Modular matrices as topological order parameter by a gauge-symmetry-preserved tensor renormalization approach

Huan He, Heidar Moradi, and Xiao-Gang Wen

Phys. Rev. B 90, 205114 – Published 10 November 2014

Abstract

Topological order has been proposed to go beyond Landau symmetry breaking theory for more than 20 years. But it is still a challenging problem to generally detect it in a generic many-body state. In this paper, we will introduce a systematic numerical method based on tensor network to calculate modular matrices in two-dimensional systems, which can fully identify topological order with gapped edge. Moreover, it is shown numerically that modular matrices, including $S$ and $T$ matrices, are robust characterization to describe phase transitions between topologically ordered states and trivial states, which can work as topological order parameters. This method only requires local information of one ground state in the form of a tensor network, and directly provides the universal data ( $S$ and $T$ matrices), without any nonuniversal contributions. Furthermore, it is generalizable to higher dimensions. Unlike calculating topological entanglement entropy by extrapolating, in which numerical complexity is exponentially high, this method extracts a much more complete set of topological data (modular matrices) with much lower numerical cost.

Received 24 February 2014
Revised 10 October 2014

DOI:https://doi.org/10.1103/PhysRevB.90.205114

Authors & Affiliations

Huan He¹, Heidar Moradi¹, and Xiao-Gang Wen^1,2

¹Perimeter Institute for Theoretical Physics, 31 Caroline St. N, Waterloo, Ontario, Canada N2L 2Y5
²Department of Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA

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Vol. 90, Iss. 20 — 15 November 2014

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Images

Figure 1
Illustration for symmetry-preserved tensor renormalization group. First (a) before SVD, block diagonalize double tensor $T$ according to the $Z_{2}$ symmetry rule; $α + β + γ + δ$ and $α^{'} + β^{'} + γ^{'} + δ^{'}$ are both even numbers. Therefore, the indices of each block matrices $B_{e e}$ , $B_{e o}$ , $B_{o e}$ , $B_{o o}$ represent whether $α + β$ and $α^{'} + β^{'}$ are even or odd. (b) Perform SVD in each block matrices and recombine the tensors coming out of SVD into tensor $T_{1}$ and $T_{2}$ , according to the rule $α + β + ε^{'}$ , $α^{'} + β^{'} + ε$ , $γ + δ + ε^{'}$ , and $γ^{'} + δ^{'} + ε$ are all even numbers. That is, tensor $T_{1}$ and $T_{2}$ both obey $Z_{2}$ gauge symmetry. Figures 1(c) and 1(d) are the same procedures as TRG. Figure 1(c) is to use SVD to decompose $T$ into $T_{1}$ and $T_{2}$ . Only $D_{cut}$ numbers of singular values will be kept. Figure 1(d) is coarse graining. The four tensors on the small square will form a new double tensor $T^{'}$ . Note that $T_{3}$ and $T_{4}$ are outcoming tensors that are cut in another direction.
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Figure 2
Modular matrices from the fixed point double tensor $T_{fp}$ . Eight legs of $T_{fp}$ will all be traced over because of torus geometry. (a) By inserting $Z_{2}$ gauge tensors $g, h, g^{'}, h^{'}$ into $T_{fp}$ , $T_{fp} (g, h | g^{'}, h^{'})$ is obtained; and tracing over eight legs of $T_{fp} (g, h | g^{'}, h^{'})$ will give rise to overlaps of $〈 ψ (g^{'}, h^{'}) | ψ (g, h) 〉$ , where $| ψ (g, h) 〉$ labels different ground states with gauge symmetry on boundary. The elements of $T$ and $S$ matrices are just reshuffling of $〈 ψ (g^{'}, h^{'}) | ψ (g, h) 〉$ , as illustrated in (b) and (c). Figure 2(b) represents $90^{\circ}$ rotation and (c) represents twist.
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Figure 3
Trace of modular matrices $S$ and $T$ as functions of $g$ display a very sharp phase transition at critical point $g_{c}$ as increasing RG steps, for both $Z_{2}$ and double-semion topological order. The $Z_{2}$ topological order transition point coincides exactly with the results in Ref. [13] by another characterization.
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Figure 4
$T$ tensor and the $G^{m}$ tensor that describes the ground state wave function of the double semion model. The “virtual qubits” are in the “1” state in the shaded squares and in the “0” state in other squares. The red line is the domain wall (string) between 0 and 1 states of the virtual qubits. The blue (black) dots represent $t_{α β γ^{'} δ^{'}} = - 1$ ( $t_{α β γ^{'} δ^{'}} = 1$ ).
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Figure 5
Phase diagram under perturbation.
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Figure 6
Illustration of gauge-symmetry-preserved simple update. Figure 6(a) shows that tenor $T_{1}$ and $T_{2}$ are contracted and act with a local imaginary evolution operator represented by the blue box. The legs with arrows are physical indices while legs without arrows are internal indices. (b) Block diagonalization according to internal indices. $B_{e e}$ and $B_{o o}$ represent the matrices with both legs even and odd. (c) $B_{e e}$ and $B_{o o}$ are SVD-ed. (d) The outcoming matrices are recombined into the original form as in (a).
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