systems with the discovery of materials such as volborthite and herbertsmithite that have kagome-like structures. In the presence of an external magnetic field, these frustrated systems can give rise to magnetization plateaus of which the plateau at \( m=1/3 \) is considered to be the most prominent. Here, we study the problem of the antiferromagnetic spin-1/2 quantum XXZ Heisenberg model on a kagome lattice by using a Jordan-Wigner transformation that maps the spins onto a problem of fermions coupled to a Chern-Simons gauge field. This mapping relies on being able to define a consistent Chern-Simons term on the lattice. Such a lattice Chern-Simons term had previously only been written for the square lattice and was used to successfully study the unfrustrated Heisenberg antiferromagnet on the square lattice. At a mean-field level, these ideas have also been applied to frustrated systems by ignoring the details of the Chern-Simons term. However, fluctuations are generally strong in these models and are expected to affect the mean-field physics. Using a recently developed method to rigorously extend the Chern-Simons term to the frustrated kagome lattice, we can now formalize the Jordan-Wigner transformation on the kagome lattice. We then discuss the possible phases that can arise at the mean-field level from this mapping and focus specifically on the case of \(1/3 \) filling (\(m=1/3\) plateau) and analyze the effects of fluctuations in our theory. We show that in the regime of XY anisotropy, the ground state at the \(1/3 \) plateau is equivalent to a bosonic fractional quantum Hall Laughlin state with filling fraction \( 1/2 \) and that at the \(5/9 \) plateau it is equivalent to the first bosonic Jain daughter state at filling fraction \(2/3\).

Eduardo Fradkin, Steven A. Kivelson, and John M. Tranquada, Theory of Intertwined Orders in High Temperature Superconductors, Reviews of Modern Physics 87, 457 (2015).

The electronic phase diagrams of many highly correlated systems, and, in particular, the cuprate high temperature superconductors, are complex, with many different phases appearing with similar (sometimes identical) ordering temperatures even as material properties, such as dopant concentration, are varied over wide ranges. This complexity is sometimes referred to as "competing orders." However, since the relation is intimate, and can even lead to the existence of new phases of matter such as the putative "pair-density wave," the general relation is better thought of in terms of "intertwined orders." Some of the experiments in the cuprates which suggest that essential aspects of the physics are reflected in the intertwining of multiple orders, not just in the nature of each order by itself, are selectively analyzed. Several theoretical ideas concerning the origin and implications of this complexity are also summarized and critiqued.

Yizhi You, Gil Young Cho and Eduardo Fradkin, Theory of the Nematic Fractional Quantum Hall State, Physical Review X 4, 041050 (2014).

We derive an effective field theory for the isotropic-nematic quantum phase transition of fractional quantum Hall states. We demonstrate that for a system with an isotropic background the low-energy effective theory of the nematic order parameter has \( z=2\) dynamical scaling exponent, due to a Berry phase term of the order parameter, which is related to the non-dissipative Hall viscosity. Employing the composite fermion theory with a quadrupolar interaction between electrons, we show that a sufficiently attractive quadrupolar interaction triggers a phase transition from the isotropic fractional quantum Hall fluid into a nematic fractional quantum Hall phase. By investigating the spectrum of collective excitations, we demonstrate that the mass gap of the Girvin-MacDonald-Platzman mode collapses at the isotropic-nematic quantum phase transition. On the other hand, Laughlin quasiparticles and the Kohn collective mode remain gapped at this quantum phase transition, and Kohn's theorem is satisfied. The leading couplings between the nematic order parameter and the gauge fields include a term of the same form as the Wen-Zee term. A disclination of the nematic order parameter carries an unquantized electric charge. We also discuss the relation between nematic degrees of freedom and the geometrical response of the fractional quantum Hall fluid.

Andrey Gromov, Gil Young Cho, Yizhi You, Alexander G. Abanov, and Eduardo Fradkin, Framing Anomaly in the Effective Theory of Fractional Quantum Hall Effect, Physical Review Letters 114, 016805 (2015).

