Search

The interplay between quantum and thermal fluctuations in the presence of quenched random disorder is a long-standing open theoretical problem which has been made more urgent by advances in modern experimental techniques. The fragility of charge density wave order to impurities makes this problem of particular interest in understanding a host of real materials, including the cuprate high-temperature superconductors. To address this question, we consider the quantum version of an exactly solvable classical model of two-dimensional randomly pinned incommensurate charge density waves first introduced by us in a recent work, and use the large-$N$ technique to obtain the phase diagram and order parameter correlations. Our theory considers quantum and thermal fluctuations and disorder on equal footing by accounting for all effects non-perturbatively, which reveals a novel crossover between under-damped and over-damped dynamics of the fluctuations of the charge density wave order parameter., Comment: 46 pages, 7 figures

The long-sought pair density wave (PDW) is an exotic phase of matter in which charge density wave (CDW) order is intertwined with the amplitude or phase of coexisting, superconducting order \cite{Berg2009,Berg2009b}. Originally predicted to exist in copper-oxides, circumstantial evidence for PDW order now exists in a variety of materials. Recently, scanning tunneling microscopy (STM) studies have reported evidence for a three-component charge density wave (CDW) at the surface of the heavy-fermion superconductor, UTe$_2$, persisting below its superconducting transition temperature. Here, we use hard x-ray diffraction measurements on crystals of UTe$_2$ at $T = 1.9$ K and $12$ K to search for a bulk signature of this CDW. Using STM measurements as a constraint, we calculate the expected locations of CDW superlattice peaks, and sweep a large volume of reciprocal space in search of a signature. We failed to find any evidence for a CDW near any of the expected superlattice positions in many Brillouin zones. We estimate an upper bound on the CDW lattice distortion of $u_{max} \lesssim 4 \times 10^{-3} \mathrm{\AA}$. Our results suggest that the CDW observed in STM is either purely electronic, somehow lacking a signature in the structural lattice, or is restricted to the material surface.

Pair density Waves (PDW) are a inhomogeneous superconducting states whose Cooper pairs posses a finite momentum resulting in a oscillatory gap in space, even in the absence of an external magnetic field. There is growing evidence for the existence of PDW superconducting order in many strongly correlated materials particularly in the cuprate superconductios and in several other different types of systems. A feature of the PDW state is that inherently it has a CDW as a composite order associated with it. Here we study the structure of the electronic topological defects of the PDW, paying special attention to the half-vortex and its electronic structure that can be detected in STM experiments. We discuss tell-tale signatures of the defects in violations of inversion symmetry, in the excitation spectrum and their spectral functions in the presence of topological defects. We discuss the "Fermi surface" topology of Bogoliubov quasi-particle of the PDW phases and we briefly discuss the role of quasi-particle-interference., Comment: We added plots of the argument of the order parameter in fig 3. The STM plots better indicate the low bias particle-hole symmetry (Figs. 8,9,14 and 15) and the layouts modified. A few typos were corrected as well as some additional clarifications incuded throughout text. Fig 12 had a mistake (wrong data) and has been updated and figure 16 has one less plot present

Geometric fluctuations of the density mode in a fractional quantum Hall (FQH) state can give rise to a nematic FQH phase, a topological state with a spontaneously broken rotational symmetry. While experiments on FQH states in the second Landau level have reported signatures of putative FQH nematics in anisotropic transport, a realistic model for this state has been lacking. We show that the standard model of particles in the lowest Landau level interacting via the Coulomb potential realizes the FQH nematic transition, which is reached by a progressive reduction of the strength of the shortest-range Haldane pseudopotential. Using exact diagonalization and variational wave functions, we demonstrate that the FQH nematic transition occurs when the system's neutral gap closes in the long-wavelength limit while the charge gap remains open. We confirm the symmetry-breaking nature of the transition by demonstrating the existence of a "circular moat" potential in the manifold of states with broken rotational symmetry, while its geometric character is revealed through the strong fluctuations of the nematic susceptibility and Hall viscosity., Comment: Four figures in main text with supplementary information

In this work we study the interaction between vortices and nematic domain walls within the framework of a Ginzburg Landau approach. The free energy of the system is written in terms of a complex order parameter characteristic of $s$-wave superconductivity and a real (Ising type) order parameter associated to nematicity. The interaction between both order parameters is described by a biquadratic and a trilinear derivative term. To study the effects of these interactions we solve the time-dependent dissipative Ginzburg Landau equations using a highly performant pseudospectral method by which we calculate the trajectories of a vortex that, for different coupling parameters, is either attracted or repelled by a wall, as well as of the wall dynamics. We show that despite its simplicity, this theory displays many phenomena observed experimentally in Fe-based superconductors. In particular we find that the sign of the biquadratic term determines the attractive (pining) or repulsive (antipining) character of the interaction, as observed in FeSe and BaFeCoAs compounds respectively. The trilinear term is responsible for the elliptical shape of vortex cores as well as for the orientation of the axes of the ellipses and vortex trajectories with respect to the axes of the structural lattice. For the case of pining, we show that the vortex core is well described by a heart-shaped structure in agreement with STM experiments performed in FeSe.

