Construction and analysis method of square root topology circuit for realizing continuum bound state
By constructing a tight-bound model and square root topological circuit of SSH double-stranded SSH double-stranded, the problems of narrow application range and low efficiency of continuous bound state research in the prior art are solved, and the application of bound states and circuit robustness in multi-bands are achieved.
Patent Information
- Application Number
- CN202410936574.3
- Authority / Receiving Office
- CN · China
- Patent Type
- Applications(China)
- Current Assignee / Owner
- Filing Date
- 2024-07-12
- Publication Date
- 2024-12-13
AI Technical Summary
In the prior art, the research on continuum bound states has problems such as narrow application range and low research efficiency, especially in the case of multi-frequency.
By constructing a tightly bound model of SSH double-stranded SSH double-stranded and building a square root topological circuit based on it, the square root of coupling strength is used for circuit design, and the continuum bound state of multi-band is realized, and the characteristic frequency and Eigenfunction analysis confirms the existence of the bound state.
It realizes the application of continuum bound states in multiple frequency bands, expands the application range and efficiency of the research, and at the same time, the circuit has strong robustness and stability.
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Figure CN119129508A_ABST
Abstract
Description
Technical Field
[0001] The present invention relates to the field of topological circuits, and more specifically, to a construction and analysis method of a square root topological circuit for realizing a continuum bound state. Background Art
[0002] As a branch of mathematics, topology originated from Euler's research on connectivity and Euler's formula in the 18th century. Circuit systems have only been used in the study of topological states in recent years, and the concept of topological circuits has gradually been formed.
[0003] In the field of circuits, a topological circuit is a graphical representation of the connection form and interconnection method of components in a circuit. It focuses on the topological structure between components rather than the specific component properties and values. This representation is concise and clear, which helps to analyze the working principle of the circuit, optimize the circuit structure, and judge the reliability of the circuit. The working principle of a topological circuit is to combine different electronic devices in a specific connection method to achieve the transfer of current and voltage, thereby completing the function of the circuit. Typical topological structures include series, parallel, bridge, and anti-phase.
[0004] The design and implementation of topological circuits fully consider the characteristics of components in the circuit, operating voltage and signal requirements, thereby ensuring the stability and reliability of the circuit. In complex circuit systems, topological circuits can effectively resist external interference and ensure the normal operation of the circuit. The design of topology also helps to optimize the cost of the circuit. Through reasonable component selection and layout, manufacturing costs can be reduced and economic benefits can be improved. At the same time, the design of topological circuits also helps to reduce unnecessary energy waste and achieve energy conservation and emission reduction. The connection method of the topological circuit can be regarded as an abstract grid structure, which can clearly display important information such as the connection method, direction, path transmission method, and the existence or non-existence of the loop of each component. This makes the topological circuit easy to understand and maintain, reduces maintenance costs, and improves maintenance efficiency.
[0005] In summary, the advantages of topological circuits are mainly reflected in their high reliability, economy and easy maintenance, and different types of topological circuits also have their own unique advantages. These advantages make topological circuits widely used in power electronics, communications, computers and other fields.
[0006] The square root topological circuit is a new type of topological structure, which was developed under the inspiration of Dirac's square root operation on equations. This circuit structure has unique advantages and application potential in circuit design. The emergence of square root topological circuits provides new perspectives and tools for circuit design and the study of topological states. It not only enriches the application field of topology, but also provides more possibilities for the development of future circuit design and electronic technology. However, the specific application and performance optimization of square root topological circuits still need further research and exploration.
[0007] A continuum bound state is a wave that can exist in a continuous spectrum of radiative waves of energy while remaining localized. The continuum bound state has properties such as good localization, discrete energy levels, and shape dependence, which make it very attractive in optical applications such as sensors, lasers, and optical waveguide devices. A continuum bound state also refers to a stationary state solution that appears in a waveguide structure with a continuous spectrum, which falls in the energy band gap of the continuous spectrum in frequency and has an infinite lifetime. It has attracted increasing attention due to its potential applications in photonic crystal fibers and lasers, sensors, and surface acoustic wave devices. Bound states in the continuum are localized states that can exist in the bulk state, and continuum bound states can coexist with radiative waves while remaining perfectly localized and unaffected by radiation. To date, several mechanisms for the generation of continuum bound states in periodic systems have been reported, including symmetry-guaranteed mode mismatch, destructive interference of topological origin, and Friedrich–Wintgen mechanism. Bound states in the continuum have been studied in photonic, electronic, and phononic systems.