We consider the geometric part of the effective action for the fractional quantum Hall effect (FQHE). It is shown that accounting for the framing anomaly of the quantum Chern-Simons theory is essential to obtain the correct gravitational linear response functions. In the lowest order in gradients, the linear response generating functional includes Chern-Simons, Wen-Zee, and gravitational Chern-Simons terms. The latter term has a contribution from the framing anomaly which fixes the value of thermal Hall conductivity and contributes to the Hall viscosity of the FQH states on a sphere. We also discuss the effects of the framing anomaly on linear responses for non-Abelian FQH states.

Xiao Chen, Gil Young Cho, Thomas Faulkner, and Eduardo Fradkin, Scaling of entanglement in 2+1-dimensional scale-invariant field theories, Journal of Statistical Mechanics (JSTAT) 2015, P02010 (2015).

We study the universal scaling behavior of the entanglement entropy of critical theories in 2+1 dimensions. We specially consider two fermionic scale-invariant models, free massless Dirac fermions and a model of fermions with quadratic band touching, and numerically study the two-cylinder entanglement entropy of the models on the torus. We find that in both cases the entanglement entropy satisfies the area law and has the subleading term which is a scaling function of the aspect ratios of the cylindrical regions. We test the scaling of entanglement in both the free fermion models using three possible scaling functions for the subleading term derived from (a) the quasi-1D conformal field theory, (b) the bosonic quantum Lifshitz model and (c) the holographic AdS/CFT correspondence. For the later case we construct an analytic scaling function using holography, appropriate for critical theories with a gravitational dual description. We find that the subleading term in the fermionic models is well described, for a range of aspect ratios, by the scaling form derived from the quantum Lifshitz model as well as that derived using the AdS/CFT correspondence (in this case only for the Dirac model). For the case where the fermionic models are placed on a square torus we find the fit to the different scaling forms is in agreement to surprisingly high precision.

Kai Sun, Krishna Kumar, and Eduardo Fradkin, discretized Chern-Simons gauge theory on arbitrary graphs, Physical Review B 92, 115148 (2015).

In this paper, we show how to discretize the Abelian Chern-Simons gauge theory on generic planar lattices/graphs (with or without translational symmetries) embedded in arbitrary two-dimensional closed orientable manifolds. We find that, as long as a one-to-one correspondence between vertices and faces can be defined on the graph such that each face is paired up with a neighboring vertex (and vice versa), a discretized Abelian Chern-Simons theory can be constructed consistently. We further verify that all the essential properties of the Chern-Simons gauge theory are preserved in the discretized setup. In addition, we find that the existence of such a one-to-one correspondence is not only a sufficient condition for discretizing a Chern-Simons gauge theory but, for the discretized theory to be nonsingular and to preserve some key properties of the topological field theory, this correspondence is also a necessary one. A specific example will then be provided, in which we discretize the Abelian Chern-Simons gauge theory on a tetrahedron.

Jeffrey C. Y. Teo, Taylor L. Hughes, and Eduardo Fradkin, Theory of Twist Liquids: Gauging and Anyonic Symmetry, Annals of Physics 360, 349 (2015).

Topological phases in 2+1 dimensions are frequently equipped with global symmetries, like conjugation, bilayer or electric-magnetic duality, that relabel anyons without affecting the topological structures. Twist defects are static point-like objects that permute the labels of orbiting anyons. Gauging these symmetries by quantizing defects into dynamical excitations leads to a wide class of more exotic topological phases referred as twist liquids, which are generically non-Abelian. We formulate a general gauging framework, characterize the anyon structure of twist liquids and provide solvable lattice models that capture the gauging phase transitions. We explicitly demonstrate the gauging of the \(\mathbb{Z}_2\)-symmetric toric code, \( SO(2N)_{1}\) and \( SU(3)_{1} \) state as well as the \(SO(8)_{1}\)-symmetric state and a non-Abelian chiral state we call the "4-Potts" state.

