Topology meets symmetry breaking: Hidden order, intrinsically gapless topological states and finite-temperature topological transitions

Abstract

Since the discovery of phase transitions driven by topological defects, the classification of phases of matter has been significantly extended beyond Ginzburg and Landau's paradigm of spontaneous symmetry breaking (SSB). In particular, intrinsic and symmetry-protected topological (SPT) orders have been discovered in (mostly gapped) quantum many-body ground states. However, these are commonly viewed as zero-temperature phenomena, and their robustness in a gapless ground state or against thermal fluctuations remains challenging to tackle. Here we introduce an explicit construction for SPT-type states with hidden order associated with SSB: They feature (quasi) long-range correlations along appropriate edges, but short-range order in the bulk; ground state degeneracy associated with SSB; and non-local string order in the bulk. We apply our construction to predict two types of finite-temperature SPT transitions, in the Ising and BKT class respectively, where the usual signs of criticality appear despite the absence of a diverging correlation length in the bulk. While the state featuring hidden Ising order is gapped, the other SPT state associated with the BKT-SPT transition has hidden $U(1)$, or XY-order and constitutes an intrinsically gapless SPT state, associated with a gapless Goldstone mode. Specifically, in this work we discuss spins with global $\mathbb{Z}_2$ or $U(1)$ symmetry coupled to link variables constituting a loop gas model. By mapping this system to an Ising-gauge theory, we demonstrate that one of the SPT phases we construct corresponds to the Higgs-SPT phase at $T=0$ -- which we show here to remain stable at finite temperature. Our work paves the way for a more systematic search for hidden order SPT phases, including in gapless systems, and raises the question if a natural (finite-$T$) spin liquid candidate exists that realizes hidden order in the Higgs-SPT class.

Figures (7)

