Gauging time-reversal symmetry in three dimensions and a bulk diagnostic for bosonic SPT phases protected by time reversal alone

The statistical Witten effect - the transmutation of a magnetic monopole from boson to fermion in the bulk of the three-dimensional bosonic topological insulator protected by $U(1)\rtimes\mathbb{Z}_2^T$ - was identified by Metlitski, Kane, and Fisher (2013) as the bulk diagnostic obtained by gauging the unitary charge-conservation part of the symmetry. The same authors acknowledged that the complementary tool was missing: no general method was known to gauge the anti-unitary time-reversal symmetry itself, and the nontrivial three-dimensional bosonic symmetry-protected topological (SPT) phases protected by time reversal alone had no known bulk signature. Later work supplied microscopic constructions of the bosonic topological insulator and anomaly machinery for symmetry groups containing $U(1)$, but a bulk diagnostic of the same concrete, physical character for the two nontrivial $\mathbb{Z}_2^T$-protected root phases - the group-cohomology eTmT phase and the beyond-cohomology phase with E8-type surface thermal response - has not been delivered. The problem is to construct a general gauging of time reversal for three-dimensional bosonic SPT phases and to derive from it bulk diagnostics, demonstrable in explicit microscopic models, that distinguish these phases from the trivial phase and from each other.

Background

A symmetry-protected topological (SPT) phase is a gapped, short-range-entangled phase of interacting bosons or fermions that has no intrinsic topological order - a unique ground state on every closed spatial manifold and no fractionalized bulk excitations - yet cannot be adiabatically deformed to a trivial product state without either breaking the protecting global symmetry or closing the bulk gap. Its nontrivial character is enforced at a boundary: a surface of a three-dimensional SPT phase must either break the symmetry, remain gapless, or develop an intrinsic two-dimensional topological order in which the symmetry is realized in a way impossible in any purely two-dimensional system.

For three-dimensional bosons protected by charge conservation $U(1)$ together with time reversal $\mathbb{Z}_2^T$ (symmetry group $G=U(1)\rtimes\mathbb{Z}_2^T$, the semidirect product reflecting that time reversal flips the sign of the conserved charge), the group-cohomology classification of bosonic SPT phases - which organizes these phases by cohomology classes of the symmetry group - predicts a $\mathbb{Z}_2\times\mathbb{Z}_2$ structure: three nontrivial phases $g_1$, $g_2$, $g_3=g_1+g_2$, where $g_1$ reduces to the SPT protected by time reversal alone, $g_2$ is destroyed if either $U(1)$ or $\mathbb{Z}_2^T$ is broken, and $g_3$ is their composite. The phase $g_2$ is the three-dimensional bosonic topological insulator (BTI), whose physical properties were developed by Vishwanath and Senthil through its surface states: a gapped, symmetry-preserving surface must carry a two-dimensional topological order (for example a toric-code-like order, i.e. a $\mathbb{Z}_2$ gauge theory with pointlike electric and magnetic anyons - an anyon being a pointlike quasiparticle of a two-dimensional topological order whose exchange statistics can be fractional - in which the electric anyon carries half the microscopic boson charge) whose symmetry fractionalization is anomalous in strictly two dimensions, and the surface can also realize gapless phases.

A systematic way to distinguish SPT phases is to gauge the protecting symmetry - to let it fluctuate as a local gauge field and inspect the excitations of the resulting gauge theory. A familiar two-dimensional example: gauging the $\mathbb{Z}_2$ symmetry of the trivial bosonic paramagnet yields toric-code order with bosonic fluxes, whereas gauging the nontrivial $\mathbb{Z}_2$ SPT yields double-semion order (fluxes that acquire a phase $e^{i\pi/2}$ under exchange, i.e. semions). For fermionic band insulators in three dimensions the analogous bulk signature is the Witten effect: because the electromagnetic response of a time-reversal-invariant band topological insulator contains a topological $\theta$-term with $\theta=\pi$, a magnetic monopole inserted into the bulk binds a half-odd-integer electric charge.

Metlitski, Kane, and Fisher showed in 2013 that the bosonic topological insulator has a corresponding effect of a genuinely statistical nature. In a bosonic insulator the effective $\theta$ angle is periodic modulo $4\pi$ rather than $2\pi$: the trivial and topological bosonic insulators correspond to $\theta=0$ and $\theta=2\pi$, and a monopole can remain electrically neutral in both. What distinguishes them is exchange statistics - in the trivial insulator the monopole is a boson, while in the BTI it is a fermion - the statistical Witten effect. Gauging (or weakly fluctuating) the unitary $U(1)$ part of the symmetry while keeping time reversal global thus provides the bulk diagnostic of the BTI, and it explains the surface anomaly: a strictly two-dimensional theory with the BTI surface properties would have instanton events in which a single fermion is created, violating locality. Metlitski, Kane, and Fisher promised a companion paper with an explicit field-theoretic, lattice-regularized construction of the BTI via a parton decomposition (splitting each microscopic boson into fractionally charged constituents) and condensation of dyons (bound states of electric charge and monopole); that companion paper never appeared, but the construction gap was closed by other routes: explicit microscopic constructions of the bosonic topological insulator have since been given.

