The bond-algebraic approach to dualities

Abstract

An algebraic theory of dualities is developed based on the notion of bond algebras. It deals with classical and quantum dualities in a unified fashion explaining the precise connection between quantum dualities and the low temperature (strong-coupling)/high temperature (weak-coupling) dualities of classical statistical mechanics (or (Euclidean) path integrals). Its range of applications includes discrete lattice, continuum field, and gauge theories. Dualities are revealed to be local, structure-preserving mappings between model-specific bond algebras that can be implemented as unitary transformations, or partial isometries if gauge symmetries are involved. This characterization permits to search systematically for dualities and self-dualities in quantum models of arbitrary system size, dimensionality and complexity, and any classical model admitting a transfer matrix or operator representation. In particular, special dualities like exact dimensional reduction, emergent, and gauge-reducing dualities that solve gauge constraints can be easily understood in terms of mappings of bond algebras. As a new example, we show that the 2 Higgs model is dual to the extended toric code model in any number of dimensions. Non-local transformations like dual variables and Jordan-Wigner dictionaries are algorithmically derived from the local mappings of bond algebras. This permits to establish a precise connection between quantum dual and classical disorder variables. Our bond-algebraic approach goes beyond the standard approach to classical dualities, and could help resolve the long standing problem of obtaining duality transformations for lattice non-Abelian models. As an illustration, we present new dualities in any spatial dimension for the quantum Heisenberg model. Finally, we discuss various applications including location of phase boundaries, spectral behavior and, notably, we show how bond-algebraic dualities help constrain and realize fermionization in an arbitrary number of spatial dimensions.

Equivalence of topological insulators and superconductors

“…11,12 In what follows we will set up the foundations to establish equivalences via dualities. Particularly, we will characterize operator maps relating Hermitian quadratic forms of fermions, i.e., Gaussian dualities, including the connection between such many-body dualities and the associated transformation of the single-particle Hamiltonian.…”

“…For systems without gauge symmetries, duality transformations are implemented by unitary mappings, 11,12 and so they preserve symmetries; the symmetries of a system are in one-to-one correspondence with the symmetries of its dual partner. However, the physical interpretation of a symmetry and its dual image can be markedly different.…”

A different kind of string

“…[26,27] for a discussion) and obtain a simple spin model with global Z symmetry and integer-valued s variables, defined on the sites of the dual lattice (note that, in D = 3 + 1 dimensions, the same transformation leads to a model with local Z symmetry [28][29][30][31][32]). More precisely, the duality transformation is an exact map of the original partition function to…”

Berezinskii–Kosterlitz–Thouless Transition Through the Eyes of Duality

“…[3][4][5][6] BKT transitions, notably characterized by essential singularities in the free energy, emerge in many physical situations including screening in Coulomb gases, 7 surface roughening, 8 melting in D = 2 solids, 4,9 and many other classical and quantum problems, such as deconfinement in D = 3 + 1 lattice gauge theories. [10][11][12][13] We study these models by invoking a bond-algebraic approach that we have recently developed. [12][13][14][15] Within our duality-based method, one relates singularities in the free energy at one temperature (or coupling constants) to those at a dual temperature (or dual coupling constants).…”