Topological charge pumping in tetramerized Kitaev chains with nodal loop in parameter space

We investigated the topological pumping charge of a tetramerized Kitaev chain with spatially modulated chemical potential, which hosts nodal loops in parameter space and violates particle number conservation. In the simplest case, with alternatively assigned hopping and pairing terms, we show that the model can be mapped into the Rice–Mele model by a partial particle-hole transformation and subsequently supports topological charge pumping as a demonstration of the Chern number for the ground state. Beyond this special case, analytic analysis shows that the nodal loops are conic curves. Numerical simulation of a finite-size chain indicates that the pumping charge is zero for a quasiadiabatic loop within the nodal loop and is ±1 for a quasiadiabatic passage enclosing the nodal loop. Our findings unveil the topology of Kitaev chains in parameter space.


Introduction
Thouless pumping [1] has received much attention over a long period of time.It is the quantum version of matter pumping by a mechanical device in our everyday life.The intriguing features are that (i) the total probability of the transferred particles is precisely quantized for a cyclic adiabatic passage without a bias voltage and (ii) a nonzero pumping charge for the ground state is shown to relate a degenerate point and then acts as a topological invariant [2].
In this work, we studied the topological pumping charge of a tetramerized Kitaev chain model with spatially modulated chemical potential.In general, the topological pumping charge in a one-dimensional Kitaev model refers to the corresponding Majorana lattice rather than the transport of spinless fermions [2,31,32].Technically, the origin of this topological feature is the degenerate point.In contrast, the present tetramerized Kitaev model hosts nodal loops in parameter space, and the pumping charge is obtained directly from the ground state of the model, which does not support particle number conservation.The main motivation of this work arises from the simplest case in which the hopping and pairing terms are completely assigned to different dimers alternatively.We show that this model can be mapped into a Rice-Mele (RM) model [33] by a partial particle-hole transformation under certain constraints.Then, the ground state of such a Kitaev model supports the topological charge pumping as a demonstration of the Chern number because the current operator is invariant under the transformation.Beyond this special case, analytic analysis has shown that the degenerate points become nodal loops.Numerical simulations for finite-size chains are employed to answer the question of what happens to the pumping charge.The results indicate that the pumping charge remains unchanged as the nodal point evolves into a loop; specifically, it is ±1 for the passage loop that encloses the nodal loop.Additionally, the pumping charge is zero for a quasiadiabatic loop within the nodal loop.
Here, we would like to address the main points of the present work.(i) The pumping charge refers to the probability of fermions in a Kitaev model, where the total number of fermions is not a conserved quantity.(ii) The pumping charge arises from a degeneracy loop rather than from isolated degeneracy points within the parameter space.To our knowledge, the research related to this topic has focused on the complex fermionic pumping charge in topological insulators [32,34] or the Majorana pumping charge in superconductors [31] and the origin of the pumping charge from nodal points [21,23,26,27,34].
Several experimental schemes have been proposed [35][36][37] to realize effective spinless superconductivity, and recent papers [38,39] noted that a class of long-range Kitaev tie models can be realized in condensed material.
A one-dimensional spinless Kitaev model can be mapped to a spin model by the Jordan-Winger transformation.The recently emerging cold atom technology may provide another promising avenue to realize the Kitaev model because the spin system can be experimentally stimulated by cold atoms [40][41][42][43].
The rest of the paper is organized as follows.We begin section 2 by introducing the Hamiltonian and its symmetry.We also analytically deduce the equations of nodal lines in parameter space.In section 3 we demonstrate that the model under some constraints can be mapped into the RM model via a partial particle-hole transformation, which inspires us to investigate the pumping charge of this model.In section 4, we numerically calculate the pumping charges of the ground state for different quasiadiabatic passages in parameter space.The results indicate that the pumping charge can be treated as a topological invariant to characterize the topology of the model.Finally, we draw conclusions in section 5. Some detailed derivations are given in the appendix.