The same authors stated the limitation that motivates the present problem: they did not know how to gauge time reversal, which is an anti-unitary symmetry, so no general tool existed to diagnose SPT phases whose symmetry group involves time reversal in an essential way. In particular, a bulk signature was lacking for the nontrivial three-dimensional SPT protected by $\mathbb{Z}_2^T$ alone. The interacting classification of such phases is $\mathbb{Z}_2\times\mathbb{Z}_2$: one root lies inside the group-cohomology classification and is known as eTmT - when its $\mathbb{Z}_2^T$ symmetry is gauged, the surface $\mathbb{Z}_2$ topological order has both the electric anyon $e$ and the magnetic anyon $m$ as Kramers doublets (states transforming with $\mathcal T^2=-1$), a fractionalization pattern that is anomalous in strictly two dimensions - while the second root is a beyond-group-cohomology phase whose surface, upon spontaneous time-reversal breaking, carries a thermal Hall response such that a domain wall between opposite time-reversal-broken regions supports a chiral edge with eight gapless modes, i.e. the edge of the two-dimensional E8 topological state - a gapped bosonic phase with no nontrivial anyons but a chiral edge theory of central charge $c=8$. These two phases are the independent roots of the $\mathbb{Z}_2\times\mathbb{Z}_2$ classification of $\mathbb{Z}_2^T$-protected phases; the group's remaining nontrivial element is their composite, the phase obtained by stacking the two. The question is what bulk measurement - the analog of monopole statistics for the BTI - distinguishes these phases, and what general method of gauging time reversal would produce it.

Prior work bears on the gauging of time reversal. Chen and Vishwanath (2015) developed a tensor-network route to gauging time reversal - a tensor network is a many-body wavefunction built by contracting local tensors on a lattice, which lets complex conjugation act locally so that time-reversal twists play the role of gauge fluxes - and completed it in one and two dimensions, where it reproduces the known SPT invariants. In three dimensions they constructed time-reversal twist membranes for the group-cohomology $\mathbb{Z}_2^T$ phase but left the extraction of topological invariants from them to future work, left a fully dynamical time-reversal gauge theory open, and found that the beyond-cohomology phase obstructs their approach, its surface symmetry transformation requiring a quantum circuit whose depth grows linearly with the surface size. Continuum field theory supplies a parallel treatment in which time-reversal gauge fields appear as topological terms on non-orientable spacetime manifolds (Kapustin 2014; Kapustin and Thorngren 2014; Metlitski 2015), a framework that has yielded surface-side anomaly indicators and continuum invariants rather than a bulk signature for the nontrivial three-dimensional SPT protected by $\mathbb{Z}_2^T$ alone. As in the statistical Witten effect itself, a weakly fluctuating gauge field suffices for a bulk probe.

Problem Statement

Metlitski, Kane, and Fisher identified the statistical Witten effect - the transmutation of a magnetic monopole from boson to fermion in the bulk of the three-dimensional bosonic topological insulator - by gauging the unitary $U(1)$ part of its $U(1)\rtimes\mathbb{Z}_2^T$ symmetry, and stated that no analogous tool existed for symmetries acting through time reversal: no general method to gauge the anti-unitary $\mathbb{Z}_2^T$ symmetry, and no bulk signature for the nontrivial three-dimensional bosonic SPT phases protected by time reversal alone. The later literature delivered microscopic constructions of the bosonic topological insulator and gauging and anomaly machinery for symmetry groups containing $U(1)$, but the time-reversal-only case in three dimensions still lacks a diagnostic of the same physical character.

The problem is to complete the gauging program for time reversal in three dimensions, with one overarching objective - a bulk diagnostic for three-dimensional bosonic SPT phases whose symmetry acts through time reversal, on the same footing as the statistical Witten effect for the bosonic topological insulator. Its jointly supporting parts are:

(1) Construct a general method to gauge the anti-unitary $\mathbb{Z}_2^T$ symmetry of gapped three-dimensional bosonic (short-range-entangled) phases.

(2) From this method, derive a physical bulk diagnostic, directly analogous to the monopole-statistics statement of the statistical Witten effect, that distinguishes the trivial phase from the two independent nontrivial $\mathbb{Z}_2^T$-protected three-dimensional bosonic SPT root phases - the group-cohomology eTmT phase and the beyond-cohomology phase with E8-type surface thermal response - and the two roots from each other. In the spirit of the source's request for a bulk signature, the diagnostic should be a statement about physical or lattice-computable properties of the gauged theory (for example statistics or Kramers structure of its particle and loop excitations, or topological terms in its action), not merely a formal classification label.