Parafermion zero modes can arise in hybrid structures composed of $\nu=1/m$ fractional quantum Hall edges proximitized with an s-wave superconductor. Here we consider parafermion and Cooper pair tunneling, and backscattering in a junction formed in such hybrid structures. We find that the $4\pi m$ periodicity due to parafermion-only tunneling reduces, in the presence of backscattering, to $4\pi$-periodic at zero temperature and $2\pi$-periodic at finite temperature unless the fermion parity is fixed. Nevertheless, a clear signature of parafermion tunneling remains in the shape of the current-phase relation., Comment: v1: 6 pages, 4 figures; v2: 8 pages, 5 figures, appendix added

The nature of the interplay between fluctuations and quenched random disorder is a long-standing open problem, particularly in systems with a continuous order parameter. This lack of a full theoretical treatment has been underscored by recent advances in experiments on charge density wave materials. To address this problem, we formulate an exactly solvable model of a two-dimensional randomly pinned incommensurate charge density wave, and use the large-$N$ technique to map out the phase diagram and order parameter correlations. Our approach captures the physics of the Berezinskii-Kosterlitz-Thouless phase transition in the clean limit at large $N$. We pay particular attention to the roles of thermal fluctuations and quenched random field disorder in destroying long-range order, finding a novel crossover between weakly- and strongly-disordered regimes., Comment: 40 pages, 4 figures

$GdTe_{3}$ is a layered antiferromagnet which has attracted attention due to its exceptionally high mobility, distinctive unidirectional incommensurate charge density wave (CDW), superconductivity under pressure, and a cascade of magnetic transitions between 7 and 12 K, with as yet unknown order parameters. Here, we use spin-polarized scanning tunneling microscopy to directly image the charge and magnetic orders in $GdTe_{3}$. Below 7 K, we find a striped antiferromagnetic phase with twice the periodicity of the Gd lattice and perpendicular to the CDW. As we heat the sample, we discover a spin density wave with the same periodicity as the CDW between 7 and 12 K; the viability of this phase is supported by our Landau free energy model. Our work reveals the order parameters of the magnetic phases in $GdTe_{3}$ and shows how the interplay between charge and spin can generate a cascade of magnetic orders., Comment: 46 pgs.; 4 main figures, 20 supplementary figures

Topological defects are singularities in an ordered phase that can have a profound effect on phase transitions and serve as a window into the order parameter. In this work we use scanning tunneling microscopy to visualize the role of topological defects in the novel magnetic field induced disappearance of an intertwined charge density wave (CDW) in the heavy fermion superconductor, UTe2. By simultaneously imaging the amplitude and phase of the CDW order, we reveal pairs of topological defects with positive and negative phase winding. The pairs are directly correlated with a zero CDW amplitude and increase in number with increasing magnetic field. These observations can be captured by a Ginzburg Landau model of a uniform superconductor coexisting with a pair density wave. A magnetic field generates vortices of the superconducting and pair density wave order which can create topological defects in the CDW and induce the experimentally observed melting of the CDW at the upper critical field. Our work reveals the important role of magnetic field generated topological defects in the melting the CDW order parameter in UTe2 and provides support for the existence of a parent pair density wave order on the surface of UTe2., Comment: 29 Pages, includes manuscript and supplemental information, 4 Main figures and 8 Supplemental Figures

In this chapter I discuss the impact of concepts of Quantum Field Theory in modern Condensed Physics. Although the interplay between these two areas is certainly not new, the impact and mutual cross-fertilization has certainly grown enormously with time, and Quantum Field Theory has become a central conceptual tool in Condensed Matter Physics. In this chapter I cover how these ideas and tools have influenced our understanding of phase transitions, both classical and quantum, as well as topological phases of matter, and dualities., Comment: Revised version to appear as a chapter in the Encyclopedia of Condensed Matter Physics 2e. 126 pages, including 380 references

The intense interest in triplet superconductivity partly stems from theoretical predictions of exotic excitations such as non-abelian Majorana modes, chiral supercurrents, and half-quantum vortices. However, fundamentally new, and unexpected states may emerge when triplet superconductivity appears in a strongly correlated system. In this work we use scanning tunneling microscopy to reveal an unusual charge density wave (CDW) order in the heavy fermion triplet superconductor, UTe2. Our high-resolution maps reveal a multi-component incommensurate CDW whose intensity get weaker with increasing field, eventually disappearing at the superconducting critical field, Hc2. To explain the origin and phenomenology of this unusual CDW, we construct a Ginzburg-Landau theory for a uniform triplet superconductor coexisting with three triplet pair density wave (PDW) states. This theory gives rise to daughter CDWs which would be sensitive to magnetic field due to their origin in a triplet PDW state, and naturally explains our data. Our discovery of a CDW sensitive to magnetic fields and strongly intertwined with superconductivity, provides important new information for understanding the order parameter of UTe2 and uncovers the possible existence of a new kind of triplet PDW order which has not been previously explored., Comment: 12 pages, 4 figures, Supplementary information

