Spin-Peierls, valence-bond solid, and Néel ground states of low-dimensional quantum antiferromagnets

Challenges for density functional theory in simulating metal-metal singlet bonding: A case study of dimerized VO2

VO2 is renowned for its electric transition from an insulating monoclinic (M1) phase, characterized by V-V dimerized structures, to a metallic rutile (R) phase above 340 K. This transition is accompanied by a magnetic change: the M1 phase exhibits a non-magnetic spin-singlet state, while the R phase exhibits a state with local magnetic moments. Simultaneous simulation of the structural, electric, and magnetic properties of this compound is of fundamental importance, but the M1 phase alone has posed a significant challenge to the density functional theory (DFT). In this study, we show none of the commonly used DFT functionals, including those combined with on-site Hubbard U to treat 3d electrons better, can accurately predict the V-V dimer length. The spin-restricted method tends to overestimate the strength of the V-V bonds, resulting in a small V-V bond length. Conversely, the spin-symmetry-breaking method exhibits the opposite trends. Each of these two bond-calculation methods underscores one of the two contentious mechanisms, i.e., Peierls lattice distortion or Mott localization due to electron-electron repulsion, involved in the metal-insulator transition in VO2. To elucidate the challenges encountered in DFT, we also employ an effective Hamiltonian that integrates one-dimensional magnetic sites, thereby revealing the inherent difficulties linked with the DFT computations.

Deconfinement transitions in three-dimensional compact lattice Abelian Higgs models with multiple-charge scalar fields

We investigate the nature of the deconfinement transitions in three-dimensional lattice Abelian Higgs models, in which a complex scalar field of integer charge Q≥2 is minimally coupled with a compact U(1) gauge field. Their phase diagram presents two phases separated by a transition line where static charges q, with q Collapse

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Affiliation(s)

Claudio Bonati Dipartimento di Fisica dell'Università di Pisa and INFN Sezione di Pisa, I-56127 Pisa, Italy
Andrea Pelissetto Dipartimento di Fisica dell'Università di Roma Sapienza and INFN Sezione di Roma I, I-00185 Roma, Italy
Ettore Vicari Dipartimento di Fisica dell'Università di Pisa, I-56127 Pisa, Italy

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Nodal band-off-diagonal superconductivity in twisted graphene superlattices

The superconducting state and mechanism are among the least understood phenomena in twisted graphene systems. Recent tunneling experiments indicate a transition between nodal and gapped pairing with electron filling, which is not naturally understood within current theory. We demonstrate that the coexistence of superconductivity and flavor polarization leads to pairing channels that are guaranteed by symmetry to be entirely band-off-diagonal, with a variety of consequences: most notably, the pairing invariant under all symmetries can have Bogoliubov Fermi surfaces in the superconducting state with protected nodal lines, or may be fully gapped, depending on parameters, and the band-off-diagonal chiral p-wave state exhibits transitions between gapped and nodal regions upon varying the doping. We demonstrate that band-off-diagonal pairing can be the leading state when only phonons are considered, and is also uniquely favored by fluctuations of a time-reversal-symmetric intervalley coherent order motivated by recent experiments. Consequently, band-off-diagonal superconductivity allows for the reconciliation of several key experimental observations in graphene moiré systems.

Coulomb-Higgs phase transition of three-dimensional lattice Abelian Higgs gauge models with noncompact gauge variables and gauge fixing

We study the critical behavior of three-dimensional (3D) lattice Abelian Higgs (AH) gauge models with noncompact gauge variables and multicomponent complex scalar fields, along the transition line between the Coulomb and Higgs phases. Previous works that focused on gauge-invariant correlations provided evidence that, for a sufficiently large number of scalar components, these transitions are continuous and associated with the stable charged fixed point of the renormalization-group flow of the 3D AH field theory (scalar electrodynamics), in which charged scalar matter is minimally coupled with an electromagnetic field. Here we extend these studies by considering gauge-dependent correlations of the gauge and matter fields, in the presence of two different gauge fixings, the Lorenz and the axial gauge fixing. Our results for N=25 are definitely consistent with the predictions of the AH field theory and therefore provide additional evidence for the characterization of the 3D AH transitions along the Coulomb-Higgs line as charged transitions in the AH field-theory universality class. Moreover, our results give additional insights on the role of the gauge fixing at charged transitions. In particular, we show that scalar correlations are critical only if a hard Lorenz gauge fixing is imposed.

