Study of anyon condensation and topological phase transitions from a ${\mathbb{Z}}_4$ topological phase using the projected entangled pair states approach

We use projected entangled pair states (PEPS) to study topological quantum phase transitions. The local description of topological order in the PEPS formalism allows us to set up order parameters which measure condensation and deconfinement of anyons and serve as substitutes for conventional order parameters. We apply these order parameters, together with anyon-anyon correlation functions and some further probes, to characterize topological phases and phase transitions within a family of models based on a ${\mathbb{Z}}_4$ symmetry, which contains ${\mathbb{Z}}_4$ quantum double, toric code, double semion, and trivial phases. We find a diverse phase diagram which exhibits a variety of different phase transitions of both first and second order which we comprehensively characterize, including direct transitions between the toric code and the double semion phase.

Figure 1

(a) Graphical notation of an on-site tensor $A$. (b) Tensor network representation of a many-body wave function, where connected legs are being contracted. (c) Tensor network representation of the wave function norm. (d) The on-site transfer operator $\mathbb{E}$. It is obtained by contracting the physical indices of $A$ and its conjugate. (e) The transfer operator $\mathbb{T}$ obtained by blocking $\mathbb{E}$ tensors in one direction.

Figure 2

Graphical representation of wave function overlaps in the thermodynamic limit. (a) Condensate fraction of $|g,\alpha \rangle$. (b) Deconfinement fraction of $|g,\alpha \rangle$.

Figure 3

(a) Left and right fixed point of the transfer operator (we assume the largest eigenvalue to be 1); a possible MPS structure, found, e.g., for fixed point models and used in the numerics, is indicated by the yellow bonds. (b) Mapping of anyonic wave function overlaps, Fig. 2, to the expectation value of string order parameters.

Figure 4

Patterns of symmetry breaking in the fixed points of the transfer operator for ${\mathbb{Z}}_4$-invariant tensors, where TC/DS denote toric code/double semion phases. Arrows indicate phase transitions breaking a ${\mathbb{Z}}_2$ symmetry.

Figure 5

Computation of anyonic order parameter $\langle g,\alpha |g^{\prime },{\alpha }^{\prime }\rangle$. The computation of the 2D order parameter (a) is carried out by dimensional reductions to string order parameters in the 1D iMPS boundaries (b), which are then evaluated by using the fixed points of the channel operators defined in (f). In order to fix normalization, the problem can be related (d), (e) to the vacuum expectation value by using the local representation of the symmetry string in the boundary iMPS (g). See text for further details.

Figure 6

Analysis of phase transitions between (a) ${\mathbb{Z}}_4⊠{\mathbb{Z}}_2$ DS and ${\mathbb{Z}}_4⊠{\mathbb{Z}}_2$ TC, and (b) $D\left({\mathbb{Z}}_4\right)$ QD and ${\mathbb{Z}}_4⊠{\mathbb{Z}}_2$ TC. The tables on the bottom left describe the color coding: Rows/columns label flux $g$/charge $\alpha$ of anyons, and colored dots/lines denote the color used for plotting the order parameters $|\langle g,\alpha |g,\alpha \rangle |$ and $|\langle g,\alpha |g^{\prime },{\alpha }^{\prime }\rangle |$, respectively. The insets (i) and (ii) show the behavior of the order parameter indicated on the $y$ axis in the vicinity of the phase transition, either (i) as a function of $\theta$ for different MPO bond dimensions $\chi$, or (ii) as a function of $\chi$ for different $\theta$. The sharp transition in (a) indicates a first-order transition, the smooth change in (b) a second-order transition. In the latter case, a critical exponent $\beta$ can be extracted from the data; cf. inset (iii).

Figure 7

Comparison of data on correlation lengths for the $D\left({\mathbb{Z}}_4\right)$ QD to TC transition; cf. Fig. 5. (a) Comparison of correlation length different methods for obtaining correlation lengths; see (b) for the color coding. (b) Correlation length for anyon-anyon correlations with flux (green) and without flux (black); the latter includes trivial two-point correlations. The data shows that in the $D\left({\mathbb{Z}}_4\right)$ QD phase, the transition is dominated by flux condensation. (c), (d) Extraction of the critical exponent from (c) the iMPS ansatz for the boundary state, and (d) from extrapolation of finite cylinders.

