In condensed matter physics, Kramers-Wannier duality implies that the state disordered by quantum fluctuations or temperature actually corresponds to an ordered state formed from the topological excitations of the ``original'' ordered state. At first sight it might appear to be impossible to observe this dual order using means associated with the original order. Although true for Ising models, we demonstrate in this paper that this is not a general statement by considering the well-known vortex duality, in particular in the quantum interpretation in $2+1\mathrm{D}$ where it is associated with the quantum phase transition from a superfluid to a Bose Mott insulator. Here, the disordered Mott insulating state is at the same time a dual superconductor corresponding to a Bose condensate of vortices. We present a simple formalism making it possible to compute the velocity propagator associated with the superfluid in terms of the degrees of freedom of the dual theory. The Mott insulator is characterized by a doublet of massive modes, and we demonstrate that one of these modes is nothing else than the longitudinal photon (gauged second sound) of the dual superconductor. For increasing momenta, the system rediscovers the original order, and the effect on the velocity correlator is that the longitudinal photon loses its pole strength. The quantum critical regime as probed by the velocity correlator is most interesting. We demonstrate that at infinite wavelength the continua of critical modes associated with second sound and the dual longitudinal photon are indistinguishable. However, at finite momenta they behave differently, tracking the weight reshuffling found in the quasiparticle spectrum of the disorder state closely.
In condensed matter physics, Kramers-Wannier duality implies that the state disordered by quantum fluctuations or temperature actually corresponds to an ordered state formed from the topological excitations of the ``original'' ordered state. At first sight it might appear to be impossible to observe this dual order using means associated with the original order. Although true for Ising models, we demonstrate in this paper that this is not a general statement by considering the well-known vortex duality, in particular in the quantum interpretation in 2+1D where it is associated with the quantum phase transition from a superfluid to a Bose Mott insulator. Here, the disordered Mott insulating state is at the same time a dual superconductor corresponding to a Bose condensate of vortices. We present a simple formalism making it possible to compute the velocity propagator associated with the superfluid in terms of the degrees of freedom of the dual theory. The Mott insulator is characterized by a doublet of massive modes, and we demonstrate that one of these modes is nothing else than the longitudinal photon (gauged second sound) of the dual superconductor. For increasing momenta, the system rediscovers the original order, and the effect on the velocity correlator is that the longitudinal photon loses its pole strength. The quantum critical regime as probed by the velocity correlator is most interesting. We demonstrate that at infinite wavelength the continua of critical modes associated with second sound and the dual longitudinal photon are indistinguishable. However, at finite momenta they behave differently, tracking the weight reshuffling found in the quasiparticle spectrum of the disorder state closely.
This paper reviews a recent non-Landau-Ginzburg-Wilson (LGW) approach to superfluid to Mott insulator transitions in two dimensional bosonic lattice systems, using a dual vortex field theory [L. Balents, L. Bartosch, A. Burkov, S. Sachdev and K. Sengupta, Phys. Rev. B 71 (2005), 144508; Phys. Rev. B 71 (2005), 144509]. The physical interpretation of conventional LGW theory of quantum criticality is re-examined and similarities and differences with the vortex picture are discussed. The “unification” of various competing (insulating) orders, and the coincidence of these orders with the Mott transition are readily understood formulation. Some aspects of the recent theory of “deconfined” quantum criticality, which are to an extent subsumed approach, are discussed. A pedagogical presentation of the “nuts and bolts” of boson-vortex duality at the hamiltonian level is included, tailored to a condensed matter audience.
