A non-hermitian quasi-periodic circuit simulation and localization characterization method

Abstract

The application discloses a non-Hermite quasi-periodic circuit simulation and localization characterization method, and belongs to the field of linear circuit and non-Hermite physical simulation. A non-reciprocal quasi-periodic circuit model composed of chain circuit nodes, a ground admittance adjustment module and a non-reciprocal coupling module is constructed, so that a node admittance matrix corresponds to a non-Hermite Aubry-André-Harper model; AC excitation is applied to selected nodes, normalized spatial weight, spatial distribution entropy and inverse participation rate are calculated according to complex voltage; pulse excitation is applied to a plurality of initial nodes in sequence, transient responses are aligned to a unified time axis and a dynamic response matrix is formed, and time-varying equivalent density distribution is formed. In combination with the spatial distribution entropy, the inverse participation rate and the propagation direction and the boundary accumulation position of the density distribution, non-Hermite skin localization, Anderson localization and extended state are distinguished, the problem that a single spatial index is difficult to distinguish between the two types of localization is solved, and two types of intensity parameters can be independently adjusted.

Description

Technical Field

[0001] This invention relates to the fields of classical linear circuit simulation, non-reciprocal circuit networks, and non-Hermitian quasi-periodic system characterization. Specifically, it relates to a method and its implementation system for establishing the circuit correspondence of a non-Hermitian Aubry-André-Harper model using chain circuit nodes, quasi-periodic ground admittance, and direction-dependent coupling, and for distinguishing different localization mechanisms by combining AC spatial response and point-by-point transient response. Background Technology

[0002] Anderson localization is a localization phenomenon in vivo caused by disordered or quasi-periodic modulation that inhibits propagation; the non-Hermitian skin effect, on the other hand, is usually caused by non-reciprocal transitions, causing a large number of states under open boundary conditions to cluster towards the same side boundary. In non-Hermitian quasi-periodic systems, these two mechanisms can coexist and compete with each other. The Aubry-André-Harper model can describe the change from extended states to local states in a one-dimensional system using quasi-periodic potentials. Introducing non-reciprocal transitions into this model allows for further research on skin localization, Anderson localization, and the transition between the two.

[0003] Existing experimental research can employ optical structures, ultracold atoms, microwave resonators, acoustic structures, or circuit networks. While these platforms can verify different physical models, optical and ultracold atom platforms typically require precise optical paths, vacuum, or cryogenic conditions, and microwave and acoustic platforms may also require specialized resonant structures. The component parameters and number of nodes in classical linear circuits are relatively easy to adjust, and the node voltages can be directly measured or obtained through circuit simulation, making them suitable for forming scalable non-Hermitian model simulation platforms.

[0004] Chinese invention patent application CN115598749A, published on January 13, 2023, entitled "A Method and System for Topological Flat-Band Control Based on Non-Hermitian Microring Resonator Array," discloses a method that periodically arranges a microring resonator array into a one-dimensional rhombic structure and achieves non-Hermitian topological flat-band control by adjusting the positions and relative gains of the gain loop and loss loop. This scheme employs an optical microring array and focuses on the non-Hermitian flat-band and boundary states. It does not form a circuit network corresponding to the non-Hermitian Aubry-André-Harper model with the directional correlation node coupling of a chain-like linear circuit and the quasi-periodic ground admittance, nor does it establish a joint localization discriminant relationship between the AC space index and the transient response matrix of multiple initial nodes.

[0005] Chinese invention patent application CN117273161A, published on December 22, 2023, entitled "Reconstruction Method and System for Eigenstates of Non-Hermitian Systems," discloses a non-Hermitian system composed of resonant coils, which achieves chiral reversal between different singularities through external excitation and coupling coefficient adjustment. This scheme focuses on the manipulation of singularity eigenstates, but does not configure the ground admittance of the chain nodes according to a quasi-periodic function, nor does it establish a joint discriminant relationship between spatial distribution entropy, inverse participation rate, and the transient response matrix of multiple initial nodes.

[0006] For non-reciprocal quasi-periodic circuits, simply observing whether the node voltage is concentrated at a certain frequency is insufficient to distinguish the source of localization: both non-Hermitian skin effect and Anderson localization may exhibit low spatial expansion, but the former has directional propagation and boundary accumulation towards a fixed open boundary, while the latter mainly exhibits suppressed propagation and is confined to a finite region within the circuit. Single transient excitations are also easily affected by the excitation location and different simulation time sampling points. Therefore, a circuit simulation and localization characterization method is needed that combines steady-state spatial indices and dynamic responses at multiple initial positions under the same circuit mapping relationship, and can separately adjust the non-reciprocal intensity and quasi-periodic modulation intensity. Summary of the Invention

[0007] The present invention aims to solve the problems of complex experimental simulation conditions for non-Hermitian quasi-periodic systems, the inconvenience of independently adjusting non-reciprocal parameters and quasi-periodic modulation parameters, and the difficulty in distinguishing between non-Hermitian skin localization and Anderson localization by relying solely on a single spatial lumped index.

[0008] This invention provides a method for simulating and localizing non-Hermitian quasi-periodic circuits, comprising: establishing a non-reciprocal quasi-periodic circuit model corresponding to a node admittance matrix and a non-Hermitian Aubry-André-Harper model; the non-reciprocal quasi-periodic circuit model includes L chained circuit nodes, each circuit node corresponding to a lattice position and connected to a ground admittance adjustment module; adjacent circuit nodes are connected through two non-reciprocal coupling modules with different effective coupling coefficients in the propagation direction; the grounding capacitance of each ground admittance adjustment module is configured according to a quasi-periodic function; applying AC current excitation to selected circuit nodes, forming normalized spatial weights based on the complex voltage responses of all circuit nodes, and calculating spatial distribution entropy; applying pulse current excitation to multiple initial circuit nodes respectively, mapping each transient voltage response to a unified time axis and forming a dynamic response matrix according to the excitation position; forming a spatial correlation matrix and a time-varying equivalent density distribution based on the dynamic response matrix; and distinguishing between non-Hermitian skin localization and André-Harper localization by combining the spatial distribution entropy and the propagation direction and open boundary accumulation position of the time-varying equivalent density distribution, thereby obtaining localization characterization results.

[0009] Optionally, in the aforementioned method, the Hamiltonian of the non-Hermitian Aubry-André-Harper model is The transition strengths in two opposite directions satisfy , No. The grounding capacitance of each circuit node satisfies ;in, Represents the system Hamiltonian. This represents the total number of lattice positions, which is equal to the total number of circuit nodes. Indicates the lattice position number; in the first summation term, This represents the summation of transition terms between adjacent lattice positions. Indicates the relationship with the first The next lattice position adjacent to each lattice position; in the second summation term, This represents the summation of quasi-periodic in-situ terms over all lattice positions; and These represent the transition intensities in two opposite propagation directions; and They represent the first The and the first The generation operator for each lattice position. and They represent the first The and the first Annihilation operator at each lattice position, Represents the Hermitian conjugate operation; Indicates the reference coupling strength. Indicates non-reciprocal strength. Indicates the quasi-periodic modulation intensity. This represents an irrational number used as a quasi-periodic modulation parameter. Represents the modulation phase constant. Represents pi (π). Represents the cosine function; Indicates the first Equivalent grounding capacitance of each circuit node. Indicates the background grounding capacitance. This indicates the amplitude of the quasi-periodic modulation capacitor; it can be changed by adjusting the capacitor, resistor, or feedback network parameters of the non-reciprocal coupling module. The corresponding degree of non-reciprocity, and according to Sure .

