Superuniversality from disorder at two-dimensional topological phase transitions

We investigate the effects of quenched randomness on topological quantum phase transitions in strongly interacting two-dimensional systems. We focus first on transitions driven by the condensation of a subset of fractionalized quasiparticles (“anyons”) identified with “electric charge” excitations of a phase with intrinsic topological order. All other anyons have nontrivial mutual statistics with the condensed subset and hence become confined at the anyon condensation transition. Using a combination of microscopically exact duality transformations and asymptotically exact real-space renormalization group techniques applied to these two-dimensional disordered gauge theories, we argue that the resulting critical scaling behavior is “superuniversal” across a wide range of such condensation transitions and is controlled by the same infinite-randomness fixed point as that of the 2D random transverse-field Ising model. We validate this claim using large-scale quantum Monte Carlo simulations that allow us to extract zero-temperature critical exponents and correlation functions in $(2+1)\mathrm{D}$ disordered interacting systems. We discuss generalizations of these results to a large class of ground-state and excited-state topological transitions in systems with intrinsic topological order as well as those where topological order is either protected or enriched by global symmetries. When the underlying topological order and the symmetry group are Abelian, our results provide prototypes for topological phase transitions between distinct many-body localized phases.

Figure 1

Terms in the Hamiltonian (1) and the gauge constraints acting on group elements $g\in G$ on links. (a) The vertex term $A_v={\sum }_g\frac{1}{\left|G\right|}{A}_v^g$ symmetrically permutes group elements around vertex $v$. (b) The plaquette operator $B_p$ projects onto “zero flux” configurations in which the oriented product of group elements around plaquette $p$ is the identity $e$. (c) The link term $C_l$ favors $g_l=e$ on link $l$. We impose gauge constraints that restrict to zero-flux configurations (${\mathrm{\Phi }}_p=e$ on all plaquettes) and take $H=-{\sum }_v{J}_v{A}_v-{\sum }_l{h}_l{C}_l$.

Figure 2

Our RSRG scheme involves two types of decimations for strongly random $J_v,h_l$: (a) decimating link $l$ adjacent to vertices $v,v^{\prime }$ yields a new vertex $\tilde{v}$ with $J_{\tilde{v}}=\frac{2}{\left|G\right|}\frac{J_v{J}_{v^{\prime }}}{h_l}$; (b) decimating vertex $v$ adjacent to links $l_1,...,l_4$ yields $h_{l_{ij}}=\frac{2}{\left|G\right|}\frac{h_{l_i}{h}_{l_j}}{J_v}$ on both blue and cyan links. While blue links ($l_{12},l_{13},\cdots ,l_{42}$) are short links in the renormalized planar graph ${\mathcal{G}}^{\prime }$, the cyan link ($l_{34}$) is a long link and appears only through the term in the renormalized Hamiltonian. While the specific assignment of short versus long links is not unique, the specific choice does not affect the resulting RG flow.

Figure 3

Duality mapping between a spin configuration in the $Q$-state Potts model (with $Q=\left|G\right|$) (left) and a link configuration (gauge field configuration) in the gauge theory (right) under the duality. (a) The mapping automatically preserves the zero-flux gauge constraint. (b) The transverse-field term on vertex $v$ in the $Q$-state Potts model maps to a vertex term $A_v=\frac{1}{\left|G\right|}{\sum }_{h\in G}{A}_v^h$ under the duality. (c) The ferromagnetic interaction in the $Q$-state Potts model corresponds to the link term under the duality.

Figure 4

The RSRG of the $Q$-state Potts model and the $G$-gauge theory and the duality mappings (where $Q=\left|G\right|$). Here we denote the Ising-type interaction and the transverse field term in the Potts model as ${\widehat{\delta }}_{v,v^{\prime }}$ and ${\widehat{\sigma }}_v$. (a) Link decimation and (b) vertex decimation both commute with the duality mapping where the corresponding changes in the graph $\mathcal{G}\mapsto {\mathcal{G}}^{\prime }$ can be found in Fig. 2. A long-link term $C_l$ such as $C_{l_{34}}$ in Fig 2 (b) corresponds to a dual higher-neighbor ferromagnetic interaction term between spin states at vertex 3 and 4 with respect to a planar graph ${\mathcal{G}}^{\prime }$.

