Time-Reversal Symmetric $U(1)$ Quantum Spin Liquids

We study possible quantum $U(1)$ spin liquids in three dimensions with time-reversal symmetry. We find a total of seven families of such $U(1)$ spin liquids, distinguished by the properties of their emergent electric or magnetic charges. We show how these spin liquids are related to each other. Two of these classes admit nontrivial protected surface states which we describe. We show how to access all of the seven spin liquids through slave particle (parton) constructions. We also provide intuitive loop gas descriptions of their ground-state wave functions. One of these phases is the “topological Mott insulator,” conventionally described as a topological insulator of an emergent fermionic “spinon.” We show that this phase admits a remarkable dual description as a topological insulator of emergent fermionic magnetic monopoles. This results in a new (possibly natural) surface phase for the topological Mott insulator and a new slave particle construction. We describe some of the continuous quantum phase transitions between the different $U(1)$ spin liquids. Each of these seven families of states admits a finer distinction in terms of their surface properties, which we determine by combining these spin liquids with symmetry-protected topological phases. We discuss lessons for materials such as pyrochlore quantum spin ices which may harbor a $U(1)$ spin liquid. We suggest the topological Mott insulator as a possible ground state in some range of parameters for the quantum spin ice Hamiltonian.

Popular Summary

Spin ice materials have emerged as promising candidates of “$U(1)$ quantum spin liquids,” a class of exotic quantum phases of matter. One fascinating feature of these $U(1)$ spin liquids is the emergence of artificial photons that couple to emergent charges and monopoles. These spin liquids can accordingly be viewed as a type of quantum ether, mimicking real-life electrodynamics in laboratory materials. Contrary to real-life electrodynamics, the emergent electrodynamics in quantum spin liquids may have many different variants. Here, we show that these variants are sharply distinguished as different quantum phases when time-reversal symmetry exists in the physical systems.

We focus on seven families of $U(1)$ spin liquids with 22 phases. Each phase is associated with different electric and magnetic particles. We recover many interesting consequences, including exotic behaviors on the surfaces of some of these spin liquids and unconventional quantum phase transitions between different spin liquids. Our results are not merely theoretical exercises; they may also be relevant to real materials such as pyrochlore spin ice. Typically, time reversal is the only symmetry in these systems that can survive both spin-orbit coupling and disorder. It is then important to determine which kind of $U(1)$ spin liquid is indeed realized if a $U(1)$ spin liquid arises in such systems. We provide some suggestive arguments to this question, and we discuss some experimental probes that may be able to identify different phases.

We expect that our findings will motivate future experiments that may help distinguish these different $U(1)$ spin liquids.

Figure 1

Charge-monopole lattice at $\theta =n\pi$ with $n$ even.

Figure 2

Charge-monopole lattice at $\theta =n\pi$ with $n$ odd.

Figure 3

Relationship between different $U(1)$ spin liquids. Two phases connected through a line share a common fundamental particle ($E$ or $M$) and can be viewed as different SPT phases formed by the common particle. In Sec. 11, we describe some interesting continuous phase transitions between the phases connected through thick red lines.

Figure 4

Charge-monopole lattice obtained by gauging the $n=1$ $M_f$ topological insulator. It is identical to Fig. 2 after rescaling the two axes as explained in the text.

Figure 5

The “wall” between a $U(1)$ spin liquid and a Higgsed vacuum. A particle ($E$ or $M$) tunnels through the wall and becomes a trivial boson, which subsequently condenses and forms the vacuum. Some nontrivial excitation must be left behind on the wall after the tunneling process.

Figure 6

Electriclike loop wave function of the $E_{bT}{M}_f$ $U(1)$ spin liquid. (a) The amplitude changes sign whenever a blue loop links with a red one. (b) A double blue line can be converted to a double red line and form a double two-segment loop. (c) The red and blue loops are switched under time reversal $\mathcal{T}$.

Figure 7

The same loop wave function as in Fig. 6, but with different time-reversal actions. (a) Time reversal keeps both the color and the direction of each loop, which describes $E_b{M}_f$ in the electric picture. (b) Time reversal keeps the color but inverts the direction of each loop, which describes $E_f{M}_b$ in the magnetic picture.

Figure 8

Loop wave functions of the $E_{bT}{M}_b$, $E_{fT}{M}_b$ and other phases: The amplitude changes sign whenever a directed loop links with an undirected one.