[0008] However, in the prior art, the research on continuum bound states is conducted at a single frequency, which has the problems of narrow application range and low research efficiency. Therefore, how to invent a square root topological circuit to realize continuum bound states is a technical problem that urgently needs to be solved in this technical field. Summary of the invention
[0009] In order to solve the problems of narrow application scope and low research efficiency of the prior art, the present invention provides a construction and analysis method for a square root topological circuit realizing a continuum bound state, which has the characteristics of strong robustness.
[0010] In order to achieve the above-mentioned purpose of the present invention, the technical scheme adopted is as follows:
[0011] A method for constructing a square root topological circuit for realizing a continuum bound state comprises the following specific steps:
[0012] Construct a tight binding model of SSH double strands including several somata; each somata has several original sites;
[0013] The square root SSH double-stranded tight binding model is constructed by inserting additional sites corresponding to both sides of each original site of each cell body in the SSH double-stranded tight binding model to separate the original sites and taking the square root of the coupling strength;
[0014] A square root topology circuit is constructed by taking each site in the tight binding model of the square root SSH double chain as a circuit node, using capacitance to represent the coupling between each circuit node, and grounding each circuit node through an inductor.
[0015] Preferably, the tight binding model of the SSH double chain consists of two identical SSH chains; in each cell body, the two SSH chains are coupled together through an intermediate chain; between adjacent cell bodies, an identical site of each intermediate chain is connected by a connecting chain.
[0016] Furthermore, each cell body of the tight binding model of the SSH double-chain specifically includes 5 sites, namely: site 1 located at the connection between the first SSH chain and the intermediate chain, site 2 located on the first SSH chain, site 3 located at the connection between the intermediate chain and the second SSH chain, site 4 located on the second SSH chain, and site 5 located on the intermediate chain; between adjacent cell bodies, the sites 5 of each intermediate chain are connected by a connecting chain.
[0017] Furthermore, the original sites are separated by inserting additional sites on both sides of each original site in each cell body in the tight binding model of the SSH double-strand, specifically:
[0018] On the first SSH chain, sites 6 and 7 are added between sites 1 and 2 and on the side of site 2 away from site 1 respectively; on the middle chain, sites 10 and 11 are added on both sides of site 5 respectively; on the second SSH chain, sites 8 and 9 are added between sites 3 and 4 and on the side of site 4 away from site 1 respectively.
[0019] Furthermore, taking the square root of the coupling strength, the specific steps are:
[0020] According to the intracellular and intercellular coupling, the Hamiltonian of the tight binding model of the SSH double chain is obtained:
[0021]
[0022] Among them, t1 and t2 represent the intracellular and intercellular coupling of the square root model, t3 is the interchain coupling parameter of the square root model, and k is the momentum;
[0023] Taking the square root of the Hamiltonian of the tight-binding model of the SSH double chain, we get the Hamiltonian of the tight-binding model of the square root SSH double chain:
[0024]
[0025] Furthermore, the current equation of the site of the square root topology circuit is specifically:
[0026]
[0027] The superscript n indicates the specific unit cell position, n (n=1,2,..,11) represents the external excitation current to the nth node, V n (n=1,2,..,11) represents the response voltage of the corresponding node, C1, C2, C3, C4, C a , C b , C c They represent different capacitance values respectively; different couplings are represented by different capacitance values.
[0028] A method for analyzing a square root topology circuit includes the following specific steps:
[0029] Constructing a square root topology circuit with N cells by using the square root topology circuit construction method;
[0030] Calculate the eigenfrequencies and eigenfunctions of the Laplacian operator for square root topology circuits;
[0031] Analyze the characteristic frequencies and characteristic functions and observe the state of the continuum bound state.
[0032] Preferably, the characteristic frequency and characteristic function are analyzed, and the specific steps are:
[0033] Calculate the characteristic frequency and characteristic function of the Laplace operator of the square root topology circuit, and construct the admittance spectrum of the circuit model; construct the eigenvalue spectrum based on the admittance spectrum of the circuit model;
[0034] Observe the eigenvalue spectrum to confirm the existence and distribution of continuum bound states.