Figure 5

Disorder-averaged zero-temperature binder ratio ${\text{B}}_{\text{av}}$ as a function of $h_{\text{max}}$ for (a) $Q=2$ and (b) $Q=3$. Different curves correspond the different system sizes. In the insets, we plot the universal scaling functions ${\tilde{B}}_{\text{av}}(x=(h_{\text{max}}-h_c)L^{1/\nu })$ obtained from a curve collapse analysis; see main text. Solid lines depicts a numerical fit to second-order polynomial in the scaling variable $x$.

Figure 6

The universal scaling function ${\tilde{S}}_{\text{av}}[x=(h_{\text{max}}-h_c)L^{1/\nu }]$ (see main text) obtained from a curve collapse analysis for (a) $Q=2$ (b) $Q=3$. We use this analysis to extract the critical exponent $\eta$ controlling the scaling of the magnetization squared.

Figure 7

(a) Disorder averaged spin-spin correlation function at the critical point for various system sizes for (a) $Q=2$ and (b) $Q=3$. (inset) Comparison between averaged and typical spin-spin correlation functions for $L=18$. Typical spin-spin correlations exhibit a faster decay as expected at an infinite-randomness fixed point.

Figure 8

(a) To compute the $U\left(1\right)$ phase factor of $A_v^{g,\omega }$ when acting at the vertex $v$, we construct a three dimensional “tent” manifold by adding a new vertex $v^{-}$ and links connecting $v^{-}$ and other vertices. The index of $v^{-}$ is set by a number that is infinitesimally smaller than the index of $v$, which induces the orientation of links in the tent. (Orientations of links other than the yellow link connecting $v^{-}$ and $v$ are not drawn explicitly.) (b) The $U\left(1\right)$ phase factor $e^{i{\varphi }_g\left(\omega \right)}$ of $A_v^{g,\omega }$ operator is given by the $U\left(1\right)$ phase factor $\omega \left[M\right]$ associated with the “tent” manifold $M$. The computation of this phase factor is explained in Appendix pp1; a crucial point is that for a lattice of triangular plaquettes, any tent manifold $M$ can be decomposed in terms of tetrahedrons $T$ so that $M={\cup }_{T\in M}T$, and $\omega \left[M\right]={\prod }_{T\in M}\omega \left[T\right]$. The tetrahedrons are the fundamental objects that are employed to properly track phase factors under decimations. The remaining operator in $A_v^{g,\omega }$ is $A_v^g$, which is identical to the one in Fig. 1.

Figure 9

Summary of the Fibonacci anyonic system. (a) The fusion rule of the Fibonacci anyonic system, which has two anyons: the trivial/vacuum anyon (1) and a non-Abelian “Fibonacci” anyon ($\tau$), which are (b) respectively denoted by a dashed line (equivalent to an empty line) and by a solid red line. (c) Possible fusion channels in the Fibonacci anyonic system, which are allowed set of configurations set by $A_v=1$ constraint in the Levin-Wen model.

Figure 10

Change of basis in the fusion tree of the Fibonacci anyonic system from which the nontrivial $F$-matrix $\left[F_{\tau \tau }^{\tau \tau }\right]$ can be read off.

Figure 11

Definition of the $B_p^s$ term in the Levin-Wen model where we use the Einstein summation convention, i.e., summation is assumed for repeated indices. $B_p^s$ inserts an anyonic flux $s$ around the plaquette $p$; we use a series of $F$-matrices to compute the matrix element of $B_p^s$. $B_p^s$ is a 12-spin interaction for a hexagonal plaquette $p$, and it acts diagonally on states on external legs denoted as states $i,j,\cdots ,h$.

Figure 12

(a) For a link decimation $C_l=1$, we polarize the spin state at $l$ to the trivial state $|1\rangle$ and remove link $l$ from the lattice. Such link decimation naturally identifies the states $b=e^{*}$ and $c=d^{*}$, where ${}^{*}$ appears due to the direction of links. Accordingly, four adjacent links to $l$ become two links after the identifications as described in the RHS. (b) Computation of $C_l\left(B_p^s{B}_{p^{\prime }}^{s^{\prime }}\right)C_l$ on the $C_l^a(\equiv |a\rangle \langle a{|}_l)=1$ subspace. The summation over $a^{\prime }$ is implicit in the RHS of the second equality. As a result of computation, we get the plaqutte term $B_{p{p}^{\prime }}^s$ associated with the larger plaquette $p{p}^{\prime }$.