A long-standing problem in the study of topological phases of matter has been to understand the types of fractional topological insulator (FTI) phases possible in 3+1 dimensions. Unlike ordinary topological insulators of free fermions, FTI phases are characterized by fractional $\Theta$-angles, long-range entanglement, and fractionalization. Starting from a simple family of $\mathbb{Z}_N$ lattice gauge theories due to Cardy and Rabinovici, we develop a class of FTI phases based on the physical mechanism of oblique confinement and the modern language of generalized global symmetries. We dub these phases oblique topological insulators. Oblique TIs arise when dyons -- bound states of electric charges and monopoles -- condense, leading to FTI phases characterized by topological order, emergent one-form symmetries, and gapped boundary states not realizable in 2+1-D alone. Based on the lattice gauge theory, we present continuum topological quantum field theories (TQFTs) for oblique TI phases involving fluctuating one-form and two-form gauge fields. We show explicitly that these TQFTs capture both the generalized global symmetries and topological orders seen in the lattice gauge theory. We also demonstrate that these theories exhibit a universal "generalized magnetoelectric effect" in the presence of two-form background gauge fields. Moreover, we characterize the possible boundary topological orders of oblique TIs, finding a new set of boundary states not studied previously for these kinds of TQFTs., Comment: 67 pages, 4 figures, 5 appendices. v2: new appendix, added references, and minor changes. v3: light revisions, additional discussion in Section 6 and Appendix E, and updated acknowledgements and references

In this work we explore the interplay between superconductivity and nematicity in the framework of a Ginzburg Landau theory with a nematic order parameter coupled to the superconductor order parameter, often used in the description of superconductivity of Fe based materials. In particular, we focus on the study of the vortex-vortex interaction in order to determine the way nematicity affects its attractive or repulsive character. To do so, we use a dynamical method based on the solutions of the Time Dependent Ginzburg Landau equations in a bulk superconductor. An important contribution of our work is the implementation of a pseudo-spectral method to solve the dynamics, known to be highly efficient and of very high order in comparison to the usual finite differences/elements methods. The coupling between the superconductor and the (real) nematic order parameters is represented by two terms in the free energy: a biquadratic term and a coupling of the nematic order parameter to the covariant derivatives of the superconductor order parameter. Our results show that there is a competing effect: while the former independently of its competitive or cooperative character generates an attractive vortex-vortex interaction, the latter always generates a repulsive interaction., Comment: 37 pages; 12 figures; corrected typos; added references

Understanding the nature of charge density waves (CDW) in cuprate superconductors has been complicated by material specific differences. A striking example is the opposite doping dependence of the CDW ordering wavevector in La-based and Y-based compounds, the two families where charge ordering is strongest and best characterized. Here we report a combined resonant soft X-ray scattering (RSXS) and neutron scattering study of charge and spin density waves in isotopically enriched La$_{1.8-x}$ Eu$_{0.2}$ Sr$_{x}$ CuO$_{4}$ over a range of doping $0.07 \leq x \leq 0.20$. For all dopings studied by RSXS, we find that the CDW amplitude is approximately temperature-independent and develops well above experimentally accessible temperatures. Surprisingly, the CDW ordering wavevector shows a non-monotonic temperature dependence, with a sudden change occurring at temperatures near the SDW onset temperature. We describe this behavior with a Landau-Ginzburg theory for an incommensurate CDW in a metallic system with a finite charge compressibility and CDW-SDW coupling. Our Landau-Ginzburg analysis suggests that the ordering wavevector at high temperatures decreases with increased doping. This behavior is opposite to the trend at low temperatures and highly reminiscent of the doping dependence seen in YBa$_2$ Cu$_3$ O$_{6+\delta}$ , suggesting a common origin of the CDW in hole-doped cuprate superconductors., Comment: Manuscript submitted to PNAS; ArXiv version formatted in Revtex4-2; Supplementary Material included as appendices; 20 pages, 14 figures (6 in the main text), and 41 references