[0035] Furthermore, when analyzing the characteristic frequencies and characteristic functions and observing the state of the continuum bound state, the impedance of each node in the circuit is calculated, a simulation impedance result graph is constructed, and the existence and distribution of topological bound states and mediocre bound states are confirmed by observing the simulation impedance result graph.
[0036] Furthermore, after confirming the existence of topological bound states and mediocre bound states, the square root topological circuit is further simulated and analyzed. The specific steps are as follows:
[0037] The square root topology circuit is simulated to obtain the simulated impedance distribution at the topological bound state and the mediocre bound state at a specific frequency;
[0038] An intermediate coupling capacitor is added to the cell body of the square root topology circuit to transform it from a bound state system to a continuum bound state system;
[0039] By observing the impedance distribution at this time, the robustness test of the square root topology circuit is performed at this time.
[0040] The beneficial effects of the present invention are as follows:
[0041] The present invention constructs a square root SSH double chain tight binding model based on the SSH double chain tight binding model, and constructs a topological inductor-capacitor square root topological circuit based on the square root SSH double chain tight binding model, realizing the bound state in the continuum in multiple frequency bands. It brings more possibilities for the application of the continuum bound state in the topological circuit, and has more important theoretical value and broad scientific research application prospects. At the same time, the circuit also has strong robustness, stable use and long service life. BRIEF DESCRIPTION OF THE DRAWINGS
[0042] Figure 1 It is a flow chart of a method for constructing a square root topological circuit for realizing a continuum bound state according to the present invention.
[0043] Figure 2 It is a schematic diagram of the tight binding model of the SSH double chain and its corresponding coupling relationship.
[0044] Figure 3 It is a schematic diagram of the tight binding model of the square root SSH double chain and its corresponding coupling relationship.
[0045] Figure 4 This is a schematic diagram of the energy level bands of the tight-binding model of the SSH double chain and the tight-binding model of the square root SSH double chain.
[0046] Figure 5 It is a schematic diagram of the square root topology circuit structure.
[0047] Figure 6 This is a schematic diagram of the topological phase energy band of the square root topological circuit under different coupling strengths.
[0048] Figure 7 This is a schematic diagram of the topological phase energy band of the square root topology circuit after adding the intermediate coupling capacitor under different coupling strengths.
[0049] Figure 8 Schematic diagram of the admittance spectrum of the square root topology circuit in Example 2.
[0050] Fig. 9 It is a schematic diagram of the eigenvalue spectrum and bound states in the band gap in Example 2.
[0051] Fig.101 is a schematic diagram of the admittance spectrum of the square root topology circuit in which the intermediate coupling capacitor is introduced in Example 2.
[0052] Fig.11 The figure is a schematic diagram showing the characteristic value spectrum and bound states in the band gap when the intermediate coupling capacitor is introduced in Example 2.
[0053] Fig.12 This is a schematic diagram of circuit node impedance simulation using NI Multisim software in Example 3.
[0054] Fig.13 Schematic diagram of the test arrangement of the circuit impedance distribution in Example 3.
[0055] Fig.14 Schematic diagram of the square root topology circuit used for measuring bound states in the band gap in Example 3.
[0056] Fig.15 3 is a diagram of impedance simulation results of the bound state and the bulk state of the square root topology circuit in Example 3.
[0057] Fig.16 3 is a diagram of impedance experiment results of the bound state and bulk state of the square root topology circuit in Example 3.
[0058] Fig.17 Schematic diagram of impedance distribution of the square root topology circuit at f_01 in Example 3.
[0059] Fig.18 Schematic diagram of impedance distribution of the square root topology circuit at f_02 in Example 3.
[0060] Fig.19 This is a schematic diagram of a square root topology circuit for measuring a continuum bound state in Example 3.
[0061] Fig. 20 3 is a diagram of impedance simulation results of the bound state and the bulk state of the square root topology circuit in Example 3.
[0062] Fig.21 3 is a diagram of impedance experiment results of the bound state and bulk state of the square root topology circuit in Example 3.
[0063] Fig. 22 Schematic diagram of impedance distribution of the square root topology circuit at f_01 in Example 3.
[0064] Fig.23 Schematic diagram of impedance distribution of the square root topology circuit at f_02 in Example 3.