Model and nodal ellipses
The Kitaev chain model describes a spin-polarized p-wave superconductor in a one-dimensional system and has received much attention since its simple form also includes a rich phase diagram.This system has a topological phase, realizing Majorana zero modes at the ends of the chain [44].On the other hand, it is the fermionized version of the well-known one-dimensional transverse-field Ising model [45], which is one of the simplest solvable models that exhibit quantum criticality and phase transition with spontaneous symmetry breaking [46].Several studies have been conducted with a focus on long-range Kitaev chains, in which the superconducting pairing term decays with distance as a power law [47][48][49][50][51][52][53].
In this work, we investigate the topological features of the Kitaev chain from an alternative perspective, referred to as the hidden topology.Such a feature does not emerge in the usual Kitaev chain considered in the literature.Considering a tetramerized one-dimensional Kitaev model, the Hamiltonian contains two parts with and The tetramerized Kitaev model is schematically illustrated in figure 1(a).This seems to be somewhat complicated, and the purpose of this separation is to provide a better presentation for the following discussions.Here, c j denotes the fermion annihilation operator at site j.(1 ± δ) are the hopping amplitudes and the strength of the pairing operator between neighboring sites with tetramerized factors 1, v, and w, and the real number V is the chemical potential.When the periodic boundary is taken, we define c 4N+j = c j .For arbitrary parameters, the unit cell includes four sites.Then, the Hamiltonian H can be written in block diagonal form by introducing Fourier transformation Here, the operator vector is defined as and the two 4 × 4 matrices A and B are and where the k-dependent factor Γ k = (1 + δ) e ik .The symmetry of the matrix h k contains the core information of the system, which allows us to obtain the conclusion without solving the 8 × 8 matrix h k explicitly.Introducing the 8 × 8 matrix we readily have which ensures that the spectrum ε k of h k is symmetric with respect to the zero-energy point.This can be demonstrated by numerical simulations for finite-size chains.The spectra of the system with representative parameters are plotted in figure 1(b2).Furthermore, |ε k | can be obtained directly from the diagonalization of the 4 × 4 matrix (A + B) (A − B) (details are shown in the appendix).
In the following, we focus on the degeneracy points of ε k in the parameter space.The derivations in the appendix show that the band degenerate points always lay in the subspace with k = 0 or π, i.e. the degenerate zero-energy points (nodal line) can be determined by Then, the corresponding nodal lines obey the equations and which are obviously an ellipse and a hyperbola, respectively.Here, the shapes of two conic curves are determined by the system parameters We note that when taking w + v → 0 (see also the exact solution for w + v = 0 in the following section), the first equation reduces to a point (δ, V) = (0, 0) if |w| < 1 .In this work, we focus on the nodal line around the point (δ, V) = (0, 0).The corresponding nodal lines and energy band edges are plotted in figures 2(a) and (b).In the following, we will investigate the topological features related to the nodal lines that are in the 2D parameter space mentioned in the previous work [29,30,54] rather than in the 3D parameter space [31].

