Reflections on time-reversal in the Symmetry Topological Field Theory

Abstract

Symmetry under time-reversal appears in the microscopic description of many physical systems. In a quantum mechanical setting it acts as an anti-unitary operator, so does not fall under general analyses based on unitary symmetries. In classifying zero temperature phases of matter in (1+1)d lattice models, the role of anti-unitary symmetries is, however, well-understood. In recent years, the Symmetry Topological Field Theory (SymTFT) approach to this classification has given a general framework to understand symmetries as topological defects, but does not naturally include anti-unitary symmetries. Following recent proposals in the literature, we adopt a symmetry-enriched SymTFT for a theory with both internal and time-reversal symmetry. In particular, we take a standard SymTFT associated with an internal unitary symmetry that is then enriched by a background time-reversal symmetry. A detailed analysis of the topological boundary conditions of this enriched SymTFT allows us to characterize the corresponding (1+1)d gapped phases that preserve the enriching symmetry (i.e. those that do not spontaneously break this symmetry in the ground state). Line operators in the SymTFT approach are related to non-local string-order parameters (with charged end-point operators) for SPT phases. These are subtle in the anti-unitary case and we explore them both on the lattice and in the continuum. We include an analysis of unitary string order parameters that reveal the Klein bottle SPT invariant. On the lattice, we show that the correct end-point charge coincides with the time-reversal-charge only when the end-point operator is hermitian.

Figures (6)

• Figure 1: Graphical analysis of unitary string-order parameters in an MPS. The MPS tensor, in canonical form, is a hollow diamond, while its complex conjugate is a filled diamond. (a) Symmetry fractionalization of a unitary representation of $G$ on physical indices, $U(g) = \prod_j u_j(g)$. A transformation by $g$ is equivalent (up to a phase) to a projective representation $V_g$ acting adjointly on the bond indices. (b) The transfer matrix with unique dominant left and right eigenvectors. The orange square (an eigenvector containing the entanglement eigenvalues) commutes with $V_g$ for all $g$. (c) Define $\mu_k(g) = \prod_{j=-\infty}^{k-1}u_j(g) \mathcal{O}_k$; the two-point function $\langle \mu_1^\dagger(g) \mu_k^{}(g)\rangle$, for large $k$, factorizes as two local tensor terms $\rho_L$ and $\rho_R$. The phase $e^{i \theta_0}$ includes phases from the symmetry fractionalization, as well as from replacing $V_g^\dagger$ by $V_{g^{-1}}$, but does not play a role in any selection rules. (d) Consider $h$ such that $U(g)U(h)=U(h)U(g)$. The string order obeys a selection rule. The local tensor term $\rho_R$ vanishes unless the charge $e^{i\chi_{g,h}^\mathcal{O}}$ in $U(h) \mathcal{O} U(h)^\dagger = e^{i \chi_{g,h}^{\mathcal{O}}}\mathcal{O}$ is equal to $\varepsilon(g,h)$, the discrete torsion phase of the projective representation. An identical condition holds for $\rho_L$.
• Figure 2: The basic SymTFT sandwich: the $(1+1)$-dimensional theory $\mathfrak{T}$ on the right-hand-side is constructed as the interval compactification of a $(2+1)$-dimensional SymTFT $\mathfrak{Z}(\mathcal{C})$ on the left-hand-side, with two boundary conditions. The gapped (topological) boundary $\mathfrak{B}^{\text{sym}}_{\mathcal{C}}$ is on the left and the physical, possibly non-topological, boundary $\mathfrak{B}^{\text{phys}}_{\mathfrak{T}}$ is on the right. If the physical boundary $\mathfrak{B}^{\text{phys}}_\mathfrak{T}$ is also topological, the resulting theory $\mathfrak{T}$ is a $(1+1)$d TQFT.
• Figure 3: Order parameters are characterized by anyons ending on the physical boundary $\mathfrak{B}^{\text{phys}}$. Anyons also ending on $\mathfrak{B}^{\text{sym}}$ give rise to local order parameters ($\mathcal{O}$), while anyons becoming a symmetry line (red) on $\mathfrak{B}^{\text{sym}}$ give rise to string-order parameters ($\mathcal{O}'$). The charge of an order parameter (obtained by linking a symmetry line (blue) with the operator) descends from the braiding properties of the corresponding bulk SymTFT anyons. Local order parameters $\mathcal{O}$ are in 1-to-1 correspondence with ‘vacua’ $v$ of the resulting TQFT. While the $\mathcal{O}$ diagonalize the symmetry action, with a change of basis we can map them to operators satisfying $v_a v_b = \delta_{ab} v_{ab}$. These $v_a$ are the usual vacua, on which the symmetry acts by permutation Bhardwaj:2023idu.
• Figure 4: (a) A $G$-symmetric gapped boundary (dotted) of the enriched topological order $\mathcal{Z}(\mathcal{C}_1)$ provides in general an interface to a $G$-SPT. (b) If the SPT on the right hand side is the trivial SPT, then there is no obstruction to terminate the $G$-twists on the boundary.
• Figure 5: The pentagon equation satisfied by $M=b(1_T,1_T)$.