Dualities help theorists describe complex systems using the language of systems that are easier to understand, but keeping track of symmetries and constructing concrete models is often nontrivial. A new analysis presents exact mappings for dual theories used to describe Dirac fermions in two dimensions.
We explore an exact duality in (2+1) dimensions [(2+1)D] between the fermionization of a bosonic theory with a Z2 subsystem symmetry and a fermionic theory with a Z2 subsystem fermion parity symmetry. A typical example is the duality between the fermionization of the plaquette Ising model and the plaquette fermion model. We first revisit the standard boson-fermion duality in (1+1) dimensions with a Z2 0-form symmetry, presenting in a way generalizable to (2+1)D. We proceed to (2+1)D with a Z2 subsystem symmetry and establish the exact duality on the lattice by using the generalized Jordan-Wigner map, with a careful discussion on the mapping of the twist and symmetry sectors. This motivates us to introduce the subsystem Arf invariant, which exhibits a foliation structure.
We describe an n-dimensional (n≥2) analog of the Jordan-Wigner transformation, which maps an arbitrary fermionic system to Pauli matrices while preserving the locality of the Hamiltonian. When the space is simply-connected, this bosonization gives a duality between any fermionic system in arbitrary n spatial dimensions and a new class of (n-1)-form Z₂ gauge theories in n dimensions with a modified Gauss’s law. We describe several examples of 2d bosonization, including free fermions on square and honeycomb lattices and the Hubbard model, and 3d bosonization, including a solvable Z₂ lattice gauge theory with Dirac cones in the spectrum. This bosonization formalism has an explicit dependence on the second Stiefel-Whitney class and a choice of spin structure on the manifold, a key feature for defining fermions. A new formula for Stiefel-Whitney homology classes on lattices is derived. We also derive the Euclidean actions for the corresponding lattice gauge theories from the bosonization. The topological actions contain Chern-Simons terms for (2+1)D or Steenrod Square terms for general dimensions. Finally, we apply the bosonization to construct various bosonic or fermionic symmetry-protectedtopological (SPT) phases. It has been shown that supercohomology fermionic SPT phases are dual to bosonic higher-group SPT phases.
We grow the web of dualities for conformal field theories (CFTs) in 2+1 spacetime dimensions to include quantum critical transitions of Dirac fermions. Our construction uses the seed duality, equating a free Dirac fermion with a complex boson coupled to a Chern-Simons gauge field, to express various Gross-Neveu-Yukawa (GNY) critical points of N fermions in terms of the same number of complex bosons and partner gauge fields. In some instances, monopole operators appear explicitly in the dual action, such as for the superconducting transition of a single Dirac fermion, which is a CFT with emergent supersymmetry. We match phase diagrams, symmetries and their anomalies. By further gauging certain symmetries, we derive offspring dualities. For example, starting with the duality for the regular GNY transition of 2 Dirac fermions, we arrive at the duality between a Neel-to-VBS deconfined quantum critical point and a gauged-Gross-Neveu transition. Finally, we use a large-N expansion for monopole operators, and compare with known scaling dimensions in the simplest single-flavour GNY transition.
We construct a dual-bosonized description of a massless Majorana fermion in (2+1)d. In contrast to Dirac fermions, for which a bosonized description can be constructed using a flux attachment procedure, neutral Majorana fermions call for a different approach.We argue that the dual theory is an SO(N)1 Chern-Simons gauge theory with a critical SO(N) vector bosonic matter field ($N\ensuremath{\ge}3$). The monopole of the SO(N) gauge field is identified with the Majorana fermion. We provide evidence for the duality by establishing the correspondence of adjacent gapped phases and by a parton construction. We also propose a generalization of the duality to Nf flavors of Majorana fermions, and discuss possible resolutions of a caveat associated with an emergent global Z2 symmetry. Finally, we conjecture a dual description of an N=1 supersymmetric fixed point in (2+1)d, which is realized by tuning a single flavor of Majorana fermions to an interacting (Gross-Neveu) critical point.
“Bosonization” refers to the possibility of describing a theory of relativistic Dirac fermions obeying standard anticommutation rules by a boson field theory. While this may be possible in all dimensions, it has so far proved most useful in d = 1, where the bosonic version of the given fermionic theory is local and simple, and often simpler than the Fermi theory. This chapter should be viewed as a stepping stone toward a more thorough approach, for which references are given at the end. In this chapter I will set up the bosonization machine, explaining its basic logic and the dictionary for transcribing a fermionic theory to a bosonic theory. The next chapter will be devoted to applications. To my knowledge, bosonization, as described here, was first carried out by Lieb and Mattis [1] in their exact solution of the Luttinger model [2]. Later, Luther and Peschel [3] showed how to use it to find asymptotic (low momentum and energy) correlation functions for more generic interacting Fermi systems. It was independently discovered in particle physics by Coleman [4], and further developed by Mandelstam [5]. Much of what I know and use is inspired by the work of Luther and Peschel. Preamble Before getting into any details, I would first like to answer two questions. First, if bosonization applies only to relativistic Dirac fermions, why is it of any interest to condensed matter theory where relativity is not essential? Second, what is the magic by which bosonization helps us tame interacting field theories? As for the first question, there are two ways in which Dirac fermions enter condensed matter physics. The first is in the study of two-dimensional Ising models, where we have already encountered them. Recall that if we use the transfer matrix approach and convert the classical problem on an N × N lattice to a quantum problem in one dimension we end up with a 2 N -dimensional Hilbert space, with a Pauli matrix at each of N sites. The two dimensions at each site represent the twofold choice of values open to the Ising spins. Consider now a spinless fermion degree of freedom at each site. Here too we have two choices: the fermion state is occupied or empty.
We present a path-integral bosonization approach for systems out of equilibrium based on a duality transformation of the original Dirac fermion theory combined with the Schwinger-Keldysh time closed contour technique, to handle the non-equilibrium situation. The duality approach to bosonization that we present is valid for D 2 space-time dimensions leading for D = 2 to exact results. In this last case we present the bosonization rules for fermion currents, calculate current-current correlation functions and establish the connection between the fermionic and bosonic distribution functions in a generic, nonequilibrium situation.