We investigate the thermalization of Sachdev-Ye-Kitaev (SYK) models coupled via random interactions following quenches from the perspective of entanglement. Previous studies have shown that when a system of two SYK models coupled by random two-body terms is quenched from the thermofield double state with sufficiently low effective temperature, the R\'enyi entropies do not saturate to the expected thermal values in the large-$N$ limit. Using numerical large-$N$ methods, we first show that the R\'enyi entropies in a pair SYK models coupled by two-body terms can thermalize, if quenched from a state with sufficiently high effective temperature, and hence exhibit state-dependent thermalization. In contrast, SYK models coupled by single-body terms appear to always thermalize. We provide evidence that the subthermal behavior in the former system is likely a large-$N$ artifact by repeating the quench for finite $N$ and finding that the saturation value of the R\'enyi entropy extrapolates to the expected thermal value in the $N \to \infty$ limit. Finally, as a finer grained measure of thermalization, we compute the late-time spectral form factor of the reduced density matrix after the quench. While a single SYK dot exhibits perfect agreement with random matrix theory, both the quadratically and quartically coupled SYK models exhibit slight deviations., Comment: v1: 41 pages single column, 14 figures; v2: Additional references; v3: Minor revisions

How superconductivity interacts with charge or nematic order is one of the great unresolved issues at the center of research in quantum materials. Ba$_{1-x}$Sr$_{x}$Ni$_{2}$As$_{2}$ (BSNA) is a charge ordered pnictide superconductor recently shown to exhibit a six-fold enhancement of superconductivity due to nematic fluctuations near a quantum phase transition (at $x_c=0.7$). The superconductivity is, however, anomalous, with the resistive transition for $0.4 < x< x_c$ occurring at a higher temperature than the specific heat anomaly. Using x-ray scattering, we discovered a new charge density wave (CDW) in BSNA in this composition range. The CDW is commensurate with a period of two lattice parameters, and is distinct from the two CDWs previously reported in this material. We argue that the anomalous transport behavior arises from heterogeneous superconductivity nucleating at antiphase domain walls in this CDW. We also present new data on the incommensurate CDW, previously identified as being unidirectional, showing that is a rotationally symmetric, "4$Q$" state with $C_4$ symmetry. Our study establishes BSNA as a rare material containing three distinct CDWs, and an exciting testbed for studying coupling between CDW, nematic, and SC orders., Comment: 5 pages, 5 figures + supplemental material document

The Fibonacci topological order is the simplest platform for a universal topological quantum computer, consisting of a single type of non-Abelian anyon, $\tau$, with fusion rule $\tau\times\tau=1+\tau$. While it has been proposed that the anyon spectrum of the $\nu=12/5$ fractional quantum Hall state includes a Fibonacci sector, a dynamical picture of how a pure Fibonacci state may emerge in a quantum Hall system has been lacking. Here we use recently proposed non-Abelian dualities to construct a Fibonacci state of bosons at filling $\nu=2$ starting from a trilayer of integer quantum Hall states. Our parent theory consists of bosonic "composite vortices" coupled to fluctuating $U(2)$ gauge fields, which is related to the standard theory of Laughlin quasiparticles by duality. The Fibonacci state is obtained by clustering the composite vortices between the layers, along with flux attachment, a procedure reminiscent of the clustering picture of the Read-Rezayi states. We further use this framework to motivate a wave function for the Fibonacci fractional quantum Hall state., Comment: 12 pages, 3 figures, and 5 appendices

The recent proposal of non-Abelian boson-fermion dualities in 2+1 dimensions, which morally relate $U(k)_N$ to $SU(N)_{-k}$ Chern-Simons-matter theories, presents a new platform for exploring the landscape of non-Abelian quantum Hall states accessible from theories of Abelian composite particles. Here we focus on dualities relating theories of Abelian quantum Hall states of bosons or fermions to theories of non-Abelian "composite fermions" partially filling Landau levels. We show that these dualities predict special filling fractions where both Abelian and non-Abelian composite fermion theories appear capable of hosting distinct topologically ordered ground states, one Abelian and the other a non-Abelian, $U(k)_2$ Blok-Wen state. Rather than being in conflict with the duality, we argue that these results indicate unexpected dynamics in which the infrared and lowest Landau level limits fail to commute across the duality. In such a scenario, the non-Abelian topological order can be destabilized in favor of the Abelian ground state, suggesting the presence of a phase transition between the Abelian and non-Abelian states that is likely to be first order. We also generalize these constructions to other non-Abelian fermion-fermion dualities, in the process obtaining new derivations of a variety of paired composite fermion phases using duality, including the anti-Pfaffian state. Finally, we describe how, in multilayer constructions, excitonic pairing of the composite fermions across $N$ layers can also generate the family of Blok-Wen states with $U(k)_2$ topological order., Comment: 34 pages, including two appendices, one figure, and three tables. New version contains light revisions, as well as some additional results and discussion in Section 3.6

We investigate the problem of intertwined orders in fractional Chern insulators by considering lattice fractional quantum Hall (FQH) states arising from pairing of composite fermions in the square-lattice Hofstadter model. At certain filling fractions, magnetic translation symmetry ensures the composite fermions form Fermi surfaces with multiple pockets, leading to the formation of finite-momentum Cooper pairs in the presence of attractive interactions. We obtain mean-field phase diagrams exhibiting a rich array of striped and topological phases, establishing paired lattice FQH states as an ideal platform to investigate the intertwining of topological and conventional broken symmetry order., Comment: 15 pages, 11 figures. v2: Updated references and acknowledgments. v3: Published version

