The Bond-Algebraic Approach to Dualities

TL;DR

The paper introduces a bond-algebraic framework that recasts dualities as structure-preserving maps between bond algebras, unifying classical and quantum dualities through unitary (or projective-unitary) implementations. By focusing on bonds rather than elementary degrees of freedom, it provides a systematic method to derive dual variables, classify dualities (including emergent and gauge-reducing types), and apply these ideas across lattice models, quantum field theories, and classical transfer-matrix formalisms. It delivers numerous new dualities and self-dual examples (e.g., Z2 Higgs/extended toric code, Heisenberg-dual constructions, XY/SoS, and gauge-theory reductions), and clarifies the relation between dualities, gauge symmetries, and quantum symmetries, with implications for fermionization, topological order, and dimensional reduction. The framework also bridges quantum and classical dualities via transfer matrices and the path-integral (STL) viewpoint, enabling exact finite-size dualities and providing practical tools for locating phase boundaries and understanding spectral properties in diverse dimensions.

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 representation. 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 (\mathbb{Z}_2) Higgs model is dual to the extended toric code model {\it in any number of dimensions}. Non-local dual variables and Jordan-Wigner dictionaries are derived from the local mappings of bond algebras. 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.

Figures (20)

• Figure 1: A graphic representation of two quantum Ising chains, connected by the self-duality isomorphism $\Phi_{{\sf d}}$ of Equation \ref{['aut_ising1']}. The crosses $\times$ represent the bonds $\sigma^x_i$, and the thick lines between crosses represent the bonds $\sigma^z_i\sigma^z_{i+1}$. $\Phi_{{\sf d}}$ exchanges the two while preserving all algebraic relations.
• Figure 2: (Left panel) Convention to denote vertices ${\bm{r}}=(r^1,r^2)=r^1 \bm{e_1}+ r^2\bm{e_2}$ in a two-dimensional square lattice with unit vectors $\bm{e_1}, \bm{e_2}$, and (right panel) links, attached to a vertex ${\bm{r}}$, $({\bm{r}},\nu)$ with $\nu=1,2$.
• Figure 3: Kitaev's honeycomb model features $S=1/2$ spins represented by a Pauli matrices $\vec{\sigma}_{k}$. The model has three types of bonds, indicated by the letter $\mu=x,y,z$, that represent the bond operators $\sigma^\mu_i \sigma^\mu_j$. $\Phi_{\sf d}$ stands for the duality mapping that realizes the exchange $\sigma^x_i \sigma^x_j\leftrightarrow\sigma^y_i \sigma^y_j$, and that will be denoted in what follows as $P_{yxz}$.
• Figure 4: Two finite-size ($N=4$ sites) quantum Ising chains with self-dual BCs that break $Z_2$ invariance, connected by the self-duality isomorphism $\Phi_{{\sf d}}$ of Equation \ref{['automorphismfi']}. The big circle at the rightmost end of the chains represents the boundary correction $\sigma^z_4$.
• Figure 5: Quantum Ising chain featuring $N$ sites, with periodic BCs.

Theorems & Definitions (4)

• definition 1
• definition 2
• theorem 1
• definition 3