I study the zero-temperature phase transition between the superfluid and the insulating ground states of the Bose-Hubbard model in a random chemical potential and at large integer average number of particles per site. Duality transformation maps the pure Bose-Hubbard model onto the sine-Gordon theory in one dimension (1D), and onto the three dimensional Higgs electrodynamics in two dimensions (2D). In 1D the random chemical potential in the dual theory couples to the space derivative of the dual field, and appears as a random magnetic field along the imaginary time direction in 2D. I show that the transition from the superfluid state both in 1D and 2D is always controlled by the random critical point. This arises due to a coupling constant in the dual theory with replicas that becomes generated at large distances by the random chemical potential, and represents a relevant perturbation at the pure superfluid-Mott insulator fixed point. At large distances the dual theory in 1D becomes equivalent to Haldane's macroscopic representation of a disordered quantum fluid, where the generated term is identified with the random backscattering. In 2D the generated coupling corresponds to the random mass of the complex field that represents vortex loops. I calculate the critical exponents at the superfluid-Bose-glass fixed point in 2D to be $\ensuremath{\nu}=1.38$ and z=1.93, and the universal conductivity at the transition ${\ensuremath{\sigma}}_{c}{=0.25e}_{*}^{2}/h$, using the one-loop field-theoretic renormalization group in fixed dimension.
Many of the most vexing phenomena observed in condensed matter physics involve strongly correlated systems near quantum critical points in two spatial dimensions. Examples range from the sharing of critical exponents (superuniversality) among quantum Hall plateau transitions to the emergence of a charge-vortex symmetry near the field-tuned superconductor-insulator transition in thin films to the appearance of “anomalous” metallic states which evade localization. These problems elude theoretical understanding because they lack small parameters; standard perturbative approaches constitute poor, often uncontrolled approximations. This thesis concerns itself with the development of techniques reaching beyond the perturbative paradigm, focusing especially on duality, the idea that two seemingly different theories are actually one and the same. While a duality is most useful when it relates a strongly interacting theory to a weakly interacting one, even dualities between strongly coupled theories can reveal emergent symmetries and exotic gapped phases that may be obscure in one theory but not its dual. Recently, starting from a relativistic analogue of flux attachment, an entire web of quantum field theory dualities was proposed connecting a wide variety of quantum critical states in two spatial dimensions. This web of dualities has led to a surge of progress on a wide variety of condensed matter problems, several examples of which are presented in this thesis. One of the major challenges that comes with the proposal of new dualities is to “derive” them. In Chapter 2, we construct simple, explicit derivations of many members of the web of dualities in (particle-hole symmetric) models of relativistic current loops. In relativistic theories, flux attachment necessarily involves transmutation of both statistics and spin, an operation which can be made precise in the context of loop models. We further show that while non-relativistic theories are invariant under attachment of even numbers of flux quanta due to the periodicity of statistics, this symmetry is completely lost in relativistic theories due to the presence of spin, clarifying the interpretation of earlier loop models in which this symmetry appeared to be present. Motivated in part by this lack of statistical periodicity in relativistic flux attachment, in Chapter 3 we turn our attention to the metallic, “composite Fermi liquid” states occuring in quantum Hall systems at filling fraction ν = 1/2n. Famously, the state at ν = 1/2 is known to display particle-hole symmetry, which has recently led to the proposal of a manifestly particle-hole symmetric theory of relativistic, or Dirac, composite fermions that features prominently in the web of dualities. Surprisingly, however, an analogous “reflection symmetry” has also been observed in transport experiments about ν = 1/4. To explain this symmetry, we propose a series of relativistic composite fermion theories for the compressible states at ν = 1/2n, in which the reflection symmetry is incorporated as a mean field time-reversal symmetry of the composite fermions. 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 ν = 1/2n, and we show that such states can be related by the observed reflection symmetry, at least at mean field level. In Chapter 4, we describe how duality can be used to access exotic gapped phases, in particular non- Abelian quantum Hall states. Using proposed non-Abelian bosonization dualities in two spatial dimensions, which morally relate U(N)k and SU(k)−N Chern-Simons-matter theories, we present pairing scenarios for which non-Abelian quantum Hall states can be obtained starting from theories of Abelian composite particles. 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 ν = k/(kM +2) by starting from k layers of bosons at ν = 1/2 with M Abelian fluxes attached and then condensing k-clusters of the dual non-Abelian bosons. We further extend this constructions to obtain generalizations of the Read-Rezayi states with emergent global symmetries. In Chapter 5, we describe a context in which non-perturbative ideas from duality and perturbative results in the language of the renormalization group (RG) can inform one another: the interplay of quenched disorder and strong interaction effects near quantum critical points. In particular, we focus on the problem of quenched disorder at the superfluid-insulator transition of the O(N) model in the large-N limit. While a random mass is strongly relevant at the free fixed point, its effect is screened by the strong interactions of the Wilson-Fisher fixed point in this model, enabling a perturbative RG study of the interplay of disorder and interactions about this fixed point. In contrast to the spiralling flows obtained in earlier double-ε expansions, we show that the theory flows directly to a quantum critical point characterized by finite disorder and interactions, with critical exponents in remarkable agreement with numerical studies of the superfluid-Mott glass transition. With these results in hand, we apply duality to discuss the possible implications of this result for the dual Abelian Higgs and Chern-Simons-Dirac fermion theories when N = 1.
