Entanglement phases as holographic duals of anyon condensates

Anyon condensation forms a mechanism that allows us to relate different topological phases. We study anyon condensation in the framework of projected entangled pair states (PEPS) where topological order is characterized through local symmetries of the entanglement. We show that anyon condensation is in one-to-one correspondence to the behavior of the virtual entanglement state at the boundary (i.e., the entanglement spectrum) under those symmetries, which encompasses both symmetry breaking and symmetry protected (SPT) order, and we use this to characterize all anyon condensations for Abelian double models through the structure of their entanglement spectrum. We illustrate our findings with the ${\mathbb{Z}}_4$ double model, which can give rise to both toric code and doubled semion order through condensation, distinguished by the SPT structure of their entanglement. Using the ability of our framework to directly measure order parameters for condensation and deconfinement, we numerically study the phase diagram of the model, including direct phase transitions between the doubled semion and the toric code phase, which are not described by anyon condensation.

Figure 1

(a) PEPS tensor $A$ with five indices. (b) The PEPS wave function is built up by contracting the virtual indices $\alpha ,\cdots ,\delta$ of the PEPS tensors as indicated by connected lines.

Figure 2

(a) Replacing one tensor by a different tensor $B$ results in a localized excitation, i.e., it is not detected by the parent Hamiltonian anywhere else. (b) An excitation is topologically trivial if $B$ can be obtained from $A$ by acting with a map $L$ on the physical system.

Figure 3

A string of $U_g$ actions on the lattice can be freely moved [Eq. (5)], making it invisible to the parent Hamiltonian except at its endpoints. It therefore describes a pair of topological excitations.

Figure 4

Tensor networks for detecting (a) confinement (the network for the norm of the state evaluates to zero) and (b) condensation (the network for the overlap with the ground state evaluates to nonzero, both as $\ell \to \infty$) for a general anyon pair of the form Eq. (6).

Figure 5

Possible symmetries in the single-site tensor of the transfer operator. (a) Joint ket+bra single-site tensor. (b) Double-layer symmetry. (c) Single-layer symmetry, related to topological order, Eq. (1). (d) Local encoding of a global physical symmetry $u_g$ related to the symmetry (b). Degeneracy of the transfer operator under such a joint symmetry implies breaking of the physical symmetry [34, 35].

Figure 6

(a) Anyon table for the toric code. Here, $\underline{e}$ and $\underline{m}$ are self-bosons and can condense. (b) Condensation of $\underline{e}$ results in confinement of particles $\left[\right[1;*\left]\right]$ (gray) and thus yields a trivial model. (c) Condensation scheme for the toric code. The diagram lists the preserved symmetry $\mathit{H}$ and the corresponding phases. Note that the two trivial phases correspond to condensing either $\underline{e}$ (left) or $\underline{m}$ (right).

Figure 7

(a) Anyon table for the $D\left({\mathbb{Z}}_4\right)$ quantum double model. The first row, first column, and the dyon $\underline{d}$ have bosonic self-statistics and can be condensed. (b) Effective anyon model obtained by condensing $\underline{d}$: the gray anyons become confined, while the remaining ones become identified as indicated. The resulting anyon theory is a doubled semion model with semions $\underline{s}$ and $\underline{\overline{s}}$ with self-statistics $i$. (c) Full condensation scheme for $D\left({\mathbb{Z}}_4\right)$. The diagram lists the preserved symmetry $\mathit{H}$ and the corresponding phase (TC=toric code, DS=double semion). The case $\mathit{H}={\mathbb{Z}}_4⊠{\mathbb{Z}}_2$ can describe two different topogical phases (TC and DS), depending on the SPT phase of the fixed point.

Figure 8

Three surfaces in the three-parameter phase diagram described in Sec. 5c. The RGB values of each point give the expectation value of the string order parameters indicated in (b), cf. also Fig. 9: red $=\langle S_{\left[\right[2;1\left]\right]}\otimes {\overline{S}}_{\left[\right[0;-1\left]\right]}\rangle$, green $=\langle S_{\left[\right[1;i\left]\right]}\otimes {\overline{S}}_{\left[\right[1;i\left]\right]}\rangle$, blue $=\langle S_{\left[\right[0;i\left]\right]}\otimes {\overline{S}}_{\left[\right[0;-i\left]\right]}\rangle$, the three of which jointly allow us to discriminate all the phases observed. QD, TC, DS, and TP denote the $D\left({\mathbb{Z}}_4\right)$ double model, toric code, double semion, and trivial phases, respectively. Note that the phase diagram exhibits two distinct toric code phases (${\mathbb{Z}}_2⊠{\mathbb{Z}}_1$, blue, and ${\mathbb{Z}}_4⊠{\mathbb{Z}}_2$, black), while all three trivial phases have symmetry ${\mathbb{Z}}_2⊠{\mathbb{Z}}_2$.