Xiao Chen, Xiongjie Yu, Gil Young Cho, Bryan K. Clark, and Eduardo Fradkin, Many-body Localization Transition in Rokhsar-Kivelson-type wave functions, Physical Review B 92, 212204 (2015).

We construct a family of many-body wave functions to study the many-body localization phase transition. The wave functions have a Rokhsar-Kivelson form, in which the weight for the configurations are chosen from the Gibbs weights of a classical spin glass model, known as the random energy model, multiplied by a random sign structure to represent a highly excited state. These wave functions show a phase transition into an MBL phase. In addition, we see three regimes of entanglement scaling with the subsystem size: scaling with the entanglement corresponding to an infinite temperature thermal phase, constant scaling, and a sub-extensive scaling between these limits. Near the phase transition point, the fluctuations of the Rényi entropies are non-Gaussian. We find that Rényi entropies with different Rényi index transition into the MBL phase at different points and have different scaling behavior, suggesting a multifractal behavior.

Yizhi You, Gil Young Cho, and Eduardo Fradkin, Nematic quantum phase transition of composite Fermi liquids in half-filled Landau levels and their geometric response, Physical Review B 93, 205401 (2016).

We present a theory of the isotropic-nematic quantum phase transition in the composite Fermi liquid arising in half-filled Landau levels. We show that the quantum phase transition between the isotropic and the nematic phase is triggered by an attractive quadrupolar interaction between electrons, as in the case of conventional Fermi liquids. We derive the theory of the nematic state and of the phase transition. This theory is based on the flux attachment procedure, which maps an electron liquid in half-filled Landau levels into the composite Fermi liquid close to a nematic transition. We show that the local fluctuations of the nematic order parameters act as an effective dynamical metric interplaying with the underlying Chern-Simons gauge fields associated with the flux attachment. Both the fluctuations of the Chern-Simons gauge field and the nematic order parameter can destroy the composite fermion quasiparticles and drive the system into a non-Fermi liquid state. The effective-field theory for the isotropic-nematic phase transition is shown to have \( z=3\) dynamical exponent due to the Landau damping of the dense Fermi system. We show that there is a Berry-phase-type term that governs the effective dynamics of the nematic order parameter fluctuations, which can be interpreted as a non-universal "Hall viscosity" of the dynamical metric. We also show that the effective-field theory of this compressible fluid has a Wen-Zee-type term. Both terms originate from the time-reversal breaking fluctuation of the Chern-Simons gauge fields. We present a perturbative (one-loop) computation of the Hall viscosity and also show that this term is also obtained by a Ward identity. We show that the topological excitation of the nematic fluid, the disclination, carries an electric charge. We show that a resonance observed in radio-frequency conductivity experiments can be interpreted as a Goldstone nematic mode gapped by lattice effects.

Xiao Chen, Tianci Zhou, David A. Huse, and Eduardo Fradkin, Out-of-time-order correlations in many-body localized and thermal phases, Annalen der Physik 529, 1600332 (2017).

We use the out-of-time-order (OTO) correlators to study the slow dynamics in the many-body localized (MBL) phase. We investigate OTO correlators in the effective ("l-bit") model of the MBL phase, and show that their amplitudes after disorder averaging approach their long-time limits as power-laws of time. This power-law dynamics is due to dephasing caused by interactions between the localized operators that fall off exponentially with distance. The long-time limits of the OTO correlators are determined by the overlaps of the local operators with the conserved l-bits. We demonstrate numerically our results in the effective model and three other more "realistic" spin chain models. Furthermore, we extend our calculations to the thermal phase and find that for a time-independent Hamiltonian, the OTO correlators also appear to vanish as a power law at long time, perhaps due to coupling to conserved densities. In contrast, we find that in the thermal phase of a Floquet spin model with no conserved densities the OTO correlator decays exponentially at long times.

