Complete characterization of non-Abelian topological phase transitions and detection of anyon splitting with projected entangled pair states

It is well known that many topological phase transitions of intrinsic Abelian topological phases are accompanied by condensation and confinement of anyons. However, for non-Abelian topological phases, more intricate phenomena can occur at their phase transitions, because the multiple degenerate degrees of freedom of a non-Abelian anyon can change in different ways after phase transitions. In this paper, we study these new phenomena, including partial condensation, partial deconfinement, and especially anyon splitting (a non-Abelian anyon splits into different kinds of the new anyon species) using projected entangled pair states (PEPS). First, we show that anyon splitting can be observed from the topologically degenerate ground states. Next, we construct a set of PEPS describing all possible degrees of freedom of the same non-Abelian anyon. From the overlaps of this set of PEPS of a given non-Abelian anyon with the ground state, we can extract the information of partial condensation. Then, we construct a central object, a matrix defined by the norms and overlaps among the PEPS in that set. The information of partial deconfinement can be extracted from this matrix. In particular, we use it to construct an order parameter which can directly detect anyon splitting. We demonstrate the power of our approach by applying it to a range of non-Abelian topological phase transitions: From $D\left(S_3\right)$ quantum double to toric code, from $D\left(S_3\right)$ quantum double to $D\left(Z_3\right)$ quantum double, from $\mathrm{Rep}\left(S_3\right)$ string-net to toric code, and finally from double Ising string-net to toric code.

Figure 1

(a) Graphical representation of a PEPS on a square lattice; local tensor depicted on the bottom right. (b) The local tensor of a PEPS describing a topologically ordered state satisfies the pulling-through condition, where the blue lines denotes the MPO $O_a$. (c) Inserting an MPO $O_a$ at the virtual level of the PEPS results in a new ground state. (d) General form of a ground state on the cylinder for topologically ordered PEPS with MPOs in the two directions that cross, forming the enlarged MPO $O_{bd}^{ac}$; the green dots are tensors at the crossing.

Figure 2

(a) Inserting a simple idempotent $P_{ac}^{\mathit{\alpha }}$ and MPOs $O_a$ and $O_c$ into the PEPS on a cylinder, where the up and down sides connect. (b) A simple idempotent ($s^{\prime }=s$) or nilpotent $(s^{\prime }\ne s)$ acts on the tensor $E_{sx}^{\mathit{\alpha }}$ carrying an anyon $\mathit{\alpha }$, where $x$ labels the internal DOFs. (c) A PEPS $|{\mathit{\alpha }}_x^{†},{\mathit{\alpha }}_y{\rangle }_s$ carrying two $\mathit{\alpha }$ excitations. (d) A PEPS $|{\mathit{\alpha }}_x{\rangle }_s$ carrying an anyon excitation $\mathit{\alpha }$, with a semi-infinite MPO string $O_s$. (e) The transfer operator ${\mathbb{T}}_{\mathit{\alpha }}^{\mathit{\alpha }}$ of the MES norm $\langle {\mathrm{\Psi }}^{\mathit{\alpha }}|{\mathrm{\Psi }}^{\mathit{\alpha }}\rangle$; notice that we use the simple idempotents to define the transfer operator. (f) The overlap $\langle {\mathit{\alpha }}_x|{\mathit{\gamma }}_y\rangle$ of the PEPS.

Figure 3

Phase diagram of the deformed $D\left(S_3\right)$ quantum double PEPS. Each phase is labeled by $H⊠Q$, which is the symmetry of the transfer operator fixed points, and TC denotes toric code. The phase diagram is symmetric about the plane $k_1=k_2$. The phase transition points on three axes are $S=(log(1+\sqrt{3})/2,0,0), N=(0,log(1+\sqrt{3})/2,0)$, and $L=(0,0,log(1+\sqrt{2})/2)$. $SZ, WL, LY$, and $NX$ are straight lines parallel to the axes.

Figure 4

Anyonic order parameters for the phase transition between $D\left(S_3\right)$ phase and $Z_2⊠Z_1$ toric code phase, and their critical exponents. The vertical dashed lines in (a) and (d) give the exact location of the critical point $k_c=log(1+\sqrt{3})/2$. The dashed lines in the insets have the slope indicated in each inset. $\chi$ is the bond dimension of the VUMPS. (a) Left: The deconfinement order parameters of $\mathit{B}, \mathit{D}$, or $\mathit{E}$; $\mathrm{eig}(M_{\text{str}}$) denotes the eigenvalues of $M_{\text{str}}\left(\mathit{B}\right), M_{\text{str}}\left(\mathit{D}\right)$, or $M_{\text{str}}\left(\mathit{E}\right)$. Right: The condensate (identification) order parameter, where $\mathrm{svd}\left(N^{\mathit{1}}\right)$ and $\mathrm{svd}\left(N^{\mathit{A}}\right)$ denote the singular values of $N^{\mathit{1}}\left(\mathit{C}\right)$ and $N^{\mathit{A}}\left(\mathit{C}\right)$. The insets (b) and (c) show the scaling of the corresponding anyonic order parameters. (d) Left: The deconfinement order parameters of $\mathit{F}$ or $\mathit{G}$; $\mathrm{eig}(M_{\text{str}}$) denotes the eigenvalues of $M_{\text{str}}\left(\mathit{F}\right)$ or $M_{\text{str}}\left(\mathit{G}\right)$. Right: The splitting order parameter of $\mathit{C}$. The insets (e) and (f) show the scaling of the corresponding anyonic order parameters.

