Constraints on bosonization in higher dimensions

Edward Witten

Arkadiusz Bochniak, Błażej Ruba, J. Wosiek +1 more

Despite being less known, local bosonizations of fermionic systems exist in spatial dimensions higher than 1. Interestingly, the dual bosonic systems are subject to local constraints, as in theories with gauge freedom. These constraints effectively implement long distance exchange interactions. In this work, we study in detail one such system, proposed a long time ago. Properties of the constraints are elaborated for two-dimensional, rectangular lattices of arbitrary sizes. For several small systems, the constraints are solved analytically. It is checked that spectra of reduced spin Hamiltonians agree with the original fermionic ones. The equivalence is extended to fermions in the presence of background Wegner ${\mathbb{Z}}_{2}$ fields coupling to fermionic parity. This is illustrated by an explicit calculation for a particular configuration of Wegner's variables. Finally, a possible connection with the recently proposed web of dualities is discussed.

Manuel Fuentes, Ana Lopez, Eduardo Fradkin +1 more

1. Exact bosonization in two spatial dimensions and a new class of lattice gauge theories Authors: Yu-An Chen, Anton Kapustin, and Djordje Radicevic Annals of Physics 393, 234 (2018); DOI: 10.1016/j.aop.2018.03.024 arXiv:1711.00515 2. Constraints of kinematic bosonization in two and higher dimensions Authors: Arkadiusz Bochniak, Blazej Ruba, Jacek Wosiek, and Adam Wyrzykowski Phys. Rev. D […]

Benedikt Fauseweh, Joachim Stolze, Götz Silvester Uhrig

The dynamics in quantum magnets can often be described by effective models with bosonic excitations obeying a hard-core constraint. Such models can be systematically derived by renormalization schemes such as continuous unitary transformations or by variational approaches. Even in the absence of further interactions the hard-core constraint makes the dynamics of the hard-core bosons nontrivial. Here, we develop a systematic diagrammatic approach to the spectral properties of hard-core bosons at finite temperature. Starting from an expansion in the density of thermally excited bosons in a system with an energy gap, our approach leads to a summation of ladder diagrams. Conceptually, the approach is not restricted to one dimension, but the one-dimensional case offers the opportunity to gauge the method by comparison to exact results obtained via a mapping to Jordan-Wigner fermions. In particular, we present results for the thermal broadening of single-particle spectral functions at finite temperature. The line shape is found to be asymmetric at elevated temperatures and the bandwidth of the dispersion narrows with increasing temperature. Additionally, the total number of thermally excited bosons is calculated and compared to various approximations and analytic results. Thereby, a flexible approach is introduced that can also be applied to more sophisticated and higher dimensional models.

Edward Witten