Method for measuring Berry phase capable of being carried by topological system body

Abstract

The invention discloses a method for measuring the Berry phase of the physical band of a topological system, and the method comprises the steps: enabling each lattice point in cells of the topological system to correspond to an acoustic cavity, constructing the coupling between the acoustic cavities through a feedback circuit system, and constructing an acoustic physical model of the cells; a transient phase modulator TPM is utilized to introduce time delay of time-dependent variation to a coupling signal, time-dependent evolution of a coupling phase is achieved, then adiabatic evolution of a system wave function in a Brillouin region k space is achieved, and k is a Brillouin region wave vector parameter; by stripping a dynamic phase from a total phase evolved by a system wave function, direct measurement of a physical band Berry phase is realized. An effective experimental measurement scheme is provided for measurement of the Berry phase of the topological system body, and the method is not only suitable for various one-dimensional or high-dimensional topological systems, but also can be popularized to a multi-energy-band non-Arbel topological system, and has good universality.

Description

Technical Field

[0001] The invention belongs to the technical field of acoustics and condensed matter physics, and relates to a method for measuring the Berry phase of a topological system body energy band. Background Art

[0002] Topological physics classifies matter through the theory of topology. One of the core points of topological physics theory is that the specific topological classification of matter can be characterized by the topological invariants of the bulk energy band, among which the Berry phase of the bulk energy band is the main observable of the topological invariant. The topological invariant of the bulk energy band directly determines the number of boundary states at the interface of the system, that is, it satisfies the bulk-edge correspondence. Although the Berry phase has a clear mathematical definition, its direct measurement in the Brillouin zone has always been a difficult problem, and there is still a lack of feasible experimental measurement schemes in acoustic systems. Summary of the invention

[0003] Purpose of the invention: The purpose of the invention is to provide a method for directly measuring the topological invariant of a topological system (i.e., the Berry phase of the bulk energy band).

[0004] Technical solution: A method for measuring the Berry phase of a topological system body energy band of the present invention comprises:

[0005] Each grid point in the topological system cell is corresponded to an acoustic cavity, and the feedback circuit system is used to construct the coupling between the acoustic cavities, thereby constructing the acoustic physical model of the cell;

[0006] A transient phase modulator (TPM) is used to introduce a time-dependent time delay into the coupling signal, so as to realize the time-dependent evolution of the coupling phase and further realize the adiabatic evolution of the system wave function in the Brillouin zone k space, where k is the Brillouin zone wave vector parameter.

[0007] By stripping the dynamical phase from the total phase of the system wave function evolution, direct measurement of the Berry phase of the bulk energy band is achieved.

[0008] Furthermore, the feedback circuit system includes a microphone, a power amplifier and a speaker; each acoustic cavity further includes a self-feedback circuit to provide gain to compensate for the loss of the acoustic cavity.

[0009] Furthermore, the transient phase modulator TPM is designed based on a digital signal processing (DSP) method, including sampling, processing and output steps that are executed in sequence, to achieve programmed time-dependent modulation of the coupling phase.

[0010] Furthermore, for a topological system with spatial translation periodicity, define its Brillouin zone wave vector parameter k and write out its cell Hamiltonian H(k); by solving the eigenvalue, we get its energy eigenvalue ε n (k) and the corresponding state function |En (k)>, where n represents the energy band number;

[0011] Introduce the time-dependent evolution of the system Hamiltonian H(k) and set k(t) = k0+k v *t, where k0 is the initial value of the Brillouin zone wave vector parameter k, v is the speed of change of the Brillouin zone wave vector parameter k, and t is time;

[0012] System wave function |ψ n The evolution of (t)> is solved by the time-dependent Schrödinger equation:

[0013]

[0014] Where i is the imaginary unit, represents the derivative with respect to time t, and H(t) represents the Hamiltonian that changes with time;

[0015] When the initial wave function of the system is the nth eigenstate |E n (0)>and the time-dependent modulation satisfies the adiabatic evolution requirement, the final state wave function of the system can be written as |ψ n (t)>=e i[α(t)+β(t)] |E n (t)>, where |E n (t)> is the state function that changes with time, e is a natural constant; α(t) is the dynamic phase of the accumulated wave function, which is related to the system and the details of the evolution; β(t) is the geometric phase of the accumulated wave function, which is only related to the properties of the system energy band itself; direct measurement of the geometric phase β(t) of the body energy band can be achieved through adiabatic evolution.

