Towards Gauging Time-Reversal Symmetry: A Tensor Network Approach

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 antiunitary symmetry. Here, we discuss a route to gauging time-reversal symmetry that 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 antiunitary symmetries. Based on that, we show how time reversal can be applied locally and also describe time-reversal symmetry twists that 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.

Popular Summary

Time-reversal symmetry is responsible for some of the most exciting topological phenomena in condensed matter systems, such as topological insulators and topological superconductors. The exotic topological nature of such systems is deeply rooted in the way in which the systems transform when time is reversed. Identifying potential topological orders in an arbitrary quantum system requires a generic tool to extract the topological feature of the time-reversal operation. For unitary symmetries, such a tool could include coupling the system to a corresponding gauge field—which functions as degrees of freedom—and observing how the system responds. Researchers are interested in “gauging” time reversal in a similar way, which requires a local definition of time-reversal symmetry action and seems to be impossible because of the antiunitary nature of time reversal. Here, we show that such a local definition is, in fact, possible if we consider the tensor network representation of a quantum state.

We use a tensor network representation to provide locality for the wave function coefficient, and we show how time reversal can be applied locally. We also describe how time-reversal symmetry twists can be inserted. With these twists, we demonstrate how topological invariants of certain time-reversal symmetric topological phases with short- and long-range entanglement in one and two dimensions can be readily extracted. Our findings open the door to studying gauging time-reversal symmetry in general tensor network states. However, many open questions still remain: Can we design a probing procedure for identifying all kinds of time-reversal-related topological orders, including those in topological insulators and superconductors? Can we introduce quantum dynamics for the time-reversal gauge field? Can we gauge time reversal based on the Hamiltonian rather than the ground state of the system?

Future investigations into these questions will shed light on how “gauging” time reversal can be applied to more generic systems.

Figure 1

Local action of time-reversal symmetry (gray and yellow region) on a short-range correlated symmetric state changes only the state at the border of the region (yellow region).

Figure 2

(a) Local time-reversal symmetry action on a symmetric MPS (b) changes only matrices near the two ends. (c) Coupling a MPS to a time-reversal flux corresponds to changing the matrices at only one place. Vertical links represent physical indices of the MPS and horizontal links represent the inner indices of the MPS.

Figure 3

Wave function of 2D states on the honeycomb lattice with $Z_2$ SPT order: each lattice site (either $A$ site or $B$ site) contains three spin-$1/2$’s; the six spin-$1/2$’s around each plaquette are either all in the $|0⟩$ state or all in the $|1⟩$ state. The $Z_2$ symmetry flips between the $|0⟩$ and $|1⟩$ state.

Figure 4

Tensors representing the trivial $Z_2$ SPT wave function in Eq. (20) and the nontrivial $Z_2$ SPT wave function in Eq. (22). The labels in circles are physical indices and the labels at the end of the links are inner indices.

Figure 5

Local $Z_2$ symmetry transformation on the tensors representing the state in Eqs. (20) and (22).

Figure 6

(a) Local $Z_2$ symmetry action on a symmetric TPS (b) changes only tensors near the border by the inner symmetry operators. (c) Coupling a TPS to a $Z_2$ symmetry flux corresponds to changing the tensors by inserting the inner symmetry operators along the twist line. For clarity, physical indices are omitted in this figure. All links represent inner indices of the TPS.

Figure 7

In the nontrivial $Z_2$ SPT state, composing two $Z_2$ twist lines in the $x$ direction is equivalent to having no twist lines in the state. The tensor product representation of the state follows from that given in Fig. 4. Physical indices are omitted in the drawing for clarity.

Figure 8

In the nontrivial $Z_2$ SPT state, composing two $Z_2$ twist lines in the $x$ direction in the presence of a $Z_2$ twist line in the $y$ direction is equivalent to a state with only a $y$ direction twist line up to a $-1$ phase factor. The tensor product representation of the state follows from that given in Fig. 4. Physical indices are omitted in the drawing for clarity.