Figure 13

The proof of Eq. (40) on $C_l^s(\equiv |s\rangle \langle s{|}_l)=1$ subspace. Since $s$ is arbitrary, Eq. (40) holds in general. In the proof, we use the fact that $F_{a^{*}as}^{b^{*}b1}=\frac{v_s}{v_a{v}_b}$ where $v_s=\sqrt{d_s}$ and employ a handle slide from 3-manifold theory [73] in the second line.

Figure 14

A configuration with a square plaquette $p$ together with four boundary legs $l_1$, $l_2$, $l_3$, and $l_4$, and four external legs. This configuration serves as a minimal example where long-link terms do not commute; see main text for more details.

Figure 15

We assign $U\left(1\right)$ phase factors for tetrahedrons with (a) positive orientation and (b) negative orientation according to the group elements on links. We denote each vertex as its vertex index, which induces the link orientations and the tetrahedron orientation. (c) The $U\left(1\right)$ phase factor associated with a triangulated 3-manifold is given by the product of the $U\left(1\right)$ phase factors of individual tetrahedrons in the triangulation.

Figure 16

(a) 1-4 Pachner move. The 3-manifold on the left is a single tetrahedron $[v_1,v_2,v_3,v_4]$ and the 3-manifold on the right is obtained by gluing four tetrahedrons $[w,v_1,v_2,v_3]$, $[w,v_1,v_2,v_4]$, $[w,v_1,v_3,v_4]$, and $[w,v_2,v_3,v_4]$. (b) 2-3 Pachner move. The 3-manifold on the left is obtained by gluing two tetrahedrons $[v_1,v_2,v_3,v_4]$ and $[v_1,v_2,v_3,v_5]$ and the 3-manifold on the right is obtained by gluing three tetrahedrons $[v_1,v_2,v_4,v_5]$, $[v_1,v_3,v_4,v_5]$, and $[v_2,v_3,v_4,v_5]$. The $U\left(1\right)$ phase factors associated with the 3-manifolds on the left and right are identical in both 1-4 and 2-3 Pachner moves.

Figure 17

(a) Lattice isomorphism ${\widehat{T}}_1$ amounts to inverting the direction of a link. Accordingly, the basis element of the corresponding link gets inverted. (b) Lattice isomorphism ${\widehat{T}}_2$ amounts to removing/adding a link. Due to gauge constraints, ${\widehat{T}}_2$ does not change the dimension of the Hilbert space. For example, the basis element $g$ is completely fixed by $g_1$ and $g_2$, the states on neighboring links, via $g=g_1·g_2$. (c) Lattice isomorphism ${\widehat{T}}_3$ relates the lattice with a vertex $v$ (left column) and the lattice without the vertex $v$ and the links connected to $v$ (right column). ${\widehat{T}}_3$ induces a mapping between $A_v=1$ subspace of the Hilbert space on the left lattice and the Hilbert space on the right lattice. To find such mapping, we pick a representative link (colored link) and define the preferred basis in the left lattice. The preferred basis is mapped to the computational basis on the right lattice via ${\widehat{T}}_3$.

Figure 18

A series of support deformations of a long-link term, where the red links denote the support of (or equivalently, the directed path associated with) the link term. The deformations can be implemented by invoking the zero-flux constraint on each plaquette. Note that the end points are fixed during the deformation.

Figure 19

A procedure which maps a long-link term to a short-link term. We employ a series of support deformation of a link term and a series of lattice isomorphism ${\widehat{T}}_2$ during the procedure.

Figure 20

${\widehat{T}}_2$ lattice isomorphism acting on a computational basis state. Here group elements (which are not shown explicitly) are placed on each link and satisfy the zero-flux constraint on each plaquette. Note that group elements on colored links are completely fixed by the group elements on neighboring links. Compared to the untwisted case, ${\widehat{T}}_2$ carries additional $U\left(1\right)$ phase factor in the twisted case.