[0016] Furthermore, the dynamical phase α(t) of the wave function satisfies:

[0017]

[0018] Among them, ε n (t′) is the nth eigenvalue of the system at time t′.

[0019] Furthermore, the geometric phase β(t) of the wave function satisfies:

[0020]

[0021] Among them, |E n (t′)> is the eigenstate corresponding to the nth eigenvalue of the system at time t′.

[0022] Furthermore, for a one-dimensional topological system, the cell contains N grid points, and the cellular Hamiltonian can be written as:

[0023]

[0024] Among them, v is the intercellular coupling strength; k is the wave vector of the cell in the Brillouin zone. The k representing the reciprocal space information corresponds to the coupling phase between the outermost cavity No. 1 and cavity No. N of the cell that satisfies the complex conjugate relationship, that is, this coupling phase constructs the synthetic Brillouin zone of the system Hamiltonian.

[0025] In order to realize the time-dependent evolution of the system wave function in the Brillouin zone, a transient phase modulator TPM is used to introduce a time-dependent time delay into the coupled signal to realize the time-dependent evolution of the coupled phase, that is, ±k(t)=k0±k v *t.

[0026] Furthermore, when the coupled signal is designed to traverse the Brillouin zone within 0.5 s, the two signals satisfy the relative phase change of ±k(t)=±4πt.

[0027] Furthermore, for the coupled modulation AAH model, in addition to the Brillouin zone wave vector parameter k, this topological model also introduces an additional coupling modulation parameter The topological properties of the energy band of this topological model can be characterized by Wilson loop, that is, observing the Berry phase of the energy band with the parameter changes;

[0028] For an AAH model with a period of three, its cellular Hamiltonian is written as:

[0029]

[0030] Where f0 is the resonance frequency of the lattice point, v j is the grid-point coupling and satisfies j=1,2,3, v0 is the average value of coupling amplitude, v m is the modulation intensity of the coupling amplitude; when v o =v m =5Hz, for any energy band n and parameter The average value of the band eigenvalues ​​satisfies When taking |E n (k=-0.5π)>is the initial state of the system, and the dynamic phases obtained by the following two time-dependent modulation methods are the same:

[0031] First, the system remains unchanged within 0.5s;

[0032] Second, the coupling phase k changes continuously by 2π within 0.5 s;

[0033] Among them, the first steady-state case does not produce any Berry phase. By comparing the final state phases of the two cases, the energy band is extracted in the parameter The Berry phase β along the k direction.

[0034] Furthermore, the topological invariant of the AAH model body energy band is the Chern number, which is calculated as:

[0035]

[0036] in, is the Berry curvature of the band.

[0037] Beneficial effects: Compared with the prior art, the present invention has the following significant advantages:

[0038] The present invention realizes the measurement of the Berry phase of the bulk energy band of the topological system by the adiabatic evolution method in the acoustic system. The present invention is not only applicable to various one-dimensional or high-dimensional topological systems, but also can be extended to non-Abelian topological systems with multiple energy bands, and has good universality. BRIEF DESCRIPTION OF THE DRAWINGS

[0039] Figure 1 (a) is a one-dimensional topological system, where the dotted box shows a cell containing N grid points; Figure 1 (b) is the acoustic physical model of the unit cell;

[0040] Figure 2 (a) is a schematic diagram of a transient phase modulator (TPM) that implements programmed time-dependent modulation of the coupling phase; Figure 2 (b) is the relative phase change of the two coupled signals within 0.5s measured experimentally, that is, the phase modulation of ±k(t) = ±4πt is achieved;

[0041] Figure 3 (a) is the grid connection mode of the cells of the coupled modulation AAH model with a period of three; Figure 3 (b) is the energy band of the AAH model with the Brillouin zone wave vector parameter k and the coupling modulation parameter changes; Figure 3 (c) is the Berry phase of the energy band The change reflects the body Chen number C1 = C3 = 1 and C2 = -2. DETAILED DESCRIPTION

[0042] The present invention will be further described below in conjunction with the accompanying drawings.

