Phase transitions and remnants of fractionalization at finite temperature in the triangular lattice quantum loop model

Abstract

The quantum loop and dimer models are archetypal correlated systems with local constraints. With natural foundations in statistical mechanics, they are of direct relevance to various important physical concepts and systems, such as topological order, lattice gauge theories, geometric frustrations, or more recently Rydberg arrays quantum simulators. However, how the thermal fluctuations interact with constraints has not been explored in the important class of non-bipartite geometries. Here we study, via unbiased quantum Monte Carlo simulations and field theoretical analysis, the finite-temperature phase diagram of the quantum loop model on the triangular lattice. We discover that the recently identified, "hidden" vison plaquette (VP) quantum crystal [1] experiences a finite-temperature continuous transition, which smoothly connects to the (2+1)d Cubic* quantum critical point separating the VP and $\mathbb{Z}_{2}$ quantum spin liquid phases. This finite-temperature phase transition acquires a unique property of ``remnants of fractionalization" at finite temperature, in that, both the cubic order parameter -- the plaquette loop resonance -- and its constituent -- the vison field -- exhibit independent criticality signatures. This phase transition is connected to a 3-state Potts transition between the lattice nematic phase and the high-temperature disordered phase. We discuss the relevance of our results for current experiments on quantum simulation platforms.

Figures (11)

• Figure 1: Finite-temperature phase diagram of the QLM on the triangular lattice which exhibits different phases: Lattice Nematic (LN, green region), Vison Plaquette (VP, blue region), and disordered (red) phases. A comprehensive summary of the data can be found in Table. S1 and S3 of the Supplemental Material (SM) suppl. The solid gray horizontal line marks the $\mathbb{Z}_{2}$ spin liquid phase only existing in the ground-state. The green $(V \simeq 0.05)$ and blue $(V\simeq 0.6)$ points correspond to the LN-VP and VP-QSL quantum phase transitions. The red star indicates a possible multicritical point at $(V, T)=(0.07(2), 0.22(2))$. The two dashed gray lines correspond to $T=0.23$ (upper) and $T=0.1$ (lower) used in Fig. \ref{['fig:fig4']}. The inset zooms in at low-$T$ near the VP-QSL Cubic* QCP, highlighting that the finite-$T$ phase boundary scales as $T_c(V) \sim |V-V_{c}|^{\nu_{C^*}}$, with $\nu_{C^*}\simeq 0.7$ the correlation length exponent for the 3d Cubic* universality class.
• Figure 2: Left panels: data collapse for (a) the nematic order parameter $\langle D \rangle$, (b) its Binder ratio $B_{D}$, (c) the vison parameter $\phi$ and (d) its Binder ratio $B_{|\phi|}$ for system sizes from $L=8$ to $L=24$. Simulations are performed at $V=-0.5$, where the ground state is deep within the LN phase. All data collapses for $\langle D \rangle$, $B_{D}$, $\phi$ and $B_{|\phi|}$ use the critical exponents $\beta_{3P}=\frac{1}{9}$ and $\nu_{3P}=\frac{5}{6}$ of the 3-state Potts model. Right panels: (e) Binder ratio $B_{|\phi|}$ of the vison order parameter and (f) Correlation ratio $R_{C{\cal T}}$ of the $t$-terms for system sizes $L=4$ to $L=32$ for $V=0.3$ (where the ground state is located within the VP phase). The inset illustrates the crossing between system sizes $L$ and $2L$ for both $B_{|\phi|}$ and $R_{C{\cal T}}$. Lines are fits to $T_{c}(L)=T_{c}(\infty)+aL^{-(\omega+1/\nu)}$, where $\omega$ is the correction exponent and $\nu$ the correlation length critical exponent. For $V=0.3$, we obtain $T_{c}(\infty)=0.14(1)$ and $\omega+1/\nu\sim1.4$. Similar analysis for other $V$ values are presented in SM suppl.
• Figure 3: Three-dimensional histograms of ($\phi_{1}, \phi_{2}, \phi_{3}$) (first row) and $(t_{1}, t_{2}, t_{3})$ (second row) within the VP phase ($T=0.1$), near the transition point ($T=0.2$) and in the disordered phase ($T=0.3$). System size is $L=24$. They illustrate the phase transition process, where the corner-cubic order in the VP phase, the cube in ($\phi_{1}, \phi_{2}, \phi_{3}$) and tetrahedron shapes in $(t_{1}, t_{2}, t_{3})$, shrink to a point in the disordered phase.
• Figure 4: (a) Kinetic energy density $K=\sum_j {\cal}\langle T_j \rangle/L^2$ for $L=8$ to $L=24$ as a function of $V$ at $T=0.1$. The vertical line indicates the transition at $V=0.07(2)$. (b) Histograms of $\phi$ at $T=0.1$ and $T=0.23$ (dashed gray lines in Fig. \ref{['fig:fig1']}) for $L=24$, with three rows corresponding to $V=0$, $V=0.07$, $V=0.1$. The disappearance of the VP phase at all values of $V$ as the temperature increases to $0.23$ suggests the LN phase directly transitions to a disordered phase.
• Figure S1: (a) The QLM and the string operator between two visons on a triangular lattice. The open circles represent the lattice sites, while solid black bonds indicate the presence of loop segments on those bonds. Dashed bonds represents the absence of loop segments. The reference vison is marked by a green star at the upper triangle of the first unit cell. Two visons $v_i$ and $v_j$, located on the triangles, are connected by a string, represented by the dashed green line. (b) Ground-state phase diagram of the QLM on the triangular lattice as obtained in Ref. ran2024hidden. In the left panel, we display three possible loop configurations within the LN phase, with the corresponding vison patterns indicated by the green numbers in each triangle. In the middle subfigure, we present a schematic representation of the real-space vison correlation functions. The color scheme employs green to denote positive correlation values and grey for negative values, while the intensity of the colors reflects the relative magnitudes of these correlations.