• Figure 1: We predict a class of HO-SPT phases featuring SSB without a local bulk order parameter, in systems where domain walls of a spin order parameter couple to a loop gas of fluctuating link variables. a) In the hidden-Ising order (HIO) model, the sign of Ising interactions $\propto \hat{S}^z_{\mathbf{i}} \hat{S}^z_{\mathbf{j}}$ between spins is flipped across strings defined by link variables $\tau^z_{\langle \mathbf{i}, \mathbf{j}\rangle}=-1$ (blue, wiggly lines). Fluctuations of the strings around sites (highlighted in yellow) are accompanied by flips of the central spin. In addition, spin-flips driven by a transverse field can be included. The 1D HIO model can be understood as an excerpt of the 2D model (dark gray box). b) Deep in the HO-SPT phase, the ground state satisfies the hidden order rule, illustrated in the top panel: spins flip sign only across a flipped link. The bottom panel shows an excitation, corresponding to a spin-flip without a link-flip. Spins and link variables in the HIO model at $\lambda=0$ can be exactly decoupled by the non-local unitary transformation $\hat{U}$ illustrated in both panels, which flips all spins between two strings with $\tau^z=-1$. c) The phase diagrams of the HIO model in 1D, Eq. \ref{['eq:1d_ham']} for $\lambda = 0$, and in 2D, Eq. \ref{['eq:TC_meets_Ising']}, have the same structure [only the critical values in 2D, $(h_\tau/J_\tau)_{\rm c,2D}$ and $(h_S/J_S)_{\rm c,2D}$, differ from their shown value $0.5$ in 1D]. The conventional SSB phase with bulk long-range order (LRO) and non-zero magnetization $M_S>0$ is located next to a symmetric phase with short-range order (SRO) and the SPT phase with hidden order (HO-SPT), which features hSSB. As explained in the text, the HO can be detected by the magnetization of Ising spins in squeezed space, $M_S^*$. A hidden quantum critical point (hQCP), or HO-SPT transition, separates the HO-SPT from the SRO phase. The orange arrow in c) indicates the corresponding scan in our numerics shown in Fig. \ref{['fig:pd_lamda']} a).
• Figure 2: Numerical study of the HO-SPT phase in the 1D HIO model. a) We perform DMRG simulations using the SyTen toolkit and take snapshots of the many-body wavefunction via the perfect sampling approach Ferris2012Buser2022, allowing us to evaluate magnetizations $M_S$ and $M_S^*$ in real and squeezed space; the latter are indicated by two overlaid color maps. For the exactly solvable case $\lambda = 0$, the hQCP found in Fig. \ref{['fig:1d_pd']} c) is located at a critical value $(J_S/J_\tau)_c = 0.2$, indicated by a short blue line. The orange arrow corresponds to the same scan in Fig. \ref{['fig:1d_pd']} c). We find that for non-zero values of $\lambda$, the HO phase persists until eventually it transitions to the LRO phase (with $M_S>0$, for large $J_S/J_\tau$) or the disordered phase (with $M_S=M_S^*=0$, for small $J_S/J_\tau$). Areas with $\frac{1}{(L-1)}\langle | \sum_j \hat{\tau}^z_{\langle j,j+1 \rangle} | \rangle > 0.3$ are indicated by hatched, grey lines. b) We define standard spin-spin correlations $\langle |\hat{S}^z_0 \hat{S}^z_x|\rangle$ as a function of distance $x$, computed along the transition from HO to LRO at $J_S/J_\tau=2.25$ in c). Colors in c) correspond to different values of $\lambda$, as highlighted by data points of the same color in a). We find that the bulk order disappears around $\lambda = 0.7$, whereas long-range edge-to-edge correlations continue to indicate the presence of SSB in the HO phase for smaller values of $\lambda$. In a) and c) we considered a chain of $L = 51$ spins and $50$ links in between, and set $h_\tau/J_\tau = h_S/J_\tau = 0.1$; gray circles indicate the underlying data points. See Appendix \ref{['sec:appB']} for more details on our numerical simulations.
• Figure 3: Schematic phase diagram of the 2D HIO model, Eq. \ref{['eq:TC_meets_Ising']}, at finite temperature and in the closed-loop limit, $\mu_\tau \to \infty$. The decoupling of spin and link degrees of freedom in squeezed space, after applying the unitary transformation in Eq. \ref{['eq:1d_HO_unitary']}, leads to a factorization of the phase diagram. The insets illustrate the respective system configurations. The TFIM of spins $\hat{\mathbf{S}}$ features a finite-$T$ Ising transition, describing where SSB takes place. The toric code in a field describing links $\hat{\mathbf{\tau}}$ is dual to a TFIM, with an associated ${\rm Ising}^*$ transition characterizing the deconfinement of the loop gas. In regimes where the loop gas is deconfined (percolating), through quantum or thermal fluctuations, spin order is hidden (blue region) in real space. For large $J_\tau / h_\tau$, a finite-$T$ SPT transition at $T_c$ is obtained. In the confined (non-percolating) region of the loop gas, for small $J_\tau / h_\tau$, and when $h_S/J_S$ is small, low-$T$ LRO gives way to a HO phase at $T_c^{(1)}$ where the loop gas thermally deconfines, before entering the fully symmetric, disordered phase at $T_c^{(2)}$. In this regime, the critical temperature separating the ordered and disordered phase is $T/J_S \approx 2.27$ at elevated temperatures.
• Figure 4: Zero-temperature phase diagram of the double-Higgs $\mathbb{Z}_2$ gauge theory, Eq. \ref{['eqIGT']}. a) On the left vertical plane, for $h_X=0$, the exactly solvable phases of the 2D HIO model are obtained, including HO of both Higgs fields. The latter regimes connect directly to the respective Higgs-SPT phases of $S$ ($\sigma$) on the bottom horizontal (back vertical) plane. The phase boundaries of these SPT phases - which we conjecture to coincide with our HO-SPT phases - are taken from Ref. Verresen2024. The topological toric code phase (green) corresponds to the disordered, PM phases of the Higgs fields. b) The HO phase remains stable upon including open $\tau^z$ strings, by $h_X \neq 0$, because domain walls of $\hat{S}^z$ spins lead to a linear confining force between open ends of the strings, $B_\square = -1$.
• Figure 5: Phase diagram of the HXYO model, Eq. \ref{['eq:U1_meets_tc']}, in the closed-loop limit $\mu_\tau \to \infty$. a) At zero temperature, long-range $U(1)$ (or XY) order associated with a spontaneously broken $U(1)$ symmetry develops in squeezed space when $|\Delta|<|J_S|$. This manifests in a gapless SPT phase with $U(1)$ hidden order (HO) for small $h_\tau/J_\tau$, and turns into a conventional $U(1)$-ordered phase when $\tau^z$ strings confine at large $h_\tau$. For $|\Delta|>|J_S|$, a $\mathbb{Z}_2$ ordered phase in the original basis is obtained, independent of the loop gas. b) At finite temperature, the hidden $U(1)$ order turns into hidden quasi-long range order, with power-law correlations in squeezed space. At higher temperatures $T^*_{\rm BKT}$ a hCP / finite-$T$ SPT transition into a symmetric, paramagnetic phase is found. The $\mathbb{Z}_2$ order remains stable up to $T_{\rm Ising}$ where it disappears in a finite-$T$ symmetry-breaking transition of Ginzburg-Landau type, in the Ising universality class. At $|\Delta|=|J_S|$, a hidden $SU(2)$ symmetry precludes any finite-$T$ phase transitions.