The Kondo-Heinsberg chain is an interesting model of a strongly correlated system which has a broad superconducting state with pair-density wave (PDW) order. Some of us have recently proposed that this PDW state is a symmetry-protected topological (SPT) state, and the gapped spin sector of the model supports Majorana zero modes. In this work, we reexamine this problem using a combination of numeric and analytic methods. In extensive density matrix renormalization group calculations, we find no evidence of a topological ground state degeneracy or the previously proposed Majorana zero modes in the PDW phase of this model. This result motivated us to reexamine the original arguments for the existence of the Majorana zero modes. A careful analysis of the effective continuum field theory of the model shows that the Hilbert space of the spin sector of the theory does not contain any single Majorana fermion excitations. This analysis shows that the PDW state of the doped 1D Kondo-Heisenberg model is not an SPT with Majorana zero modes., Comment: 13 pages, 8 figures

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 $SU(N_f+1)_1$ 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 $\nu = kN_f / (N_f + 1 + kMN_f)$., Comment: 34 pages, 1 figure. Minor typos corrected

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 meting 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., Comment: 43 pages, 16 figures, 236 references. Mildly edited final version to appear in Annual Reviews of Condensed Matter Physics (with improved figures)

The properties of the isotropic incompressible $\nu=5/2$ fractional quantum Hall (FQH) state are described by a paired state of composite fermions in zero (effective) magnetic field, with a uniform $p_x+ip_y$ pairing order parameter, which is a non-Abelian topological phase with chiral Majorana and charge modes at the boundary. Recent experiments suggest the existence of a proximate nematic phase at $\nu=5/2$. This finding motivates us to consider an inhomogeneous paired state - a $p_x+ip_y$ pair-density-wave (PDW) - whose melting could be the origin of the observed liquid-crystalline phases. This state can viewed as an array of domain and anti-domain walls of the $p_x+i p_y$ order parameter. We show that the nodes of the PDW order parameter, the location of the domain walls (and anti-domain walls) where the order parameter changes sign, support a pair of symmetry-protected counter-propagating Majorana modes. The coupling behavior of the domain wall Majorana modes crucially depends on the interplay of the Fermi energy $E_{F}$ and the PDW pairing energy $E_{\textrm{pdw}}$. The analysis of this interplay yields a rich set of topological states. The pair-density-wave order state in paired FQH system provides a fertile setting to study Abelian and non-Abelian FQH phases - as well as transitions thereof - tuned by the strength of the paired liquid crystalline order., Comment: 27 pages, 11 figures; Published version

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., Comment: 25 pages, minor revisions

We study the thermoelectric response of a device containing a pair of helical edge states contacted at the same temperature $T$ and chemical potential $\mu$ and connected to an external reservoir, with different chemical potential and temperature, through a side quantum dot. Different operational modes can be induced by applying a magnetic field $B$ and a gate voltage $V_g$ at the quantum dot. At finite $B$, the quantum dot acts simultaneously as a charge and a spin filter. Charge and spin currents are induced, not only through the quantum dot, but also along the edge states. We focus on linear response and analyze the regimes, which we identify as charge heat engines or refrigerator, spin heat engine and spin refrigerator., Comment: 11 pages, 9 figures

Novel controlled non-perturbative techniques are a must in the study of strongly correlated systems, especially near quantum criticality. One of these techniques, bosonization, has been extensively used to understand one-dimensional, as well as higher dimensional electronic systems at finite density. In this paper, we generalize the theory of two-dimensional bosonization of Fermi liquids, in the presence of a homogeneous weak magnetic field perpendicular to the plane. Here, we extend the formalism of bosonization to treat free spinless fermions at finite density in a uniform magnetic field. We show that particle-hole fluctuations of a Fermi surface satisfy a {\em covariant Schwinger algebra}, allowing to express a fermionic theory with forward scattering interactions as a quadratic bosonic theory representing the quantum fluctuations of the Fermi surface. By means of a coherent-state path integral formalism we compute the fermion propagator as well as particle-hole bosonic correlations functions. We analyze the presence of de Haas-van Alphen oscillations and show how the quantum oscillations of the orbital magnetization, the Lifshitz-Kosevich theory, are obtained by means of the bosonized theory. We also study the effects of forward scattering interactions. In particular, we obtain oscillatory corrections to the Landau zero sound collective mode., Comment: 20 pages, 6 figures. Improved version. Two figures added. References updated. New section on forward scattering interactions added

