Boson condensation and instability in the tensor network representation of string-net states

The tensor network representation of many-body quantum states, given by local tensors, provides a promising numerical tool for the study of strongly correlated topological phases in two dimensions. However, the representation may be vulnerable to instabilities caused by small variations in the local tensors. For example, the topological order in the tensor network representations of the toric code ground state has been shown in Chen, Zeng, Gu, Chuang, and Wen, Phys. Rev. B 82, 165119 (2010) to be unstable if the variations break certain $Z_2$ symmetry of the tensor. In this work, we ask whether other types of topological orders suffer from similar kinds of instability and if so, what is the underlying physical mechanism and whether we can protect the order by enforcing certain symmetries on the tensor. We answer these questions by showing that the tensor network representations of all string-net models are indeed unstable, but the matrix product operator (MPO) symmetries of the tensors identified in Şahinoğlu, Williamson, Bultinck, Mariën, Haegeman, Schuch, and Verstraete, arXiv:1409.2150 can help to protect the order. In particular, we show that a subset of variations that break the MPO symmetries lead to instability by inducing the condensation of bosonic quasiparticles, which destroys the topological order in the wave function. Therefore such variations must be forbidden for the encoded topological order to be reliably extracted from the local tensors. On the other hand, if a tensor network based variational algorithm is used to simulate the phase transition due to boson condensation, such variation directions may prove important to access the continuous transition correctly.

Figure 1

Vertex and plaquette terms of the toric code Hamiltonian.

Figure 2

Single-line TNR of the toric code state: we double the local Hilbert space on each edge and take $|i\rangle \to |ii\rangle , i=0,1$. So the state has a $Z_2$ topological order but on a bigger Hilbert space. We associate to each vertex the tensor product of three Hilbert spaces closest to it. We can now place a tensor, $T^0$ on each vertex with three out of plane physical legs, $i,j,\text{and}k$, and three in-plane virtual legs, $\alpha ,\beta ,\text{and}\gamma$ with values as given in Eq. (4).

Figure 3

Numerical calculation of topological entanglement entropy $S_{\text{topo}}\left(\epsilon \right)$ of states represented by toric code fixed point single-line tensors, $T^0$, varied with an infinitesimal random tensor in different subspaces. $\epsilon$ value is kept fixed at $\epsilon =0.01$. Blue dot corresponds to $S_{\text{topo}}$ with no variation. $I_V=I^{\otimes 3}$ is the projector on to the full virtual space. $\mathbb{M}=\frac{1}{2}(I^{\otimes 3}+Z^{\otimes 3})$ is the projector onto the space of variations that respect the $Z^{\otimes 3}$ symmetries. So, $I_V-\mathbb{M}$ is a projector onto the space of variations that break $Z^{\otimes 3}$ symmetries. We see that variations in $I_V-\mathbb{M}$ subspace are unstable while variations in $\mathbb{M}$ are stable. Details of this numerical calculation are given in Appendix pp9-s1.

Figure 4

Double-line TNR of the toric code state. We double the local Hilbert spaces in the same way as we do for single-line TNRs (Fig. 2). We associate to each vertex a tensor $T_{\alpha ,{\alpha }^{\prime };\beta ,{\beta }^{\prime };\gamma ,{\gamma }^{\prime }}^{i,j,k}$, where the out of plane legs, $i,j,\text{and}k$, correspond to the three physical indices, and the in-plane legs $\alpha ,{\alpha }^{\prime },\beta ,{\beta }^{\prime },\gamma ,\text{and}{\gamma }^{\prime }$ are the virtual indices. The virtual indices of the tensors contract along the shared edges to produce the toric code state on the physical indices.

