Abstract The composite particle duality extends the notions of both flux attachment and statistical transmutation in spacetime dimensions beyond 2 + 1D. It constitutes an exact correspondence that can be understood either as a theoretical framework or as a dynamical physical mechanism. The immediate implication of the duality is that an interacting quantum system in arbitrary dimensions can experience a modification of its statistical properties if coupled to a certain gauge field. In other words, commutation relations of quantum fields can be effectively modified by a dynamical physical process. For instance, an originally bosonic quantum fluid in d spatial dimensions can feature composite fermionic (or anyonic) excitations when coupled to a statistical gauge field. In 3 + 1D the mechanism of flux attachment induces a dynamical formation of dyons as higher-dimensional analogues of Laughlin quasiparticles. In 1 + 1D there is lack of flux attachment but a remnant in the form of a statistical gauge field can be explicitly constructed. We also introduce a family of interacting quantum many-body systems that undergo statistical transmutation as indicated by the duality. This opens the door to a new realm of topological phases across dimensions both in lattice and continuum systems.
Abstract The composite particle duality extends the notions of both flux attachment and statistical transmutation in spacetime dimensions beyond 2 + 1D. It constitutes an exact correspondence that can be understood either as a theoretical framework or as a dynamical physical mechanism. The immediate implication of the duality is that an interacting quantum system in arbitrary dimensions can experience a modification of its statistical properties if coupled to a certain gauge field. In other words, commutation relations of quantum fields can be effectively modified by a dynamical physical process. For instance, an originally bosonic quantum fluid in d spatial dimensions can feature composite fermionic (or anyonic) excitations when coupled to a statistical gauge field. In 3 + 1D the mechanism of flux attachment induces a dynamical formation of dyons as higher-dimensional analogues of Laughlin quasiparticles. In 1 + 1D there is lack of flux attachment but a remnant in the form of a statistical gauge field can be explicitly constructed. We also introduce a family of interacting quantum many-body systems that undergo statistical transmutation as indicated by the duality. This opens the door to a new realm of topological phases across dimensions both in lattice and continuum systems.
We unveil a duality that extends the notions of both flux attachment and statistical transmutation in spacetime dimensions beyond (2+1)D. Thus, a quantum system in arbitrary dimensions can experience a modification of its statistical properties if coupled to a certain gauge field. For instance, a bosonic quantum fluid can feature composite fermionic (or anyonic) excitations when coupled to a statistical gauge field. We compute the explicit form of the aforementioned synthetic gauge fields in $\text{D}\ensuremath{\le}3+1$. We introduce a bosonic liquid and its composite dual in (1+1)D as proof of principle. We recover well-known results, resolve old controversies, and suggest a microscopic mechanism for the emergence of such a gauge field. We also outline potential directions for experimental realizations in ultracold atom platforms.
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.
The commutation relations of composite fields are studied in the 3, 2 and 1 spatial dimensions. It is shown that the field of an atom consisting of a nucleus field and an electron field satisfies, in the space-like asymptotic limit, the canonical commutation relations within the sub-Fock space of the atom. The field-particle duality in the bound state is discussed from a statistical point of view. Then, the commutation relations of the scalar object in the Schwinger (Thirring) model are briefly discussed and they are shown to be consistent with its interpretation as the Nambu-Goldstone boson. The composite anyon fields are shown to satisfy the proper anyonic commutation relations with additive phase exponents. Then, quasiparticle picture of the anyons is clarified under the restriction of this additivity. The difference between field and particle characteristics becomes more prominent in the 2-dimensional space. It is argued that the hierarchy of the fractional quantum Hall effect can be rather simply understood by utilizing the quasiparticle characters of the anyons in the case that the background-boson gauge is assumed. To contrast with this case, the composite fermion theories are critically reviewed.
The Fibonacci topological order is the simplest platform for a universal topological quantum computer. While the $\ensuremath{\nu}=12/5$ fractional quantum Hall (QH) state has been proposed to support a Fibonacci sector, a dynamical picture of how a pure Fibonacci state may emerge in a QH system has been lacking. We use non-Abelian dualities to construct a Fibonacci state of bosons at filling $\ensuremath{\nu}=2$ starting from a trilayer of integer QH states. Our parent theory consists of bosonic composite vortices coupled to fluctuating U(2) gauge fields, which is dual to the theory of Laughlin quasiparticles. The Fibonacci state is obtained by interlayer clustering of the composite vortices, along with flux attachment. We use this framework to motivate a wave function for the Fibonacci state.
I consider general interacting systems of quantum particles in one spatial dimension. These consist of bosons or fermions, which can have any number of components, arbitrary spin, or a combination thereof, featuring low-energy two- and multiparticle interactions. The single-particle dispersion can be Galilean (nonrelativistic), relativistic, or have any other form that may be relevant for the continuum limit of lattice theories. Using an algebra of generalized functions, statistical transmutation operators that are genuinely unitary are obtained, putting bosons and fermions in a one-to-one correspondence without the need for a short-distance hard core. In the nonrelativistic case, low-energy interactions for bosons yield, order by order, fermionic dual interactions that correspond to the standard low-energy expansion for fermions. In this way, interacting fermions and bosons are fully equivalent to each other at low energies. While the Bose-Fermi mappings do not depend on microscopic details, the resulting statistical interactions heavily depend on the kinetic energy structure of the respective Hamiltonians. These statistical interactions are obtained explicitly for a variety of models, and regularized and renormalized in the momentum representation, which allows for theoretically and computationally feasible implementations of the dual theories. The mapping is rewritten as a gauge interaction, and one-dimensional anyons are also considered.
We have seen in the previous chapter how the existence of an order-disorder duality structure allows the obtainment of a full quantum theory of topological excitations. In the present chapter, conversely, we will see how the same structure is at the very roots of a method by which one can generate, out of bosonic fields, new composite fields with different statistics, either fermionic or generalized. In the first case, the method is usually called bosonization and allows a full description of fermions within the bosonic theory, whereas in the second, the method provides a complete description of anyons, as the particles with generalized statistics have been called [59], also in the framework of the bosonic theory. From their inception, both bosonization and the construction of fields with generalized statistics are deeply related to the order-disorder duality structure, since both fermion and anyon fields can be expressed as products of order and disorder operators, which respectively carry charge and topological charge [31, 32, 45]. The statistics of the resulting composite field, then, is proportional to the product of the charge and topological charge borne by this field. In this chapter we will expose the basic features of bosonization, as well as its relation with generalized statistics, and apply it to different quantum field theory systems in D = 2, 3 and 4, thereby verifying it is an extremely powerful tool for solving interacting field theories. Indeed, in some cases, this method can lead to the exact solution of highly nonlinear systems. The Symmetrization Postulate and Its Violation: Bosons, Fermions and Anyons Physical systems most frequently possess identical objects. Such is the case, for instance, of electrons, photons, phonons, etc. Also billiard balls, perhaps. The description of identical objects, however, changes dramatically according to whether we use a classical or quantum-mechanical approach. In the first case, one can always distinguish identical objects because we can follow their trajectories, whereas in the second case, this is no longer possible. Indeed, within a quantummechanical framework the wave-functions of different identical particles may overlap, thus causing us at the end of a physical process, to completely loose track of which particle corresponds to each of the initial ones.