[0010] Optionally, in the aforementioned method, the ground admittance adjustment module includes a ground capacitance branch, a ground inductance branch, and a ground resistance branch connected in parallel between the circuit node and the reference ground. The ground capacitance branch includes at least one fixed capacitor and at least one adjustable or replaceable capacitor. The parallel equivalent capacitance of the ground capacitance branch is the first... The grounding capacitance of each circuit node, and satisfying ;in, Indicates the background grounding capacitance. This indicates the amplitude of the quasi-periodic modulation capacitor. This represents the absolute value of the modulation capacitor amplitude; the grounding inductor branch and the grounding capacitor branch together determine the target response frequency band; the grounding resistor branch is used to limit the voltage increase caused by active non-reciprocal coupling and improve the convergence of the circuit solution.

[0011] Optionally, in the aforementioned method, the non-reciprocal coupling module includes a reference coupling capacitor branch and a current-mode negative impedance converter branch connected in parallel between adjacent circuit nodes. The current-mode negative impedance converter branch includes an operational amplifier, a feedback impedance network connected to the operational amplifier, and a non-reciprocal parameter capacitor connected to the adjacent circuit node. The reference coupling capacitor branch determines the reference coupling strength J. The feedback impedance network and the non-reciprocal parameter capacitor together make the off-diagonal admittances corresponding to the two propagation directions of the adjacent circuit node unequal. The non-reciprocal strength α is adjusted by changing the parameters of the feedback impedance network or the non-reciprocal parameter capacitor.

[0012] Optionally, in the aforementioned method, the AC measurement mode for applying AC current excitation to selected circuit nodes includes: selecting circuit nodes in or near the middle of the chain as AC excitation nodes, performing a pre-scan within a preset wide frequency range, performing a fine scan after determining the target frequency band based on the resonant response of the non-reciprocal quasi-periodic circuit model, or performing fixed-frequency analysis at the target frequency; recording the real and imaginary parts of the complex voltage response of each circuit node in Cartesian form, and maintaining consistency in excitation amplitude, AC excitation nodes, acquisition nodes, and data format under different non-reciprocal intensity α and quasi-periodic modulation intensity λ.

[0013] Optionally, in the aforementioned method, for a given non-reciprocal strength Quasi-periodic modulation intensity and angular frequency The complex voltage response is used to form a spatial response vector. and in accordance with Calculate the first The normalized spatial weights of each circuit node, according to Calculate the spatial distribution entropy, according to Calculate the inverse participation rate; where, Indicates non-reciprocal strength. Indicates the quasi-periodic modulation intensity. Represents the angular frequency of the AC excitation; This represents the spatial response column vector composed of the complex voltages of all circuit nodes, ordered by node number. Indicates the first Complex voltages at each circuit node In the process of summation, the first... Complex voltages at each circuit node; This represents the node number from which the normalized spatial weights are to be calculated. This indicates the summation index in the denominator. , Indicates the total number of circuit nodes. Indicates the transpose operation; and This represents the modulus of the corresponding complex voltage. and This represents the square of the modulus of the corresponding complex voltage; Indicates the first Normalized spatial weights of each circuit node, Represents the spatial distribution entropy. Represents the natural logarithm. Indicates the inverse participation rate. Represents the normalized space weights The square of the product of the normalized spatial weights and their natural logarithm is treated as the limit value of zero when the normalized spatial weights are zero. The corresponding frequency points are excluded when the normalized denominator is not greater than the lower limit of numerical precision. Scanning and The spatial distribution entropy phase diagram and the inverse participation rate phase diagram are obtained.

[0014] Optionally, in the aforementioned method, the transient measurement mode of successively applying pulse current excitation to multiple initial circuit nodes includes: from Select from each circuit node The initial circuit node, for the first Each initial circuit node is connected to only one pulse current source, and the transient voltage response of each circuit node is recorded. After completing one transient analysis, the pulse current source is switched to the next initial circuit node, until the analysis is complete. The transient analysis is performed point-by-point; a unified time series is established within the common effective time range of each transient analysis, and the transient voltage response is mapped to the unified time axis by interpolation to form a response column vector. And form a dynamic response matrix. ;in, Indicates the total number of circuit nodes. Indicates the number of independent stimulus positions. Indicates the excitation sequence number of the initial circuit node and , Indicates the circuit node number and , Represents a point in time on a unified timeline; Indicates the first Under the second incentive Each circuit node at time... transient voltage, Indicates the first The column vector of total node responses formed when each initial circuit node is excited. Indicates the transpose operation; Indicates by The dynamic response matrix, composed of response columns arranged in the order of excitation, has a dimension of . .

[0015] Optionally, in the aforementioned method, each response column of the dynamic response matrix is ​​normalized using the second norm to form a normalized response matrix; when it is necessary to maintain the linear independence of the response subspace, the normalized response matrix is ​​subjected to QR orthogonalization and taken as the normalized response matrix. Zhongyu The columns corresponding to the initial circuit nodes; the normalized response matrix or the The corresponding columns in the matrix are used as the processed response matrix, according to... Form a spatial correlation matrix, and with As the first The time-varying equivalent density of each circuit node, according to Forming the centroid trajectory, according to Form a set of boundary nodes The boundary weights; where, Indicates the number of independent stimulus positions. This represents the orthogonal factor matrix obtained by QR orthogonalization; Indicates time The processed response matrix, This represents the conjugate transpose of the processed response matrix. This represents the conjugate transpose operation; Represents the spatial incidence matrix. Represents the first... One diagonal element; Indicates the first Each circuit node at time... The time-varying equivalent density, The location of the centroid of the time-varying equivalent density; This represents a pre-selected set of boundary nodes. This means summing only by traversing the set. Nodes within, Indicates boundary weights; Indicates the node sequence number and , Indicates the total number of circuit nodes. It represents a point in time on a unified time axis; under the same system size and measurement configuration, when the spatial distribution entropy is low, the boundary weight is high, and the centroid moves toward the same open boundary, it is determined to be non-Hermitian skin localization; when the spatial distribution entropy is low and the time-varying equivalent density is limited to a finite region within the body, it is determined to be Anderson localization; when the spatial distribution entropy is high and the inverse participation rate is low, it is determined to be an extended state.

[0016] This invention also provides a simulation and localization system for non-Hermitian quasi-periodic circuits, including a non-reciprocal quasi-periodic circuit network, an excitation and acquisition unit, and a data processing unit. The non-reciprocal quasi-periodic circuit network includes L chained circuit nodes, a ground admittance adjustment module connecting each circuit node to a reference ground, and a non-reciprocal coupling module connecting adjacent circuit nodes. The ground admittance adjustment module configures grounding capacitors according to a quasi-periodic function, and the non-reciprocal coupling module makes the effective coupling coefficients different in the two propagation directions, so that the node admittance matrix corresponds to a non-Hermitian Aubry-André-Harper model. The excitation and acquisition unit is used to apply AC current excitation to selected circuit nodes in AC measurement mode and acquire the complex voltage response of each circuit node, and to apply pulse current excitation to multiple initial circuit nodes in transient measurement mode and acquire the transient voltage response of each circuit node. The data processing unit is used to calculate the spatial distribution entropy based on the complex voltage response, align the transient voltage response to a unified time axis to form a dynamic response matrix and a time-varying equivalent density distribution, and output the localized characterization result based on the spatial distribution entropy and the propagation direction and open boundary accumulation position of the time-varying equivalent density distribution.

[0017] Optionally, in the aforementioned system scheme, the excitation and acquisition unit includes an AC current source, a pulse current source, a node selection switch network, and a voltage acquisition interface. The node selection switch network is used to connect the AC current source to circuit nodes in or near the middle of the chain, and to connect the pulse current source to multiple initial circuit nodes in sequence. The non-reciprocal quasi-periodic circuit network adopts an open boundary, and L is configured as a number in the Fibonacci sequence of 13, 21, 34, or 55. The data processing unit is also used to calculate the inverse participation rate, boundary weights, and centroid trajectory, and output a spatial distribution entropy phase diagram and a time-varying equivalent density distribution heatmap.