Frozen Deconfined Quantum Criticality

There is a number of contradictory findings with regard to whether the theory describing easy-plane quantum antiferromagnets undergoes a second-order phase transition. The traditional Landau-Ginzburg-Wilson approach suggests a first-order phase transition, as there are two different competing order parameters. On the other hand, it is known that the theory has the property of self-duality which has been connected to the existence of a deconfined quantum critical point (DQCP). The latter regime suggests that order parameters are not the elementary building blocks of the theory, but rather consist of fractionalized particles that are confined in both phases of the transition and only appear-deconfine-at the critical point. Nevertheless, many numerical Monte Carlo simulations disagree with the claim of a DQCP in the system, indicating instead a first-order phase transition. Here we establish from exact lattice duality transformations and renormalization group analysis that the easy-plane CP^{1} antiferromagnet does feature a DQCP. We uncover the criticality starting from a regime analogous to the zero temperature limit of a certain classical statistical mechanics system which we therefore dub frozen. At criticality our bosonic theory is dual to a fermionic one with two massless Dirac fermions, which thus undergoes a second-order phase transition as well.

Scalar gauge-Higgs models with discrete Abelian symmetry groups

We investigate the phase diagram and the nature of the phase transitions of three-dimensional lattice gauge-Higgs models obtained by gauging the Z_{N} subgroup of the global Z_{q} invariance group of the Z_{q} clock model (N is a submultiple of q). The phase diagram is generally characterized by the presence of three different phases, separated by three distinct transition lines. We investigate the critical behavior along the two transition lines characterized by the ordering of the scalar field. Along the transition line separating the disordered-confined phase from the ordered-deconfined phase, standard arguments within the Landau-Ginzburg-Wilson framework predict that the behavior is the same as in a generic ferromagnetic model with Z_{p} global symmetry, p being the ratio q/N. Thus, continuous transitions belong to the Ising and to the O(2) universality class for p=2 and p≥4, respectively, while for p=3 only first-order transitions are possible. The results of Monte Carlo simulations confirm these predictions. There is also a second transition line, which separates two phases in which gauge fields are essentially ordered. Along this line we observe the same critical behavior as in the Z_{q} clock model, as it occurs in the absence of gauge fields.

Hilbert Space Structure of the Low Energy Sector of U(N) Quantum Hall Ferromagnets and Their Classical Limit

Using the Lieb–Mattis ordering theorem of electronic energy levels, we identify the Hilbert space of the low energy sector of U(N) quantum Hall/Heisenberg ferromagnets at filling factor M for L Landau/lattice sites with the carrier space of irreducible representations of U(N) described by rectangular Young tableaux of M rows and L columns, and associated with Grassmannian phase spaces U(N)/U(M)×U(N−M). We embed this N-component fermion mixture in Fock space through a Schwinger–Jordan (boson and fermion) representation of U(N)-spin operators. We provide different realizations of basis vectors using Young diagrams, Gelfand–Tsetlin patterns and Fock states (for an electron/flux occupation number in the fermionic/bosonic representation). U(N)-spin operator matrix elements in the Gelfand–Tsetlin basis are explicitly given. Coherent state excitations above the ground state are computed and labeled by complex (N−M)×M matrix points Z on the Grassmannian phase space. They adopt the form of a U(N) displaced/rotated highest-weight vector, or a multinomial Bose–Einstein condensate in the flux occupation number representation. Replacing U(N)-spin operators by their expectation values in a Grassmannian coherent state allows for a semi-classical treatment of the low energy (long wavelength) U(N)-spin-wave coherent excitations (skyrmions) of U(N) quantum Hall ferromagnets in terms of Grasmannian nonlinear sigma models.

Cluster Expansion and Resurgence in the Polyakov Model

In the Polyakov model, a nonperturbative mass gap is formed at leading-order semiclassics by instanton effects. By using the notions of critical points at infinity, cluster expansion, and Lefschetz thimbles, we show that a third-order effect in semiclassics gives an imaginary ambiguous contribution to the mass gap, which is supposed to be real and unambiguous. This is troublesome for the original analysis, and it is difficult to resolve this issue directly in quantum field theory (QFT). However, we find a new compactification of the Polyakov model to quantum mechanics, by using a background 't Hooft flux. The compactification has the merit of remembering the monopole instantons of the full QFT within Born-Oppenheimer approximation, while the periodic compactification does not. In the quantum mechanical limit, we prove the resurgent cancellation of the ambiguity in three-instanton sector against ambiguity in the Borel resummation of the perturbation theory around one instanton. Assuming that this result holds in QFT, we provide a large-order asymptotics of perturbation theory around perturbative vacuum and instanton.

Emergent Kondo Behavior from Gauge Fluctuations in Spin Liquids

The Kondo effect is a prominent quantum phenomenon describing the many-body screening of a local magnetic impurity. Here, we reveal a new type of nonmagnetic Kondo behavior generated by gauge fluctuations in strongly correlated baths. We show that a nonmagnetic bond defect not only introduces the potential scattering but also locally enhances the gauge fluctuations. The local gauge fluctuations further mediate a pseudospin exchange interaction that produces an asymmetric Kondo fixed point in low energy. The gauge-fluctuation-induced Kondo phenomena do not exhibit the characteristic resistivity behavior of the conventional Kondo effect, but display a nonmonotonous temperature dependence of thermal conductivity as well as an anisotropic pseudospin correlation. Moreover, with its origin from gauge fluctuations, the Kondo features can be regarded as promising indicators for identifying quantum spin liquids. Our work advances fundamental knowledge of novel Kondo phenomena in strongly correlated systems, which have no counterparts in thermal baths within the single-particle description.