[0065] Fig.24 It is a flow chart of an analysis method of a square root topology circuit of the present invention. Fig.25 This is a schematic diagram of the second unit cell of the square root topology circuit in Example 3. Fig.26 This is a schematic diagram of the eighth unit cell of the square root topology circuit in Example 3. Fig. 27 It is the second unit cell node of the square root topology circuit in Example 3 The impedance diagram at Fig.28 It is the eighth unit cell node of the square root topology circuit in Example 3 Impedance diagram at . DETAILED DESCRIPTION
[0066] The present invention is described in detail below with reference to the accompanying drawings and specific embodiments.
[0067] Example 1
[0068] like Figure 1 As shown, a method for constructing a square root topological circuit for realizing a continuum bound state includes the following specific steps:
[0069] Construct a tight binding model of SSH double strands including several somata; each somata has several original sites;
[0070] The square root SSH double-stranded tight binding model is constructed by inserting additional sites corresponding to both sides of each original site of each cell body in the SSH double-stranded tight binding model to separate the original sites and taking the square root of the coupling strength;
[0071] A square root topology circuit is constructed by taking each site in the tight binding model of the square root SSH double chain as a circuit node, using capacitance to represent the coupling between each circuit node, and grounding each circuit node through an inductor.
[0072] like Figure 2 As shown in the figure, the tight binding model of the SSH double chain consists of two identical SSH chains; in each cell body, the two SSH chains are coupled together through an intermediate chain; between adjacent cell bodies, an identical site of each intermediate chain is connected by a connecting chain, which is marked by a dotted line in the figure. t4 represents a jump between two adjacent positions
[0073] In a specific embodiment, each cell body of the tight binding model of the SSH double-stranded chain specifically includes 5 sites, specifically: site 1 located at the connection between the first SSH chain and the intermediate chain, site 2 located on the first SSH chain, site 3 located at the connection between the intermediate chain and the second SSH chain, site 4 located on the second SSH chain, and site 5 located on the intermediate chain; between adjacent cell bodies, the sites 5 of the respective intermediate chains are connected by a connecting chain.
[0074] In a specific embodiment, Figure 3 As shown, the original sites are separated by inserting additional sites corresponding to both sides of each original site in each cell body in the tight binding model of the SSH double strand, specifically:
[0075] On the first SSH chain, sites 6 and 7 are added between sites 1 and 2 and on the side of site 2 away from site 1 respectively; on the middle chain, sites 10 and 11 are added on both sides of site 5 respectively; on the second SSH chain, sites 8 and 9 are added between sites 3 and 4 and on the side of site 4 away from site 1 respectively.
[0076] In a specific embodiment, taking the square root of the coupling strength, the specific steps are:
[0077] According to the intracellular and intercellular coupling, the Hamiltonian of the tight binding model of the SSH double chain is obtained:
[0078]
[0079] Among them, t1 and t2 represent the intracellular and intercellular coupling of the square root model, t3 is the interchain coupling parameter of the square root model, and k is the momentum;
[0080] Taking the square root of the Hamiltonian of the tight-binding model of the SSH double chain, we get the Hamiltonian of the tight-binding model of the square root SSH double chain:
[0081]
[0082] like Figure 4 As shown, based on the above Hamiltonian, the energy level band diagrams under the two models are calculated, where t1 / t2<1. At this point, it can be found that after squaring the Hamiltonian of the square root SSH double chain tight binding model, the Hamiltonian of the original tight binding model can be obtained, so this lattice model with square root properties is called the square root SSH double chain.
[0083] The structure of the obtained square root topology circuit is as follows Figure 5 As shown, there is a one-to-one correspondence between the coupling of the circuit nodes and the coupling of the tight binding model. That is, capacitors C1, C2, C3 and C4 are used to achieve coupling, and each circuit node is grounded by the same inductor L. In addition, capacitor grounding should be introduced at nodes other than 5, 10 and 11. According to Kirchhoff's law and Ohm's law, the current and voltage equations of the 11 nodes can be given. The current equation of the site of the square root topology circuit is specifically:
[0084]
[0085] The superscript n indicates the specific unit cell position,n (n=1,2,..,11) represents the external excitation current to the nth node, V n (n=1,2,..,11) represents the response voltage of the corresponding node, C1, C2, C3, C4, C a , C b , C c They represent different capacitance values respectively; different couplings are represented by different capacitance values.