[0018] Compared with the prior art, the present invention realizes non-reciprocal coupling and quasi-periodic grounding admittance as adjustable circuit parameters by the correspondence between the node admittance matrix and the non-Hermitian Aubry-André-Harper model, so that the two types of effects can be changed separately in the same node chain.

[0019] This invention uses AC complex voltage to calculate spatial distribution entropy and inverse participation rate, which quantitatively reflect the spatial expansion of the node response. It also constructs a dynamic response matrix, spatial correlation matrix, and time-varying equivalent density distribution from the point-by-point impulse responses of multiple initial circuit nodes, reflecting the propagation direction, local location within the circuit, and open-boundary accumulation process. Combining these two methods reduces the likelihood of making the same judgment about two types of localization based solely on low spatial expansion.

[0020] This invention uses a point-by-point rather than simultaneous application of multiple pulses to obtain the response at different input positions, and maps each set of data under the adaptive time step to a unified time axis, which can reduce input control interference caused by multiple sources working at the same time, and enable the responses at different excitation positions to be combined and compared in columns.

[0021] The ground admittance adjustment module and the non-reciprocal coupling module of the present invention can be repeatedly set along the chain nodes. The number of nodes, boundary conditions and scanning parameters can be configured according to research needs, which facilitates the formation of spatial distribution entropy phase diagrams and time-varying equivalent density distribution heat maps under different system sizes. Attached Figure Description

[0022] Figure 1 This is a schematic diagram of the chain connection of a multi-node non-reciprocal quasi-periodic circuit.

[0023] Figure 2 A flowchart for the dual-mode measurement of spatial distribution entropy and transient density distribution.

[0024] Figure 3 This is a schematic diagram of a ground admittance adjustment module for a single node.

[0025] Figure 4 This is a circuit diagram of a non-reciprocal coupling module between adjacent nodes. Detailed Implementation

[0026] The term "non-reciprocal" as used below refers to the fact that the effective coupling coefficient between adjacent circuit nodes is different when the signal is transmitted from one direction to another and when the signal is transmitted in the opposite direction; "quasi-periodic" refers to the fact that the node grounding parameter changes with the node number according to a modulation relationship that does not form a short-period repetition with the finite node chain; "open boundary" refers to the fact that the beginning and end of the node chain are not closed to each other. Figure 2 In this context, skin represents non-Hermitian skin localization, AL represents Anderson localization, and IPR represents inverse participation rate.

[0027] Non-reciprocal quasi-periodic circuit models can be formed either from physical linear circuits or from SPICE-compatible circuit simulation netlists with the same topology and component parameters as physical circuits. In physical circuit implementation, AC current sources, pulse current sources, and multi-channel acquisition devices handle excitation and measurement functions, respectively; in simulation implementation, the circuit solver outputs the complex or transient voltages of each circuit node. Both implementations use the same nodes, branches, parameters, and data processing relationships.

[0028] Example 1 This embodiment provides a method for simulating and localizing non-Hermitian quasi-periodic circuits, including: establishing a non-reciprocal quasi-periodic circuit model corresponding to a node admittance matrix and a non-Hermitian Aubry-André-Harper model. The non-reciprocal quasi-periodic circuit model includes L chained circuit nodes, each circuit node corresponding to a lattice position and connected to a ground admittance adjustment module. Adjacent circuit nodes are connected through two non-reciprocal coupling modules with different effective coupling coefficients in the propagation direction. The grounding capacitance of each ground admittance adjustment module is configured according to a quasi-periodic function. AC current excitation is applied to selected circuit nodes, and normalized spatial weights are formed based on the complex voltage responses of all circuit nodes, and spatial distribution entropy is calculated. Pulse current excitation is applied to multiple initial circuit nodes respectively, and each transient voltage response is mapped to a unified time axis and a dynamic response matrix is ​​formed according to the excitation position. A spatial correlation matrix and a time-varying equivalent density distribution are formed based on the dynamic response matrix. Combining the spatial distribution entropy and the propagation direction and open boundary accumulation position of the time-varying equivalent density distribution, non-Hermitian skin localization and André-Harper localization are distinguished to obtain localization characterization results.

[0029] like Figure 1 As shown, the non-reciprocal quasi-periodic circuit model has multiple circuit nodes arranged along one spatial direction. Figure 1 The diagram illustrates the connection configuration of several consecutive nodes. Each circuit node is connected downwards to a ground admittance adjustment module. Reference coupling branches and non-reciprocal coupling branches are connected in parallel between adjacent circuit nodes, and open boundaries are used at both ends of the node chain. Figure 1 The component reference designator in the netlist is used to identify specific components, and the symbols within curly braces represent the specific values ​​of parameterizable components.

[0030] At angular frequency ω, the node injection current, the node admittance matrix, and the node complex voltage satisfy the following equation: (1) In equation (1), Represents the angular frequency of the AC excitation. Indicates the total number of circuit nodes; Represents angular frequency The following is composed of the current injected into each circuit node. Column vectors It represents the combination of ground admittance and inter-node coupling admittance. Nodal admittance matrix Represents angular frequency The following is composed of complex voltages at each circuit node. Column vectors. The adjacent off-diagonal elements of the nodal admittance matrix are determined by the coupling branches between adjacent nodes, while the main diagonal elements include admittances generated by connections to adjacent nodes and ground admittances. By making the adjacent off-diagonal elements unequal in both directions and by making the main diagonal elements vary with the node index according to a quasi-periodic function, a non-Hermitian Aubry-André-Harper model can be mapped within the allowable range of the common scaling factor and frequency-dependent diagonal bias.

[0031] like Figure 2 As shown, the AC measurement mode starts with a single-point AC excitation, acquires the complex voltages of all circuit nodes through AC frequency sweep or fixed-frequency analysis, constructs the normalized voltage intensity, and calculates the spatial distribution entropy. The transient measurement mode applies pulse excitations to multiple target nodes successively, constructs the node-excitation response matrix after transient analysis and time axis unification, and calculates the time-varying equivalent density distribution. The AC measurement mode provides the spatial expansion degree, while the transient measurement mode provides the direction and positional changes of the response propagating from the initial circuit node to other circuit nodes.

[0032] When the total voltage intensity used for AC response normalization is lower than the numerical accuracy limit of the circuit solver, the spatial index is not output for that frequency point; instead, the adjacent effective frequency point is returned or the excitation frequency is adjusted. When a pulse excitation does not form a usable response within a common effective time range, the data is marked as invalid and reacquired, and the zero column is not used in the response matrix calculation. Through the above data boundary processing, the misinterpretation of numerical zero values ​​as strong localization can be avoided. In this embodiment, complementary information is formed by two types of responses from the same circuit, thereby establishing a complete input, processing, and output link for localization representation.

[0033] Example 2 Based on Example 1, this example further specifies that the Hamiltonian of the non-Hermitian Aubry-André-Harper model is The transition strengths in two opposite directions satisfy , No. The grounding capacitance of each circuit node satisfies ;in, Represents the system Hamiltonian. This represents the total number of lattice positions, which is equal to the total number of circuit nodes. Indicates the lattice position number; in the first summation term, This represents the summation of transition terms between adjacent lattice positions. Indicates the relationship with the first The next lattice position adjacent to each lattice position; in the second summation term, This represents the summation of quasi-periodic in-situ terms over all lattice positions; and These represent the transition intensities in two opposite propagation directions; and They represent the first The and the first The generation operator for each lattice position. and They represent the first The and the first Annihilation operator at each lattice position, Represents the Hermitian conjugate operation; Indicates the reference coupling strength. Indicates non-reciprocal strength. Indicates the quasi-periodic modulation intensity. This represents an irrational number used as a quasi-periodic modulation parameter. Represents the modulation phase constant. Represents pi (π). Represents the cosine function; Indicates the first Equivalent grounding capacitance of each circuit node. Indicates the background grounding capacitance. This indicates the amplitude of the quasi-periodic modulation capacitor; it can be changed by adjusting the capacitor, resistor, or feedback network parameters of the non-reciprocal coupling module. The corresponding degree of non-reciprocity, and according to Sure .