Deconfined Criticality and Bosonization Duality in Easy-Plane Chern-Simons Two-Dimensional Antiferromagnets

Two-dimensional quantum systems with competing orders can feature a deconfined quantum critical point, yielding a continuous phase transition that is incompatible with the Landau-Ginzburg-Wilson scenario, predicting instead a first-order phase transition. This is caused by the LGW order parameter breaking up into new elementary excitations at the critical point. Canonical candidates for deconfined quantum criticality are quantum antiferromagnets with competing magnetic orders, captured by the easy-plane CP^{1} model. A delicate issue however is that numerics indicates the easy-plane CP^{1} antiferromagnet to exhibit a first-order transition. Here we show that an additional topological Chern-Simons term in the action changes this picture completely in several ways. We find that the topological easy-plane antiferromagnet undergoes a second-order transition with quantized critical exponents. Further, a particle-vortex duality naturally maps the partition function of the Chern-Simons easy-plane antiferromagnet into one of massless Dirac fermions.

Higher-charge three-dimensional compact lattice Abelian-Higgs models

We consider three-dimensional higher-charge multicomponent lattice Abelian-Higgs (AH) models, in which a compact U(1) gauge field is coupled to an N-component complex scalar field with integer charge q, so that they have local U(1) and global SU(N) symmetries. We discuss the dependence of the phase diagram, and the nature of the phase transitions, on the charge q of the scalar field and the number N≥2 of components. We argue that the phase diagram of higher-charge models presents three different phases, related to the condensation of gauge-invariant bilinear scalar fields breaking the global SU(N) symmetry, and to the confinement and deconfinement of external charge-one particles. The transition lines separating the different phases show different features, which also depend on the number N of components. Therefore, the phase diagram of higher-charge models substantially differs from that of unit-charge models, which undergo only transitions driven by the breaking of the global SU(N) symmetry, while the gauge correlations do not play any relevant role. We support the conjectured scenario with numerical results, based on finite-size scaling analyses of Monte Carlo simuations for doubly charged unit-length scalar fields with small and large number of components, i.e., N=2 and N=25.

Multicritical Deconfined Quantum Criticality and Lifshitz Point of a Helical Valence-Bond Phase

The S=1/2 square-lattice J-Q model hosts a deconfined quantum phase transition between antiferromagnetic and dimerized (valence-bond solid) ground states. We here study two deformations of this model-a term projecting staggered singlets, as well as a modulation of the J terms forming alternating "staircases" of strong and weak couplings. The first deformation preserves all lattice symmetries. Using quantum Monte Carlo simulations, we show that it nevertheless introduces a second relevant field, likely by producing topological defects. The second deformation induces helical valence-bond order. Thus, we identify the deconfined quantum critical point as a multicritical Lifshitz point-the end point of the helical phase and also the end point of a line of first-order transitions. The helical-antiferromagnetic transitions form a line of generic deconfined quantum-critical points. These findings extend the scope of deconfined quantum criticality and resolve a previously inconsistent critical-exponent bound from the conformal-bootstrap method.

Emergent Fermi Surface in a Triangular-Lattice SU(4) Quantum Antiferromagnet

Motivated by multiple possible physical realizations, we study the SU(4) quantum antiferromagnet with a fundamental representation on each site of the triangular lattice. We provide evidence for a gapless liquid ground state of this system with an emergent Fermi surface of fractionalized fermionic partons coupled with a U(1) gauge field. Our conclusions are based on numerical simulations using the density matrix renormalization group method, which we support with a field theory analysis.

Three-dimensional monopole-free CP^{N-1} models

We investigate the phase diagram and the nature of the phase transitions of three-dimensional monopole-free CP^{N-1} models, characterized by a global U(N) symmetry, a U(1) gauge symmetry, and the absence of monopoles. We present numerical analyses based on Monte Carlo simulations for N=2, 4, 10, 15, and 25. We observe a finite-temperature transition in all cases, related to the condensation of a local gauge-invariant order parameter. For N=2 we are unable to draw any definite conclusion on the nature of the transition. The results may be interpreted in terms of either a weak first-order transition or a continuous transition with anomalously large scaling corrections. However, the results allow us to exclude that the transition belongs to the O(3) vector universality class, as it occurs in the standard three-dimensional CP^{1} model without monopole suppression. For N=4,10,and15, the transition is of first order, and significantly weaker than that observed in the presence of monopoles. For N=25 the results are consistent with a conventional continuous transition. We compare our results with the existing literature and with the predictions of different field-theory approaches. They are consistent with the scenario in which the model undergoes continuous transitions for large values of N, including N=∞, in agreement with analytic large-N calculations for the N-component Abelian-Higgs model.

