Quantum Restored Symmetry Protected Topological Phases

TL;DR

This work addresses how symmetry-protected topological (SPT) phases can persist when the protecting symmetry is dynamically broken by fluctuations by introducing quantum restored SPTs (QRSPTs). It analyzes a one-dimensional spinful SSH chain with fluctuating s-wave superconductivity using a combination of mean-field reasoning, bosonization, RG, semiclassical edge analysis, and DMRG to map the phase diagram and identify a QRSPT phase with edge states protected by symmetry. The results reveal a spin-edge to charge-edge transition, a BKT-like superconducting boundary, and robust edge degeneracies tied to an antiunitary symmetry C, highlighting a rich interplay between bulk topology and fluctuating order. This QRSPT framework provides a new design principle for topological phases via time-averaged symmetry and motivates exploration in higher dimensions and interacting regimes, with potential implications for engineered quantum materials and devices.

Abstract

Symmetry protected topological (SPT) phases are fundamental quantum many-body states of matter beyond Landau's paradigm. Here, we introduce the concept of quantum restored SPTs (QRSPTs), where the protecting symmetry is spontaneously broken at each instance in time, but restored after time average over quantum fluctuations, so that topological features re-emerge. To illustrate the concept, we study a one-dimensional fermionic Su-Schrieffer-Heeger model with fluctuating superconducting order. We solve this problem in several limiting cases using a variety of analytical methods and compare them to numerical (density matrix renormalization group) simulations which are valid throughout the parameter regime. We thereby map out the phase diagram and identify a QRSPT phase with topological features which are reminiscent from (but not identical to) the topology of the underlying free fermion system. The QRSPT paradigm thereby stimulates a new perspective for the constructive design of novel topological quantum many-body phases.

Figures (10)

• Figure 1: a) Consider a free fermion SPT where the symmetry group G prevents the admixture of distinct topological sectors of the Bloch Hamiltonian and of zero energy boundary states with distinct quantum numbers. b) A mean-field SSB of G trivializes such a quantum state, but c) strong quantum fluctuations of the order parameter may destroy long-range order even if the local expectation value of the order parameter amplitude is non-zero. Thereby, topological features reemerge, and the SPT is quantum restored.
• Figure 2: a) Schematic representation of the model Eq. \ref{['Eq: Hamiltonian for the total model.']}. The rectangular boxes denote the Cooper pair boxes. b) Mean-field phase diagram: $\Delta$ breaks the symmetry protecting the free-fermion topology (cf Fig. \ref{['Fig:QRSPT']} b)) so that topological and trivial phase may be adiabatically connected. TI and triv.I. represent topological and trivial insulator respectively. Note that there is a gap closure (as expected from standard SSH physics) at $t=t^{\prime}$ (for $\Delta = 0$). c) Schematic phase diagram of Eq. \ref{['Eq: Hamiltonian for the total model.']}. Note the QRSPT phase at small $\Delta$ and $t < t'$.
• Figure 3: a) Eigenvalues of the edge Hamiltonian, Eq. \ref{['Eq: Effective edge Hamiltonian']}, as a function of $\Delta$ for $t=0, N_{g}=2$. Note the edge transition from gapless spin edge modes to gapless charge edge modes at $\Delta = E_{\rm{C}}$. $N_{1}$ represents the number of bosons on the first bosonic site. $N_{\rm{total}} = 2N_{1} + n_{1}$ represents the total charge on the left edge. b) Edge states correspond to kink-like field configurations within the effective field theory near the free fermion critical point, Eq. \ref{['Eq: Bosonized Equation']}. c) RG flow obtained using the flow equations given in Eq:\ref{['Eq: RG flow equations']}.
• Figure 4: a) Color plot of the central charge obtained using finite DMRG for a system size of 40 and $N_{g} = 4$. The phase diagram is obtained for $\frac{E_{\rm{C}}}{\sqrt{t^{2}+ t^{\prime2}}} = 0.01$. Red lines denote the analytically obtained position of the Berezinskii-Kosterlitz-Thouless transition line. The grey dot (grey dashed line) correspond to the locations in parameter space at which the data in panels b) and c) are taken. b) Log-Log plot of the bosonic and fermionic correlator for $\Delta = 50 E_{\rm{C}}$ and $\arctan(\frac{t}{t^{\prime}}) = 0.47$. The power-law fit for the bosonic and fermionic correlator gives a $K_{\rm{SC}}$ value of $0.071 \pm 0.001$ and $0.070 \pm 0.001$ respectively. c) Entanglement spectrum obtained using iDMRG for $\Delta = 4 E_{\rm{C}}$ and $\frac{E_{\rm{C}}}{\sqrt{t^{2}+t^{\prime2}}} = 0.01$ and $N_{g} = 2$. The green-shaded regime corresponds to criticality. Consistently with analytical expectations, we observe even degeneracy throughout the regime corresponding to QRSPT in Fig. \ref{['Fig:Model']}(c). Note that we only show the entanglement spectrum values upto $\epsilon_{i} = 12$.
• Figure S1: Shaded region represents a segment with unique value of the bosonic variables. The translation from the shaded region for the bosonic variables $(\Phi_{\uparrow},\Phi_{\downarrow})$ to $(\Phi_{s},\Phi_{\rho})$ is done using the transformation identity in Eq. \ref{['SM:Eq: Definitions of Parameters in the bosoniized equation']}.