Figure 5

Numerical calculation of topological entanglement entropy $S_{\text{topo}}\left(\epsilon \right)$ of the states represented by toric code fixed point double-line tensors, $T^0$, varied with an infinitesimal random tensor in different subspaces. $\epsilon$ value is kept fixed at $\epsilon =0.01$. Blue dot corresponds to $S_{\text{topo}}$ with no variation. $I_V$ is the projector onto the full virtual space. $M_0$ is the projector on the stand-alone subspace. $\mathbb{M}$ is the MPO-injective subspace projector. We take a random tensor and apply the projectors to generate random tensors in respective subspaces. Variations in $I_V-M_0$ violate $Z\otimes Z$ symmetry. Variations in $M_0-\mathbb{M}$ violate $X^{\otimes 6}$ but not $Z\otimes Z$. Variations in $\mathbb{M}$ violate no virtual symmetry. The details of this numerical calculation are given in Appendix pp1.

Figure 6

Calculation of stand-alone space. We put the fixed point double tensor network on a large disk with a hole at the origin (one double tensor removed). This tensor network has dangling virtual indices (red legs) at the outer and inner boundaries. The upper/lower virtual legs are shown as dotted/solid red lines. We trace out the virtual indices at the outer boundary, and the support space of the remaining tensor at the inner boundary gives us the stand-alone space.

Figure 7

$X$ and $Z$ operators appear differently in the toric code double-line double tensor contracted on a disk with a hole. $X$ operators on the inner boundary only appear with $X$ operators on the outer boundary, which vanish upon taking the trace. However, $Z$ operators on the inner boundary appear with identity on the outer boundary. So these terms survive the trace. This is why $Z^{\otimes 2}$ symmetry is imposed on the stand-alone space but not the $X^{\otimes 6}$ symmetry.

Figure 8

A pictorial representation of relevant vector spaces. A tensor $T^0$ is a linear map from the virtual space to the physical space. We denote the full virtual space as $I_V$ and the full physical space as $I_P$. $M_0$ (region in red and blue) is the stand-alone subspace of the virtual space. $\mathbb{M}$ (region in blue) is the MPO-injective subspace of the stand-alone space. MPO-injective subspace is isomorphic to the local ground-state physical subspace (also in blue), denoted as $P_{\mathrm{GS}}$, which is a subspace of the full physical space.

Figure 9

Calculation of topological spin. We create two pairs (shown as red and blue) of particle, antiparticle pairs $a-\overline{a}$, with $a$ situated at site 1 and 2. We apply the following procedure in this order: (a) move first $a$ (red) from 2 to 3, (b) move second $a$ (blue) from 1 to 2, (c) move first $a$ (red) from 3 to 1. Finally, (d) we annihilate each $a$ with the antiparticles of the other anyon (i.e., red $a$ with blue $\overline{a}$ and vice versa). When the propagation of $a$ happens through a gauge-string operator, which disappears along the path, this order of process becomes irrelevant, as the second string-operator does not interact with the first one, and the whole process is equivalent to creating and annihilating two pairs of $a-\overline{a}$, which has amplitude 1. It implies $a$ has a trivial topological spin.

Figure 10

Numerical calculation of the topological entanglement entropy $S_{\text{topo}}\left(\epsilon \right)$ of the states represented by double-semion fixed point double-line tensors $T^0$ varied with an infinitesimal random tensor in different subspaces. $\epsilon$ value is kept fixed at $\epsilon =0.01$. Blue dot corresponds to $S_{\text{topo}}$ with no variation. $I_V$ is the projector onto the full virtual space. $M_0$ is the projector onto the stand-alone subspace. $\mathbb{M}$ is the MPO-injective subspace projector. We take a random tensor and apply the projectors to generate random tensors in respective subspaces. Variations in $I_V-M_0$ violate $Z\otimes Z$ symmetry. Variations in $M_0-\mathbb{M}$ violate $i^{n\left(d\right)}{X}^{\otimes 6}$ but not $Z\otimes Z$. Variations in $\mathbb{M}$ violate no virtual symmetry. The details of this numerical calculations are given in Appendix pp1.

Figure 11

Numerical calculation of topological entanglement entropy $S_{\text{topo}}\left(\epsilon \right)$ of states represented by toric code fixed point triple-line tensors $T^0$ varied with an infinitesimal random tensor in different subspaces. $\epsilon$ value is kept fixed at $\epsilon =0.1$. Blue dot corresponds to $S_{\text{topo}}$ with no variation. $I_V$ is the projector onto the full virtual space. $M_0$ is the projector onto the stand-alone subspace. $\mathbb{M}$ is the MPO-injective subspace projector. We take a random tensor and apply the projectors to generate random tensors in respective subspaces. Details of this numerical calculation are given in Appendix pp1.