[0034] The non-Hermitian Aubry-André-Harper model used in this embodiment is written as follows: (2) In equation (2), Represents the system Hamiltonian. This represents the total number of lattice positions, which is equal to the total number of circuit nodes. Indicates the lattice position number; the index range of the first summation term. Corresponding to the transition terms between all adjacent lattice positions, Indicates the relationship with the first The next lattice position adjacent to the first lattice position; the index range of the second summation term. Quasi-periodic in-situ terms corresponding to all lattice positions; and These represent the transition intensities in two opposite propagation directions; and They represent the first The and the first The generation operator for each lattice position. and They represent the first The and the first Annihilation operator at each lattice position, Represents the Hermitian conjugate operation; Indicates the quasi-periodic modulation intensity. This represents an irrational number used as a quasi-periodic modulation parameter. Represents the modulation phase constant. Represents pi (π). This represents the cosine function. The left and right transition strengths are set using the following formula: (3) In equation (3), and These represent the transition intensities of the two opposite propagation directions in equation (2), respectively. Indicates the reference coupling strength. Indicates non-reciprocal strength. This represents an equal, opposite offset applied in both directions relative to half the reference coupling strength. When At that time, the transition strength in both directions is half of the reference coupling strength; when When the transition intensities are different in the two directions, Equation (3) gives the transition intensities according to the defined propagation directions, and the circuit admittance and transition intensities are related by a common scaling factor.

[0035] The grounding capacitor of the j-th circuit node is configured as follows: (4) In equation (4), Indicates the first Equivalent grounding capacitance of each circuit node. Indicates the circuit node number and , Indicates the total number of circuit nodes; This represents the background grounding capacitor used to ensure that the grounding capacitance of each node is positive. This indicates the amplitude of the quasi-periodic modulation capacitor. This represents an irrational number used as a quasi-periodic modulation parameter. Represents the modulation phase constant. Represents pi (π). Represents the cosine function. The quasi-periodic modulation intensity in equation (2) is expressed as follows. For a given... , and First, calculate the quasi-periodic function value corresponding to each node, and then determine the value based on the proportional relationship between the circuit admittance and the model in the potential. and the grounding capacitance at each node. Change Without altering the non-reciprocal coupling structure between adjacent nodes, the equivalent quasi-periodic modulation intensity can be changed; by changing the parameters of the non-reciprocal coupling module while keeping the grounding capacitor array of each node unchanged, the equivalent non-reciprocal intensity can be altered. This allows for scanning of both types of parameters separately within the same node chain.

[0036] Example 3 Based on Embodiment 2, this embodiment further specifies that: the ground admittance adjustment module includes a ground capacitor branch, a ground inductance branch, and a ground resistance branch connected in parallel between the circuit node and the reference ground. The ground capacitor branch includes at least one fixed capacitor and at least one adjustable or replaceable capacitor. The parallel equivalent capacitance of the ground capacitor branch is the first... The grounding capacitance of each circuit node, and satisfying ;in, Indicates the background grounding capacitance. This indicates the amplitude of the quasi-periodic modulation capacitor. This represents the absolute value of the modulation capacitor amplitude; the grounding inductor branch and the grounding capacitor branch together determine the target response frequency band; the grounding resistor branch is used to limit the voltage increase caused by active non-reciprocal coupling and improve the convergence of the circuit solution.

[0037] like Figure 3 As shown, a parallel branch is set between a single circuit node and the reference ground. In the figure, C206 and C186 represent the capacitor elements in the grounding capacitor branch of this node, L33 represents the grounding inductor element, and R89 represents the grounding resistor element; Cd, L0, C34, and R in curly braces are parameter names in the circuit netlist. The parallel capacitors in the figure together form the equivalent grounding capacitance of this node. When the node number changes, the capacitor parameters related to the node are replaced with the configuration values ​​of the corresponding node.

[0038] To ensure that the grounding capacitance of all nodes remains positive, the amplitudes of the background grounding capacitance and the quasi-periodic modulation capacitance must satisfy: (5) In equation (5), Indicates the background grounding capacitance. This indicates the amplitude of the quasi-periodic modulation capacitor. express The absolute value of the grounding capacitance is given by this formula, which provides a conservative condition for ensuring that the grounding capacitance of each node is positive within the range of cosine modulation values. The grounding inductance branch and the grounding capacitance branch together define the resonant and effective response frequency bands of the circuit; the grounding resistance branch provides a dissipation path for each node, which is used to suppress the unbounded voltage growth caused by active non-reciprocal branches and improve the convergence of AC and transient solutions. The value of the grounding resistance is selected so as not to mask the spatial distribution differences caused by inter-node coupling and to maintain a stable solution.

[0039] In a specific configuration, the node chain uses 34 circuit nodes, numbered n1 to n34 sequentially. The grounding capacitance of each node is calculated according to equation (4) and written into the corresponding netlist parameters. The number of nodes can also be configured to 13, 21, 55, or other values ​​that meet the model research scale. When changing the number of nodes, the branch type and quasi-periodic configuration rules of the grounding admittance adjustment module remain unchanged. In this embodiment, quasi-periodic in-situ modulation is implemented as replaceable or adjustable capacitance parameters through modular grounding branches.

[0040] Example 4 Based on Embodiment 2, this embodiment further specifies that: the non-reciprocal coupling module includes a reference coupling capacitor branch and a current-type negative impedance converter branch connected in parallel between adjacent circuit nodes. The current-type negative impedance converter branch includes an operational amplifier, a feedback impedance network connected to the operational amplifier, and a non-reciprocal parameter capacitor connected to the adjacent circuit node. The reference coupling capacitor branch determines the reference coupling strength J. The feedback impedance network and the non-reciprocal parameter capacitor together make the off-diagonal admittances corresponding to the two propagation directions of the adjacent circuit node unequal. The non-reciprocal strength α is adjusted by changing the parameters of the feedback impedance network or the non-reciprocal parameter capacitor.

[0041] like Figure 4 As shown, the non-reciprocal coupling module between adjacent circuit nodes includes the upper reference coupling capacitor C197 and the lower current-mode negative impedance converter branch. The current-mode negative impedance converter branch consists of operational amplifier U34, feedback resistors R67 and R68, feedback capacitors C183 and C182, and non-reciprocal parameter capacitor C181. In the figure, the resistance values ​​of feedback resistors R67 and R68 are both 8 kΩ, and the capacitance values ​​of feedback capacitors C183 and C182 are both 0.6 μF. C0 and inicc in the figure are circuit netlist parameter names used to configure the reference coupling branch and the non-reciprocal branch, respectively.

[0042] The reference coupling capacitor produces the same reciprocal admittance component in both propagation directions. The current-mode negative impedance converter branch changes the sign or magnitude of the equivalent admittance observed from one node according to the feedback connection of the operational amplifier, thereby making the two adjacent off-diagonal elements in the node admittance matrix unequal. When adjusting the feedback resistor, feedback capacitor, or non-reciprocal parameter capacitor, the equivalent coupling admittance in both directions is first obtained in the target response frequency band, and then the non-reciprocal intensity α is corresponding to the proportional relationship of Equation (3).

[0043] When implementing the physical circuit, the operational amplifier's supply voltage, closed-loop bandwidth, and output swing should cover the target response frequency band and node voltage range; during simulation implementation, the operational amplifier model and power supply ports should be consistent with... Figure 4 Consistency. If the scanning parameters cause the output to saturate or lead to numerical divergence, the excitation amplitude is reduced or the non-reciprocal branch value at that parameter point is decreased, and the admittance relationship in both directions is re-verified. In this embodiment, the directional correlation coupling can be continuously changed while keeping the number of nodes constant, thereby allowing the non-reciprocal action and quasi-periodic grounding modulation to be controlled separately.