Figure 8

Comparison of correlations for first-order (left column) vs second-order (right column) transition. (a), (b) Data obtained with iMPS, extrapolated in $\chi$, showing convergence to constant $\xi$ (first order) vs divergent (second order) behavior. (c), (d) Scaling of $\xi$ in the vicinity of the phase transition obtained with iMPS, showing convergence vs critical scaling with $\nu =1$ as the transition point is approached. (e), (f) Inverse correlation lengths obtained by diagonalizing the transfer operator using an excitation ansatz.

Figure 9

Fidelity susceptibility, Eq. (39), for (a) first- and (b) second-order phase transition computed using iMPS with $\chi =32$ for different step sizes $\delta$ (cf. Appendix pp1). In the first-order case (a), ${\chi }_F$ approaches a delta function, while in the second-order case, it diverges as $log|\theta -{\theta }_c|$. The insets show the scaling of ${\chi }_F$ with the step size $\delta$, which also exhibits distinct behaviors.

Figure 10

(a) Phase diagram of the three-parameter family of topological models with ${\mathbb{Z}}_4$ symmetry introduced in Sec. 4a which exhibits all topological phases, shown along three hyperplanes. The color coding uses RGB values given by the anyon wave function norms/overlaps shown in (b); cf. Appendix pp3. $\chi =16$ has been used for the approximation of fixed points in the iMPS calculations.

Figure 11

Condensate and deconfinement fractions for the phase transition between $D\left({\mathbb{Z}}_4\right)$ QD and ${\mathbb{Z}}_4⊠{\mathbb{Z}}_2$ TC constructed by local filtering [line (I) in Fig. 10], computed with $\chi =24$. The color coding is given by the table in the top right corner as explained in Fig. 6. (i) Scaling of the deconfinement fraction $\langle 0,i|\langle 0,i\rangle$ in the vicinity of the transition; we find a critical exponent $\beta =0.12\left(1\right)$. (ii) Critical exponent $\gamma =0.178\left(6\right)$ obtained from the scaling of the susceptibility of the deconfinement, where $\delta$ denotes the step sizes used for approximating the derivative. (iii) Correlation length $\xi$ determined from finite cylinders, yielding a critical exponent $\nu =1.06\left(9\right)$. See text and Fig. 6 for further discussion of methods.

Figure 12

Order parameters for the phase transitions between (a) $D\left({\mathbb{Z}}_4\right)$ QD and ${\mathbb{Z}}_2⊠{\mathbb{Z}}_1$ TC and (b) $D\left({\mathbb{Z}}_4\right)$ QD and ${\mathbb{Z}}_4⊠{\mathbb{Z}}_2$ DS. The colors are defined through the anyon tables in the lower left corners; cf. Fig. 6. Insets (i) and (ii) in (a) and (b) give the extraction of the critical exponents $\beta$ and $\nu$ of an order parameter and correlation length, respectively. Inset (iii) in (a) and (b) shows the inverse correlation length extracted from the transfer operator using an iMPS excitation ansatz. Here, black (green) lines correspond to correlations of anyons with trivial (nontrivial) flux. See text for a discussion.

Figure 13

Order parameters for the phase transition between DS and ${\mathbb{Z}}_4⊠{\mathbb{Z}}_2$ TC along the line (IV) in Fig. 10, computed with $\chi =24$. The colors are defined through the table on the right; cf. Fig. 6. (i) Scaling of the correlation length $\xi$ of the boundary phase, indicative of a second-order transition. (ii) Correlation length determined from the transfer operator using an iMPS excitation ansatz. (iii) Order parameter for SPT order in the fixed point of the transfer operator, Eq. (43), in the vicinity of the phase transition. It exhibits a sharp jump which allows us to accurately determine the transition point.