Flat-band ferromagnetism of SU(N) Hubbard model on Tasaki lattices

We investigate the para-ferro magnetic transition of the repulsive SU(N) Hubbard model on a type of one- and two-dimensional decorated cubic lattices, referred as Tasaki lattices, which feature massive single-particle ground state degeneracy. Under certain restrictions for constructing localized many-particle ground states of flat-band ferromagnetism, the quantum model of strongly correlated electrons is mapped to a classical statistical geometric site-percolation problem, where the nontrivial weights of different configurations must be considered. We prove rigorously the existence of para-ferro transition for the SU(N) Hubbard model on one-dimensional Tasaki lattice and determine the critical density by the transfer-matrix method. In two dimensions, we numerically investigate the phase transition of SU(3), SU(4) and SU(10) Hubbard models by Metropolis Monte Carlo simulation. We find that the critical density exceeds that of standard percolation, and increases with spin degrees of freedom, implying that the effective repulsive interaction becomes stronger for larger N. We further rigorously prove the existence of flat-band ferromagnetism of the SU(N) Hubbard model when the number of particles equals to the degeneracy of the lowest band in the single-particle energy spectrum.

Unifying description of competing orders in two-dimensional quantum magnets

Quantum magnets provide the simplest example of strongly interacting quantum matter, yet they continue to resist a comprehensive understanding above one spatial dimension. We explore a promising framework in two dimensions, the Dirac spin liquid (DSL) — quantum electrodynamics (QED₃) with 4 Dirac fermions coupled to photons. Importantly, its excitations include magnetic monopoles that drive confinement. We address previously open key questions — the symmetry actions on monopoles on square, honeycomb, triangular and kagome lattices. The stability of the DSL is enhanced on triangular and kagome lattices compared to bipartite (square and honeycomb) lattices. We obtain the universal signatures of the DSL on triangular and kagome lattices, including those of monopole excitations, as a guide to numerics and experiments on existing materials. Even when unstable, the DSL helps unify and organize the plethora of ordered phases in correlated two-dimensional materials.

The Dirac spin liquid is a candidate description for the strongly correlated behaviour of some quantum magnets. Song et al. study the symmetry dependence physics of monopole excitations and argue that the lattice-dependent consequences for magnetic ordering may provide a unifying picture for 2D quantum magnetism.

Spin-S Designer Hamiltonians and the Square Lattice S=1 Haldane Nematic

We introduce a strategy to write down lattice models of spin rotational symmetric Hamiltonians with arbitrary spin S that are Marshall positive and can be simulated efficiently using world-line Monte Carlo methods. As an application of our approach we consider a square lattice S=1 model for which we design a (3×3)-spin plaquette interaction. By numerical simulations we establish that our model realizes a novel "Haldane nematic" phase that breaks lattice rotational symmetry by the spontaneous formation of Haldane chains, while preserving spin rotations, time reversal, and lattice translations. By supplementing our model with a two-spin Heisenberg interaction, we present a study of the transition between Néel and Haldane nematic phase, which we find to be of first order.

Three-dimensional ferromagnetic CP^{N-1} models

We investigate the critical behavior of three-dimensional ferromagnetic CP^{N-1} models, which are characterized by a global U(N) and a local U(1) symmetry. We perform numerical simulations of a lattice model for N=2, 3, and 4. For N=2 we find a critical transition in the Heisenberg O(3) universality class, while for N=3 and 4 the system undergoes a first-order transition. For N=3 the transition is very weak and a clear signature of its discontinuous nature is only observed for sizes L≳50. We also determine the critical behavior for a large class of lattice Hamiltonians in the large-N limit. The results confirm the existence of a stable large-NCP^{N-1} fixed point. However, this evidence contradicts the standard picture obtained in the Landau-Ginzburg-Wilson (LGW) framework using a gauge-invariant order parameter: The presence of a cubic term in the effective LGW field theory for any N≥3 would usually be taken as an indication that these models generically undergo first-order transitions.

Superconductivity from the condensation of topological defects in a quantum spin-Hall insulator

The discovery of quantum spin-Hall (QSH) insulators has brought topology to the forefront of condensed matter physics. While a QSH state from spin-orbit coupling can be fully understood in terms of band theory, fascinating many-body effects are expected if it instead results from spontaneous symmetry breaking. Here, we introduce a model of interacting Dirac fermions where a QSH state is dynamically generated. Our tuning parameter further allows us to destabilize the QSH state in favour of a superconducting state through proliferation of charge-2e topological defects. This route to superconductivity put forward by Grover and Senthil is an instance of a deconfined quantum critical point (DQCP). Our model offers the possibility to study DQCPs without a second length scale associated with the reduced symmetry between field theory and lattice realization and, by construction, is amenable to large-scale fermion quantum Monte Carlo simulations.