We study signatures of chaos in the quantum Lifshitz model through out-of-time ordered correlators (OTOC) of current operators. This model is a free scalar field theory with dynamical critical exponent $z=2$. It describes the quantum phase transition in 2D systems, such as quantum dimer models, between a phase with an uniform ground state to another one with a spontaneously translation invariance. At the lowest temperatures the chaotic dynamics are dominated by a marginally irrelevant operator which induces a temperature dependent stiffness term. The numerical computations of OTOC exhibit a non-zero Lyapunov exponent (LE) in a wide range of temperatures and interaction strengths. The LE (in units of temperature) is a weakly temperature-dependent function; it vanishes at weak interaction and saturates for strong interaction. The Butterfly velocity increases monotonically with interaction strength in the studied region while remaining smaller than the interaction-induced velocity/stiffness., Comment: 15 pages + appendices. 12 figures

We study the $2+1$ dimensional boson-fermion duality in the presence of background curvature and electromagnetic fields. The main players are, on the one hand, a massive complex $|\phi|^4$ scalar field coupled to a $U(1)$ Maxwell-Chern-Simons gauge field at level $1$, representing a relativistic composite boson with one unit of attached flux, and on the other hand, a massive Dirac fermion. We show that, in a curved background and at the level of the partition function, the relativistic composite boson, in the infinite coupling limit, is dual to a short-range interacting Dirac fermion. The coupling to the gravitational spin connection arises naturally from the spin factors of the Wilson loop in the Chern-Simons theory. A non-minimal coupling to the scalar curvature is included on the bosonic side in order to obtain agreement between partition functions. Although an explicit Lagrangian expression for the fermionic interactions is not obtained, their short-range nature constrains them to be irrelevant, which protects the duality in its strong interpretation as an exact mapping at the IR fixed point between a Wilson-Fischer-Chern-Simons complex scalar and a free Dirac fermion. We also show that, even away from the IR, keeping the $|\phi|^4$ term is of key importance as it provides the short-range bosonic interactions necessary to prevent intersections of worldlines in the path integral, thus forbidding unknotting of knots and ensuring preservation of the worldline topologies., Comment: Final version published in Annals of Physics

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., Comment: 14 pages, 10 figures; minor corrections in the numerical results

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., Comment: 41 pages, including two appendices

We study the linear thermoelectric response of a quantum dot embedded in a constriction of a quantum Hall bar with fractional filling factors nu=1/m within Laughlin series. We calculate the figure of merit ZT for the maximum efficiency at a fixed temperature difference. We find a significant enhancement of this quantity in the fractional filling in relation to the integer-filling case, which is a direct consequence of the fractionalization of the electron in the fractional quantum Hall state. We present simple theoretical expressions for the Onsager coefficients at low temperatures, which explicitly show that ZT and the Seebeck coefficient increase with m., Comment: 5 pages, 4 figures, suplemental material

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., Comment: 14 pages, 7 figures. Expanded version in PRB format with revised mean field analysis (conclusions are essentially unchanged)

We study gapless quantum spin chains with spin 1/2 and 1: the Fredkin and Motzkin models. Their entangled groundstates are known exactly but not their excitation spectra. We first express the groundstates in the continuum which allows for the calculation of spin and entanglement properties in a unified fashion. Doing so, we uncover an emergent conformal-type symmetry, thus consolidating the connection to a widely studied family of Lifshitz quantum critical points in 2d. We then obtain the low lying excited states via large-scale DMRG simulations and find that the dynamical exponent is z = 3.2 in both cases. Other excited states show a different z, indicating that these models have multiple dynamics. Moreover, we modify the spin-1/2 model by adding a ferromagnetic Heisenberg term, which changes the entire spectrum. We track the resulting non-trivial evolution of the dynamical exponents using DMRG. Finally, we exploit an exact map from the quantum Hamiltonian to the non-equilibrium dynamics of a classical spin chain to shed light on the quantum dynamics., Comment: For a special issue of Journal of Physics A celebrating John Cardy's 70th birthday. 19+6 pages, 7+3 figures, 2+1 tables. v2: final version with minor modifications

Many-body systems with multiple emergent time scales arise in various contexts, including classical critical systems, correlated quantum materials, and ultra-cold atoms. We investigate such non-trivial quantum dynamics in a new setting: a spin-1 bilinear-biquadratic chain. It has a solvable entangled groundstate, but a gapless excitation spectrum that is poorly understood. By using large-scale DMRG simulations, we find that the lowest excitations have a dynamical exponent $z$ that varies from 2 to 3.2 as we vary a coupling in the Hamiltonian. We find an additional gapless mode with a continuously varying exponent $2\leq z <2.7$, which establishes the presence of multiple dynamics. In order to explain these striking properties, we construct a continuum wavefunction for the groundstate, which correctly describes the correlations and entanglement properties. We also give a continuum parent Hamiltonian, but show that additional ingredients are needed to capture the excitations of the chain. By using an exact mapping to the non-equilibrium dynamics of a classical spin chain, we find that the large dynamical exponent is due to subdiffusive spin motion. Finally, we discuss the connections to other spin chains and to a family of quantum critical models in 2d., Comment: 4+5 pages, 2+4 figures, 1 table. v2: minor changes to reflect the published version

