arXiv subjects
Publications and source records attributed to Djordje Radicevic.
High energy physics features many ingenious tools for extracting finite results from formally divergent expressions. This brief note argues from a new perspective that all such formal infinities are meaningful markers of new physics. As such, they deserve to be explored in detail -- and even the simplest quantum field theories have more of them than is commonly thought.
This paper revisits standard calculations of free field entanglement entropy in light of the newly developed lattice-continuum correspondence. This correspondence prescribes an explicit method to extract an approximately continuum quantum field theory out of a fully regularized lattice theory. This prescription will here be extended to subregion algebras, and it will be shown how entropies of continuum boson and fermion theories can be computed by working purely with lattice quantities. This gives a clear picture of the origin of divergences in entanglement entropy while also presenting a concise and detailed recipe for calculating this important quantity in continuum theories.
Flux-attached theories are a novel class of lattice gauge theories whose gauge constraints involve both electric and magnetic operators. Like ordinary gauge theories, they possess confining phases. Unlike ordinary gauge theories, their confinement does not imply a trivial gapped vacuum. This paper will offer three lessons about the confining phases of flux-attached $\mathbb Z_K$ theories in two spatial dimensions. First, on an arbitrary orientable lattice, flux attachment that satisfies a simple, explicitly derived criterion leads to a confining theory whose low-energy behavior is captured by an action of a general Chern-Simons form. Second, on a square lattice, this criterion can be solved, and all theories that satisfy it can be enumerated. The simplest such theory has an action given by a difference of two Chern-Simons terms, and it features a kind of subsystem symmetry that causes its topological entanglement entropy to behave pathologically. Third, the simplest flux-attached theory on a square lattice that does not satisfy the above criterion is exactly solvable when the gauge group is $\mathbb Z_2$. On a torus, its confined phase possesses a twofold topological degeneracy that stems from a sum over spin structures in a dual fermionic theory. This makes this flux-attached $\mathbb Z_2$ theory an appealing candidate for a microscopic description of a $\mathrm U(1)_2$ Chern-Simons theory.
This paper fires the opening salvo in the systematic construction of the lattice-continuum correspondence, a precise dictionary that describes the emergence of continuum quantum theories from finite, nonperturbatively defined models ("lattice theories"). Here the focus will be on quantum field theory in (0+1)D, i.e. quantum mechanics. The main conceptual achievement is an explicit and systematic procedure for reducing a theory with a large but finite Hilbert space to a subtheory in which wavefunctions satisfy prescribed smoothness and compactness constraints. This reduction, here named taming, in effect defines quantum mechanics on a continuum target space. When appropriate lattice theories are tamed, many familiar continuum notions explicitly emerge, e.g. canonical commutation relations, contact terms in correlation functions, continuous spacetime symmetries, and supersymmetry algebras. All of these are thus "put on the lattice" using the present framework. This analysis also leads to further insights into old subjects: for example, it is proven that any supersymmetric lattice theory must have a vanishing Witten index.
This paper proposes a general framework for nonperturbatively defining continuum quantum field theories. Unlike most such frameworks, the one offered here is finitary: continuum theories are defined by reducing large but finite quantum systems to subsystems with conserved entanglement patterns at short distances. This makes it possible to start from a lattice theory and use rather elementary mathematics to isolate the entire algebraic structure of the corresponding low-energy continuum theory. The first half of this paper illustrates this approach through a persnickety study of (1 + 1)D continuum theories that emerge from the $\mathbb{Z}_K$ clock model at large $K$. This leads to a direct lattice derivation of many known continuum results, such as the operator product expansion of vertex operators in the free scalar CFT. Many new results are obtained too. For example, self-consistency of the lattice-continuum correspondence leads to a rich, novel proposal for the symmetry breaking structure of the clock model at weak coupling, deep in the BKT regime. This also makes precise what one means by "continuous" when saying that continuous symmetries cannot be broken in (1+1)D. The second half of this paper is devoted to path integrals for continuum theories of bosons and fermions defined in this finitary formalism. The path integrals constructed here have both nonperturbative lattice definitions and manifest continuum properties, such as symmetries under infinitesimal rotations or dilatations. Remarkably, this setup also makes it possible to generalize Noether's theorem to discrete symmetries.