Topological order, emergent gauge fields, and Fermi surface reconstruction

This review describes how topological order associated with the presence of emergent gauge fields can reconstruct Fermi surfaces of metals, even in the absence of translational symmetry breaking. We begin with an introduction to topological order using Wegner's quantum [Formula: see text] gauge theory on the square lattice: the topological state is characterized by the expulsion of defects, carrying [Formula: see text] magnetic flux. The interplay between topological order and the breaking of global symmetry is described by the non-zero temperature statistical mechanics of classical XY models in dimension D  =  3; such models also describe the zero temperature quantum phases of bosons with short-range interactions on the square lattice at integer filling. The topological state is again characterized by the expulsion of certain defects, in a state with fluctuating symmetry-breaking order, along with the presence of emergent gauge fields. The phase diagrams of the [Formula: see text] gauge theory and the XY models are obtained by embedding them in U(1) gauge theories, and by studying their Higgs and confining phases. These ideas are then applied to the single-band Hubbard model on the square lattice. A SU(2) gauge theory describes the fluctuations of spin-density-wave order, and its phase diagram is presented by analogy to the XY models. We obtain a class of zero temperature metallic states with fluctuating spin-density wave order, topological order associated with defect expulsion, deconfined emergent gauge fields, reconstructed Fermi surfaces (with 'chargon' or electron-like quasiparticles), but no broken symmetry. We conclude with the application of such metallic states to the pseudogap phase of the cuprates, and note the recent comparison with numerical studies of the Hubbard model and photoemission observations of the electron-doped cuprates. In a detour, we also discuss the influence of Berry phases, and how they can lead to deconfined quantum critical points: this applies to bosons on the square lattice at half-integer filling, and to quantum dimer models.

Quantum Spin Liquid Intertwining Nematic and Superconducting Order in Fese

Despite its seemingly simple composition and structure, the pairing mechanism of FeSe remains an open problem due to several striking phenomena. Among them are nematic order without magnetic order, nodeless gap and unusual inelastic neutron spectra with a broad continuum, and gap anisotropy consistent with orbital selection of unknown origin. Here we propose a microscopic description of a nematic quantum spin liquid that reproduces key features of neutron spectra. We then study how the spin fluctuations of the local moments lead to pairing within a spin-fermion model. We find the resulting superconducting order parameter to be nodeless s±d wave within each domain. Further we show that orbital dependent Kondo-like coupling can readily capture observed gap anisotropy. Our prediction calls for inelastic neutron scattering in a detwinned sample.

Interaction Effects with Varying N in SU(N) Symmetric Fermion Lattice Systems

The interaction effects in ultracold Fermi gases with SU(N) symmetry are studied nonperturbatively in half filled one-dimensional lattices by employing quantum Monte Carlo simulations. We find that, as N increases, weak and strong interacting systems are driven to a crossover region, but from opposite directions as a convergence of itinerancy and Mottness. In the weak interaction region, particles are nearly itinerant, and interparticle collisions are enhanced by N, resulting in the amplification of interaction effects. In contrast, in the strong coupling region, increasing N softens the Mott-insulating background through the enhanced virtual hopping processes. The crossover region exhibits nearly N-independent physical quantities, including the relative bandwidth, Fermi distribution, and the spin structure factor. The difference between even-N and odd-N systems is most prominent at small N's with strong interactions, since the odd case allows local real hopping with an energy scale much larger than the virtual one. The above effects can be experimentally tested in ultracold atom experiments with alkaline-earth(-like) fermions such as ^{87}Sr (^{173}Yb).

Topological order in the pseudogap metal

We compute the electronic Green's function of the topologically ordered Higgs phase of a SU(2) gauge theory of fluctuating antiferromagnetism on the square lattice. The results are compared with cluster extensions of dynamical mean field theory, and quantum Monte Carlo calculations, on the pseudogap phase of the strongly interacting hole-doped Hubbard model. Good agreement is found in the momentum, frequency, hopping, and doping dependencies of the spectral function and electronic self-energy. We show that lines of (approximate) zeros of the zero-frequency electronic Green's function are signs of the underlying topological order of the gauge theory and describe how these lines of zeros appear in our theory of the Hubbard model. We also derive a modified, nonperturbative version of the Luttinger theorem that holds in the Higgs phase.

Continuous Easy-Plane Deconfined Phase Transition on the Kagome Lattice

We use large scale quantum Monte Carlo simulations to study an extended Hubbard model of hard core bosons on the kagome lattice. In the limit of strong nearest-neighbor interactions at 1/3 filling, the interplay between frustration and quantum fluctuations leads to a valence bond solid ground state. The system undergoes a quantum phase transition to a superfluid phase as the interaction strength is decreased. It is still under debate whether the transition is weakly first order or represents an unconventional continuous phase transition. We present a theory in terms of an easy plane noncompact CP^{1} gauge theory describing the phase transition at 1/3 filling. Utilizing large scale quantum Monte Carlo simulations with parallel tempering in the canonical ensemble up to 15552 spins, we provide evidence that the phase transition is continuous at exactly 1/3 filling. A careful finite size scaling analysis reveals an unconventional scaling behavior hinting at deconfined quantum criticality.