We construct the symmetric-gapped surface states of a fractional topological insulator with electromagnetic $\theta$-angle $\theta_{em} = \frac{\pi}{3}$ and a discrete $\mathbb{Z}_3$ gauge field. They are the proper generalizations of the T-pfaffian state and pfaffian/anti-semion state and feature an extended periodicity compared with their of "integer" topological band insulators counterparts. We demonstrate that the surface states have the correct anomalies associated with time-reversal symmetry and charge conservation., Comment: 5 pages, 33 references and 2 pages of supplemental material

We construct an effective field theory, a two-dimensional two-component metallic system described by a model with two Fermi surfaces ("pockets"). This model describes a translationally invariant metallic system with two types of fermions, each with its own Fermi surface, with forward scattering interactions. This model, in addition to the $O(2)$ rotational invariance, has a $U(1) \times U(1)$ symmetry of separate charge conservation for each Fermi surface. For sufficiently attractive interactions in the $d$-wave (quadrupolar) channel this model has an interesting phase diagram that includes a spontaneously generated anomalous Hall metal phase. We derive the Landau-Ginzburg effective action of quadrupolar order parameter fields which enjoys an $O(2)\times U(1)$ global symmetry associated to spatial isotropy and the internal $U(1)$ relative phase symmetries respectively. We show that the order parameter theory is dynamically local with a dynamical scaling of $z=2$ and perform a one-loop renormalization group analysis of the Landau-Ginzburg theory. The electronic liquid crystal phases that result from spontaneous symmetry breaking are studied and we show the presence of Landau damped Nambu-Goldstone modes at low momenta that is a signature of non-Fermi liquid behavior. Electromagnetic linear response is also analyzed in both the normal and symmetry broken phases from the point of view of the order parameter theory. The nature of the coupling of electromagnetism to the order parameter fields in the normal phase is non-minimal and decidedly contains a precursor to the anomalous Hall response in the form of a order-parameter-dependent Chern-Simons term in the effective action., Comment: 31 pages, 12 figures

Frustrated quantum magnets are a central theme in condensed matter physics due to the richness of their phase diagrams. They support a panoply of phases including various ordered states and topological phases. Yet, this problem has defied a solution for a long time due to the lack of controlled approximations which make it difficult to distinguish between competing phases. Here we report the discovery of a special quantum macroscopically degenerate point in the XXZ model on the spin-1/2 kagome quantum antiferromagnet for the ratio of Ising to antiferromagnetic transverse coupling Jz/J=-1/2. This point is proximate to many competing phases explaining the source of the complexity of the phase diagram. We identify five phases near this point including both spin-liquid and broken-symmetry phases and give evidence that the kagome Heisenberg antiferromagnet is close to a transition between two phases., Comment: 5 pages, including 4 figures + 1 table. 7 pages of supplementary information with 4 figures + 1 table. v2 closely matches the version accepted in PRL (Editor's suggestion). PRL title is "Macroscopically Degenerate Exactly Solvable Point in the Spin-1/2 Kagome Quantum Antiferromagnet"

The pair density wave (PDW) superconducting state has been proposed to explain the layer- decoupling effect observed in the compound La$_{2-x}$Ba$_x$CuO$_4$ at $x=1/8$ (Phys. Rev. Lett. 99, 127003). In this state the superconducting order parameter is spatially modulated, in contrast with the usual superconducting (SC) state where the order parameter is uniform. In this work, we study the properties of the amplitude (Higgs) modes in a unidirectional PDW state. To this end we consider a phenomenological model of PDW type states coupled to a Fermi surface of fermionic quasiparticles. In contrast to conventional superconductors that have a single Higgs mode, unidirectional PDW superconductors have two Higgs modes. While in the PDW state the Fermi surface largely remains gapless, we find that the damping of the PDW Higgs modes into fermionic quasiparticles requires exceeding an energy threshold. We show that this suppression of damping in the PDW state is due to kinematics. As a result, only one of the two Higgs modes is significantly damped. In addition, motivated by the experimental phase diagram, we discuss the mixing of Higgs modes in the coexistence regime of the PDW and uniform SC states. These results should be observable directly in a Raman spectroscopy, in momentum resolved electron energy loss spectroscopy, and in resonant inelastic X-ray scattering, thus providing evidence of the PDW states., Comment: 11 pages, 6 figures; replaced with published version