[0043] The embodiment of the present invention provides a method for measuring the Berry phase of the bulk energy band of a topological system. For a one-dimensional or two-dimensional topological system, its cell is constructed based on an acoustic coupling cavity. By time-dependently regulating the coupling phase k between the outermost acoustic cavities, the adiabatic evolution of the system wave function in the Brillouin zone k space is achieved. Furthermore, by stripping the dynamic phase from the total phase of the system wave function evolution, the direct measurement of the Berry phase β of the bulk energy band is achieved.

[0044] The present invention is described in detail below from the perspectives of mathematical principles, device design and operation instructions.

[0045] For a topological system with spatial translation periodicity, its Brillouin zone wave vector parameter k can be defined and its cell Hamiltonian H(k) can be written. By solving the eigenvalue, its energy eigenvalue ε can be obtained. n (k) and the corresponding state function |E n (k)>, where n represents the energy band number.

[0046] Introduce the time-dependent evolution of the system Hamiltonian H(k) and set k(t) = k0+k v *t, where k0 is the initial value of the Brillouin zone wave vector parameter k, v is the speed of change of the wave vector parameter k in the Brillouin zone, and t is time. Because of the translation periodicity of k space, k changes continuously within a range of 2π, which means that the system Hamiltonian traverses the entire Brillouin zone.

[0047] System wave function |ψ n The evolution of (t)> can be solved by the time-dependent Schrödinger equation:

[0048]

[0049] Where i is the imaginary unit, represents the derivative with respect to time t, and H(t) represents the Hamiltonian that changes with time.

[0050] When the initial wave function of the system is the nth eigenstate |E n (0)>and the time-dependent modulation satisfies the adiabatic evolution requirement, the final state wave function of the system can be written as |ψ n (t)>=e i[α(t)+β(t)] |E n (t)>, where |E n (t)> is a state function that changes with time, and the α(t) and β(t) in the e index satisfy:

[0051]

[0052] Among them, ε n (t′) and |E n (t′)> are the nth eigenvalue and corresponding eigenstate of the system at time t′ respectively.

[0053] Obviously, α(t) is the dynamic phase of wave function accumulation, which is related to the system and the details of evolution, while β(t) is the geometric phase of wave function accumulation, which is only related to the properties of the system energy band itself. Therefore, through adiabatic evolution, direct measurement of the geometric phase of the body energy band βt) can be achieved.

[0054] like Figure 1 As shown in (a), for a one-dimensional topological system, its cell contains N grid points, and the cellular Hamiltonian can be written as:

[0055]

[0056] Among them, v is the intercellular coupling strength; k is the wave vector of the cell in the Brillouin zone. Obviously, k, which represents the reciprocal space information, corresponds to the coupling phase between the outermost cavity No. 1 and cavity No. N of the cell that satisfies the complex conjugate relationship, that is, this coupling phase constructs the synthetic Brillouin zone of the system Hamiltonian.

[0057] By corresponding each grid point to an acoustic cavity and using the feedback circuit system to construct the coupling between the acoustic cavities, the acoustic physical model of the cell can be directly constructed, such as Figure 1 (b) The feedback circuit system includes a microphone, a power amplifier and a speaker, and each acoustic cavity further includes a self-feedback circuit to provide gain to compensate for the loss of the acoustic cavity (the acoustic cavity circuit coupling without TPM belongs to the prior art, and reference can be made to the inventor's published articles, such as Communication Physics 7, 339 (2024), Physical Review B 109, L020302 (2024)).

[0058] In order to realize the time-dependent evolution of the system wave function in the Brillouin zone, a transient phase modulator (TPM) can be designed based on digital signal processing (DSP). By introducing a time-dependent time delay to the coupled signal, the time-dependent evolution of the coupled phase can be realized, that is, ±k(t) = k0±k v *t. Combined Figure 1 , the inter-cell coupling is equivalent to the phase e between cavity 1 and cavity N ±ik(t) The coupling is time-dependently modulated via a dual-channel TPM, allowing the system to “traverse the Brillouin zone”.