Figure 12

Numerical calculation of topologiccal entanglement entropy $S_{\text{topo}}\left(\epsilon \right)$ of states represented by double semion model fixed point triple-line tensors, $T^0$, varied with an infinitesimal random tensor in different subspaces. $\epsilon$ value is kept fixed at $\epsilon =0.1$. Blue dot corresponds to $S_{\text{topo}}$ with no variation. $I_V$ is the projector onto the full virtual space. $M_0$ is the projector onto the stand-alone subspace. $\mathbb{M}$ is the MPO projector. We take a random tensor and apply the projectors to generate random tensors in respective subspaces. Details of this numerical calculation are given in Appendix pp1.

Figure 13

Numerical calculation of topological entanglement entropy $S_{\text{topo}}\left(\epsilon \right)$ of states represented by Fibonacci model fixed point triple-line tensors $T^0$ varied with an infinitesimal random tensor in different subspaces. $\epsilon$ value is kept fixed at $\epsilon =0.1$. Blue dot corresponds to $S_{\text{topo}}$ with no variation. $I_V$ is the projector onto the full virtual space. $M_0$ is the projector onto the stand-alone subspace. $\mathbb{M}$ is the MPO-injective subspace projector. We take a random tensor and apply the projectors to generate random tensors in respective subspaces. The exact numerical values on this plot can be found in Appendix pp1.

Figure 14

The action of a generic (simple and nonsimple) open-end string operator corresponding to anyon $\alpha$ on tensors can be calculated in a similar fashion as that of simple-string operator Wilson loops. (a) We start by applying the string operator on the “loop state” on the fattened lattice. (b) The string operator becomes a superposition of operations ${\sum }_s{n}_s{\mathrm{\Phi }}_{t_{1}s{t}_2}$. ${\mathrm{\Phi }}_{t_{1}s{t}_2}$ acts as follows: at the ends, the string operator acts as ${\mathrm{\Omega }}_{\alpha ;t_{1}s}$ and ${\omega }_{\alpha ;s{t}_2}$ matrices on the plaquette loops, while in the middle, it is simply an $s$-type string to be fused with the nearby plaquette loops. (c) We fuse all strings in the previous step to get the physical state. The effect of the string operator can be absorbed into redefining the tensors along the path. A generic string operator changes the tensors along its path. The only case where it does not change the tensors is for simple-string operators of type 0.

Figure 15

The honeycomb lattice is put on a cylinder with some boundary tensors $T_r$. We calculate the topological entanglement entropy by calculating the entanglement entropy of the right half of the cylinder.

Figure 16

A loop state on the fat lattice. Fat lattice means strings are allowed to move away from the edges, as long as they do not cross the center of the plaquettes.

Figure 17

We calculate entanglement entropy of the right-half of the cylinder with a certain boundary condition ${\mathbb{T}}_r$. The entanglement cut is in the middle of the cylinder.

Figure 18

MESs are eigenstates of different Wilson loop operators at the entanglement cut. (a) For a fixed point single-line TNR, the state on the cylinder is always in +1 eignestate of $X$ loop, as it identically disappears. (b) The state is also in +1 eigenstate of a simultaneous operation of two $Z$ loops, one at the entanglement cut, other at the right-most boundary. It implies, we can be in two MESs depending on the boundary tensor choice. If the boundary tensor is in +1 eigenstate of the boundary $Z$ loop, then the state is in +1 eigenstate of the entanglement-cut $Z$ loop. Similarly, if the boundary tensor is in $-1$ eigenstate of the boundary $Z$ loop, then the state is in $-1$ eigenstate of the entanglement-cut $Z$ loop.

