Topology and Criticality

Gapless Symmetry-Protected Topological Order

While the classification of gapped topological systems is fairly exhaustive, it is natural to ask about the prospect of topological effects in gapless systems. This is not easy: the very definition of topological phase, as used so far, relies crucially on the bulk being gapped. Nevertheless, in Refs [1], we proposed a systematic construction to generate gapless systems with topologically protected edge modes --- called gapless symmetry-protected topological states (gSPT) --- and showed that they are as generic as their gapped counterpart. They actually occur at the phase transition between gapped SPTs and symmetry-broken states, and can also be understood as symmetry-enriched quantum criticality. They have the same bulk criticality as in the "untwisted" case, but an anomalous surface criticality that is robust against any symmetry-preserving edge perturbation. 

Interestingly, a lot of the tools developed for gapped bosonic SPTs can be transposed to the gapless case. In particular, our 2D gapless SPT wavefunctions can be written as correlators in a conformal field theory (CFT). Zohar Ringel and I had already introduced this relation between gapped SPT wavefunctions and CFTs in a previous paper [3].

• [1] Thomas Scaffidi, Daniel E. Parker, Romain Vasseur, "Gapless Symmetry Protected Topological Order", Phys. Rev. X 7, 041048 (2017)

• [2] Daniel E Parker, Thomas Scaffidi, Romain Vasseur, "Topological Luttinger Liquids from Decorated Domain Walls", Phys. Rev. B 97, 165114 (2018)

• [3] Thomas Scaffidi, Zohar Ringel, "Wavefunctions of Symmetry Protected Topological Phases from Conformal Field Theories", Phys. Rev. B 93, 115105 (2016)

Quantum phase transition between toric code and double semion

Phase transitions between different topological phases are ideal candidates to study phase transitions beyond the Landau-Ginzburg-Wilson paradigm of symmetry-breaking. Such direct phase transitions between topological states are also closely related to the gapless SPTs described above.

The two simplest phases with topological order are arguably the two flavors of Z2 gauge theories, namely the toric code and the double semion. The electric field lines form deconfined "loop soups" in these phases, the only difference between them being that the loops have fugacity (-1) in the double semion case. Studying the phase transition between them is thus a natural place to look for non-trivial quantum phase transitions between topological phases. In [1], thanks to a judicious mapping to an SPT model, we were able to design a sign-problem-free quantum Monte Carlo algorithm to study this transition. This enabled us to study much larger systems than previously reached with exact diagonalization.

Unexpectedly, we discovered an intermediate gapless phase in which the loops form stripes. This stripe phase admits a description in terms of a deconfined U(1) gauge theory, which evades Polyakov's result on the confinement of U(1) gauge theories in 2D thanks to the incommensurability of the stripe order. This deconfinement occurs over a Cantor set along the parameter space of stripe wavevector, a phenomenon previously predicted by Eduardo Fradkin and collaborators [2] in the context of quantum dimer models and dubbed "Cantor deconfinement". We have thus uncovered yet another example of the close connection between topological phase transitions and deconfined quantum criticality.

• [1] Maxime Dupont, Snir Gazit, Thomas Scaffidi, "Evidence for deconfined U(1) gauge theory at the transition between toric code and double semion", Phys. Rev. B 103, 140412 (2021)

• [2] Fradkin et al, "On bipartite Rokhsar-Kivelson points and Cantor deconfinement"

Quantum phase transitions between Z2 SPTs

The algorithm we developed for the above project is very general and can be used for a broad class of SPTs. In this follow-up work [1], we studied with quantum Monte Carlo 3 different quantum phase transitions between different models for Z2 bosonic SPTs. Although we found a variety of intermediate symmetry-broken phases in this case, this work paves the way for a study of a wider phase diagram in which these ordered phases can be frustrated by adding extra terms, making it possible to find direct transitions between different topological states. Indeed, in a follow-up work from the group of Ashvin Vishwanath [2], a direct transition with an enlarged symmetry group was discovered in a variation of the SPT models studied in [1].

• [1] Maxime Dupont, Snir Gazit, Thomas Scaffidi, "From trivial to topological paramagnets: The case of Z2 and Z2^3 symmetries in two dimensions", Phys. Rev. B 103, 144437 (2021)

• [2] Tantivasadakarn et al, "Pivot Hamiltonians as generators of symmetry and entanglement", arXiv (2021)