Several simple models of strongly correlated bosons on three-dimensional lattices have been shown to possess exotic fractionalized Mott insulating phases with a gapless ``photon'' excitation. In this paper we show how to view the physics of this ``Coulomb'' state in terms of the excitations of proximate superfluid. We argue for the presence of ordered vortex cores with a broken discrete symmetry in the nearby superfluid phase and that proliferating these degenerate but distinct vortices with equal amplitudes produces the Coulomb phase. This provides a simple physical description of the origin of the exotic excitations of the Coulomb state. The physical picture is formalized by means of a dual description of three-dimensional bosonic systems in terms of fluctuating quantum mechanical vortex loops. Such a dual formulation is extensively developed. It is shown how the Coulomb phase (as well as various other familiar phases) of three-dimensional bosonic systems may be described in this vortex loop theory. For bosons at half-filling and the closely related system of spin-1∕2 quantum magnets on a cubic lattice, fractionalized phases as well as bond- or ``box''-ordered states are possible. The latter are analyzed by an extension of techniques previously developed in two spatial dimensions. The relation between these ``confining'' phases with broken translational symmetry and the fractionalized Coulomb phase is exposed.
We show that the nature of quantum phases around the superconductor-insulator transition (SIT) is controlled by charge-vortex topological interactions and does not depend on the details of material parameters and disorder. We find three distinct phases, superconductor, superinsulator and bosonic topological insulator. The superinsulator is a state of matter with infinite resistance in a finite temperature range, which is the S-dual of the superconductor and in which charge transport is prevented by electric strings binding charges of opposite sign. The electric strings ensuring linear confinement of charges are generated by instantons and are dual to superconducting Abrikosov vortices. Material parameters and disorder enter the London penetration depth of the superconductor, the string tension of the superinsulator and the quantum fluctuation parameter driving the transition between them. They are entirely encoded in four phenomenological parameters of a topological gauge theory of the SIT. Finally, we point out that, in the context of strong coupling gauge theories, the many-body localization phenomenon that is often referred to as an underlying mechanism for superinsulation is a mere transcription of the well-known phenomenon of confinement into solid state physics language and is entirely driven by endogenous disorder embodied by instantons with no need of exogenous disorder.
Inspired by the duality picture between superconductivity (SC) and insulator in two spatial dimension (2D), we conjecture that the order parameter, suitable for characterizing 2D fermionic insulating state, is the disorder operator, usually known in the context of statistical transformation. Namely, the change of the phase of the disorder operator along a closed loop measures the particle density accommodating inside this loop. Thus, identifying this (doped) particle density with the dual counterpart of the magnetic induction in 2D SC, we can naturally introduce the disorder operator as the dual order parameter of 2D insulators. The disorder operator has a branch cut emitting from this ``vortex'' to the single infinitely far point. To test this conjecture against arbitrary two-dimensional lattice models, we have chosen this branch cut to be compatible with the periodic boundary condition and obtain a general form of its expectation value for noninteracting metal and/or insulator wave function, including gapped mean-field order wave function. Based on this expression, we observed analytically that it indeed vanishes for a wide class of band metals in the thermodynamic limit. On the other hand, it takes a finite value in insulating states, which is quantified by the localization length or the real-valued gauge invariant two form dubbed as the quantum metric tensor. When successively applied along a closed loop, our disorder operator plays the role of twisting the boundary condition of a periodic system. We argue this point by highlighting the Aharonov-Bohm phase associated with this nonlocal operator.