Figure 5

Anyonic order parameters of the $\mathrm{Rep}\left(S_3\right)$ model and their critical exponents. The vertical dashed lines in (a) and (d) represent the exact location of the critical point $k_c=log(1+\sqrt{3})/2$. The dashed lines in the insets have the slope given in each inset. $\chi$ is the bond dimension of the VUMPS. (a) Left: The deconfinement order parameters, where $\mathit{\alpha }$ can be $\mathit{C}, \mathit{D}$, or $\mathit{E}$. Right: The condensate (identification) order parameter, where $\mathrm{svd}\left(N^{\mathit{1}}\right)$ and $\mathrm{svd}\left(\mathit{A}\right)$ are the singular values of $N^{\mathit{1}}\left(\mathit{B}\right)$ and $N^{\mathit{A}}\left(\mathit{B}\right)$. The insets (b) and (c) show the critical scaling of the corresponding anyonic order parameters. (d) Left: The partial deconfinement order parameters of $\mathit{F}$ or $\mathit{G}$; $\mathrm{eig}(M_{\text{str}}$) denotes the eigenvalues of $M_{\text{str}}\left(\mathit{F}\right)$ or $M_{\text{str}}\left(\mathit{G}\right)$. Right: The splitting order parameter of $\mathit{B}$. The insets (e) and (f) show the critical scaling of the corresponding anyonic order parameters.

Figure 6

Anyonic order parameters for the phase transition between the $D\left(S_3\right)$ phase and the $D\left(Z_3\right)$ phase, and their critical exponents. The vertical dashed line in (a) indicates the location of the critical point $k_c=log(1+\sqrt{2})/2$. The slope of the dashed lines in the insets is $1/8$. $\chi$ is the bond dimension of the VUMPS. (a) Left: The deconfinement order parameters of $\mathit{F}$ and $\mathit{G}$; $\mathrm{eig}\left(M_{\text{str}}\right)$ denotes the eigenvalues of $M_{\text{str}}\left(\mathit{F}\right)$ or $M_{\text{str}}\left(\mathit{G}\right)$. Top right: The condensate order parameter of $\mathit{A}$. Bottom right: The splitting order parameters for $\mathit{C}, \mathit{B}, \mathit{D}$, or $\mathit{E}$. The insets (b), (c), and (d) show the scaling of the corresponding anyonic order parameters.

Figure 7

Phase diagram of the deformed DIsing PEPS. The two dashed lines are $k_{\sigma }=k_{\psi }$ and $k_{\psi }=0.35$. If we express the coordinates using $(k_{\sigma },k_{\psi })$, the special points are $A=(log(1+\sqrt{2})/2,0), B=(+\infty ,log(1+\sqrt{2})/4), C=(log\sqrt{3},log\sqrt{3})$, and $D=(log\sqrt{2},+\infty )$.

Figure 8

Anyonic order parameters for the phase transition between the DIsing phase and the toric code phase, and their critical exponents. The slope of dashed lines in the insets is $1/8$. $\chi$ is the bond dimension of VUMPS. (a) The condensate order parameter $|\langle \mathit{1}|\mathit{\psi }\overline{\mathit{\psi }}\rangle |$, identification order parameter $|\langle \mathit{\psi }|\overline{\mathit{\psi }}\rangle |$, deconfinement order parameters $|\langle \mathit{\sigma }|\mathit{\sigma }\rangle |$ and $|\langle \mathit{\sigma }\overline{\mathit{\psi }}|\mathit{\sigma }\overline{\mathit{\psi }}\rangle |$, and splitting order parameter ${\tilde{M}}_{\text{off}}\left(\mathit{\sigma }\overline{\mathit{\sigma }}\right)$. The insets (b) and (c) shows their critical scaling.

Figure 9

(a) A string-net model and (b) a quantum double model on a honeycomb lattice. Red squares represent the physical DOFs on which the deformation operators act, and red circles are the physical DOFs on which deformation operators do not act. Blue circles are the virtual loops of the string-net PEPS. Blue triangles are the virtual DOFs of the quantum double PEPS.

Figure 10

(a) A tensor network generated by the tensor in Eq. (E19). (b) The DOFs of the tensor network are represented by dots. The red, green, and black lines form the primal, dual, and medial lattices, respectively.