(3) Demonstrate the diagnostic in explicit microscopic realizations - lattice Hamiltonians or exactly solvable models for each phase - in which it can be computed, and show consistency with the known anomalous surface topological orders of these phases.

The task is to define the gauging generally for three-dimensional bosonic SPT phases with $\mathbb{Z}_2^T$ symmetry, to obtain invariants that reproduce the known $\mathbb{Z}_2\times\mathbb{Z}_2$ classification of $\mathbb{Z}_2^T$-protected phases (separating the trivial phase and the two independent roots, eTmT and the beyond-cohomology phase, from one another, which fixes the remaining nontrivial element, their composite), and to evaluate the invariants in at least one explicit microscopic model per phase.

References

1. M. A. Metlitski, C. L. Kane, and M. P. A. Fisher, "Bosonic topological insulator in three dimensions and the statistical Witten effect", Phys. Rev. B 88, 035131 (2013). DOI: 10.1103/PhysRevB.88.035131, https://doi.org/10.1103/PhysRevB.88.035131 (preprint arXiv:1302.6535, https://arxiv.org/abs/1302.6535).
2. A. Vishwanath and T. Senthil, "Physics of Three-Dimensional Bosonic Topological Insulators: Surface-Deconfined Criticality and Quantized Magnetoelectric Effect", Phys. Rev. X 3, 011016 (2013). DOI: 10.1103/PhysRevX.3.011016, https://doi.org/10.1103/PhysRevX.3.011016 (preprint arXiv:1209.3058, https://arxiv.org/abs/1209.3058).
3. Scott Geraedts and Olexei Motrunich, "Model Realization and Numerical Studies of a Three-Dimensional Bosonic Topological Insulator and Symmetry-Enriched Topological Phases", Phys. Rev. X 4, 041049 (2014). DOI: 10.1103/PhysRevX.4.041049, https://doi.org/10.1103/PhysRevX.4.041049 (preprint arXiv:1408.1096, https://arxiv.org/abs/1408.1096).
4. Peng Ye and Zheng-Cheng Gu, "Vortex-Line Condensation in Three Dimensions: A Physical Mechanism for Bosonic Topological Insulators", Phys. Rev. X 5, 021029 (2015). DOI: 10.1103/PhysRevX.5.021029, https://doi.org/10.1103/PhysRevX.5.021029 (preprint arXiv:1410.2594, https://arxiv.org/abs/1410.2594).
5. Xie Chen and Ashvin Vishwanath, "Towards Gauging Time-Reversal Symmetry: A Tensor Network Approach", Phys. Rev. X 5, 041034 (2015). DOI: 10.1103/PhysRevX.5.041034, https://doi.org/10.1103/PhysRevX.5.041034 (preprint arXiv:1401.3736, titled "'Gauging' time reversal symmetry in tensor network states", https://arxiv.org/abs/1401.3736).
6. Anton Kapustin, "Symmetry Protected Topological Phases, Anomalies, and Cobordisms: Beyond Group Cohomology", arXiv:1403.1467 (2014), https://arxiv.org/abs/1403.1467.
7. Anton Kapustin and Ryan Thorngren, "Anomalies of Discrete Symmetries in Various Dimensions and Group Cohomology", arXiv:1404.3230 (2014), https://arxiv.org/abs/1404.3230.
8. Max A. Metlitski, "S-duality of u(1) gauge theory with theta=pi on non-orientable manifolds: Applications to topological insulators and superconductors", arXiv:1510.05663 (2015), https://arxiv.org/abs/1510.05663.
9. Xie Chen, Fiona J. Burnell, Ashvin Vishwanath, and Lukasz Fidkowski, "Anomalous Symmetry Fractionalization and Surface Topological Order", Phys. Rev. X 5, 041013 (2015). DOI: 10.1103/PhysRevX.5.041013, https://doi.org/10.1103/PhysRevX.5.041013 (preprint arXiv:1403.6491, https://arxiv.org/abs/1403.6491).
10. Xie Chen, Yuan-Ming Lu, and Ashvin Vishwanath, "Symmetry protected topological phases from decorated domain walls", Nature Communications 5, 3507 (2014). DOI: 10.1038/ncomms4507, https://doi.org/10.1038/ncomms4507 (preprint arXiv:1303.4301, https://arxiv.org/abs/1303.4301).