[0059] like Figure 2 As shown in (a), the transient phase modulator TPM uses DSP technology (such as TMS320F28335 chip, DSP is a mature technology in acoustic signal processing), and sequentially realizes programmed time-dependent modulation of the coupling phase through sampling, processing, and output steps. For example, when the coupling signal is designed to traverse the Brillouin zone within 0.5s, the two signals satisfy the relative phase change of ±k(t) = ±4πt. Figure 2 (b) The experimental results are completely consistent with the theoretical design.

[0060] Taking the classic coupled modulation type Aubry-Andre-Harper (AAH) as an example, in addition to the Brillouin zone wave vector parameter k, this topological model also introduces an additional coupled modulation parameter Therefore, the model is a two-dimensional topological model.

[0061] The topological invariant of the AAH model body energy band is the Chern number, which is calculated as follows:

[0062]

[0063] in, is the Berry curvature of the band.

[0064] In addition, the topological properties of the bulk energy band of this topological model can also be characterized by Wilson loop, that is, observing the Berry phase of the bulk energy band with the parameter changes.

[0065] For an AAH model with a period of three, its cellular Hamiltonian is written as:

[0066]

[0067] Where f0 is the resonance frequency of the lattice point, v j is the grid-point coupling and satisfies j=1,2,3, v0 is the average value of coupling amplitude, v m is the modulation intensity of the coupling amplitude. The physical connection method corresponding to this cell is as follows Figure 3 (a). When v o =v m =5Hz, its energy band is as follows Figure 3 (b) For any energy band n and parameter The average value of the band eigenvalues ​​satisfies This means that when taking |E n (k=-0.5π)>is the initial state of the system, and the dynamic phases obtained by the following two time-dependent modulation methods are the same:

[0068] First, the system remains unchanged within 0.5s;

[0069] Second, the coupling phase k changes continuously by 2π within 0.5s.

[0070] Among them, the first steady state does not produce any Berry phase. Therefore, by comparing the final state phases in the two cases, the energy band in the parameter The Berry phase β along the k direction. Figure 3 (c) shows the Berry phase of the body energy band The direction and range of change reflect the Chern numbers of the energy band C1=C3=1 and C2=-2.

[0071] In summary, the present invention first constructs coupling between acoustic cavities by means of feedback circuits (including microphones, power amplifiers, and speakers), and reflects the Brillouin zone parameter k of the topological model in the coupling phase of the two outermost cavities of the cell, thereby directly constructing the cellular model. Then, the DSP method is used to realize the time-dependent modulation of the acoustic coupling phase, so as to achieve the goal of driving the system to "traverse the Brillouin zone", stimulate the eigenwave function of the cellular system and drive its time-dependent adiabatic evolution. Finally, the Berry phase of the topological body energy band is extracted by stripping the dynamic phase.

[0072] The present invention provides an effective experimental measurement scheme for measuring the Berry phase of the body energy band of a topological system.

Claims

1. A method for measuring the Berry phase of a topological system body energy band, characterized in that: include: Each grid point in the topological system cell is corresponded to an acoustic cavity, and the feedback circuit system is used to construct the coupling between the acoustic cavities, thereby constructing the acoustic physical model of the cell; A transient phase modulator (TPM) is used to introduce a time-dependent time delay into the coupling signal, so as to realize the time-dependent evolution of the coupling phase and further realize the adiabatic evolution of the system wave function in the Brillouin zone k space, where k is the Brillouin zone wave vector parameter. By stripping the dynamical phase from the total phase of the system wave function evolution, direct measurement of the Berry phase of the bulk energy band is achieved.

2. The method for measuring the Berry phase of the topological system body energy band according to claim 1, characterized in that: The feedback circuit system includes a microphone, a power amplifier and a speaker; each acoustic cavity further includes a self-feedback circuit to provide gain to compensate for the loss of the acoustic cavity.

3. The method for measuring the Berry phase of the topological system body energy band according to claim 2, characterized in that: The transient phase modulator TPM is designed based on the digital signal processing DSP method, including sampling, processing and output steps executed in sequence to achieve programmed time-dependent modulation of the coupling phase.