Equivalent RM model
In this section, we start with a Kitaev model that connects to a well-known model possessing topological features.We consider the Hamiltonian H with v = −w, at which the Hamiltonian becomes a simplified form and obey the commutative relations The relations can be checked by a straightforward derivation for the following explicit forms of four matrices.Here, two k-independent 4 × 4 matrices are and while the other two k-dependent ones are and Obviously, H and H 1 have common eigenstates.Importantly, the derivation in the appendix shows that H and H 1 have the same ground state within the region |w| < 1.This means that one can focus on the investigation and analysis of H 1 only.Notably, we will show that H 1 has a connection to a RM model.Taking a partial particle-hole transformation with j ∈ [1, N], H 1 becomes a RM model [33] It is well-known that H RM is one of the basic models discussed in connection with ferroelectrics [55,56] and topological properties.This provides a natural platform for directly studying topological invariants through dynamics both from theoretical [2,31,32] and experimental [57] perspectives.For a RM model, it has been shown [2] that if the system adiabatically evolves along a loop enclosing the degeneracy points (0, 0) in the δ − V plane, then the polarization will change by ±1, where the sign depends on the direction of the loop.
On the other hand, if the loop does not contain the degeneracy point, then the pumped charge is zero [2].
Specifically, considering a time-dependent RM Hamiltonian H RM (t) with δ(t) = δ(T + t) and V(t) = V(T + t), the pumping charge passing site j for the evolved state |ϕ (t)⟩ from the initial ground state |ϕ (0)⟩ of H RM (0) for the time evolution period T can be expressed as where the current operator is where j = 1, 2, . . ., N. The pumping charge Q j is quantized when it is an adiabatic cycle, characterizing the topological feature.
Obviously, the current measures the flow of fermions; therefore, it may include both supercurrent and normal current.Importantly, we note that the current operator J j is invariant under the partial particle-hole transformation in equation (23).Consequently, the pumping charge represents the original electric charge in practical terms.This finding suggested that the tetramerized Kitaev chain with v = −w has the same topological features as the RM model within the region |w| < 1.
Although this conclusion was obtained rigorously, it is a little surprising because a RM model supports the conservation of the particle number, while the Kitaev model does not [29].In addition, the unveiled topology for the special case with v = −w may be extended to a more general case.This is the main goal of this work.

Topology of the nodal ellipse
Now, we turn to the question of what happens in the case with nonzero v + w, at which the degeneracy points form an ellipse.Here, we would like to emphasize that the nodal line lies in a 2D plane [29,30,54] in this work rather than in the 3D space in a previous work [31] that is very similar to the present work.For a nodal loop within a 3D space, the pumping charge is studied along a closed passage piercing the nodal loop.Therefore, we will investigate the topology associated with the nodal loop from another aspect.Specifically, how does the pumping charge when an adiabatic loop encloses the ellipse or lies inside the ellipse?In the latter case, the loop does not enclose any degeneracy points, there is no doubt that the pumping charge is zero.However, it is difficult to predict this result for the former case because the total pumping charge depends on the vortex of an individual degenerate point [2,31].A possible conjecture can be given in the spirit of the conclusion for the topology of the Fermi surface [58].It claims that the integral of the Berry vector potential in momentum space for a loop surrounding a Fermi surface remains unchanged when the surface reduces to a degenerate point.Similarly, it has been shown that the pumping charge is connected to the integral of the gradient field of polarization in parameter space for a loop surrounding a nodal point in an RM model.It is expected that the pumping charge remains unchanged when the nodal point evolves into a loop.In this situation, numerical simulation is an efficient tool for providing evidence for theoretical conjecture.
A numerical simulation is performed for the time-dependent Hamiltonian with the parameters which is an ellipse with a center at (δ 0 , V 0 ) in the δ − V plane.Here, ω controls the varying speed of the time-dependent Hamiltonian.The computation is performed using a uniform mesh in time discretization.Time t is discretized into t i , with t 0 = 0 and t M = T.For a given initial eigenstate |ϕ (0)⟩, the time-evolved state is computed using where T is the time-order operator.In the simulation, the value of M is considered sufficiently large to obtain a convergent result.The total pumping charge passing through site j during the time evolution period T can be expressed as and the corresponding average current and average pumping charge are defined as Due to the translational symmetry of the system, the current and pumping charge across the same type of dimer are identical.However, the average current and pumping charge are favorable for numerical computation.
In principle, the value of ω should be sufficiently small to fulfill the requirement of quasiadiabatic evolution.In the computation, we select ω to satisfy the quasiadiabatic condition, under which the obtained pumping charge is not sensitive to a slight change in ω.Figures 3 and 4 present the plots of the simulations for two kinds of quasiadiabatic passages with the degeneracy point and nodal line unenclosed and enclosed by a loop, respectively.The instantaneous current and total accumulated charge of the ground state are computed.According to our analysis for the case with w + v = 0, the final value of the pumping charge Q depends on whether the evolution loop encloses the degeneracy point (0, 0) in the δ − V parameter space.We first demonstrate this point in figure 3, which clearly shows that Q = 0 or −1 with high precision for the unenclosed or enclosed circle.We then plot the results for the case with w + v ̸ = 0 in figure 4. Notably, we find that the pumping charge Q = 0 or −1 is highly precise for unenclosed or enclosed loops.Hence, in this sense, the topological charge of the loop is directly related to the vorticity of the nodal point.This strongly implies that the topology of the Kitaev model with a nodal ellipse is the same as that with a nodal point.b3) and (b4) demonstrate the corresponding numerical results for the current and pumping charge.This indicates that if the encircles the nodal line, the pumping is very close to −1, and the pumping charge is very close to 0 if the nodal line is outside the adiabatic loop.