This paper develops a detailed lattice-continuum correspondence for all common examples of Abelian gauge theories, with and without matter. These rules for extracting a continuum theory out of a lattice one represent an elementary way to rigorously define continuum gauge theories. The focus is on (2+1)D but the techniques developed here work in all dimensions. The first half of this paper is devoted to pure Maxwell theory. It is precisely shown how continuum Maxwell theory emerges at low energies in an appropriate parameter regime of the $\mathbb{Z}_K$ lattice gauge theory at large $K$. The familiar features of this theory -- its OPE structure, the distinction between compact and noncompact degrees of freedom, infrared "particle-vortex" dualities, and the Coulomb law behavior of its Wilson loop -- are all derived directly from the lattice. The rest of the paper studies gauge fields coupled to either bosonic or fermionic matter. Scalar QED is analyzed at the same level of precision as pure Maxwell theory, with new comments on its phase structure and its connections to the topological BF theory. Ordinary QED (Dirac fermions coupled to Maxwell theory) is constructed with an eye toward properly defining spinors and avoiding global anomalies. The conventional continuum QED is shown to arise from a specific restriction of the starting theory to smoothly varying fields. Such smoothing also turns out to be a necessary condition for Chern-Simons theory to arise from integrating out massive fermions in a path integral framework. In an effort to find a canonical origin of Chern-Simons theory that does not rely on path integral smoothing, a simple flux-attached lattice gauge theory is shown to give rise to a Chern-Simons-like action in the confining regime.
Fracton theories possess exponentially degenerate ground states, excitations with restricted mobility, and nontopological higher-form symmetries. This paper shows that such theories can be defined on arbitrary spatial lattices in three dimensions. The key element of this construction is a generalization of higher-form gauge theories to so-called $\mathfrak{F}_p$ gauge theories, in which gauge transformations of rank-$k$ fields are specified by rank-$(k - p)$ gauge parameters. The $\mathbb{Z}_2$ rank-two theory of type $\mathfrak{F}_2$, placed on a cubic lattice and coupled to scalar matter, is shown to have a topological phase exactly dual to the well-known X-cube model. Generalizations of this example yield novel fracton theories. In the continuum, the $\mathrm{U}(1)$ rank-two theory of type $\mathfrak{F}_2$ is shown to have a perturbatively gapless fracton regime that cannot be consistently interpreted as a tensor gauge theory of any kind. The compact scalar fields that naturally couple to this $\mathfrak{F}_2$ theory also show gapless fracton behavior; on a cubic lattice they have a conserved $\mathrm{U}(1)$ charge and dipole moment, but these particular charges are not necessarily conserved on more general lattices. The construction straightforwardly generalizes to $\mathfrak{F}_2$ theories of nonabelian rank-two gauge fields, giving first examples of pure nonabelian higher-rank theories.
Starting from the operator algebra of the (1+1)D Ising model on a spatial lattice, this paper explicitly constructs a subalgebra of smooth operators that are natural candidates for continuum fields in the scaling limit. At the critical value of the transverse field, these smooth operators are analytically shown to reproduce the operator product expansions found in the Ising conformal field theory.
This paper establishes a precise mapping between lattice and continuum operators in theories of (1 + 1)D fermions. To reach the continuum regime of a lattice theory, renormalization group techniques are here supplemented by a new kind of decimation called smoothing out. Smoothing out amounts to imposing constraints that make fermion fields vary slowly in position space; in momentum space, this corresponds to introducing boundary conditions below a certain depth of the Dirac sea. This procedure necessitates the introduction of a second small parameter to describe the continuum limit. This length scale, much larger than the lattice spacing but much smaller than the macroscopic system size, controls the derivative expansions of fields, and hence plays the role of the "string scale" in quantum field theory. Smoothing out a theory of Dirac fermions on the lattice provides a transparent, fully lattice-based derivation of the operator product expansion, current algebra, and Abelian bosonization rules in the continuum. Nontrivial high-momentum effects are demonstrated in all these derivations. In particular, the radius of the compact scalar dual to the Dirac fermion is found to vary with momentum, smoothly shrinking to zero at a cutoff set by the "string scale."
This paper derives a large web of exact lattice dualities in one and two spatial dimensions. Some of the dualities are well-known, while others, such as two-dimensional boson-parafermion dualities, are new. The procedure is systematic, independent of specific Hamiltonians, and generalizes to higher dimensions. One important result is a demonstration that spin structures in arbitrary lattice fermion theories can always be simply defined as topological gauge fields whose gauge group is the fermion number parity. This definition agrees with other expected properties of spin structures, and it motivates the introduction of "paraspin structures" that serve the same role in parafermion theories.