Figure 19

Bulk double tensor is a sum of tensor products between ${\left(B_f\right)}_{ec}$ ($B_f$ on the entanglement cut) and ${\left(B_f\right)}_r$ ($B_f$ on the right boundary). So, when we contract a boundary tensor ${\mathbb{T}}_r$ with the bulk tensor, it contracts with ${\left(B_f\right)}_r$ giving a scalar $c_f$. So the resulting tensor is $\mathrm{Ev}\left(\mathbb{T}\left(R\right){\mathbb{T}}_r\right)={\sum }_f{c}_f{\left(B_f\right)}_{ec}$. Consequently, $S_{\mathrm{topo}}$ using Eq. (G5) is simply $ln\left({\sum }_f\frac{c_f^2}{c_0^2}\right)$.

Figure 20

Smooth boundary condition for triple-line tensor network. Tensors $T_b$ are used on the boundary. $T_b$ has five virtual legs, $a_1,a_1^{\prime },a_2,a_2^{\prime },i_{12}$, and one physical leg, $i_{12}$. Physical leg and the middle leg take the same values. We assign a particular value to the components of this tensor, ${\left(T_b\right)}_{i_{12}{a}_1{a}_1^{\prime };a_2{a}_2^{\prime }}^{i_{12}}={\delta }_{i_{12},0}{\delta }_{a_1,a_1^{\prime }}{\delta }_{a_2,a_2^{\prime }}{\delta }_{a_1{a}_2{i}_{12}}$.

Figure 21

Dependence of $S_{\mathrm{topo}}$ on boundary condition for toric-code double line TNR. We start with the boundary tensor, $T_b$, shown in Fig. 20. We add a random variation ${\epsilon }_b{T}_b^r$ to $T_b$ and calculate $S_{\mathrm{topo}}\left({\epsilon }_b\right)$ for random bulk variations in different subspaces. We keep $T_b^r$ fixed and increase the variation strength ${\epsilon }_b$. We see all classes of stable bulk variations have the same $S_{\mathrm{topo}}$ for each ${\epsilon }_b$ as the fixed point (no-variation) tensor. And the unstable class of bulk variation shows no dependence on ${\epsilon }_b$. It shows that stable variations indeed are in the same topological phase as the RG fixed point state, and unstable variation is a trivial phase.

Figure 22

Calculation of $S_{\mathrm{topo}}$ for a single-line toric code fixed point tensor network state. We fix the half-cylinder length as $L=500$. The circumference is varied from 50 to 110. $S$ varies linearly with $C$. This line is extrapolated back to $C=0$. Its intersection with the $y$ axis gives $S_{\mathrm{topo}}$. Right figure is a zoomed in version of the left figure to show the intersection point clearly. We find $S_{\mathrm{topo}}\approx ln\left(2\right)$.

Figure 23

$S_{\mathrm{topo}}$ was calculated for a fixed half-cylinder length, $L=500$, in Fig. 22. We now vary $L$ from 10 to 1000. We see that $S_{\mathrm{topo}}$ is converged even for small values of $L$. So one does not need a large cylinder length to get the right $S_{\mathrm{topo}}$ value. It is expected as it is an RG fixed point tensor network state.

Figure 24

Calculation of $S_{\mathrm{topo}}$ for a state represented by single-line toric code fixed point tensor varied with an MPO violating tensor. The strength of the variation is fixed at $\epsilon =0.01$. We fix the half-cylinder length at $L=500$. The circumference is varied from 50 to 110. $S$ varies linearly with $C$. This line is extrapolated back to $C=0$. Its intersection with the $y$ axis gives $S_{\mathrm{topo}}$. The right figure is a zoomed in version of the left figure to show the intersection point clearly. We find $S_{\mathrm{topo}}\approx 0$, that is, it is a trivial state.

Figure 25

$S_{\mathrm{topo}}$ was calculated for a fixed half-cylinder length, $L=500$, in Fig. 24. We now vary $L$ from 10 to 1000. We see that $S_{\mathrm{topo}}$ is close to $ln\left(2\right)$ for small cylinders but converges to zero when the cylinder length $L$ is increased from 1 to 1000. So it is indeed a topologically trivial state.

Figure 26

The variation strength $\epsilon$ affects convergence. The higher the variation strength (as long as it is below any critical points) the faster is the convergence with the length of the size of the system.