Laimei Nie, Akash V. Maharaj, Eduardo Fradkin, and Steven A. Kivelson, Vestigial nematicity from spin and/or charge order in the cuprates, Physical Review B 96, 085142 (2017).

Nematic order has manifested itself in a variety of materials in the cuprate family. We propose an effective field theory of a layered system with incommensurate, intertwined spin- and charge-density wave (SDW and CDW) orders, each of which consists of two components related by \(C_4\) rotations. Using a variational method (which is exact in a large-\(N\) limit), we study the development of nematicity from partially melting those density waves by either increasing temperature or adding quenched disorder. As temperature decreases we first find a transition to a single nematic phase, but depending on the range of parameters (e.g., doping concentration) the strongest fluctuations associated with this phase reflect either proximate SDW or CDW order. We also discuss the changes in parameters that can account for the differences in the SDW-CDW interplay between the 214 family and the other hole-doped cuprates.

Ramanjit Sohal, Luiz H. Santos and Eduardo Fradkin Chern-Simons Composite Fermion Theory of Fractional Chern Insulators, Physical Review B 97, 125131 (2018).

We formulate a Chern-Simons composite fermion theory for fractional Chern insulators (FCIs), whereby bare fermions are mapped into composite fermions coupled to a lattice Chern-Simons gauge theory. We apply this construction to a Chern insulator model on the kagome lattice and identify a rich structure of gapped topological phases characterized by fractionalized excitations including states with unequal filling and Hall conductance. Gapped states with the same Hall conductance at different filling fractions are characterized as realizing distinct symmetry fractionalization classes.

Hart Goldman and Eduardo Fradkin Loop Models, Modular Invariance, and Three Dimensional Bosonization, Physical Review B 97, 195112 (2018).

We consider a family of quantum loop models in 2+1 spacetime dimensions with marginally long-ranged and statistical interactions mediated by a \(U(1)\) gauge field, both purely in 2+1 dimensions and on a surface in a (3+1)-dimensional bulk system. In the absence of fractional spin, these theories have been shown to be self-dual under particle-vortex duality and shifts of the statistical angle of the loops by \( 2 \pi \), which form a subgroup of the modular group, PSL(2,\(\mathbb{Z}\)). We show that careful consideration of fractional spin in these theories completely breaks their statistical periodicity and describe how this occurs, resolving a disagreement with the conformal field theories they appear to approach at criticality. We show explicitly that incorporation of fractional spin leads to loop model dualities which parallel the recent web of (2+1)-dimensional field theory dualities, providing a nontrivial check on its validity.

Yuxuan Wang, Stephen D. Edkins, Mohammad H. Hamidian, J. C. Séamus Davis, Eduardo Fradkin, and Steven A. Kivelson, Pair Density Waves in Superconducting Vortex Halos, Physical Review B 97, 174510 (2018).

We analyze the interplay between a d-wave uniform superconducting and a pair-density-wave (PDW) order parameter in the neighborhood of a vortex. We develop a phenomenological nonlinear sigma model, solve the saddle-point equation for the order-parameter configuration, and compute the resulting local density of states in the vortex halo. The intertwining of the two superconducting orders leads to a charge density modulation with the same periodicity as the PDW, which is twice the period of the charge density wave that arises as a second harmonic of the PDW itself. We discuss key features of the charge density modulation that can be directly compared with recent results from scanning tunneling microscopy and speculate on the role PDW order may play in the global phase diagram of the hole-doped cuprates.

Hart Goldman and Eduardo Fradkin, Dirac composite fermions and emergent reflection symmetry about even denominator filling fractions, Physical Review B 98, 165137 (2018).