Intertwining Topological Order and Broken Symmetry in a Theory of Fluctuating Spin-Density Waves

The pseudogap metal phase of the hole-doped cuprate superconductors has two seemingly unrelated characteristics: a gap in the electronic spectrum in the "antinodal" region of the square lattice Brillouin zone and discrete broken symmetries. We present a SU(2) gauge theory of quantum fluctuations of magnetically ordered states which appear in a classical theory of square lattice antiferromagnets, in a spin-density wave mean field theory of the square lattice Hubbard model, and in a CP^{1} theory of spinons. This theory leads to metals with an antinodal gap and topological order which intertwines with the observed broken symmetries.

1T-TaS2 as a quantum spin liquid

1T-TaS2 is unique among transition metal dichalcogenides in that it is understood to be a correlation-driven insulator, where the unpaired electron in a 13-site cluster experiences enough correlation to form a Mott insulator. We argue, based on existing data, that this well-known material should be considered as a quantum spin liquid, either a fully gapped [Formula: see text] spin liquid or a Dirac spin liquid. We discuss the exotic states that emerge upon doping and propose further experimental probes.

Emergent gauge fields and the high-temperature superconductors

The quantum entanglement of many states of matter can be represented by electric and magnetic fields, much like those found in Maxwell's theory. These fields 'emerge' from the quantum structure of the many-electron state, rather than being fundamental degrees of freedom of the vacuum. I review basic aspects of the theory of emergent gauge fields in insulators in an intuitive manner. In metals, Fermi liquid (FL) theory relies on adiabatic continuity from the free electron state, and its central consequence is the existence of long-lived electron-like quasi-particles around a Fermi surface enclosing a volume determined by the total density of electrons, via the Luttinger theorem. However, long-range entanglement and emergent gauge fields can also be present in metals. I focus on the 'fractionalized Fermi liquid' (FL*) state, which also has long-lived electron-like quasi-particles around a Fermi surface; however, the Luttinger theorem on the Fermi volume is violated, and this requires the presence of emergent gauge fields, and the associated loss of adiabatic continuity with the free electron state. Finally, I present a brief survey of some recent experiments in the hole-doped cuprate superconductors, and interpret the properties of the pseudogap regime in the framework of the FL* theory.This article is part of the themed issue 'Unifying physics and technology in light of Maxwell's equations'.

Spin-Peierls Instability of Three-Dimensional Spin Liquids with Majorana Fermi Surfaces

Three-dimensional (3D) variants of the Kitaev model can harbor gapless spin liquids with a Majorana Fermi surface on certain tricoordinated lattice structures such as the recently introduced hyperoctagon lattice. Here, we investigate Fermi surface instabilities arising from additional spin exchange terms (such as a Heisenberg coupling) which introduce interactions between the emergent Majorana fermion degrees of freedom. We show that independent of the sign and structure of the interactions, the Majorana surface is always unstable. Generically, the system spontaneously doubles its unit cell at exponentially small temperatures and forms a spin liquid with line nodes. Depending on the microscopics, further symmetries of the system can be broken at this transition. These spin-Peierls instabilities of a 3D spin liquid are closely related to BCS instabilities of fermions.

Spin Nematics, Valence-Bond Solids, and Spin Liquids in SO(N) Quantum Spin Models on the Triangular Lattice

We introduce a simple model of SO(N) spins with two-site interactions which is amenable to quantum Monte Carlo studies without a sign problem on nonbipartite lattices. We present numerical results for this model on the two-dimensional triangular lattice where we find evidence for a spin nematic at small N, a valence-bond solid at large N, and a quantum spin liquid at intermediate N. By the introduction of a sign-free four-site interaction, we uncover a rich phase diagram with evidence for both first-order and exotic continuous phase transitions.

Quantum dimer model for the pseudogap metal

We propose a quantum dimer model for the metallic state of the hole-doped cuprates at low hole density, p. The Hilbert space is spanned by spinless, neutral, bosonic dimers and spin S = 1/2, charge +e fermionic dimers. The model realizes a "fractionalized Fermi liquid" with no symmetry breaking and small hole pocket Fermi surfaces enclosing a total area determined by p. Exact diagonalization, on lattices of sizes up to 8 × 8, shows anisotropic quasiparticle residue around the pocket Fermi surfaces. We discuss the relationship to experiments.