[0044] Example 5 Based on Example 2, this example further defines the AC measurement mode for applying AC current excitation to selected circuit nodes as follows: selecting circuit nodes in or near the middle of the chain as AC excitation nodes, performing a pre-scan within a preset wide frequency range, performing a fine scan after determining the target frequency band based on the resonant response of the non-reciprocal quasi-periodic circuit model, or performing fixed-frequency analysis at the target frequency; recording the real and imaginary parts of the complex voltage response of each circuit node in Cartesian form, and maintaining consistency in excitation amplitude, AC excitation nodes, acquisition nodes, and data format under different non-reciprocal intensity α and quasi-periodic modulation intensity λ.

[0045] In a 34-node configuration, n17 near the middle of the chain can be selected as the AC excitation node to reduce the additional bias of the excitation position on the left and right spatial distribution. First, perform an AC pre-scan over a wide frequency range to observe the complex voltage response and overall resonant region of all circuit nodes; then, select a target frequency band containing effective resonant responses without operational amplifier saturation or solution anomalies for fine scanning, or select a target frequency for fixed-frequency analysis.

[0046] For each group of non-reciprocal intensity and quasi-periodic modulation intensity, the amplitude, phase, and connection node of the AC current source remain consistent, and all circuit nodes serve as acquisition nodes. The output data is in Cartesian form, storing the real and imaginary parts of the complex voltage at each node, preserving not only the decibel amplitude but also the information needed to calculate the voltage magnitude square and analyze phase relationships when necessary. Different parameter groups use the same node order, frequency order, and field names.

[0047] like Figure 2As shown in the flowchart on the left, after AC excitation, frequency sweep or fixed-frequency analysis, full-node complex voltage acquisition, normalized voltage intensity construction, and spatial distribution entropy and inverse participation rate calculation are performed sequentially. If the total voltage intensity at a certain frequency point is zero or lower than the predetermined numerical accuracy limit, that frequency point is not included in the normalization calculation, and an invalid mark is retained in the results. This embodiment, by fixing the excitation and data format, provides a comparable measurement benchmark for spatial indices under different parameters.

[0048] Example 6 Based on Example 5, this example further specifies: for a given non-reciprocal strength Quasi-periodic modulation intensity and angular frequency The complex voltage response is used to form a spatial response vector. and in accordance with Calculate the first The normalized spatial weights of each circuit node, according to Calculate the spatial distribution entropy, according to Calculate the inverse participation rate; where, Indicates non-reciprocal strength. Indicates the quasi-periodic modulation intensity. Represents the angular frequency of the AC excitation; This represents the spatial response column vector composed of the complex voltages of all circuit nodes, ordered by node number. Indicates the first Complex voltages at each circuit node In the process of summation, the first... Complex voltages at each circuit node; This represents the node number from which the normalized spatial weights are to be calculated. This indicates the summation index in the denominator. , Indicates the total number of circuit nodes. Indicates the transpose operation; and This represents the modulus of the corresponding complex voltage. and This represents the square of the modulus of the corresponding complex voltage; Indicates the first Normalized spatial weights of each circuit node, Represents the spatial distribution entropy. Represents the natural logarithm. Indicates the inverse participation rate. Represents the normalized space weights The square of the product of the normalized spatial weights and their natural logarithm is treated as the limit value of zero when the normalized spatial weights are zero. The corresponding frequency points are excluded when the normalized denominator is not greater than the lower limit of numerical precision. Scanning and The spatial distribution entropy phase diagram and the inverse participation rate phase diagram are obtained.

[0049] For a given non-reciprocal intensity, quasi-periodic modulation intensity, and angular frequency, construct a spatial response vector from the complex voltage responses of the L circuit nodes according to their node indices: (6) In equation (6), Indicates at angular frequency The following is from The spatial response column vector is formed by arranging the complex voltages of each circuit node according to the node number. Represents the angular frequency of the AC excitation. Indicates the total number of circuit nodes; Indicates the first Each circuit node at angular frequency Complex voltage under, Indicates the node sequence number and , This indicates the transpose operation. To reduce the impact of differences in excitation amplitude and overall gain on the spatial distribution criterion, the normalized spatial weights are calculated using the following formula: (7) In equation (7), Represents angular frequency Next Normalized spatial weights of each circuit node, Represents the angular frequency of the AC excitation. Indicates the index of the node to be calculated and ; Indicates the first Complex voltages at each node, This represents the square of the magnitude of the complex voltage; Indicates the summation index in the denominator and , Indicates the first Complex voltages at each node, This represents the square of the magnitude of the complex voltage; The denominator represents the total number of circuit nodes. This represents the sum of the squares of the complex voltage moduli at all nodes at that frequency. All at each effective frequency... The sum of them is Calculate the spatial distribution entropy based on the normalized spatial weights: (8) In equation (8), Represents angular frequency The spatial distribution entropy under the following conditions Represents the angular frequency of the AC excitation; The first expression (7) defines the first Normalized spatial weights for each node Indicates the summation index and , Indicates the total number of circuit nodes. Represents the natural logarithm. When At that time, the corresponding According to the limit value Processing. Lower spatial distribution entropy indicates that the response is concentrated in a few nodes; higher spatial distribution entropy indicates that the response is distributed across more nodes. To further reflect the number of effective nodes participating in the response, the inverse participation rate is calculated: (9) In equation (9), Represents angular frequency The inverse participation rate below Represents the angular frequency of the AC excitation; Indicates the first Normalized spatial weights of each node, express The square of, Indicates the summation index and , This represents the total number of circuit nodes. When the normalized spatial weights are concentrated, the inverse participation rate is relatively high; when the normalized spatial weights are dispersed, the inverse participation rate is relatively low. By scanning the periodic modulation intensity point by point, the spatial distribution entropy curve and the inverse participation rate curve can be obtained. Repeating the scan for non-reciprocal intensity yields the spatial distribution entropy phase diagram and the inverse participation rate phase diagram in the two-dimensional parameter space. Furthermore, the number of nodes can be changed under the same scanning rules to compare the changes in indicators for different system sizes. This embodiment uses two complementary spatial indicators to reduce the sensitivity of a single indicator to the response of abnormal nodes.

[0050] Example 7 Based on Example 6, this example further specifies that the transient measurement mode of applying pulse current excitation to multiple initial circuit nodes successively includes: from Select from each circuit node The initial circuit node, for the first Each initial circuit node is connected to only one pulse current source, and the transient voltage response of each circuit node is recorded. After completing one transient analysis, the pulse current source is switched to the next initial circuit node, until the analysis is complete. The transient analysis is performed point-by-point; a unified time series is established within the common effective time range of each transient analysis, and the transient voltage response is mapped to the unified time axis by interpolation to form a response column vector. And form a dynamic response matrix. ;in, Indicates the total number of circuit nodes. Indicates the number of independent stimulus positions. Indicates the excitation sequence number of the initial circuit node and , Indicates the circuit node number and , Represents a point in time on a unified timeline; Indicates the first Under the second incentive Each circuit node at time... transient voltage, Indicates the first The column vector of total node responses formed when each initial circuit node is excited. Indicates the transpose operation; Indicates by The dynamic response matrix, composed of response columns arranged in the order of excitation, has a dimension of . .

[0051] like Figure 2 As shown in the flowchart on the right, transient density distribution measurement applies pulse excitation to multiple target nodes sequentially, rather than connecting multiple pulse sources simultaneously in the same measurement. For the m-th initial circuit node, a pulse current source is connected only to that node, while other circuit structures, component parameters, simulation termination time, and acquisition nodes remain unchanged. After completing one transient analysis, the pulse current source is switched to the next initial circuit node. For initial configurations requiring alternating occupancy, even-numbered nodes can be selected sequentially as initial circuit nodes.