4. The method for measuring the Berry phase of a topological system body energy band according to any one of claims 1 to 3, characterized in that: For a topological system with spatial translation periodicity, define its Brillouin zone wave vector parameter k and write out its cell Hamiltonian H(k); by solving the eigenvalue, we get its energy eigenvalue ε n (k) and the corresponding state function |E n (k)>, where n represents the energy band number; Introduce the time-dependent evolution of the system Hamiltonian H(k) and set k(t) = k0+k v *t, where k0 is the initial value of the Brillouin zone wave vector parameter k, v is the speed of change of the Brillouin zone wave vector parameter k, and t is time; System wave function |ψ n The evolution of (t)> is solved by the time-dependent Schrödinger equation: Where i is the imaginary unit, represents the derivative with respect to time t, and H(t) represents the Hamiltonian that changes with time; When the initial wave function of the system is the nth eigenstate |E n (0)>and the time-dependent modulation satisfies the adiabatic evolution requirement, the final state wave function of the system can be written as |ψ n (t)>=e i[α(t)+β(t)] |E n (t)>, where |E n (t)> is the state function that changes with time, e is a natural constant; α(t) is the dynamic phase of the wave function accumulation, which is related to the system and the details of the evolution; β(t) is the geometric phase of the wave function accumulation, which is only related to the properties of the system energy band itself; direct measurement of the geometric phase β(t) of the body energy band can be achieved through adiabatic evolution.

5. The method for measuring the Berry phase of the topological system body energy band according to claim 4, characterized in that: The dynamical phase α(t) of the wave function satisfies: Among them, ε n (t′) is the nth eigenvalue of the system at time t′.

6. The method for measuring the Berry phase of the topological system body energy band according to claim 4, characterized in that: The geometric phase β(t) of the wave function satisfies: Among them, |E n (t′)> is the eigenstate corresponding to the nth eigenvalue of the system at time t′.

7. The method for measuring the Berry phase of the topological system body energy band according to claim 4, characterized in that: For a one-dimensional topological system, the cell contains N grid points, and the cellular Hamiltonian can be written as: Among them, v is the intercellular coupling strength; k is the wave vector of the cell in the Brillouin zone. The k representing the reciprocal space information corresponds to the coupling phase between the outermost cavity No. 1 and cavity No. N of the cell that satisfies the complex conjugate relationship, that is, this coupling phase constructs the synthetic Brillouin zone of the system Hamiltonian. In order to realize the time-dependent evolution of the system wave function in the Brillouin zone, a transient phase modulator TPM is used to introduce a time-dependent time delay into the coupled signal to realize the time-dependent evolution of the coupled phase, that is, ±k(t)=k0±k v *t.

8. The method for measuring the Berry phase of the topological system body energy band according to claim 7, characterized in that: When the coupling signal is designed to traverse the Brillouin zone within 0.5 s, the two signals satisfy the relative phase change of ±k(t)=±4πt.

9. The method for measuring the Berry phase of the topological system body energy band according to claim 4, characterized in that: For the coupled modulation AAH model, in addition to the Brillouin zone wave vector parameter k, this topological model also introduces an additional coupled modulation parameter The topological properties of the energy band of this topological model can be characterized by Wilson loop, that is, observing the Berry phase of the energy band with the parameter changes; For an AAH model with a period of three, its cellular Hamiltonian is written as: Where f0 is the resonance frequency of the lattice point, v j is the grid-point coupling and satisfies j=1,2,3, v0 is the average value of coupling amplitude, v m is the modulation intensity of the coupling amplitude; when v o =v m =5Hz, for any energy band n and parameter The average value of the band eigenvalues ​​satisfies When taking |E n (k=-0.5π)>is the initial state of the system, and the dynamic phases obtained by the following two time-dependent modulation methods are the same: First, the system remains unchanged within 0.5s; Second, the coupling phase k changes continuously by 2π within 0.5 s; Among them, the first steady-state case does not produce any Berry phase. By comparing the final state phases of the two cases, the energy band is extracted in the parameter The Berry phase β along the k direction.

10. The method for measuring the Berry phase of the topological system body energy band according to claim 9, characterized in that: The topological invariant of the AAH model body energy band is the Chern number, which is calculated as follows: in, is the Berry curvature of the band.

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