English Abstract

Traditionally, we understand phases of matter by spontaneous symmetry breaking and therefore we have order parameters to characterize different phases. However, there are states of matter which fall beyond this type of characterization, we call these phases topological phases. The first experimental discovery of topological phase was at 1980s, however, we still do not have a clear understanding of topological phase nor do we know exactly what the topological orders are for characterizing these phases (in contrast with normal orders).
Recently, using the ideas based on local unitary transformations, it is shown that an topological order is associated with the pattern of longrange entanglement in gapped quantum systems with finite topological entanglement entropy. On the other hand, several nonlocal stringorder parameters are proposed for characterizing symmetryprotected topological states for different kinds of symmetries in 1D base on infinitesize matrix product state (iMPS) calculation .
iMPS is a powerful tool to represent ground state of 1D gapped system. However, accuracy decrease near critical point. Multiscale entanglement renormalization ansatz (MERA) on the other hand, is more accurate near critical point. MERA is a numerical method that use the concept of entanglement renormalization, with the help of inserting disentanglers into the system so that shortrange entanglement can be removed before renormalization. This prevents the accumulation of degrees of freedom during the renormalization so that this method can be addressed to large size of 1D or 2D system even at quantum critical points. Therefore, if one can use MERA to calculate the nonlocal stringorder parameters proposed, not only will it be a confirmation, but also better results could be get.
However, calculation of these parameters with MERA turned out to be highly nontrivial because the operator is nonlocal and it breaks the causal cone property of MERA, which makes the calculation exponentially hard. Nevertheless, we found that we can reduce the calculation by using symmetric MERA. We need Z_2 and inversion symmetry. The former is already proposed but there's no method to do inversion symmetry in any kind of tensor network until now. By inspecting the geometry of MERA, we successfully find the method to incorporate inversion symmetry in MERA with a brickandrope representation. These enable people to calculate the inversion stringorder parameter with MERA. We calculate the proposed parameters and confirmed that the Haldane phase is protected by both inversion and time reversal symmetry.
This thesis is structured as follows: In Chap. 1, we give a brief introduction to topological phase, including historical review and its relation with quantum entanglement and symmetry. Followed by the review of S = 1 Haldane chain. In Chap. 2, we review different kinds of tensor network methods and focus onMERA, with a detailed explanation and interpretation of the tensors in it. In Chap. 3, we discuss how to implement a Z2 symmetric MERA and introduce our algorithm on spatial inversion symmetric MERA based on
a brickandrope representation. In Chap. 4, we present the results of stringorder parameters using both timereversal and inversion symmetric MERA. We also observe the RG flow in the 2 phase goes to 2 different fixed point. We conclude our work in Chap. 5.

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