Towards Gauging Time-Reversal Symmetry: A Tensor Network Approach

Xie Chen,Ashvin Vishwanath

doi:10.1103/physrevx.5.041034

Abstract

It is well known that unitary symmetries can be `gauged', i.e. defined to act in a local way, which leads to a corresponding gauge field. Gauging, for example, the charge conservation symmetry leads to electromagnetic gauge fields. It is an open question whether an analogous process is possible for time reversal which is an anti-unitary symmetry. Here we discuss a route to gauging time reversal symmetry which applies to gapped quantum ground states that admit a tensor network representation. The tensor network representation of quantum states provides a notion of locality for the wave function coefficient and hence a notion of locality for the action of complex conjugation in anti-unitary symmetries. Based on that, we show how time reversal can be applied locally and also describe time reversal symmetry twists which act as gauge fluxes through nontrivial loops in the system. As with unitary symmetries, gauging time reversal provides useful access to the physical properties of the system. We show how topological invariants of certain time reversal symmetric topological phases in $D=1,2$ are readily extracted using these ideas.

Highlights

For condensed matter systems with global symmetry, coupling to the corresponding gauge field provides a useful access to the physical properties of the system
We can define the local action of complex conjugation, which is the key in defining local actions of antiunitary symmetries
By composing the time-reversal twists, we can extract topological invariants of the phase from the projective composition rules. We demonstrate how this works for 1D time-reversal symmetry-protected topological (SPT) phases, which is a reinterpretation of the procedure used to determine the SPT order from the matrix product state representation of the state

Summary

INTRODUCTION

For condensed matter systems with global symmetry, coupling to the corresponding gauge field provides a useful access to the physical properties of the system. If we apply time-reversal symmetry locally (complex conjugation and Uk on sites in a subregion), tensors both inside and outside the subregion remain effectively invariant while tensors along the border can change Intuitively, this definition of local time-reversal symmetry action changes the quantum state in a way we would expect. The local action of time-reversal symmetry on matrix product states has been discussed extensively in the study of 1D symmetry-protected topological phases [17,18,19,20]. TrðAi1 Ai2 ...AiN ÞÃU ⊗ U Á Á Á ⊗ Uji1i2...iNi; ð5Þ acting time reversal locally on a single site in the matrix product state changes the matrices to

UTijðAjÞÃ: ð6Þ j

Review

Example

CONCLUSION AND OPEN QUESTIONS