[0052] One feasible pulse source configuration is PULSE(0, 0.01, 0, 1μs, 1μs, 10ms), where each value is used only to provide a specific short pulse input. The actual pulse amplitude, rise time, fall time, pulse width, and simulation time are set according to the target response frequency band and the circuit stability range. The data file for each transient analysis must contain at least a time column and voltage columns for L circuit nodes.

[0053] Since the circuit solver can use an adaptive time step, the sampling times of each excitation output may be inconsistent. Therefore, a monotonically increasing unified time series is established within the common valid time range of all data files, and interpolation is used to map the voltages of each node to the same time point. For any time t, the total node response generated by the m-th initial circuit node is written as: (10) In equation (10), Indicates the first When the initial circuit nodes are excited at a unified time axis The resulting full node response column vector, Indicates the excitation sequence number of the initial node and , Indicates the number of independent stimulus positions. Represents a point in time on a unified timeline; Indicates the first Under the second incentive Each circuit node at time... transient voltage, Indicates the node sequence number and , Indicates the total number of circuit nodes. This represents the transpose operation. The response columns corresponding to the initial circuit nodes are combined in the order of excitation to obtain: (11) In equation (11), Indicates time The dynamic response matrix is ​​composed of the response columns arranged in the order of excitation. The first expression (10) defines the first One response column, Indicates the response column sequence number and , Indicates the number of independent stimulus positions. Indicates the total number of circuit nodes. Represents a point in time on a unified timeline; The dimension is Each column retains propagation information from one input position to all output nodes. Point-by-point excitation avoids amplitude and phase control problems when multiple sources are working simultaneously, and can utilize the superposition relationship of linear circuits to form a multi-initial-position response subspace during the data processing stage.

[0054] Example 8 Based on Example 7, this example further specifies: L2 norm normalization is performed on each response column of the dynamic response matrix to form a normalized response matrix; when it is necessary to maintain the linear independence of the response subspace, QR orthogonalization is performed on the normalized response matrix and... Zhongyu The columns corresponding to the initial circuit nodes; the normalized response matrix or the The corresponding columns in the matrix are used as the processed response matrix, according to... Form a spatial correlation matrix, and with As the first The time-varying equivalent density of each circuit node, according to Forming the centroid trajectory, according to Form a set of boundary nodes The boundary weights; where, Indicates the number of independent stimulus positions. This represents the orthogonal factor matrix obtained by QR orthogonalization; Indicates time The processed response matrix, This represents the conjugate transpose of the processed response matrix. This represents the conjugate transpose operation; Represents the spatial incidence matrix. Represents the first... One diagonal element; Indicates the first Each circuit node at time... The time-varying equivalent density, The location of the centroid of the time-varying equivalent density; This represents a pre-selected set of boundary nodes. This means summing only by traversing the set. Nodes within, Indicates boundary weights; Indicates the node sequence number and , Indicates the total number of circuit nodes. It represents a point in time on a unified time axis; under the same system size and measurement configuration, when the spatial distribution entropy is low, the boundary weight is high, and the centroid moves toward the same open boundary, it is determined to be non-Hermitian skin localization; when the spatial distribution entropy is low and the time-varying equivalent density is limited to a finite region within the body, it is determined to be Anderson localization; when the spatial distribution entropy is high and the inverse participation rate is low, it is determined to be an extended state.

[0055] To reduce the overall amplitude difference of the response columns corresponding to different initial circuit nodes, each effective response column is normalized according to the L2 norm: (12) In equation (12), Indicates the first A normalized response column, The first expression (10) defines the first One original response column, Indicates the response column sequence number and , Indicates the number of independent stimulus positions. Represents a point in time on a unified timeline; Represents response column The 2-norm, also known as the Euclidean norm, is indicated by the subscript. Used to identify the L2 norm. When If the value is below the lower limit of numerical precision, the response column will not be included in subsequent calculations and will be retrieved again. The normalized response matrix is ​​constructed; when it is necessary to construct a linearly independent response subspace, the normalized response matrix is ​​orthogonalized using QR methods. (13) In equation (13), This indicates that the expression derived from equation (12) A normalized response matrix consisting of normalized response columns. This represents the orthogonal factor matrix obtained from QR decomposition, whose column vectors form an orthogonal basis. This represents the upper triangular factor matrix obtained from QR decomposition. Represents a point in time on a unified timeline; Indicates the total number of circuit nodes. Indicates the number of independent stimulus positions. and The dimension is , The dimension is If orthogonalization is performed, then take... Zhongyu The columns corresponding to the initial circuit nodes are used as the processed response matrix; if orthogonalization is not performed, then directly... As the processed response matrix. Represent the processed response matrix and construct the spatial correlation matrix: (14) In equation (14), Indicates time Spatial correlation matrix, This represents the processed response matrix after column normalization and optional QR orthogonalization. express The conjugate transpose of . This represents the conjugate transpose operation. Represents a point in time on a unified timeline; Indicates the total number of circuit nodes. Indicates the number of independent stimulus positions. The dimension is , The dimension is , The dimension is The diagonal elements of the spatial incidence matrix are defined as the time-varying equivalent density: (15) In equation (15), Indicates the first Each circuit node at time... The time-varying equivalent density, Representing the spatial incidence matrix The diagonal elements, Indicates the node sequence number and , Indicates the total number of circuit nodes. This represents a point in time on a unified time axis. Using the node number as spatial coordinates, the centroid trajectory can be calculated. (16) In equation (16), Indicates time The time-varying equivalent density centroid location, Indicates the node number as a spatial coordinate and , Indicates the first Each node at time... The time-varying equivalent density, Indicates the total number of circuit nodes. Indicates a point in time on a unified timeline; molecule The denominator represents the sum of time-varying equivalent densities weighted by node indices. This represents the sum of the time-varying equivalent densities of all nodes. Then, a set of boundary nodes is pre-selected, consisting of consecutive nodes close to either the left-open or right-open boundary. And maintain the set in all parameter groups. The boundary weights remain unchanged and are calculated using the following formula: (17) In equation (17), Indicates time Boundary weights, This represents a pre-selected set of boundary nodes that remains unchanged across all parameter groups. The summation in the numerator only involves traversing the set. Nodes within; Indicates the first Each node at time... The time-varying equivalent density, Indicates the node sequence number. Indicates the total number of circuit nodes. Indicates a point in time on a unified timeline; molecule The denominator represents the sum of time-varying equivalent densities within the set of boundary nodes. This represents the sum of the time-varying equivalent densities of all nodes. If the denominator is... If the value is below the lower limit of numerical precision, the centroid and boundary weights will not be output at that moment. The time-varying equivalent density is plotted as a heatmap according to the node number and time. The centroid trajectory is superimposed on the heatmap or output separately.

[0056] Joint discrimination is performed under the same system size, excitation settings, and data processing configuration. When the spatial distribution entropy is low, the boundary weights are relatively high, and the centroid continuously moves towards the same open boundary, it indicates that the response has directional propagation and boundary accumulation, which is characterized as non-Hermitian skin localization. When the spatial distribution entropy is low, the time-varying equivalent density is mainly confined to a finite region within the volume, and there is no continuous unidirectional boundary accumulation, it is characterized as Anderson localization. When the spatial distribution entropy is relatively high and the inverse participation rate is relatively low, it is characterized as an extended state. The comparison quantity uses the relative change in the same parameter scan. This embodiment combines the degree of spatial concentration and dynamic direction information to distinguish between two sources of localization that both manifest as spatial response concentration.