Figure 9

String order parameters for condensation and deconfinement for the interpolation (I) in Fig. 8, describing a $D\left({\mathbb{Z}}_4\right)$ to toric code transition, obtained by approximating the fixed point of $\mathbb{T}$ with an iMPS of bond dimension $\chi =24$. The type of string order parameters is encoded by the color, as indicated in the anyon table in the lower left corner: dots correspond to the deconfinement parameter $\langle S_{\left[\right[g;\alpha \left]\right]}\otimes {\overline{S}}_{\left[\right[g;\alpha \left]\right]}\rangle \equiv \langle g,\alpha |g,\alpha \rangle$, while lines correspond to condensate fractions $\langle S_{\left[\right[g;\alpha \left]\right]}\otimes {\overline{S}}_{\left[[g^{\prime };{\alpha }^{\prime }]\right]}\rangle \equiv \langle g,\alpha |g^{\prime },{\alpha }^{\prime }\rangle$ for the pairs they connect; order parameters with the same color are (numerically) identical. Specifically, the blue line gives the condensate fraction of the $\underline{2m}=\left[\right[2;1\left]\right]$ magnetic particle, and the red line measures the deconfinement of the $\underline{e}=\left[\right[0;i\left]\right]$ particle, which becomes confined if $\underline{2m}$ condenses. The solid line is the analytical result from the mapping to the 2D Ising model, showing excellent agreement. The upper left inset (i) shows the scaling of the deconfinement parameter $\langle S_{\left[\right[0;i\left]\right]}\otimes {\overline{S}}_{\left[\right[0;i\left]\right]}\rangle$ in the vicinity of critical point, with critical exponent $\beta =0.12\left(1\right)$. The right insets (ii) give the correlation length around the critical point and the corresponding critical exponent $\nu =1.06\left(9\right)$, extracted from exact diagonalization of the transfer operator on cylinders of diameter $N_v$; the extrapolation $N_v=\infty$ has been obtained by fitting with $aexp\left(-b{N}_v\right)+C_{\infty }$. We find that the critical exponents are the same on both sides of the phase transition.

Figure 10

Phase transition between double semion model (${\mathbb{Z}}_4⊠Z_2$ symmetry) and trivial phases (${\mathbb{Z}}_2⊠{\mathbb{Z}}_2$ symmetry) along line (II) in Fig. 8. The plot shows the deconfinement/condensate order parameters (color coded as indicated by the dots/lines in the anyon table in the inset) vs the interpolation parameter ${\theta }_{\mathrm{TC},{\mathbb{Z}}_2}$; see Fig. 9 for details. Insets (i) and (ii) are the same as in Fig. 9, giving fits for the critical exponents ${\beta }_{\pm }$ and $\nu$, respectively. Inset (iii) shows the behavior condensate fraction $\langle S_{\left[\right[2;1\left]\right]}\otimes {\overline{S}}_{\left[\right[0;1\left]\right]}\rangle \equiv \langle 2,1|0,1\rangle$ (blue in the main plot) close to the phase transition for different iMPS bond dimensions $\chi$, demonstrating convergence to a smooth (albeit steep) curve. In inset (iv), the same data are plotted against $1/\chi$ for different ${\theta }_{TC,{\mathbb{Z}}_2}$, reconfirming the second-order nature of the transition [compare with inset (ii) of Fig. 11 b!], and allowing us to accurately localize the phase transition.

Figure 11

Condensation and deconfinement order parameters for two phase transitions between the double semion model (left) and the ${\mathbb{Z}}_4⊠{\mathbb{Z}}_2$ toric code (right), corresponding to an SPT transition at the boundary, cf. Fig. 9. (a) Interpolation along the line (III) in Fig. 8, cf. inset for the color coding. The order parameters for condensation of the dyon $\underline{d}=\left[\right[2;-1\left]\right]$ (blue) and deconfinement of the magnon $\underline{m}=\left[\right[1;1\left]\right]$ (red) can be mapped to the magnetization of the 2D Ising model (solid lines). (b) Interpolation obtained by interpolating the per-site transfer operator, cf. text, showing clear signs of a first-order phase transition. This is further supported by the analysis in the insets: (i) gives the behavior of the condensate fraction of the dyon $\underline{d}=\left[\right[2;-1\left]\right]$ (blue) in the vicinity of the phase transition for different $\chi$, showing convergence to a discontinuous curve. (ii) gives the same data as a function of $1/\chi$ for different $\theta$, and we observe a sharp change of the behavior at $\theta \approx 0.28970$. [(i) and (ii) should be contrasted with insets (iii) and (iv) of Fig. 10.] Finally, inset (iii) gives the fidelity susceptibility ${\chi }_F$, measuring the rate of change of the normalized wave function $|\widehat{\psi }\left(\theta \right)\rangle$ with $\theta$, $|\langle \widehat{\psi }\left(\theta \right)|\widehat{\psi }(\theta +\mathrm{d}\theta )\rangle |=1-\frac{1}{2}{\chi }_F{N}^2{\left(\mathrm{d}\theta \right)}^2$, for different step sizes $\mathrm{d}\theta =\delta \to 0$, which approaches a delta peak as expected for a first-order transition [44, 45].