Systematic study for two-dimensional Z2 topological phase transitions at high-symmetry points in all layer groups

Kevin Loo, Qing-Rui Wang

We use the pullback trivialization technique to systematically construct gapped interfaces and anomalous boundaries for fermionic symmetry-protected topological (FSPT) states by extending their symmetry group ${G}_{f}={\mathbb{Z}}_{2}^{f}{\ifmmode\times\else\texttimes\fi{}}_{{\ensuremath{\omega}}_{2}}G$ to larger groups. These FSPT states may involve decoration layers of both Majorana chains and complex fermions. We derive general consistency formulas explicitly for ($2+1$)-dimensional (D) and ($3+1$)D systems, where nontrivial twists arise from fermionic symmetric local unitaries or ``gauge transformations'' that ensure coboundaries vanish at the cochain level. Additionally, we present explicit example for a ($3+1$)D FSPT of symmetry group ${G}_{f}={\mathbb{Z}}_{2}^{f}\ifmmode\times\else\texttimes\fi{}{\mathbb{Z}}_{4}\ifmmode\times\else\texttimes\fi{}{\mathbb{Z}}_{4}$ with Majorana chain decorations.

Hong-Hao Song, Chen Peng, Rui-Zhen Huang +1 more

Wenjie Ji, Xiao-Gang Wen

For a zero-temperature Landau symmetry breaking transition in $n$-dimensional space that completely breaks a finite symmetry $G$, the critical point at the transition has the symmetry $G$. In this paper, we show that the critical point also has a dual symmetry - a $(n-1)$-symmetry described by a higher group when $G$ is Abelian or an algebraic $(n-1)$-symmetry beyond higher group when $G$ is non-Abelian. In fact, any $G$-symmetric system can be viewed as a boundary of $G$-gauge theory in one higher dimension. The conservation of gauge charge and gauge flux in the bulk $G$-gauge theory gives rise to the symmetry and the dual symmetry respectively. So any $G$-symmetric system actually has a larger symmetry called categorical symmetry, which is a combination of the symmetry and the dual symmetry. However, part (and only part) of the categorical symmetry must be spontaneously broken in any gapped phase of the system, but there exists a gapless state where the categorical symmetry is not spontaneously broken. Such a gapless state corresponds to the usual critical point of Landau symmetry breaking transition. The above results remain valid even if we expand the notion of symmetry to include higher symmetries and algebraic higher symmetries. Thus our result also applies to critical points for transitions between topological phases of matter. In particular, we show that there can be several critical points for the transition from the 3+1D $Z_2$ gauge theory to a trivial phase. The critical point from Higgs condensation has a categorical symmetry formed by a $Z_2$ 0-symmetry and its dual - a $Z_2$ 2-symmetry, while the critical point of the confinement transition has a categorical symmetry formed by a $Z_2$ 1-symmetry and its dual - another $Z_2$ 1-symmetry.

C. Morais Smith, Wouter Beugeling, Nathan Goldman

Deepak Dhar

Tsui, Lokman

The study of continuous phase transitions triggered by spontaneous symmetry breaking has brought revolutionary ideas to physics. Recently, through the discovery of symmetry protected topological phases, it is realized that continuous quantum phase transition can also occur between states with the same symmetry but different topology. The subject of this dissertation is the phase transition between these symmetry protected topological states (SPTs). There are two main parts in this dissertation.In the first part we consider spatial dimension $d$ and symmetry group $G$ so that the cohomology group, $H^{d+1}(G,U(1))$, contains at least one $Z_{2n}$ or $Z$ factor. We show that the phase transition between the trivial SPT and the root states that generate the $ Z_{2n} $ or $Z$ groups can be induced on the boundary of a d+1 dimensional $G\\times Z_2^T$-symmetric SPT by a $Z_2^T$ symmetry breaking field. Moreover we show these boundary phase transitions can be ``transplanted'' to d dimensions and realized in lattice models as a function of a tuning parameter. The price one pays is for the critical value of the tuning parameter there is an extra non-local (duality-like) symmetry. In the case where the phase transition is continuous, our theory predicts the presence of unusual (sometimes fractionalized) excitations corresponding to delocalized boundary excitations of the non-trivial SPT on one side of the transition. This theory also predicts other phase transition scenarios including first order transition and transition via an intermediate symmetry breaking phase. In the second part, we study the phase transition between bosonic topological phases protected by $Z_n\\times Z_n$ in 1+1 dimensions. We find a direct transition occurs when $ n=2,3,4 $ and in all cases the critical point possesses two gap opening relevant operators: one leads to a Landau-forbidden symmetry breaking phase transition and the other to the topological phase transition. We also obtained a constraint($ c\\geq1 $) on the central charge for general phase transitions between symmetry protected bosonic topological phases in 1+1D.