Dirac fermions coupled to gauge fields can exhibit the chiral anomaly even on a finite spatial lattice. A careful description of this phenomenon yields new insights into the nature of spin-charge relations and on-site symmetries (symmetries that are gauged by placing gauge fields on all links of the lattice). One notable result is that only sufficiently small symmetry groups can act on-site in a system with finitely many degrees of freedom. Symmetries that break this rule either cannot be gauged on an arbitrary lattice, or the gauging decouples the matter fields from gauge fields. These "anomalies" are not quantum in nature, and they are diagnosed geometrically, by the volume of the spatial manifold. The familiar particle number ${\mathrm U}(1)$ exhibits this kind of anomalous behavior in any finite fermion theory. The chiral anomaly in such a finite system instead manifests itself most simply after gauging the ${\mathbb Z}_2$ fermion parity.
We revisit the issue of defining the entropy of a spatial region in a broad class of quantum theories. In theories with explicit regularizations, working within an elementary but general algebraic framework applicable to matter and gauge theories alike, we give precise path integral expressions for three known types of entanglement entropy that we call full, distillable, and gauge-invariant. For a class of gauge theories that do not necessarily have a regularization in our framework, including Chern-Simons theory, we describe a related approach to defining entropies based on locally extending the Hilbert space at the entangling edge, and we discuss its connections to other calculational prescriptions. Based on results from both approaches, we conjecture that it is always the full entanglement entropy that is calculated by standard holographic techniques in strongly coupled conformal theories.
We describe a 2d analog of the Jordan-Wigner transformation which maps an arbitrary fermionic system on a 2d lattice to a lattice gauge theory while preserving the locality of the Hamiltonian. When the space is simply-connected, this bosonization map is an equivalence. We describe several examples of 2d bosonization, including free fermions on square and honeycomb lattices and the Hubbard model. We describe Euclidean actions for the corresponding lattice gauge theories and find that they contains Chern-Simons-like terms. Finally, we write down a fermionic dual of the gauged Ising model (the Fradkin-Shenker model).
This paper is a physicist's review of the major conceptual issues concerning the problem of spectral universality in quantum systems. Here we present a unified, graph-based view of all archetypical models of such universality (billiards, particles in random media, interacting spin or fermion systems). We find phenomenological relations between the onset of ergodicity (Gaussian-random delocalization of eigenstates) and the structure of the appropriate graphs, and we construct a heuristic picture of summing trajectories on graphs that describes why a generic interacting system should be ergodic. We also provide an operator-based discussion of quantum chaos and propose criteria to distinguish bases that can usefully diagnose ergodicity. The result of this analysis is a rough but systematic outline of how ergodicity changes across the space of all theories with a given Hilbert space dimension. As a special case, we study the SYK model and report on the transition from maximal to partial ergodicity as the disorder strength is decreased.
Bosonization dualities relate two different Chern-Simons-matter theories, with bosonic matter on one side replaced by fermionic matter on the other. We first describe a more general class of non-Abelian bosonization dualities. We then explore the non-relativistic physics of these theories in the quantum Hall regime. The bosonic theory lies in a condensed phase and admits vortices which are known to form a non-Abelian quantum Hall state. We ask how this same physics arises in the fermionic theory. We find that a condensed boson corresponds to a fully filled Landau level of fermions, while bosonic vortices map to fermionic holes. We confirm that the ground state of the two theories is indeed described by the same quantum Hall wavefunction.
This paper presents an algebraic formulation of the renormalization group flow in quantum mechanics on flat target spaces. For any interacting quantum mechanical theory, the fixed point of this flow is a theory of classical probability, not a different effective quantum mechanics. Each energy eigenstate of the UV Hamiltonian flows to a probability distribution whose entropy is a natural diagnostic of quantum ergodicity of the original state. These conclusions are supported by various examples worked out in some detail.
Using the algebraic approach to entanglement entropy, we study several dual pairs of lattice theories and show how the entropy is completely preserved across each duality. Our main result is that a maximal algebra of observables in a region typically dualizes to a non-maximal algebra in a dual region. In particular, we show how the usual notion of tracing out external degrees of freedom dualizes to a tracing out coupled to an additional summation over superselection sectors. We briefly comment on possible extensions of our results to more intricate dualities, including holographic ones.
Building on the recent progress in solving Chern-Simons-matter theories in the planar limit, we compute the scaling dimensions of a large class of disorder ("monopole") operators in $U(N)_k$ Chern-Simons-fermion theories at all 't Hooft couplings $λ= N/k$. We find that the lowest-dimension operator of this sort has dimension $\frac23 k^{3/2}$. We comment on the implications of these results to analyzing maps of fermionic disorder operators under 3D bosonization.