Hi, I just want to say that the problem has been solved.

The original model is a spinless fermion chain as H1 ----- H1 ------------- H = sum_i - t * [ cdag_i * c_(i+1) + h.c. ] - mu * n_i + D * [ c_i * c_(i+1) + h.c. ] ------------------

In mps_optim, the model is transformed into a boson chain as H2 ------ H2 ---------- H = sum_i - t * [ bdag_i * b_(i+1) + h.c. ] - mu * n_i + - D * [ b_i * b_(i+1) + h.c. ] ---------------- where in the last line, a minus appears.

This minus does not influence the ground state energy, since the spectrum is an even function of D, i.e., E(D) = E(-D). Nevertheless, it would influence the sign in the wavefunction. For instance, with D=5, my matlab code gives the wavefunction -----D=5-------- 000: 0 001: 0.4287 010: 0.0910 011: 0 100: 0.4287 101: 0 110: 0 111: 0.7900 -------------------

On the other hand, with D=-5, my matlab code gives ---------------------- 000: 0 001: -0.4287 010: 0.0910 011: 0 100: -0.4287 101: 0 110: 0 111: 0.7900 ----------------------

Sun Zhao-Yu

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From：comp-phys-alps-users-request@lists.phys.ethz.ch To：comp-phys-alps-users@lists.phys.ethz.ch Subject：Comp-phys-alps-users_Digest,_Vol_172,_Issue_1 Date：2021-08-06 18:00:01Send Comp-phys-alps-users mailing list submissions to

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Today's Topics:

1. about using mps_optim to simulate a p-wave superconductor. (=?UTF-8?B?5a2Z54Wn5a6H?=)

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Message: 1 Date: Fri, 06 Aug 2021 15:15:44 +0800 From: "=?UTF-8?B?5a2Z54Wn5a6H?=" sunzhaoyu2020@whpu.edu.cn To: "comp-phys-alps-users" comp-phys-alps-users@lists.phys.ethz.ch Subject: [ALPS-users] about using mps_optim to simulate a p-wave superconductor. Message-ID: 210806151544156532004919@whpu.edu.cn Content-Type: text/plain; charset="utf-8"

Hi, everyone,

I want to calculate the ground state of 1D p-wave superconducter, which has a Z2 symmetry,

H = sum_i - t * [ cdag_i * c_(i+1) + h.c. ] - mu * n_i + D * [ c_i * c_(i+1) + h.c. ]

My physical parameter settings are ---------------- L = 3 mu = 2 t = 1 D = 5

I have used three different methods to calculate the ground state, so as to check the reliability of the results, (1) I compile mps_optim from source with Z2 symmetry (2) I use mps_optim in anaconda without Z2 symmetry (3) I code the model with matlab in standard basis [ i.e., "empty 0" and "single-occupation 1", c|1>=|0>, cdag|0>=|1> ]

I find that (2) and (3) give the same wavefunction: --------------------- 000: 0 001: 0.4287 010: 0.0910 011: 0 100: 0.4287 101: 0 110: 0 111: 0.7900 ------------------- It indicates that (2) and (3) are using the same basis, i.e., the standard basis.

However, (1) just gives the values with extra minus sign: ---------------------- 000: 0 001: -0.4287 010: 0.0910 011: 0 100: -0.4287 101: 0 110: 0 111: 0.7900 ----------------------

Moreover, I have read the supplemental codes of this Kitaev model in the paper "Matrix product state applications for the ALPS project", i.e., the file "tsc.xml", and find that the fermion operators c and cdag are also defined in a non-standard way, which also indicates a non-standard basis:

-----------------------

<SITEBASIS name="spinless fermion"> <QUANTUMNUMBER name="P" min="0" max="1" type="fermionic"/>

<OPERATOR name="c" matrixelement="1"> <CHANGE quantumnumber="P" change="1"/> </OPERATOR>

<OPERATOR name="cdag" matrixelement="1"> <CHANGE quantumnumber="P" change="-1"/> </OPERATOR>

</SITEBASIS>

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With this two facts, I guess mps_optim code with Z2 symmetry may use a different basis rather than the standard basis { |1>, |0> }.

Thereby, my question is that, What is the basis in the mps_optim with Z2 symmetry ? Or alternatively, what is the relation between this basis with standard basis ?

This question is quite important for extracting the true wavefunction. Any comment would be appreciated. Thank you very much.

Sun Zhao-Yu Wuhan Polytechnic University

sunzhaoyu2020@whpu.edu.cn