In this paper, we propose a generalization of the $S$-duality of four-dimensional quantum electrodynamics ($\text{QED}_4$) to $\text{QED}_4$ with fractionally charged excitations, the fractional $S$-duality. Such $\text{QED}_4$ can be obtained by gauging the $\text{U(1)}$ symmetry of a topologically ordered state with fractional charges. When time-reversal symmetry is imposed, the axion angle ($\theta$) can take a nontrivial but still time-reversal invariant value $\pi/t^2$ ($t\in\mathbb{Z}$). Here, $1/t$ specifies the minimal electric charge carried by bulk excitations. Such states with time-reversal and $\text{U(1)}$ global symmetry (fermion number conservation) are fractional topological insulators (FTI). We propose a topological quantum field theory description, which microscopically justifies the fractional $S$-duality. Then, we consider stacking operations (i.e., a direct sum of Hamiltonians) among FTIs. We find that there are two topologically distinct classes of FTIs: type-I and type-II. Type-I ($t\in\mathbb{Z}_{\rm odd}$) can be obtained by directly stacking a non-interacting topological insulator and a fractionalized gapped fermionic state with minimal charge $1/t$ and vanishing $\theta$. But type-II ($t\in\mathbb{Z}_{\rm even}$) cannot be realized through any stacking. Finally, we study the Surface Topological Order of fractional topological insulators., Comment: 10 pages

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 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., Comment: 11 pages, 3 figures. Removed some repeated references

The single-layered ruthenate Sr$_2$RuO$_4$ has attracted a great deal of interest as a spin-triplet superconductor with an order parameter that may potentially break time reversal invariance and host half-quantized vortices with Majorana zero modes. While the actual nature of the superconducting state is still a matter of controversy, it has long been believed that it condenses from a metallic state that is well described by a conventional Fermi liquid. In this work we use a combination of Fourier transform scanning tunneling spectroscopy (FT-STS) and momentum resolved electron energy loss spectroscopy (M-EELS) to probe interaction effects in the normal state of Sr$_2$RuO$_4$. Our high-resolution FT-STS data show signatures of the \beta-band with a distinctly quasi-one-dimensional (1D) character. The band dispersion reveals surprisingly strong interaction effects that dramatically renormalize the Fermi velocity, suggesting that the normal state of Sr$_2$RuO$_4$ is that of a 'correlated metal' where correlations are strengthened by the quasi 1D nature of the bands. In addition, kinks at energies of approximately 10meV, 38meV and 70meV are observed. By comparing STM and M-EELS data we show that the two higher energy features arise from coupling with collective modes. The strong correlation effects and the kinks in the quasi 1D bands may provide important information for understanding the superconducting state. This work opens up a unique approach to revealing the superconducting order parameter in this compound., Comment: 17 pages, 4 figures

Bose condensation has shaped our understanding of macroscopic quantum phenomena, having been realized in superconductors, atomic gases, and liquid helium. Excitons are bosons that have been predicted to condense into either a superfluid or an insulating electronic crystal. Using the recently developed momentum-resolved electron energy-loss spectroscopy (M-EELS), we study electronic collective modes in the transition metal dichalcogenide semimetal, 1T-TiSe$_2$. Near the phase transition temperature, T$_c$ = 190 K, the energy of the electronic mode falls to zero at nonzero momentum, indicating dynamical slowing down of plasma fluctuations and crystallization of the valence electrons into an exciton condensate. Our study provides compelling evidence for exciton condensation in a three-dimensional solid and establishes M-EELS as a versatile technique sensitive to valence band excitations in quantum materials., Comment: missing figures added

We study the von Neumann and R\'enyi entanglement entropy (EE) of scale-invariant theories defined on tori in 2+1 and 3+1 spacetime dimensions. We focus on spatial bi-partitions of the torus into two cylinders, and allow for twisted boundary conditions along the non-contractible cycles. Various analytical and numerical results are obtained for the universal EE of the relativistic boson and Dirac fermion conformal field theories (CFTs), and for the fermionic quadratic band touching and the boson with $z=2$ Lifshitz scaling. The shape dependence of the EE clearly distinguishes these theories, although intriguing similarities are found in certain limits. We also study the evolution of the EE when a mass is introduced to detune the system from its scale-invariant point, by employing a renormalized EE that goes beyond a naive subtraction of the area law. In certain cases we find non-monotonic behavior of the torus EE under RG flow, which distinguishes it from the EE of a disk., Comment: 47 pages, 13 figures

I review the concept of a {\em disorder operator}, introduced originally by Kadanoff in the context of the two-dimensional Ising model. Disorder operators acquire an expectation value in the disordered phase of the classical spin system. This concept has had applications and implications to many areas of physics ranging from quantum spin chains to gauge theories to topological phases of matter. In this paper I describe the role that disorder operators play in our understanding of ordered, disordered and topological phases of matter. The role of disorder operators, and their generalizations, and their connection with dualities in different systems, as well as with Majorana fermions and parafermions, is discussed in detail. Their role in recent fermion-boson and boson-boson dualities is briefly discussed., Comment: 39 pages, 3 figures, 137 references. New expanded version. The new manuscript has an expanded introduction, a new section 7 on dualities (including particle-vortex dualities) and their relation with the concept of disorder operators. To appear in a special memorial issue for Leo Kadanoff of the Journal of Statistical Physics