Figure 21

The 3-manifold in the left (right) column appears in the phase factor of $A_v^{h,\omega }{\widehat{T}}_2$ (${\widehat{T}}_2{A}_v^{h,\omega }$) in Figs. 20 and 20. More explicitly for case (a), the left panel consists of two tetrahedrons $[v,v^{\prime },w,w^{\prime }]$ and $[v^{-},v,w,w^{\prime }]$. The former accounts from the phase factor accompanying the ${\widehat{T}}_2$ transformation and the latter arises from the tent construction. The right panel compromises three tetrahedrons, $[v^{-},v^{\prime },w,w^{\prime }]$, $[v^{-},v,v^{\prime },w]$, and $[v^{-},v,v^{\prime },w^{\prime }]$. The first tetrahedron originates from the ${\widehat{T}}_2$ transformation and the last two tetrahedrons originate from the tent construction. Similar considerations lead to the 3-manifold construction corresponding to (b). In each case, using a 2-3 Pachner move, the phase factors of $A_v^{h,\omega }{\widehat{T}}_2$ and ${\widehat{T}}_2{A}_v^{h,\omega }$ acting on a computational basis state can be shown to be identical.

Figure 22

Lattice isomorphism ${\widehat{T}}_3$ is defined for a vertex $v$. We map the preferred basis vector, shown in the left column, on $A_v^{\omega }=1$ subspace on the graph with the vertex $v$ to the computational basis $|g_1,g_2,\cdots \rangle$ on the graph where the vertex $v$ and adjacent links are removed, shown in the right column.

Figure 23

Using a series of ${\widehat{T}}_2$ lattice isomorphisms, one can isolate a vertex $v$ in such a way that it is surrounded by a triangle. We colored the links which are affected by ${\widehat{T}}_2$.

Figure 24

Graphical proof of Eq. (B3). The 3-manifold on the left is obtained by gluing the tetrahedron associated with the $U\left(1\right)$ phase factors appearing in the amplitude of $|g;g_1,g_2\rangle$ in Eq. (B2) and the 3-manifold from the tent construction of $A_v^{h,\omega }$. The 3-manifold on the right is associated with the phase factor of the amplitude of $|h·g;g_1,g_2\rangle$ in Eq. (B2). The $U\left(1\right)$ phase factors associated with the above two 3-manifolds are identified by a 1-4 Pachner move.

Figure 25

Graphical proof of Eq. (B6). The 3-manifold on the left is given by the gluing of tetrahedron $[v,w,w^{\prime },w^{″}]$, originating from the phase factor of $|g;g_1,g_2\rangle$ in Eq. (B2), with two relevant tetrahedrons, $[w^{-},w,v,w^{\prime }]$ and $[w^{-},w,v,w^{″}]$, originating from the tent construction of $A_w^{h,\omega }$ (shown as three distinct tetrahedrons, for clarity). The associated $U\left(1\right)$ phase factor for the latter two tetrahedrons corresponds to $e^{i{\varphi }_h^{\mathcal{G}}\left(\omega \right)}$ in Eq. (B5). The 3-manifold on the right is obtained by gluing the tetrahedron $[v,w^{-},w^{\prime },w^{″}]$, originating from the phase factor of $|h·g;h·g_1,g_2\rangle$ in Eq. (B2), and the tetrahedron $[w^{-},w,w^{\prime },w^{″}]$, with which the phase factor $e^{i{\varphi }_h^{{\mathcal{G}}^{\prime }}\left(\omega \right)}$ in Eq. (B4) is associated. Two 3-manifolds are related by a 2-3 Pachner move and, importantly, the group elements on links of $[w^{-},w,w^{\prime },w^{″}]$ are independent of $g$.

Figure 26

A series of ${\widehat{T}}_2$ lattice isomorphisms that moves two distant vertices $v_1$ and $v_2$ until they are nearest neighbors. We colored the links which are affected by ${\widehat{T}}_2$.

Figure 27

A configuration where the vertices $v_1$, $v_2$, $\cdots$, and $v_k$ associated with the vertex operator $A_{v_1{v}_2...v_k}^{\omega }$ are located locally in space and surrounded by the triangle formed by vertices $w$, $w^{\prime }$, and $w^{″}$. Since the vertex operator $A_{v_1{v}_2...v_k}^{\omega }$ is generated from link term projections, all the group elements on links $[v_1,v_2]$, $[v_2,v_3]$, $\cdots$, and $[v_{k-1},v_k]$ are trivial.

