論文 / 著書情報
Article / Book Information
題目(和文)
Title(English)

Exact Solution of the Lindblad Equation for Open XY spin Chain

著者(和文)

山中康平

Author(English)

Kohei Yamanaka

出典(和文)

学位:博士(理学),学位授与機関:東京工業大学,
報告番号:甲第12487号,
授与年月日:2023年6月30日,
学位の種別:課程博士,
審査員:笹本 智弘,村上 修一,西田 祐介,石塚 大晃,平原 徹

Citation(English)

Degree:Doctor (Science),Conferring organization: Tokyo Institute of Technology,
Report number:甲第12487号,
Conferred date:2023/6/30,
Degree Type:Course doctor,
Examiner:,,,,

学位種別(和文)

博士論文

Category(English)

Doctoral Thesis

種別(和文)

論文要旨

Type(English)

Summary

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（博士課程）Doctoral Program論 文 要 旨ＴＨＥＳＩＳ ＳＵＭＭＡＲＹ

系・コース：
Department of, Graduate major in

物理学
物理学

学生氏名：
Student’s Name

山中 康平

系
コース

申請学位（専攻分野）：

博士

Academic Degree Requested

Doctor of

指導教員（主）：Academic Supervisor(main)（理学笹本智弘）

指導教員（副）：Academic Supervisor(sub)要旨（英文 800 語程度）
Thesis Summary （approx.800 English Words ）Open systems are one of the important subjects in nonequilibrium physics. They show interesting
nonequilibrium behaviors, and there are many examples of open systems in various branches of physics.
Due to the development of experimental techniques recently, analytical studies are more and more
important. We focus on the open XY spin chain with boundary dissipations. In the closed system, the XY
spin chain has several phases which are characterized by the asymptotic behavior of the spin-spincorrelation. Open XY spin chain is also expected to have a phase transition and new physical phenomena
not seen in isolated systems can be observed. There are two conventional methods for studying open
quantum systems such as an analysis using the Lindblad equation and the nonequilibrium Green'sfunction (NEGF) method. We mainly use the Lindblad equation. This equation is one of the quantum
master equations. This equation has the Markov property and complete positivity. In recent years, one
has discussed about the feasibility of this equation in experiments and the validity of the theoreticalframework. In addition, transient behaviors of open systems whose dynamics is described by thisequation are not well understood, although there are many studies of phenomena in steady state.
In this thesis, we obtain the exact solution of the Open XY spin chain with boundary dissipations. Themain results are exact formulas of time-dependent physical observables such as magnetization, spincurrent, and correlation function. In order to obtain these formulas, we introduce the third quantization
whose main concept is the projection from the original Hilbert space to the Liouvillian-Fock space.
Evaluating our formulas, we observe interesting behaviors numerically. This procedure is generally
adaptable to Hamiltonians described by the quadratic form and linear dissipative operators. We haveevaluated this formula and observed interesting numerical behavior. Furthermore, our formulas allow usto discuss analytically some behaviors.
In the isotropic case (XX spin chain), the matrix describing the Lindblad dynamics is diagonalizable,
yielding an exact formula for the time-dependent physical observables. In this diagonalization, it isimportant that we can reduce the size of the matrix to be diagonalized by appropriate matrix
transformations and obtain exact formulas for eigenvalues and the corresponding eigenvectors from theanalysis of a complex symmetric tridiagonal matrix. Using this formula, several transient behaviors arestudied analytically. When the spatio-temporal behavior of the magnetization is observed for a short time
window, a light-cone structures appear. This behavior is estimated by analogy from the quench dynamics
of the corresponding isolated system. The wave fronts of the light-cones are also determined by the startand end times of the plateau regime where the magnetization does not change. This behavior can bediscussed analytically by asymptotically analyzing the integral representation of the physical observable.
On the other hand, it is known that long-time behaviors are characterized by the Liouvillian gap which isobtained as the maximum real part in the non-zero eigenvalues for the Liouvillian and the relaxation timeof a physical observable is defined by its inverse. For sufficiently large system sizes, we analytically obtain
the O(N^{-3}) scaling behavior for the Liouvillian gap including its coefficients.
Furthermore, even in the anisotropic (XY spin chain) case, the matrix describing the Lindblad dynamics
can be diagonalized by solving a five-term recurrence relation and we obtain analytical solutions for its
eigenvalues and eigenvectors. Using this diagonalization, we obtain an exact formula for thetime-dependent physical observables. By using this formula, we observe different behaviors which areobserved in the isotropic case. One of them is the overlap of light-cones with different velocities. The
second wave front of the light-cone is observed only when the magnetic field is small. As the magnetic
field increases, this wave front disappears beyond a critical point. This critical point is the same as that in
the non-equilibrium steady state. The location of the second wave front is represented as the site where
the spatial behavior of the magnetization changes. The disordered-oscillatory behavior is observed inside
the second wave fronts, and it is numerically inferred that this behavior is related to the presence ofcorrelations. On the other hand, the system size scaling behavior of the Liouvillian gap, whichcharacterizes the long-time behavior, can be studied. We examine that the Liouvillian gap is O(N^{-3})scaling except at the critical point and O(N^{-5}) scaling at the critical point. The Liouvillian gap under azero magnetic field can be calculated rigorously for the scaling behavior including its coefficient. (736)

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Note : Thesis Summary should be submitted in either a copy of 2000 Japanese Characters and 300 Words (Engｌish) or 1copy of 800
Words (English).
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Attention: Thesis Summary will be published on Tokyo Tech Research Repository Website (T2R2).