Three-dimensional antiferromagnetic CP(N-1) models

We investigate the critical behavior of three-dimensional antiferromagnetic CP(N-1) (ACP(N-1)) models in cubic lattices, which are characterized by a global U(N) symmetry and a local U(1) gauge symmetry. Assuming that critical fluctuations are associated with a staggered gauge-invariant (Hermitian traceless matrix) order parameter, we determine the corresponding Landau-Ginzburg-Wilson (LGW) model. For N=3 this mapping allows us to conclude that the three-component ACP(2) model undergoes a continuous transition that belongs to the O(8) vector universality class, with an effective enlargement of the symmetry at the critical point. This prediction is confirmed by numerical analyses of the finite-size scaling behaviors of the ACP(2) and the O(8) vector models, which show the same universal features at their transitions. We also present a renormalization-group (RG) analysis of the LGW theories for N≥4. We compute perturbative series in two different renormalization schemes and analyze the corresponding RG flow. We do not find stable fixed points that can be associated with continuous transitions.

Vector boson excitations near deconfined quantum critical points

We show that the Néel states of two-dimensional antiferromagnets have low energy vector boson excitations in the vicinity of deconfined quantum critical points. We compute the universal damping of these excitations arising from spin-wave emission. Detection of such a vector boson will demonstrate the existence of emergent topological gauge excitations in a quantum spin system.

Fate of CPN-1 fixed points with q monopoles

We present an extensive quantum Monte Carlo study of the Néel to valence-bond solid (VBS) phase transition on rectangular- and honeycomb-lattice SU(N) antiferromagnets in sign-problem-free models. We find that in contrast to the honeycomb lattice and previously studied square-lattice systems, on the rectangular lattice for small N, a first-order Néel-VBS transition is realized. On increasing N≥4, we observe that the transition becomes continuous and with the same universal exponents as found on the honeycomb and square lattices (studied here for N=5, 7, 10), providing strong support for a deconfined quantum critical point. Combining our new results with previous numerical and analytical studies, we present a general phase diagram of the stability of CPN-1 fixed points with q monopoles.

Néel-state to valence-bond-solid transition on the honeycomb lattice: evidence for deconfined criticality

We study a spin-1/2 SU(2) model on the honeycomb lattice with nearest-neighbor antiferromagnetic exchange J that favors Néel order and competing six-spin interactions Q that favor a valence-bond-solid (VBS) state in which the bond energies order at the "columnar" wave vector K=(2π/3,-2π/3). We present quantum Monte Carlo evidence for a direct continuous quantum phase transition between Néel and VBS states, with exponents and logarithmic violations of scaling consistent with those at analogous deconfined critical points on the square lattice. Although this strongly suggests a description in terms of deconfined criticality, the measured threefold anisotropy of the phase of the VBS order parameter shows unusual near-marginal behavior at the critical point.

Dimerized solids and resonating plaquette order in SU(N)-Dirac fermions

We study the quantum phases of fermions with an explicit SU(N)-symmetric, Heisenberg-like nearest-neighbor flavor exchange interaction on the honeycomb lattice at half filling. Employing projective (zero temperature) quantum Monte Carlo simulations for even values of N, we explore the evolution from a weak-coupling semimetal into the strong-coupling, insulating regime. Furthermore, we compare our numerical results to a saddle-point approximation in the large-N limit. From the large-N regime down to the SU(6) case, the insulating state is found to be a columnar valence bond crystal, with a direct transition to the semimetal at weak, finite coupling, in agreement with the mean-field result in the large-N limit. At SU(4) however, the insulator exhibits a subtly different valence bond crystal structure, stabilized by resonating valence bond plaquettes. In the SU(2) limit, our results support a direct transition between the semimetal and an antiferromagnetic insulator.

Confinement and deconfinement of spinons in two dimensions

We use Monte Carlo methods to study spinons in two-dimensional quantum spin systems, characterizing their intrinsic size λ and confinement length Λ. We confirm that spinons are deconfined, Λ→∞ and λ finite, in a resonating valence-bond spin-liquid state. In a valence-bond solid, we find finite λ and Λ, with λ of a single spinon significantly larger than the bound state-the spinon is soft and shrinks as the bound state is formed. Both λ and Λ diverge upon approaching the critical point separating valence-bond solid and Néel ground states. We conclude that the spinon deconfinement is marginal in the lowest-energy state in the spin-1 sector, due to weak attractive spinon interactions. Deconfinement in the vicinity of the critical point should occur at higher energies.

Deconfined criticality in the frustrated Heisenberg honeycomb antiferromagnet

Using the density matrix renormalization group, we determine the phase diagram of the spin-1/2 Heisenberg antiferromagnet on a honeycomb lattice with a nearest-neighbor interaction J(1) and a frustrating, next-nearest-neighbor exchange J(2). As frustration increases, the ground state exhibits Néel, plaquette, and dimer orders, with critical points at J(2)/J(1) = 0.22 and 0.35. We observe that both the spin gap and the corresponding order parameters vanish continuously at both the critical points, indicating the presence of deconfined quantum criticality.