[0086] In this embodiment, the schematic diagram of the energy band of the square root SSH double-chain coupling circuit under different coupling strengths is as follows: Figure 6 , Figure 7 As shown, Figure 6 For the topological phase, C1=1nF, C2=4.7nF, C4=0nF. Figure 7 The C4 topology phase is added, and now C1=1nF, C2=4.7nF, C4=3.3nF. Other common parameters are C3=3.3nF, L=10uH.
[0087] Example 2
[0088] like Fig.23 As shown, a method for analyzing a square root topology circuit includes the following specific steps:
[0089] Constructing a square root topology circuit with N cells by using the square root topology circuit construction method;
[0090] Calculate the eigenfrequencies and eigenfunctions of the Laplacian operator for square root topology circuits;
[0091] Analyze the characteristic frequencies and characteristic functions and observe the state of the continuum bound state.
[0092] In this embodiment, the characteristic frequency and characteristic function are analyzed, and the specific steps are as follows:
[0093] Calculate the characteristic frequency and characteristic function of the Laplace operator of the square root topology circuit, and construct the admittance spectrum of the circuit model; construct the eigenvalue spectrum based on the admittance spectrum of the circuit model;
[0094] Observe the eigenvalue spectrum to confirm the existence and distribution of continuum bound states.
[0095] In this embodiment, a finite circuit size with N = 110 nodes is constructed. The circuit includes 10 cells. The calculated admittance spectrum and its eigenvalue spectrum are as follows: Figure 8 , Fig. 9 As shown, the red circle represents the topological bound state, the blue circle and the gray circle represent the trivial bound state and the bulk state, respectively. The inset shows a local enlargement of the trivial bound state, where the red and blue shaded areas correspond to the band gap range in the topological energy band.01 =430.5kHz and f 02 =722.6kHz, four topologically bound degenerate states were numerically determined, and their density distribution is located at the right end of the model. In addition, other localized states were observed at 506kHz, 596kHz, and 784kHz in the blue circle, and their density is mainly located at the left end of the model. Fig.10 ,11 shows the admittance spectrum and eigenvalue spectrum with the introduction of the intermediate coupling capacitor C4. It can be observed that the topological bound state does not exist in the band gap, but is mixed with the continuum state.
[0096] In this embodiment, the frequency of the topological bound state is the same as that of the bound state in the bulk gap, i.e., f 01 and f 02 ; however, these states are in a continuum compared to the bound states described previously. The four degenerate states are all localized at the right end of the circuit, consistent with bound states in the band gap. Moreover, it can be found that these topological bound states in red circles still exist even in the absence of a band gap, indicating the existence of continuum bound states. Although the topological bound state is now in the bulk state, it still maintains good localization because the density of states is still localized at the two nodes on the right end, similar to the topological bound state in the band gap. In addition, it can be clearly found that the mediocre bound state undergoes a change in frequency, indicating that it is not topologically protected.
[0097] Example 3
[0098] In a specific embodiment, the characteristic frequency and characteristic function are analyzed, and when the state of the continuum bound state is observed, the impedance of each node of the circuit is calculated, a simulation impedance result graph is constructed, and the existence and distribution of the topological bound state and the mediocre bound state are confirmed by observing the simulation impedance result graph.
[0099] In this embodiment, Fig.12 , 13 As shown in the figure, NI Multisim software is used to simulate the circuit node impedance and a finite circuit with 10 cells is constructed. The specific circuit structure is as follows Fig.14 As shown in the figure on the left, a local amplification of one of the unit cells in the circuit is shown. At this time, the circuit parameters are set to C1 = 1nF, C2 = 4.7nF, C3 = 3.3nF, C4 = 0nF, L = 10uH. According to the previous energy spectrum calculation, it can be found that at f 01 =430.5kHz and f 02 =722.6kHz, topological bound states appear at two frequencies. Based on these calculation results, this embodiment calculates the impedance of each node of the circuit under theoretical and experimental conditions, and uses Fig.14 The impedance characteristics of the bound state in the band gap are specifically introduced by taking the three representative nodes marked A, B, and C as examples.
[0100] In this embodiment, the simulated impedance results of nodes A, B, and C are shown in FIG. Fig.15 As shown, the red solid line Z A , Black Z B , Blue Z C Representation, where the red and blue shaded areas represent the topological and trivial band gaps, respectively. Fig.16 The experimental impedance results of nodes A, B, and C are shown in detail. It can be found that the impedance Z A , Z B , Z C The resonance peaks all appear at f 01 and f 02 , 495kHz and 814kHz, 506kHz and 784kHz, which just correspond to the topological bound state and the mediocre bound state. 01 and f 02 The simulated impedance distribution at Fig.17 , 18 shown.