[0057] Example 9 This embodiment provides a simulation and localization system for non-Hermitian quasi-periodic circuits, including a non-reciprocal quasi-periodic circuit network, an excitation and acquisition unit, and a data processing unit. The non-reciprocal quasi-periodic circuit network includes L chained circuit nodes, a ground admittance adjustment module connecting each circuit node to a reference ground, and a non-reciprocal coupling module connecting adjacent circuit nodes. The ground admittance adjustment module configures the grounding capacitor according to a quasi-periodic function, and the non-reciprocal coupling module makes the effective coupling coefficients of the two propagation directions different, so that the node admittance matrix corresponds to a non-Hermitian Aubry-André-Harper model. The excitation and acquisition unit is used to apply AC current excitation to selected circuit nodes in AC measurement mode and acquire the complex voltage response of each circuit node, and to apply pulse current excitation to multiple initial circuit nodes in transient measurement mode and acquire the transient voltage response of each circuit node. The data processing unit is used to calculate the spatial distribution entropy based on the complex voltage response, align the transient voltage response to a unified time axis to form a dynamic response matrix and a time-varying equivalent density distribution, and output the localized characterization result based on the spatial distribution entropy and the propagation direction and open boundary accumulation position of the time-varying equivalent density distribution.

[0058] Non-reciprocal quasi-periodic circuit networks can be implemented using printed circuit boards, breadboards, or parametric circuit netlists. In physical circuit implementation, each circuit node has an accessible voltage measurement port, ground admittance adjustment modules are connected between the node and the common reference ground, and non-reciprocal coupling modules are connected between adjacent nodes; in simulation implementation, the same ports and branches are represented by netlist nodes and component models, respectively.

[0059] The excitation and acquisition unit has an AC measurement path and a transient measurement path. The AC measurement path connects an AC current source to a selected circuit node and simultaneously acquires the complex voltage response of each circuit node; the transient measurement path connects a pulse current source to multiple initial circuit nodes sequentially and acquires the transient voltage response of each circuit node during each excitation. Both paths share node numbers, component parameter groups, and acquisition fields, enabling the data processing unit to correlate AC and transient results according to the same parameter combination.

[0060] The data processing unit includes functions for AC data parsing, spatial index calculation, transient time axis alignment, response matrix construction, spatial correlation calculation, and localized result output. The AC data parsing function reads the real and imaginary parts of the complex voltage; the transient time axis alignment function obtains the common effective time range from each point-by-point excitation data; and the localized result output function combines the spatial distribution entropy with the directional propagation and boundary accumulation relationships in the time-varying equivalent density to form a localized representation result. This system creates a closed loop between circuit parameter configuration, excitation acquisition, and result representation.

[0061] Example 10 Based on Example 9, this example further specifies that: the excitation and acquisition unit includes an AC current source, a pulse current source, a node selection switch network, and a voltage acquisition interface; the node selection switch network is used to connect the AC current source to circuit nodes in or near the middle of the chain, and to connect the pulse current source to multiple initial circuit nodes in sequence; the non-reciprocal quasi-periodic circuit network adopts an open boundary, and L is configured as a number in the Fibonacci sequence such as 13, 21, 34, or 55; the data processing unit is also used to calculate the inverse participation rate, boundary weights, and centroid trajectory, and output a spatial distribution entropy phase diagram and a time-varying equivalent density distribution heatmap.

[0062] The node selection switch network can be implemented using physical analog switches, relays, or source connection configurations in a circuit simulator. During AC measurements, the node selection switch network connects only one AC current source to a circuit node in or near the middle of the chain. During transient measurements, the node selection switch network sequentially switches the pulse current source according to a predetermined initial circuit node list, switching to the next node only after one acquisition is completed. The parameters of the ground admittance adjustment module and the non-reciprocal coupling module remain unchanged during switching.

[0063] In a 34-node configuration, the AC current source can be connected to n17, and the pulse current source can be connected sequentially to n2, n4, and so on up to n34; the beginning and end of the node chain are not connected to each other, thus forming an open boundary. For Fibonacci sequence node configurations of 13, 21, 34, or 55, the data processing unit establishes complex voltage vectors and dynamic response matrices according to the actual number of nodes, while maintaining the correspondence between node numbers and data columns.

[0064] The data processing unit arranges the spatial distribution entropy and inverse participation rate under different non-reciprocal and quasi-periodic modulation intensities into a parametric phase diagram, arranges the time-varying equivalent density into a heatmap by node number and time, and simultaneously outputs boundary weights and centroid trajectories. If missing nodes, non-monotonic time series, repeated time points, or inconsistent voltage field numbers with node numbers are found, the calculation for that group is stopped and the corresponding data file is returned for re-acquisition. This embodiment enables dual-mode measurements of multi-size circuits to be performed according to a unified process through node selection, field verification, and graphical output.

Claims

1. A method for simulating and localizing non-Hermitian quasi-periodic circuits, characterized in that, include: A non-reciprocal quasi-periodic circuit model corresponding to the node admittance matrix and the non-Hermitian Aubry-André-Harper model is established. The non-reciprocal quasi-periodic circuit model includes L chain circuit nodes. Each circuit node corresponds to a lattice position and is connected to a ground admittance adjustment module. Adjacent circuit nodes are connected through two non-reciprocal coupling modules with different effective coupling coefficients in the propagation direction. The ground capacitance of each ground admittance adjustment module is configured according to a quasi-periodic function. An AC current excitation is applied to the selected circuit nodes, and a normalized spatial weight is formed based on the complex voltage response of all circuit nodes, and the spatial distribution entropy is calculated. Pulse current excitation is applied to multiple initial circuit nodes respectively, and each transient voltage response is mapped to a unified time axis and a dynamic response matrix is ​​formed according to the excitation position. A spatial correlation matrix and a time-varying equivalent density distribution are formed based on the dynamic response matrix. By combining the spatial distribution entropy with the propagation direction and open boundary accumulation position of the time-varying equivalent density distribution, non-Hermitian skin localization and Anderson localization are distinguished to obtain localization characterization results.

2. The method for simulating and localizing non-Hermitian quasi-periodic circuits according to claim 1, characterized in that, The Hamiltonian of the non-Hermitian Aubry-André-Harper model is The transition strengths in two opposite directions satisfy , No. The grounding capacitance of each circuit node satisfies ; in, Represents the system Hamiltonian. This represents the total number of lattice positions, which is equal to the total number of circuit nodes. Indicates the lattice position number; in the first summation term, This represents the summation of transition terms between adjacent lattice positions. Indicates the relationship with the first The next lattice position adjacent to each lattice position; in the second summation term, This represents the summation of quasi-periodic in-situ terms over all lattice positions; and These represent the transition intensities in two opposite propagation directions; and They represent the first The and the first The generation operator for each lattice position. and They represent the first The and the first Annihilation operator at each lattice position, Represents the Hermitian conjugate operation; Indicates the reference coupling strength. Indicates non-reciprocal strength. Indicates the quasi-periodic modulation intensity. This represents an irrational number used as a quasi-periodic modulation parameter. Represents the modulation phase constant. Represents pi (π). Represents the cosine function; Indicates the first Equivalent grounding capacitance of each circuit node. Indicates the background grounding capacitance. Indicates the amplitude of the quasi-periodic modulation capacitor; By adjusting the capacitance, resistance, or feedback network parameters of the non-reciprocal coupling module, the following changes can be made: The corresponding degree of non-reciprocity, and according to Sure .

3. The method for simulating and localizing non-Hermitian quasi-periodic circuits according to claim 2, characterized in that, The ground admittance adjustment module includes a ground capacitor branch, a ground inductance branch, and a ground resistance branch connected in parallel between the circuit node and the reference ground. The ground capacitor branch includes at least one fixed capacitor and at least one adjustable or replaceable capacitor. The parallel equivalent capacitance of the ground capacitor branch is the first... The grounding capacitance of each circuit node, and satisfying ; in, Indicates the background grounding capacitance. This indicates the amplitude of the quasi-periodic modulation capacitor. This represents the absolute value of the modulation capacitor amplitude; The grounding inductor branch and the grounding capacitor branch together determine the target response frequency band, and the grounding resistor branch is used to limit the voltage increase caused by active non-reciprocal coupling and improve the convergence of circuit solution.

