Time-reversal symmetric U(1) quantum spin liquids

TL;DR

<3-5 sentence high-level summary>We classify and characterize all time-reversal symmetric $U(1)$ spin liquids in 3D that possess a gapless emergent photon, showing seven distinct bulk families distinguished by charge–monopole content and a $ heta$-term; by stacking with SPTs, there are 22 total phases with varied surface states. The authors develop multiple complementary viewpoints—charge–monopole lattices, parton constructions, loop-wavefunction representations, and dual descriptions (notably the Topological Mott Insulator as both an $E$-particle TI and an $M$-particle TI)—to map the relationships among these phases and predict continuous quantum phase transitions between them. Two phases, $(E_{fT}M_f)_ heta$ and $E_{bT}M_f$, host protected surface states, which can be understood through wall-construction pictures or dual Dirac-cone descriptions. The results have implications for materials such as pyrochlore quantum spin ices and guide experimental signatures (neutron scattering, surface probes, and magnetic-field responses) to identify which spin-liquid phase, if any, is realized in a given system.

Abstract

We study possible quantum $U(1)$ spin liquids in three dimensions with time-reversal symmetry. We find a total of 7 families of such $U(1)$ spin liquids, distinguished by the properties of their emergent electric/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 7 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.

Figures (8)

• 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. \ref{['transitions']} we describe some intersting 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. \ref{['cmlat2']} 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.