[0101] In order to further explore the characteristics of bound states and enhance intuitive understanding thereof, in a specific embodiment, after confirming the existence of topological bound states and mediocre bound states, a simulation analysis is further performed on the square root topological circuit, and the specific steps are as follows:
[0102] The square root topology circuit is simulated to obtain the simulated impedance distribution at the topological bound state and the mediocre bound state at a specific frequency;
[0103] An intermediate coupling capacitor is added to the cell body of the square root topology circuit to transform it from a bound state system to a continuum bound state system;
[0104] By observing the impedance distribution at this time, the robustness test of the square root topology circuit is performed at this time.
[0105] In this embodiment, by adding an intermediate coupling capacitor C4 = 10nF in the unit cell of the experimental LC circuit, we transform the bound state system into a continuum bound state system; the system is as follows Fig.19 The right figure shows a local enlargement of one of the unit cells. The simulated impedance and experimental impedance results of the circuit nodes A, B, and C are shown in Fig. 20 and Fig.21 As shown, the red, black and blue solid lines represent the impedance Z A , Z B , Z CThe peaks at 430.5kHz, 722.6kHz and 665kHz correspond to BICs and trivial bound states, respectively. Compared with the previous body impedance, it can be clearly found that the body impedance at this time is completely suppressed, which further illustrates that the continuum bound state system has good localization. In this case, since the state of node A has topological properties and the state of node C is trivial, Z A The impedance peak is almost the same as the bound state in the band gap, Z C The impedance peak is significantly smaller. It is worth noting that the measured impedance value is still consistent with the simulation result. Fig. 22 and Fig.23 The circuit is described at f 01 and f 02 Impedance distribution at different frequencies.
[0106] In this embodiment, the topological state generally exhibits resilience and robustness against disorder and defects. In order to test the robustness of the continuum bound state, destructive defects are introduced into the second and eighth unit cells of the circuit structure by adding C4, that is, one inter-cell coupling capacitor in the unit cell is removed. The specific circuit diagram is shown in FIG. Fig.25 ) and Figure Fig.26 , where the inset provides a localized magnification of the cell where the defect was introduced. Fig. 27 and Fig.28 The impedance diagram at node A is described, where the actual and dotted lines represent the impedance results under the experiment and simulation respectively. It can be found that no matter what kind of defect is present, the impedance is consistent with the defect-free system and is still at the same resonant frequency, indicating that the circuit has a strong anti-interference ability. If a part is damaged during future use, the circuit function can still operate normally.
[0107] The present invention extends the bound states in the continuum in the classical system to the multi-frequency case, and verifies this through experiments in the circuit system. The one-dimensional continuum bound state observed in the LC circuit in the analysis method of the square root topological circuit of the present invention may be extended to the two-dimensional and three-dimensional continuum bound states, thereby promoting the development of electronic devices based on the continuum bound state, such as resonators, waveguides and filters. In addition, the core concept of the present invention through experimental design can be further extended to the field of high-frequency circuits by applying microstrip transmission line technology. Specifically, if metal wires are used to replace traditional capacitors and inductors, it is expected that the operating frequency of the topological circuit will be increased to the gigahertz level. At the same time, a flexible and changeable configuration method is also provided, allowing the diversification of circuit functions to be achieved by designing frequency hopping parameters and using active components such as operational amplifiers. This type of high-frequency topological circuit will show broad application prospects in future antennas, filters and other radio frequency devices.
[0108] Obviously, the above embodiments of the present invention are only examples for clearly explaining the present invention, and are not intended to limit the implementation methods of the present invention. Any modifications, equivalent substitutions and improvements made within the spirit and principles of the present invention shall be included in the protection scope of the claims of the present invention.
Claims
1. A method for constructing a square root topological circuit for realizing a continuum bound state, characterized in that: The specific steps include: Construct a tight binding model of SSH double strands including several somata; each somata has several original sites; The square root SSH double-stranded tight binding model is constructed by inserting additional sites corresponding to both sides of each original site of each cell body in the SSH double-stranded tight binding model to separate the original sites and taking the square root of the coupling strength; A square root topology circuit is constructed by taking each site in the tight binding model of the square root SSH double chain as a circuit node, using capacitance to represent the coupling between each circuit node, and grounding each circuit node through an inductor.