4. The method for simulating and localizing non-Hermitian quasi-periodic circuits according to claim 2, characterized in that, The non-reciprocal coupling module includes a reference coupling capacitor branch and a current-mode negative impedance converter branch connected in parallel between adjacent circuit nodes. The current-mode negative impedance converter branch includes an operational amplifier, a feedback impedance network connected to the operational amplifier, and a non-reciprocal parameter capacitor connected to the adjacent circuit node. The reference coupling capacitor branch determines the reference coupling strength J. The feedback impedance network and the non-reciprocal parameter capacitor together make the off-diagonal admittances corresponding to the two propagation directions of adjacent circuit nodes unequal. The non-reciprocal strength α is adjusted by changing the parameters of the feedback impedance network or the non-reciprocal parameter capacitor.

5. The method for simulating and localizing non-Hermitian quasi-periodic circuits according to claim 2, characterized in that, AC measurement modes that apply AC current excitation to selected circuit nodes include: Select circuit nodes in or near the middle of the chain as AC excitation nodes, perform a pre-scan within a preset wide frequency range, and then perform a fine scan or perform fixed-frequency analysis at the target frequency after determining the target frequency band based on the resonant response of the non-reciprocal quasi-periodic circuit model. The real and imaginary parts of the complex voltage response of each circuit node are recorded in Cartesian form, and the excitation amplitude, the AC excitation node, the acquisition node, and the data format are kept consistent under different non-reciprocal intensity α and quasi-periodic modulation intensity λ.

6. The method for simulating and localizing non-Hermitian quasi-periodic circuits according to claim 5, characterized in that, For a given non-reciprocal strength Quasi-periodic modulation intensity and angular frequency The complex voltage response is used to form a spatial response vector. and in accordance with Calculate the first The normalized spatial weights of each circuit node, according to Calculate the spatial distribution entropy, according to Calculate the inverse participation rate; in, Indicates non-reciprocal strength. Indicates the quasi-periodic modulation intensity. Represents the angular frequency of the AC excitation; This represents the spatial response column vector composed of the complex voltages of all circuit nodes, ordered by node number. Indicates the first Complex voltages at each circuit node In the process of summation, the first... Complex voltages at each circuit node; This represents the node number from which the normalized spatial weights are to be calculated. This indicates the summation index in the denominator. , Indicates the total number of circuit nodes. Indicates the transpose operation; and This represents the modulus of the corresponding complex voltage. and This represents the square of the modulus of the corresponding complex voltage; Indicates the first Normalized spatial weights of each circuit node, Represents the spatial distribution entropy. Represents the natural logarithm. Indicates the inverse participation rate. Represents the normalized space weights The square of; When the normalized spatial weight is zero, its product with its natural logarithm is treated as the limit value of zero; when the normalized denominator is not greater than the lower limit of numerical precision, the corresponding frequency point is excluded; scanning. and The spatial distribution entropy phase diagram and the inverse participation rate phase diagram are obtained.

7. The method for simulating and localizing non-Hermitian quasi-periodic circuits according to claim 6, characterized in that, Transient measurement modes that apply pulsed current excitation to multiple initial circuit nodes successively include: from Select from each circuit node The initial circuit node, for the first Each initial circuit node is connected to only one pulse current source, and the transient voltage response of each circuit node is recorded. After completing one transient analysis, the pulse current source is switched to the next initial circuit node, until the analysis is complete. Point-by-point incentives; A unified time series is established within the common effective time range of each transient analysis. The transient voltage response is then mapped to the unified time axis through interpolation, forming a response column vector. And form a dynamic response matrix. ; in, Indicates the total number of circuit nodes. Indicates the number of independent stimulus positions. Indicates the excitation sequence number of the initial circuit node and , Indicates the circuit node number and , Represents a point in time on a unified timeline; Indicates the first Under the second incentive Each circuit node at time... transient voltage, Indicates the first The column vector of total node responses formed when each initial circuit node is excited. Indicates the transpose operation; Indicates by The dynamic response matrix, composed of response columns arranged in the order of excitation, has a dimension of . .

8. The method for simulating and localizing non-Hermitian quasi-periodic circuits according to claim 7, characterized in that, The response columns of the dynamic response matrix are normalized using the second norm to form a normalized response matrix; When it is necessary to maintain the linear independence of the response subspace, QR orthogonalization is performed on the normalized response matrix and then... Zhongyu The columns corresponding to the initial circuit nodes; The normalized response matrix or the The corresponding columns in the matrix are used as the processed response matrix, according to... Form a spatial correlation matrix, and with As the first The time-varying equivalent density of each circuit node, according to Forming the centroid trajectory, according to Form a set of boundary nodes Boundary weights; in, Indicates the number of independent stimulus positions. This represents the orthogonal factor matrix obtained by QR orthogonalization; Indicates time The processed response matrix, This represents the conjugate transpose of the processed response matrix. This represents the conjugate transpose operation; Represents the spatial incidence matrix. Represents the first... One diagonal element; Indicates the first Each circuit node at time... The time-varying equivalent density, Indicates the location of the centroid of the time-varying equivalent density; This represents a pre-selected set of boundary nodes. This means summing only by traversing the set. Nodes within, Indicates boundary weights; Indicates the node sequence number and , Indicates the total number of circuit nodes. Represents a point in time on a unified timeline; Under the same system size and measurement configuration, a non-Hermitian skin localization is defined when the spatial distribution entropy is low, the boundary weight is high, and the centroid moves toward the same open boundary; an Anderson localization is defined when the spatial distribution entropy is low and the time-varying equivalent density is limited to a finite region within the body; and an extended state is defined when the spatial distribution entropy is high and the inverse participation rate is low.

9. A simulation and localization characterization system for non-Hermitian quasi-periodic circuits, characterized in that, The system includes a non-reciprocal quasi-periodic circuit network, an excitation and acquisition unit, and a data processing unit. The non-reciprocal quasi-periodic circuit network includes L chained circuit nodes, a ground admittance adjustment module connecting each circuit node to a reference ground, and a non-reciprocal coupling module connecting adjacent circuit nodes. The ground admittance adjustment module configures the grounding capacitor according to a quasi-periodic function, and the non-reciprocal coupling module makes the effective coupling coefficients of the two propagation directions different so that the node admittance matrix corresponds to a non-Hermitian Aubry-André-Harper model. The excitation and acquisition unit is used to apply AC current excitation to selected circuit nodes in AC measurement mode and acquire the complex voltage response of each circuit node, and to apply pulse current excitation to multiple initial circuit nodes successively in transient measurement mode and acquire the transient voltage response of each circuit node. The data processing unit is used to calculate the spatial distribution entropy based on the complex voltage response, align the transient voltage response to a unified time axis to form a dynamic response matrix and a time-varying equivalent density distribution, and output the localized characterization result based on the spatial distribution entropy and the propagation direction and open boundary accumulation position of the time-varying equivalent density distribution.

10. The non-Hermitian quasi-periodic circuit simulation and localization characterization system according to claim 9, characterized in that, The excitation and acquisition unit includes an AC current source, a pulse current source, a node selection switch network, and a voltage acquisition interface. The node selection switch network is used to connect the AC current source to circuit nodes in or near the middle of the chain, and to connect the pulse current source to multiple initial circuit nodes in sequence. The non-reciprocal quasi-periodic circuit network adopts an open boundary, and L is configured as a number in the Fibonacci sequence of 13, 21, 34, or 55. The data processing unit is also used to calculate the inverse participation rate, boundary weights, and centroid trajectory, and output a spatial distribution entropy phase diagram and a time-varying equivalent density distribution heatmap.

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