Bosonization and Kramers-Wannier dualities in general dimensions

TL;DR

This work presents a comprehensive, unitary bosonization framework that maps parity-gauged fermionic systems to spin systems on arbitrary polyhedral decompositions by gauging fermion parity, enforcing a flat gauge field, and applying a disentangling unitary. It reveals higher-dimensional KW dualities arising from minimal Majorana translations, with noninvertible duality operators tied to projections onto higher-form symmetry eigenspaces and dependent on discrete spin structures via Kasteleyn orientations. The framework unifies and extends known bosonization methods (e.g., Bravyi–Kitaev, Chen–Kapustin) and provides explicit constructions in 2D and generalizes to 3D and beyond, offering a systematic route to explore fermion–spin correspondences in arbitrary dimensions. The results illuminate the role of flatness constraints, Gauss laws, and higher-form symmetries in shaping dualities, with implications for parity anomalies, boundary dynamics, and potential continuum-limit connections to field theories. Overall, the paper delivers a versatile, geometric toolkit for exact lattice bosonization and dualities across dimensions.

Abstract

It is well known that the noninteracting Majorana chain is dual to the one-dimensional transverse-field Ising model, either through the Jordan-Wigner transformation or by gauging fermion parity. In this correspondence, the minimal translation of the Majorana chain maps to the celebrated Kramers-Wannier (KW) duality of the spin model, with the critical point mapped to the self-dual point. In this work, we generalize this mapping to two and higher dimensions by constructing a unitary equivalence between the parity-gauged fermionic system and a spin system defined on arbitrary polyhedral decompositions of space. Imposing the flatness condition on the gauge field yields a bosonization duality between the original (ungauged) fermionic system and a gauged spin system obeying a Gauss law. The dependence of the Gauss law in the spin system on the Kasteleyn orientation (and the discrete spin structure) of the fermionic system is made explicit. Applying this bosonization to one or two copies of Majorana fermions on translationally invariant lattices, we derive higher-dimensional analogs of KW (self-)dualities in spin systems arising from fermionic minimal translations. The KW (self-)dualities are non-invertible due to projections onto eigenspaces of higher-form symmetries in the associated symmetry operators. The bosonization framework we present is intuitive, general, and systematic, encompassing other known exact bosonization methods while offering a novel approach to establish new connections between fermionic and spin systems in arbitrary dimensions.

Figures (15)

• Figure 1: Schematic of the relations between fermionic systems and spin systems. A parity-gauged fermionic system is shown in the main text to be equivalent to a spin system via a disentangling unitary. Imposing the flatness condition on the gauge spins establishes a duality between the ungauged (parity-even) fermionic system and the gauged fermionic system. On the spin side, this flatness condition becomes a Gauss-law constraint. The combination of parity gauging (under the flatness condition) and the disentangling unitary transformation defines a bosonization map from a general fermionic system to a gauged spin system.
• Figure 2: Parity-gauging of the Majorana chain. We place one "gauge" 1/2-spin on each bond between unit cells of two Majorana fermions and impose the Gauss law $G_j \equiv Z_{j-1}(i \gamma_{2j-1}\gamma_{2j}) Z_j = 1$ for all $j$.
• Figure 3: Quartet (a) and zigzag (b) assignment schemes of edges on the square lattice. In both schemes, two edges (red) in a face are assigned to a Majorana fermion on the face. The dashed blue arrows indicate the ordering of fermions in the bilinear $S_e$: an arrow from $\gamma_{f_2(e)}$ to $\gamma_{f_1(e)}$ fixes $S_e = i \gamma_{f_1(e)} \gamma_{f_2(e)}$. The global order of edges $e_{n-1}<e_n$ used in Eq. (\ref{['eq:square_U']}) are indicated by the labels on the lower left corner of (b). The black arrow shows the increasing direction once ordering in a row is done. In other words, the edges are ordered from left to right and then from bottom to top.
• Figure 4: Bosonization of Majorana fermion bilinears on the square lattice. Dashed blue arrows indicate the ordering of the two Majorana fermions in the bilinears. Corresponding spin operators are the products of Pauli operators on the colored edges: $Z$ (red), $X$ (blue), and $Y$ (green). If there is an extra minus sign, then it is placed at the upper left corner. In (e), a loop around vertex $v$ over which concatenated fermion bilinears are multiplied is shown; the corresponding spin operator is the Gauss-law operator $G_v$. The loop in (f) is obtained from that in (e) by the minimal half-unit-cell translation $T^{1/2}_x$.
• Figure 5: Surface graphs of edges with Majorana fermions on the each face. In the assignment procedure when all edges within a face are assigned to one type of the Majorana fermions $\gamma$ or $\gamma'$ on the face, these singled-out fermions form a dual surface graph as illustrated in (a). If both fermions on a face are assigned at least one edge, all fermions may participate in a dual surface graph as illustrated in (b). Intra-face bilinears (dashed blue) are distinguished from inter-face bilinears (dashed gray). The arrows indicate the ordering of the two Majorana fermions within a fermion bilinear: the fermion at the starting (ending) point is placed to the right (left). To illustrate Kasteleyn orientations, the orientations of the dual edges are chosen purposefully so that there are an odd number of clockwise-oriented edges for the dual face at the center.