2. The method for constructing a square root topological circuit for realizing a continuum bound state according to claim 1, characterized in that: The tight binding model of the SSH double chain is composed of two identical SSH chains; in each cell body, the two SSH chains are coupled together through an intermediate chain; between adjacent cell bodies, an identical site of each intermediate chain is connected by a connecting chain.
3. The method for constructing a square root topological circuit for realizing a continuum bound state according to claim 2, characterized in that: Each cell body of the tight binding model of the SSH double-chain specifically includes 5 sites, namely: site 1 located at the connection between the first SSH chain and the intermediate chain, site 2 located on the first SSH chain, site 3 located at the connection between the intermediate chain and the second SSH chain, site 4 located on the second SSH chain, and site 5 located on the intermediate chain; between adjacent cell bodies, the sites 5 of their respective intermediate chains are connected by a connecting chain.
4. The method for constructing a square root topological circuit for realizing a continuum bound state according to claim 3, characterized in that: The original sites are separated by inserting additional sites on both sides of each original site in each cell body in the tight binding model of the SSH double strand, specifically: On the first SSH chain, sites 6 and 7 are added between sites 1 and 2 and on the side of site 2 away from site 1; on the middle chain, sites 10 and 11 are added on both sides of site 5; On the second SSH chain, sites 8 and 9 are added between sites 3 and 4 and on the side of site 4 away from site 1, respectively.
5. The method for constructing a square root topological circuit for realizing a continuum bound state according to claim 4, characterized in that: Take the square root of the coupling strength. The specific steps are: According to the intracellular and intercellular coupling, the Hamiltonian of the tight binding model of the SSH double chain is obtained: Among them, t1 and t2 represent the intracellular and intercellular coupling of the square root model, t3 is the interchain coupling parameter of the square root model, and k is the momentum; Taking the square root of the Hamiltonian of the tight-binding model of the SSH double chain, we get the Hamiltonian of the tight-binding model of the square root SSH double chain:
6. The method for constructing a square root topological circuit for realizing a continuum bound state according to claim 5, characterized in that: The current equation at the site of the square root topology circuit is specifically: The superscript n indicates the specific unit cell position, n (n=1,2,..,11) represents the external excitation current to the nth node, V n (n=1,2,..,11) represents the response voltage of the corresponding node, C1, C2, C3, C4, C a , C b , C c They represent different capacitance values respectively; different couplings are represented by different capacitance values.
7. A method for analyzing a square root topology circuit, characterized in that: The specific steps include: Constructing a square root topology circuit having N cells by the square root topology circuit construction method according to any one of claims 1 to 6; Calculate the eigenfrequencies and eigenfunctions of the Laplacian operator for square root topology circuits; Analyze the characteristic frequencies and characteristic functions and observe the state of the continuum bound state.
8. The method for analyzing a square root topology circuit according to claim 7, characterized in that: Analyze the characteristic frequency and characteristic function. The specific steps are: Calculate the characteristic frequency and characteristic function of the Laplace operator of the square root topology circuit, and construct the admittance spectrum of the circuit model; construct the eigenvalue spectrum based on the admittance spectrum of the circuit model; Observe the eigenvalue spectrum to confirm the existence and distribution of continuum bound states.
9. The method for analyzing a square root topology circuit according to claim 8, characterized in that: By analyzing the characteristic frequencies and characteristic functions and observing the state of the continuum bound state, the impedance of each node in the circuit is calculated, and a simulation impedance result graph is constructed. The existence and distribution of topological bound states and mediocre bound states are confirmed by observing the simulation impedance result graph.
10. The method for analyzing a square root topology circuit according to claim 9, characterized in that: After confirming the existence of topological bound states and mediocre bound states, the square root topological circuit is further simulated and analyzed. The specific steps are as follows: The square root topology circuit is simulated to obtain the simulated impedance distribution at the topological bound state and the mediocre bound state at a specific frequency; An intermediate coupling capacitor is added to the cell body of the square root topology circuit to transform it from a bound state system to a continuum bound state system; By observing the impedance distribution at this time, the robustness test of the square root topology circuit is performed at this time.