Detecting underlying symmetry-protected topological phases via strange correlators and edge engineering

TL;DR

The paper tackles the challenge of identifying hidden symmetry-protected topological (SPT) phases that resemble conventional states. It introduces a framework combining edge engineering and generalized strange correlators to diagnose SPT order via bulk-edge correspondence, and validates it on a dimerized spin-$\tfrac{1}{2}$ Heisenberg model that is quasi-1D in character. The results show the dimer phase is a nontrivial SPT connected to the 1D Haldane phase, featuring a ferromagnetically ordered zigzag edge and extraordinary surface critical behavior at the $(2+1)\mathrm{D}$ O(3) bulk critical point, challenging classical-quantum mappings. Together, these findings provide a standard, practical method for uncovering topological phases that masquerade as ordinary states of matter in higher dimensions.

Abstract

The vast majority of symmetry-protected topological (SPT) states are difficult to detect, which often leads to their misidentification as ordinary or topologically trivial phases. In this work, we propose a general framework for detecting these hidden topological states. We distinguish the ordinary matter state from the topological phase by exploiting the boundary effects in space (via surface behaviors on engineered edge) and time (via strange correlators) according to the principle of bulk-edge correspondence. As a concrete example, we study the dimerized spin-1/2 Heisenberg model on a square lattice using quantum Monte Carlo simulations, focusing on its paramagnetic dimer phase and edge states. The dimer phase has been widely regarded as topologically trivial due to its gapped edge state on conventional edges. However, the model can also be viewed as two-dimensional antiferromagnetically (AF) coupled usual ladders, which suggests an SPT state adiabatically connected to the one-dimensional Haldane phase. We resolve this puzzle and demonstrate that the dimer phase is indeed a quasi-one-dimensional SPT state by measuring generalized strange correlators introduced in this work and by showing that the nontrivial gapless edge state on a zigzag edge is ferromagnetically ordered, resulting from effective ferromagnetic interactions between degenerate spinons liberated on each side of the cut. Furthermore, we show that the ordered edge state gives rise to an extraordinary surface critical behavior at the (2+1)-dimensional O(3) bulk critical points of the model, which contradicts theoretical predictions based on classical-quantum mapping. Overall, we establish a standard detection method for uncovering topological phases that masquerade as ordinary states of matter.

Figures (10)

• Figure 1: (a) A usual ladder with AF rung couplings $J_1$ and (b) a diagonal ladder with AF diagonal couplings $J_1$ and $J_2$, typically, $J_1=J_2$. $J=1$ denotes the intrachain coupling. (c) A particular spin configuration matches a typical short-range valence bond (VB) state, which constitutes the ground state of the usual ladder. Dashed ovals indecate spin-1 $\mathbf{S}^{(\rm E)}_i=\mathbf{S}_{i,0}+\mathbf{S}_{i+1,1}$ After all $S_i^z=0$ removed, the remaining spins show Néel order. The thick dashed lines show the two ways to cut systems. Two spinons appear if the cut (cut1) goes through one VB connecting two spin-1s, while cutting the ladder vertically (cut2) breaks a spin-1, no spinon appear. (d) A particular spin configuration matches a typical short-range VB state that constitutes the ground state of the diagonal ladder. Dashed ovals indicate spin-1 $\mathbf{S}^{(\rm O)}_i=\mathbf{S}_{i,0}+\mathbf{S}_{i,1}$ After all $S_i^z=0$ removed, the remaining spins show Néel order. The thick dashed line represents a vertical cut that breaks a VB connecting two spin-1s; two spinons are generated. The staggered pattern (e) and the columnar pattern (f) of VBs are product states $|\Omega\rangle$ used to define strange correlators. Thick-black lines denote VBs. When computing a two-dimensional system, the patterns are repeated along the $y$ direction, covering the lattice model. (g) is the transposition graph of (c) and (e), and (h) is the transposition graph of (d) and (f).
• Figure 2: (a) and (b) show the dimerized spin-$1/2$ Heisenberg model on a bipartite square lattice with sublattices A (yellow circles) and B (blue circles). Strong bonds $J_1>0$ are marked by thick red lines, weak bonds $J>0$ by thin red lines, and interladder coupling $J_{\perp}>0$ by thin blue lines. (a) Periodic boundary conditions are applied in $x$ and $y$ directions. The dashed rectangular box encircles a usual ladder with AF rungs. (b) Cutting along the dashed lines to expose edges, generating spinons along the edges opened by cut1, shown as open circles. Cut2 opens the boundaries trivially without spinons generated. (c) and (d) show the Q1D coupled diagonal ladders, which are bipartite with sublattices A (yellow circles) and B (blue circles). A diagonal ladder is shown inside the dashed rectangular box. (c) applies periodic boundary conditions in both $x$ and $y$ directions. (d) applies open boundaries in $x$ direction to expose edges (open circles), with periodic boundary conditions $y$ direction.
• Figure 3: The $J$-$J_{\perp}$ phase diagram of the dimerized spin-1/2 Heisenberg model on a square lattice. The dimer phase is labeled by Q1DH, indicating that it is a Q1D Haldane state. The dashed lines denote $J=1$ and $J=J_{\perp}$, respectively. $J_{\perp 1c}$ lables the critical point $(J=1, J_{\perp}=0.31407(5))$ and $J_{\perp 2c}$ labels the one $(J=0.52337(3),J_{\perp}=0.52337(3))$.
• Figure 4: Finite-size behavior of the strange correlators of the 1D diagonal and usual ladder for different parameters defined in Fig. \ref{['Fig:twoladders']}. (a) $C^{\rm O}_{\rm SC}(L/2)$ of the diagonal ladder with $J=J_{1}$=1, which converges to 0.5098(2) for $J_{2}=1.0$ and to 0.6217(4) for$J_{2}=0.0$, respectively. (b) $C^{\rm E}_{\rm SC}(L/2)$ of the usual ladder with $J$=1, which converges to 0.62176(2) for $J_{1}=1.0$ and to 1.837(1) for $J_{1}=0.26$, respectively.
• Figure 5: Finite-size scaling analyses of the strange correlators of the Q1D coupled diagonal and usual ladders. (a) $C_{\rm SC}^{\rm O} (L/2)$ of the coupled diagonal ladders converges to 0.23(2) for $J_{\perp}=0.08$ and to 0.237(5) for $J_{\perp}=0.12$, with $J=J_1=J_2$=1. (b) $C_{\rm SC}^{\rm E}(L/2)$ of Q1D coupled usual ladders converges to 0.27(2) for ($J=1$, $J_{\perp}=0.16$) and to 0.72(2) for ($J=0.26$, $J_{\perp}=0.26$) with $J_1=1$.