Figure 28

The preferred basis vector for $A_{v_1{v}_2}^{\omega }=1$ subspace. As in the single-vertex case, we fix the representative link, denoted as $[v_1,w]$ in this case, at the outset of the calculation. The phase factor is given by the $U\left(1\right)$ phase factor of a 3-manifold obtained by gluing two tetrahedrons. For $k$ vertices, the corresponding 3-manifold is obtained by gluing $k$ tetrahedrons.

Figure 29

A graphical proof that the preferred basis state appearing in Eq. (B7) belongs to the subspace $A_{v_1,v_2}^{h,\omega }=1$. (a) $U\left(1\right)$ phase factor associated with a specific component of the preferred basis, $\omega \left[M\right[\{g,g_1,g_2\}\left]\right]$, where $g$ is the state on $[v_1,w]$. (b) Applying $A_{v_1}^{h,\omega }$ adds three tetrahedrons associated with the tent construction. (c) Applying a 1-4 Pachner move simplifies four tetrahedrons into a single one. (d) Applying $A_{v_2}^{h,\omega }$ adds four tetrahedrons associated with the tent construction. (e) Applying a 2-3 Pachner move to the left most tetrahedrons. (f) Detaching the leftmost tetrahedron and gluing together the remaining four tetrahedrons. (g) Applying a 1-4 Pachner move on the right most tetrahedrons, arriving at the final step, which equals $\omega \left[M\right[\{h·g,g_1,g_2\}\left]\right]$, as required. Note that while the states on links $[v_1,v_2]$ and $[v_1^{-},v_2^{-}]$ are trivial, the state on $[v_1^{-},v_2]$ is not trivial in general.

Figure 30

A graphical proof that the two operations $A_{v_1,v_2}^{h,\omega }{\widehat{T}}_3$ and ${\widehat{T}}_3{A}_{v_1,v_2}^{h,\omega }$ result in the same quantum state on ${\mathcal{G}}^{\prime }$. (a) $U\left(1\right)$ phase factor associated with a specific component of the preferred basis, $\omega \left[M\right[\{g,g_1,g_2\}\left]\right]$, where $g$ is the state on $[v_1,w]$. (b) Applying $A_w^{h,\omega }$ results in three additional (relevant) tetrahedrons. (c) Gluing together the left most three tetrahedrons. (d) Applying a 2-3 Pachner move to the three tetrahedrons mentioned in (c). (e) Gluing together the right most three tetrahedrons. (f) Applying a 2-3 Pachner move on the three tetrahedrons mentioned in (e). This concludes our proof that, as in the case of Eq. (B6), the relation $e^{i{\varphi }_h^{\mathcal{G}\prime }\left(\omega \right)}=\frac{e^{i{\varphi }_h^{\mathcal{G}}\left(\omega \right)}\omega \left[M\left[\{g,g_1,g_2\}\right]\right]}{\omega \left[M\right[\{h·g,h·g_1,g_2\}\left]\right]}$ holds.

Figure 31

Input data for the Levin-Wen model. (a) The fusion rule among anyons in $\mathcal{C}$. (b) If we invert the direction of a link, we take antiparticle of the corresponding anyon label. (c) $F$-matrix associated with the change in the fusion basis.

Figure 32

(a) Definition of the vertex term $A_v$ in the Levin-Wen model. $A_v$ projects onto the configuration where three anyons on the adjacent links are compatible with the fusion rule. (b) Definition of the plaquette term $B_p$ in the Levin-Wen model. $B_p$ is given by the summation of $\frac{d_s}{D^2}{B}_p^s$, where $B_p^s$ operator is defined in Fig. 11.

Figure 33

The disorder-average Binder ratio as a function of $\beta$ at $h_{\max }=6.19$ and $Q=3$ for various system sizes.

Figure 34

The Binder ratio calculated for system size $L=12$ and $Q=3$ at $h_{\text{max}}=6.24$ and ${\beta }_{\text{max}}=2^{12}$. Different columns correspond for several samples (disorder realizations). For each sample, we show simulation results with different total number of measurements $N_m={10}^2$, ${10}^3$, and ${10}^4$ divided 10 bins.

Figure 35

Convergence of the disorder-averaged binder ratio $B_{\text{av}}$ as a function of the number of disorder averages for $Q=3$ for various system sizes $L$ and the field strength $h_{\text{max}}$.