Dynamic generation of nonequilibrium superconducting states in Kitaev chain

The topic of non-equilibrium phenomena in quantum many-body systems is fundamental and compelling in condensed matter physics. In conventional theory, quantum phase transitions occur at zero temperature, and a phase diagram refers, in particular, to the ground state of a system. However, it is obviously that the ground state is intimately related to the excited states for a certain class of systems since they all stem from a same time-independent Hamiltonian. For instance, for a Hamiltonian written in pseudospin form $\sum_{k}\mathbf{B}_{k}\mathbf{\cdot s}_{k}$, the ground state is determined by the field distribution $\mathbf{B}_{k}/\left|\mathbf{B}_{k}\right|$ ZG , which can be extracted from any eigenstates in the tensor product form. In this sense, non-equilibrium phenomena within the prethermalization regime can probably reflect the phase diagram from an alternative aspect. A well-known probe of many-body quantum dynamics is quantum quench, in which a many-body system is initially prepared in a state that is typically the ground state of some simple Hamiltonian and subsequently evolves with a different Hamiltonian. In a conventional framework, the properties of prequench and postquench Hamiltonians are extracted from the evolved states instead of the ground state. The use of nonequilibrium many-body dynamics presents an alternative approach for accessing a new exotic quantum state with an energy level considerably different from that of the ground state Choi ; Else ; Khemani ; Lindner ; Kaneko ; Tindall ; YXMPRA ; ZXZPRB2 ; TK ; JT1 ; JT2 . Advancements in atomic physics, quantum optics, and nanoscience have allowed the development of artificial systems with high accuracy Jochim ; Greiner . Thus, theoretical models can be simulated and realized experimentally. Direct simulation of a simple model is useful not only for solving key challenges in condensed matter physics but also in the engineering design of quantum devices. Notably, the availability of an experimentally controllable fermion system provides an unprecedented opportunity to explore nonequilibrium dynamics in interacting many-body systems.

In addition, it has recently been established that experiments with various materials and excitation conditions have witnessed phenomena with no equilibrium analog or accessibility of chemical substitution, including superconducting-like phases AC ; DF ; MM ; TS , charge density waves (CDW) LS ; HM ; AK and excitonic condensation YM . Therefore, how to stabilize a system in a non-equilibrium superconducting phase with a long lifetime is a great challenge and is at the forefront of current research. Among various non-equilibrium protocols, the generation of the $\eta$-pairing-like state plays a pivotal role in which the existence of doublon and holes facilitate the superconductivity SK ; TK ; JT ; FP ; RF1 ; RF2 ; SE ; JL ; TK2 ; ZXZPRB2 ; ZXZPRB3 ; JT1 ; JT2 .

In the present paper, we report our theoretical investigation of nonequilibrium dynamics in a paradigmatic quantum many-body model—the Kitaev chain. Comparing to the previous works on the Hubbard model, the intial state is an easily prepared state YXMPRA and quenched Hamiltonian is time-independent with parameters covering rich phase diagram in the present work. The Kitaev chain is a lattice model of a p-wave superconducting wire, which realizes Majorana zero modes at the ends of the chain Kitaev . Unpaired Majorana modes have been exponentially localized at the ends of open Kitaev chains Sarma ; Stern ; Alicea . The primary feature of this model originates from the pairing term, which violates the conservation of the fermion number but preserves its parity, leading to a superconducting state. In general, most studies on this model have centered on the ground state or the dynamical quantum phase transitions related to the ground state MH ; LZ . The strength of the pairing interactions has an important role in giving rise to a gapped superconducting state. However, knowledge regarding the pair in each quantum phase remains limited. Specifically, the nearest-neighbor (NN) pair creation (annihilation) term is subtle for the formation of a pair. It results in an NN pair in real space. By contrast, the Hamiltonian in k-space indicates that it also contributes to Bardeen-Cooper-Schrieffer (BCS)-like pairing in momentum space, which involves two particles with opposite momentum. In addition, distinct features originate from the pair term in not only statics but also dynamics. To determine the features of a given state, we introduced two types of order parameters associated with two channels of pairing: local pairing in real space and BCS-like pairing in momentum space. On the basis of the exact solution, we found that the two order parameters were identical for the ground state and could help determine the quantum phase diagram. This revealed an unprecedented feature of the ground state that helps balance between two types of pairing channels. The primary aim of this study was to obtain pertinent information regarding the nonequilibrium state. In this study, acquisition of a vacuum state by the quench process, in which a prequench Hamiltonian has infinite chemical potential, was considered to be the ground state. We investigated evolved states under the postquench Hamiltonian with parameters covering the entire region and found two types of order parameters that were different but could help determine the quantum phase diagram. Our results yield two key findings: a superconducting state can be dynamically generated from a trivial state, and the difference between two types of order parameters for a nonequilibrium state indicates the generation of various superconducting states from the ground state, thus expanding the variety of the state of matter that is associated with different channels of pairing.