Figure 9

Cutting the system on a torus open along a nontrivial loop in the $x$ direction (dotted circle) and turning the torus into a cylinder. The $Z_2$ twist line in the $y$ direction creates two $Z_2$ symmetry defects (orange dots) and the two $Z_2$ twist lines in the $x$ direction at the bottom of the cylinder measure the $Z_2^2$ quantum number of one of the $Z_2$ symmetry defects.

Figure 10

Tensor representing 2D trivial SPT state with time-reversal symmetry. The labels in circles are physical indices and the labels at the end of the links are inner indices.

Figure 11

Acting time reversal on one site induces “inner symmetry operators” $X\otimes X$ on the inner indices.

Figure 12

Local symmetry action on the tensor representing the SPT state with $Z_2\times Z_2^T$ symmetry. The solid ($Z_2$) part of the tensor follows from that given in Sec. 3a and the dashed (time-reversal) part of the tensor is the same as that in Fig. 10. Local $Z_2$ and time-reversal symmetry induce changes to the tensors as shown on the left- and right-hand side of the figure.

Figure 13

In the $Z_2\times Z_2^T$ SPT states, composing two time-reversal twist lines in the $x$ direction in the presence of a $Z_2$ twist line in the $y$ direction is equivalent to a state with only a $y$ direction $Z_2$ twist line up to a $\pm 1$ phase factor, which corresponds to the ${\mathcal{T}}^2=\pm 1$ transformation law on each $Z_2$ symmetry defect. The tensor product representation of the state follows from that given in Fig. 12. Physical indices are omitted in the drawing for clarity.

Figure 14

Local time-reversal action on the tensor representing $Z_2\times Z_2^T$ SPT state with $Z_2$ twist line inserted on the top side. When $\alpha$ is trivial (right-hand side), the induced inner symmetry operator remains the same as in Fig. 12; when $\alpha$ is nontrivial (left-hand side), the induced inner symmetry operator changes by $Z^{\sigma }\otimes I^{\sigma }$.

Figure 15

The tensor product representation of (a) the toric code and (b) the double semion $Z_2$ gauge theory. The labels in circles are physical indices and the labels at the end of the links are inner indices.

Figure 16

The tensors representing the $Z_2$ gauge theories are invariant under inner symmetry operators $(X\otimes X)\alpha$ and $(X\otimes X)\overline{\alpha }$.

Figure 17

Local symmetry action on the tensor representing the $Z_2$ gauge theory with time-reversal symmetry. The solid ($Z_2$) part of the tensor follows from that in Fig. 15 and the dashed (time-reversal) part of the tensor is the same as in Fig. 10. Local time-reversal symmetry induces changes to the tensors as shown on the right-hand side of the figure. The left-hand side shows an inner invariance of the tensor related to the $Z_2$ gauge symmetry of the state.

Figure 18

3D time-reversal-related topological orders may be detected by combining time-reversal twist membranes with other gauge or symmetry twist membranes or Wilson loops.

Figure 19

The single line (bond dimension-2) tensor product representation of the toric code. The labels in circles are physical indices and the labels at the end of the links are inner indices.

Figure 20

Time-reversal-enriched single line toric code tensor and its transformation under local time-reversal symmetry action. The solid ($Z_2$) part of the tensor follows from that in Fig. 19 and the dashed (time-reversal) part of the tensor is the same as in Fig. 10. Time-reversal operation includes phase factors ${\eta }^{\prime }$ between pairs of $Z_2$ and time-reversal spins connected by a red arrow.

Figure 21

Time-reversal-enriched single line toric code tensor and its transformation along a $Z_2$ gauge twist line in the $y$ direction.

Figure 22

In the time-reversal-enriched single line toric code tensor product state, inserting a $Z_2$ gauge twist line in the $y$ direction in the presence of two time-reversal twist lines in the $x$ direction is equivalent to a state with only the time-reversal twist lines up to a $\pm 1$ phase factor, which corresponds to the ${\mathcal{T}}^2=\pm 1$ transformation law on each $Z_2$ gauge defect. The tensor product representation of the state follows from that given in Fig. 20. Physical indices are omitted in the drawing for clarity.