Figure 14

(a) Order parameters along the phase transition between ${\mathbb{Z}}_4⊠{\mathbb{Z}}_2$ DS and ${\mathbb{Z}}_2⊠{\mathbb{Z}}_1$ TC; cf. Fig. 6 for the legend. The model has an exact self-duality $\theta \leftrightarrow 1-\theta$; see text. (i) Correlation length (from iMPS) vs $\chi$ around the transition, indicative of a second-order transition. (ii) Extraction of critical exponent $\nu$ from finite cylinders, yielding $\nu =0.65\left(6\right)$. (iii) Extraction of critical exponents $\beta$, yielding two exponents ${\beta }_1=0.069\left(6\right)$ and ${\beta }_2=0.081\left(5\right)$. The observed exponents are compatible with a 4-state Potts transition. (b) Inverse correlation length extracted from the transfer matrix using an iMPS excitation ansatz. Black (green) denotes again anyon correlations with (without) flux, showing that the self-duality map exchanges flux and charge. (c) Correlation length computed using iMPS, yielding $\nu =0.66\left(2\right)$.

Figure 15

(a) ${\theta }_{\text{DS}}=1$ hyperplane of the three-parameter family constructed in Sec. 4a, Fig. 10, exhibiting a DS and a ${\mathbb{Z}}_2⊠{\mathbb{Z}}_2$ trivial phase. See Fig. 10 for the color coding. Transitions are scanned along lines with different angles $\varphi$. (b) Scaling of $\langle 0,i|0,i\rangle$ and $\langle 0,1|2,-1\rangle$ in the vicinity of the transition, as a function of $\varphi$. (c) Critical exponents ${\beta }_{\pm }$ for their scaling as a function of $\varphi$; we find a continuously varying transition with $\nu \approx 1$ constant.

Figure 16

Phase diagrams of models which are obtained by deforming (a) ${\mathbb{Z}}_4⊠{\mathbb{Z}}_2$ TC, (b) ${\mathbb{Z}}_2⊠{\mathbb{Z}}_1$ TC, and (c) ${\mathbb{Z}}_4⊠{\mathbb{Z}}_2$ DS towards trivial phases, as discussed in Sec. 4b. The corresponding anyon tables below each phase diagram explain the color coding, where the RGB values of each point are given by the corresponding anyon wave function overlaps/norms. All data have been obtained with $\chi =16$.

Figure 17

Order parameters along the interpolation DS to ${\mathbb{Z}}_4⊠{\mathbb{Z}}_4$ trivial phase (cf. table on bottom left). The left inset shows a zoom of the condensate fraction $\langle 0,1|2,1\rangle$ in the vicinity of the transition, and the right inset the scaling of the correlation length with the iMPS bond dimension $\chi$ (with a quadratic fit), both of which show clear signs of a first-order transition.

Figure 18

Comparison of fidelity per site for the second-order phase transition by $D\left({\mathbb{Z}}_4\right)$ QD and ${\mathbb{Z}}_4⊠{\mathbb{Z}}_2$ TC [(a), (c)] and first-order phase transition between ${\mathbb{Z}}_4⊠{\mathbb{Z}}_2$ DS and ${\mathbb{Z}}_4⊠{\mathbb{Z}}_2$ TC [(b), (d)].

Figure 19

Dispersion relation of the transfer operator at the data points in the vicinity of phase transition between ${\mathbb{Z}}_4⊠{\mathbb{Z}}_2$ DS and ${\mathbb{Z}}_2⊠{\mathbb{Z}}_1$ TC phase (see Sec. 6c2). The computations have been performed for the bond dimension $\chi =24$.

Figure 20

(a) Standard Ising model on a square lattice. The string of defective edges shown by colored edges indicates the presence of Pauli $x$ in the link. An edge with the diamond denotes an action corresponding to local order parameter. (b) The norm of the vacuum $|0,1\rangle$ constructed by contracting the bra and ket index. (c) More descriptive illustration for the tensor network of vacuum with the local structure of on-site tensors. (d) Ising model which emerged from the norm of the vacuum.