Motivated by the appearance of a "reflection symmetry" in transport experiments and the absence of statistical periodicity in relativistic quantum field theories, we propose a series of relativistic composite fermion theories for the compressible states appearing at filling fractions \(\nu =1/2n\) in quantum Hall systems. These theories consist of electrically neutral Dirac fermions attached to \(2n\) flux quanta via an emergent Chern-Simons gauge field. While not possessing an explicit particle-hole symmetry, these theories reproduce the known Jain sequence states proximate to \( \nu=1/2n\), and we show that such states can be related by the observed reflection symmetry, at least at mean-field level. We further argue that the lowest Landau-level limit requires that the Dirac fermions be tuned to criticality, whether or not this symmetry extends to the compressible states themselves.

Daniel F. Agterberg, J.C. Séamus Davis, Stephen D. Edkins, Eduardo Fradkin, Dale J. Van Harlingen, Steven A. Kivelson, Patrick A. Lee, Leo Radzihovsky, John M. Tranquada, and Yuxuan Wang, The Physics of Pair Density Waves: Cuprate Superconductors and Beyond, Annual Review of Condensed Matter Physics 11, 231(2020).

We review the physics of pair-density wave (PDW) superconductors. We begin with a macroscopic description that emphasizes order induced by PDW states, such as charge-density wave, and discuss related vestigial states that emerge as a consequence of partial melting of the PDW order. We review and critically discuss the mounting experimental evidence for such PDW order in the cuprate superconductors, the status of the theoretical microscopic description of such order, and the current debate on whether the PDW is a mother order or another competing order in the cuprates. In addition, we give an overview of the weak coupling version of PDW order, Fulde-Ferrell-Larkin-Ovchinnikov states, in the context of cold atom systems, unconventional superconductors, and non-centrosymmetric and Weyl materials.

Hart Goldman, Ramanjit Sohal, and Eduardo Fradkin, Landau-Ginzburg Theories of Non-Abelian Quantum Hall States from Non-Abelian Bosonization, Physical Review B 100, 115111 (2019).

It is an important open problem to understand the landscape of non-Abelian fractional quantum Hall phases which can be obtained starting from physically motivated theories of Abelian composite particles. We show that progress on this problem can be made using recently proposed non-Abelian bosonization dualities in \(2+1\) dimensions, which morally relate \(U(N)_k\) and \(SU(k)_N\) Chern-Simons-matter theories. The advantage of these dualities is that regions of the phase diagram which may be obscure on one side of the duality can be accessed by condensing local operators on the other side. Starting from parent Abelian states, we use this approach to construct Landau-Ginzburg theories of non-Abelian states through a pairing mechanism. In particular, we obtain the bosonic Read-Rezayi sequence at fillings \(\nu=k/(kM+2)\) by starting from k layers of bosons at \(\nu=1/2\) with M Abelian fluxes attached. The Read-Rezayi states arise when k clusters of the dual non-Abelian bosons condense. We extend this construction by showing that \(N_f\)-component generalizations of the Halperin \( (2,2,1) \) bosonic states have dual descriptions in terms of \(SU(Nf+1)_1\) Chern-Simons-matter theories, revealing an emergent global symmetry in the process. Clustering k layers of these theories yields a non-Abelian \(SU(N_f)\)-singlet state at filling \(\nu=kN_f/(Nf+1+kMN_f)\).

Hart Goldman, Ramanjit Sohal, and Eduardo Fradkin, A composite particle construction of the Fibonacci fractional quantum Hall state, Physical Review B 103, 235118 (2021).

The Fibonacci topological order is the simplest platform for a universal topological quantum computer. While the \(\nu=12/5\) fractional quantum Hall (QH) state has been proposed to support a Fibonacci sector, a dynamical picture of how a pure Fibonacci state may emerge in a QH system has been lacking. We use non-Abelian dualities to construct a Fibonacci state of bosons at filling \(\nu=2\) starting from a trilayer of integer QH states. Our parent theory consists of bosonic composite vortices coupled to fluctuating \(U(2)\) gauge fields, which is dual to the theory of Laughlin quasiparticles. The Fibonacci state is obtained by interlayer clustering of the composite vortices, along with flux attachment. We use this framework to motivate a wave function for the Fibonacci state.