11. Michael Hermele and Xie Chen, "Flux-Fusion Anomaly Test and Bosonic Topological Crystalline Insulators", Phys. Rev. X 6, 041006 (2016). DOI: 10.1103/PhysRevX.6.041006, https://doi.org/10.1103/PhysRevX.6.041006 (preprint arXiv:1508.00573, https://arxiv.org/abs/1508.00573).
12. Chenjie Wang, Chien-Hung Lin, and Michael Levin, "Bulk-Boundary Correspondence for Three-Dimensional Symmetry-Protected Topological Phases", Phys. Rev. X 6, 021015 (2016). DOI: 10.1103/PhysRevX.6.021015, https://doi.org/10.1103/PhysRevX.6.021015 (preprint arXiv:1512.09111, https://arxiv.org/abs/1512.09111).
13. Matthew F. Lapa and Michael Levin, "Anomaly Indicators for Topological Orders with U(1) and Time-Reversal Symmetry", Phys. Rev. B 100, 165129 (2019). DOI: 10.1103/PhysRevB.100.165129, https://doi.org/10.1103/PhysRevB.100.165129 (preprint arXiv:1905.00435, https://arxiv.org/abs/1905.00435).
14. Po-Shen Hsin and Alex Turzillo, "Symmetry-Enriched Quantum Spin Liquids in (3+1)d", J. High Energy Phys. 2020, 022 (2020). DOI: 10.1007/JHEP09(2020)022, https://doi.org/10.1007/JHEP09(2020)022 (preprint arXiv:1904.11550, https://arxiv.org/abs/1904.11550).
15. Shang-Qiang Ning, Liujun Zou, and Meng Cheng, "Fractionalization and Anomalies in Symmetry-Enriched U(1) Gauge Theories", Phys. Rev. Research 2, 043043 (2020). DOI: 10.1103/PhysRevResearch.2.043043, https://doi.org/10.1103/PhysRevResearch.2.043043 (preprint arXiv:1905.03276, https://arxiv.org/abs/1905.03276).
16. Liujun Zou, Chong Wang, and T. Senthil, "Symmetry Enriched U(1) Quantum Spin Liquids", Phys. Rev. B 97, 195126 (2018). DOI: 10.1103/PhysRevB.97.195126, https://doi.org/10.1103/PhysRevB.97.195126 (preprint arXiv:1710.00743, https://arxiv.org/abs/1710.00743).
17. Zheng-Cheng Gu, Juven C. Wang, and Xiao-Gang Wen, "Multikink topological terms and charge-binding domain-wall condensation induced symmetry-protected topological states: Beyond Chern-Simons/BF field theories", Phys. Rev. B 93, 115136 (2016). DOI: 10.1103/PhysRevB.93.115136, https://doi.org/10.1103/PhysRevB.93.115136 (preprint arXiv:1503.01768, https://arxiv.org/abs/1503.01768).
18. T. Senthil, "Symmetry Protected Topological Phases of Quantum Matter", Annual Review of Condensed Matter Physics 6, 299-324 (2015). DOI: 10.1146/annurev-conmatphys-031214-014740, https://doi.org/10.1146/annurev-conmatphys-031214-014740 (preprint arXiv:1405.4015, https://arxiv.org/abs/1405.4015).
19. Benjamin Moy, Hart Goldman, Ramanjit Sohal, and Eduardo Fradkin, "Theory of Oblique Topological Insulators", SciPost Phys. 14, 023 (2023). DOI: 10.21468/SciPostPhys.14.2.023, https://doi.org/10.21468/SciPostPhys.14.2.023 (preprint arXiv:2206.07725, https://arxiv.org/abs/2206.07725).
20. Maissam Barkeshli, Parsa Bonderson, Meng Cheng, Chao-Ming Jian, and Kevin Walker, "Reflection and Time Reversal Symmetry Enriched Topological Phases of Matter: Path Integrals, Non-orientable Manifolds, and Anomalies", Communications in Mathematical Physics 374, 1021-1124 (2020). DOI: 10.1007/s00220-019-03475-8, https://doi.org/10.1007/s00220-019-03475-8 (preprint arXiv:1612.07792, https://arxiv.org/abs/1612.07792).
21. Daniel Bulmash and Maissam Barkeshli, "Absolute Anomalies in (2+1)D Symmetry-Enriched Topological States and Exact (3+1)D Constructions", Phys. Rev. Research 2, 043033 (2020). DOI: 10.1103/PhysRevResearch.2.043033, https://doi.org/10.1103/PhysRevResearch.2.043033 (preprint arXiv:2003.11553, https://arxiv.org/abs/2003.11553).
22. Apoorv Tiwari, Xiao Chen, Ken Shiozaki, and Shinsei Ryu, "Bosonic topological phases of matter: Bulk-boundary correspondence, symmetry protected topological invariants, and gauging", Phys. Rev. B 97, 245133 (2018). DOI: 10.1103/PhysRevB.97.245133, https://doi.org/10.1103/PhysRevB.97.245133.