Summary
In summary, we investigated the topology associated with nodal loops in a system without particle number conservation.In addition, the nodal loop lies in a 2D parameter space rather than 3D space.In previous work on a nodal line in 3D space, such as that in [31], the related topology feature was essentially the same as that of an isolated degenerate point.This can be seen from a cross section in 3D space, in which the nodal line reduces to a degenerate point.As an example, we studied the topological pumping charge of a tetramerized Kitaev chain with spatially modulated chemical potential.This model has the advantage that it can be mapped into a RM model by a partial particle-hole transformation under certain constraints.This motivates us to compute the pumping charge beyond this special case.Numerical simulation of a finite-size chain indicates that the pumping charge is zero for a quasiadiabatic loop within the nodal loop and ±1 for the passage loop enclosing the nodal loop.This indicates that such a Kitaev model supports topological charge pumping as a demonstration of Chern number.Our findings unveil the topology of Kitaev chains in parameter space by exploring topological matter from an alternative perspective.

Figure 1 .
Figure 1.(a) A schematic diagram of the Hamiltonian in equation (1).A unit cell is composed of four sites with modulated chemical potential.The strengths of the hopping terms and pairing terms are stagger.1 − δ and v (1 + δ) are the strengths of the hopping terms, and w (1 − δ) and 1 + δ are the strengths of pair creation (annihilation).(b1) shows a triangular loop on the δ − V plane, and (b2) shows the corresponding energy spectra along the loop.The spectra are symmetrical with respect to zero, and there are two degenerate points because the loop has two crossing points with the degenerate line of the energy spectra.The parameters are N = 50, w = 0.6, and v = −0.3.

Figure 2 .
Figure 2. (a) Several representative nodal lines derived from equation ( 12) on the δ − V plane for fixed w and different v.(b) Color contour plots of the numerical results of one of the energy bands with the wavevector k = 0.The black loop is the nodal line from (a) with v = −0.3 and this indicates that the degenerate line on the δ − V plane is a closed loop, which accords with that in (a).The other parameters are N = 50 and w = 0.6.

Figure 3 .
Figure3.The adiabatic current in equation(30) and pumping charge in equation(31) of the ground state of the Hamiltonian with a single degenerate point for different passages.(a) The green solid dot is the degenerate point (δ, V) = (0, 0) and the red lines represent different adiabatic loops.The equations are δ = sin (ωt) + δ0, V = 3 cos (ωt) and θ = ωt; for c, d, e, and f, the parameters δ0 are 0, 0.9, 1.1, and 2, respectively; and (b1), (b2), (b3) and show the corresponding numerical results of the adiabatic current and pumping charge.The time interval is t l − t l−1 = 0.0628, and the other parameters are w = 0.6, v = −0.6,ω = 0.001, and N = 50.This implies that if the adiabatic loop encircles the degenerate point, the pumping charge is nearly -1; otherwise, the pumping charge is nearly 0.