Weak plaquette valence bond order in the S = 1/2 honeycomb J1 - J2 Heisenberg model

Using the density matrix renormalization group, we investigate the S = 1/2 Heisenberg model on the honeycomb lattice with first (J(1)) and second (J(2)) neighbor interactions. We are able to study long open cylinders with widths up to 12 lattice spacings. For J(2)/J(1) near 0.3, we find an apparently paramagnetic phase, bordered by an antiferromagnetic phase for J(2) ≲ 0.26 and by a valence bond crystal for J(2) ≳ 0.36. The longest correlation length that we find in this intermediate phase is for plaquette valence bond order. This correlation length grows strongly with cylinder circumference, indicating either quantum criticality or weak plaquette valence bond order.

Resonating valence bond wave functions and classical interacting dimer models

We relate properties of nearest-neighbor resonating valence-bond (NNRVB) wave functions for SU(g) spin systems on two-dimensional bipartite lattices to those of fully packed interacting classical dimer models on the same lattice. The interaction energy can be expressed as a sum of n-body potentials V(n), which are recursively determined from the NNRVB wave function on finite subgraphs of the original lattice. The magnitude of the n-body interaction V(n) (n>1) is of order O(g(-(n-1))) for small g(-1). The leading term is a two-body nearest-neighbor interaction V2(g) favoring two parallel dimers on elementary plaquettes. For SU(2) spins, using our calculated value of V2(g=2), we find that the long-distance behavior of the bond-energy correlation function is dominated by an oscillatory term that decays as 1/|r|α with α≈1.22. This result is in remarkable quantitative agreement with earlier direct numerical studies of the corresponding wave function, which give α≈1.20.

Topological quantum liquids with quaternion non-Abelian statistics

Noncollinear magnetic order is typically characterized by a tetrad ground state manifold (GSM) of three perpendicular vectors or nematic directors. We study three types of tetrad orders in two spatial dimensions, whose GSMs are SO(3) = S(3)/Z(2), S(3)/Z(4), and S(3)/Q(8), respectively. Q(8) denotes the non-Abelian quaternion group with eight elements. We demonstrate that after quantum disordering these three types of tetrad orders, the systems enter fully gapped liquid phases described by Z(2), Z(4), and non-Abelian quaternion gauge field theories, respectively. The latter case realizes Kitaev's non-Abelian toric code in terms of a rather simple spin-1 SU(2) quantum magnet. This non-Abelian topological phase possesses a 22-fold ground state degeneracy on the torus arising from the 22 representations of the Drinfeld double of Q(8).

Effects of the Berry phase and instantons in one-dimensional Kondo-Heisenberg model

Motivated by the global phase diagram of antiferromagnetic heavy-fermion metals, we study the Kondo effect from the perspective of a nonlinear sigma model in the one-dimensional Kondo-Heisenberg model away from half-filling. We focus on the effects of the instanton configurations of the sigma-model field and the associated Berry phase. Guided by the results derived using bosonization methods, we demonstrate that the Kondo-singlet formation is accompanied by an emergent Berry phase. This Berry phase also captures the competition between the Kondo-singlet formation and spin-Peierls correlations. Related effects are likely to be realized in Kondo lattice systems in higher dimensions.

Plaquette valence bond ordering in a J1-J2 Heisenberg antiferromagnet on a honeycomb lattice

We study an S = 1/2 Heisenberg model on the honeycomb lattice with first and second neighbor antiferromagnetic exchange (J(1)-J(2) model), employing exact diagonalization in both the S(z) = 0 basis and nearest neighbor singlet valence bond (NNVB) basis. We find that for 0.2 < J(2)/J(1) < 0.3, the NNVB basis gives a proper description of the ground state in comparison with the exact results. By analyzing the dimer-dimer as well as the plaquette-plaquette correlations and also defining appropriate structure factors, we investigate possible symmetry breaking states as the candidates for the ground state in the frustrated region. We provide a body of evidence in favor of plaquette valence bond ordering for 0.2 < J(2)/J(1) < 0.3. By further increasing the ratio J(2)/J(1), this state undergoes a transition to the staggered dimerized state.

Majorana liquids: the complete fractionalization of the electron

We describe ground states of correlated electron systems in which the electron fractionalizes into separate quasiparticles which carry its spin and its charge, and into real Majorana fermions which carry its Fermi statistics. Such parent states provide a unified theory of previously studied fractionalized states: their descendants include insulating and conducting states with neutral spin S=1/2 fermionic spinons, and states with spinless fermionic charge carriers. We illustrate these ideas on the honeycomb lattice, with field theories of such states and their phase transitions.

Spontaneous trimerization in two-dimensional antiferromagnets

In this paper, we propose an exotic quantum paramagnetic state in two-dimensional antiferromagnets-the spontaneous trimer state-which is the direct product state of the trimers of spins. Each trimer is a singlet state formed by three neighboring spins with SU(3) symmetry. A frustrated spin-1 Heisenberg model in the kagome lattice is investigated. By analogy to the pseudo-potential approach in the fractional quantum Hall effect (FQHE), we find that the trimer state provides a good description for the exact ground state of this model. Other interesting properties, such as the local excitations as well as magnetization plateaus have also been investigated.