This paper is organized as follows. In Sec. II, we present the model and introduce two types of order parameters to investigate the feature of superconducting state. In Subsecs. II.1 and II.2, we calculate the order parameters in $k$ space and real space for the ground state, respectively. In Sec. III, we investigate the features of dynamic behavior of the system. Finally, we give a summary and discussion in Section IV. Some details of the derivations are given in Appendix.

We consider the following fermionic Hamiltonian on a lattice of length $N$

where $c_{j}^{{\dagger}}$ $(c_{j})$ is a fermionic creation (annihilation) operator on site $j$, $n_{j}=c_{j}^{{\dagger}}c_{j}$, $J$ the tunneling rate, $\mu$ the chemical potential, and $\Delta$ the strength of the $p$-wave pair creation (annihilation). $c_{N+1}=c_{1}$ is defined for periodic boundary condition. The Hamiltonian in Eq. (1) has a rich phase diagram that describes a spin-polarized $p$-wave superconductor in one dimension. This system has a topological phase in which a zero- energy Majorana mode is located at each end of a long chain. It is the fermionized version of the well-known one-dimensional transverse-field Ising model Pfeuty , which is one of the simplest solvable models that exhibits quantum criticality and phase transition with spontaneous symmetry breaking SachdevBook .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 TPCHOY ; AGG ; DV1 ; DV2 ; OV ; LL ; UB .

This study centered on the dynamics of a system with NN superconducting pairing term, which warrants further-systematic studies, such as those on long-range correlations. The Hamiltonian can be exactly solved based on Fourier transformation (see Appendix A). The ground-state wave function can be expressed as

with groundstate energy

Accordingly, the phase diagram can be obtained from $\left|\text{{G}}\right\rangle$ and $E_{\mathrm{g}}$, with the phase boundaries

These lines separate four regions in the parameter space, representing different phases with winding numbers CKC ; NL $N=0$, and $\pm 1$, respectively. Fig. 1(a) depicts the phase diagram. The phase diagram indicates that the chemical potential $\mu$ has a subtle role in the ground state. It restricts the value of $\left|\mu\right|$ to maintain the non-trivial phases. The might be because the sufficiently large $\left|\mu\right|$ suppresses pair creation. Thus the number of pairs is higher in nontrivial phases than in trivial phases which leads to the following two questions. First, how to quantify the amount of pairs in each phase? Second, what are the features of the pairing in real or momentum space. The answers are the main part of our goals in this work.

To answer the questions, we explore the features of a superconducting state by introducing two types of order parameters BP . Two operators $\widehat{\eta}_{l}$ and $\widehat{\zeta}_{k}$ characterize pairing channels in real space with

and $k$ space with

For a given state $\left|\psi\right\rangle$, the quantity $|\left\langle\psi\right|\widehat{\eta}_{l}\left|\psi\right\rangle|$ helps measure the rate of transition for a pair located around the $l$th site, whereas $|\left\langle\psi\right|\widehat{\zeta}_{k}\left|\psi\right\rangle|$ helps measure the rate of transition for a pair at the $k$ channel. The corresponding order parameters are defined on the basis of the average magnitude over all channels, as expressed in

and

Nonzero $\eta$ and $\zeta$ indicate that the state $\left|\psi\right\rangle$ is a superconducting state. In general, $\eta$ and $\zeta$ have different values for a given state. In the following subsections, we describe the analytical expressions of $\eta$ and $\zeta$ for the ground states of the system in different regions and their behaviors at phase boundaries.

In this section, we present the features of the dynamic behavior of the system. It relates to the time evolution $\left|\psi(t)\right\rangle$ for a particular initial state

which is essentially an empty state $\prod_{j=1}^{N}\left|0\right\rangle_{j}$ for fermion in real space. The evolved state

may trend toward a stationary state after a sufficiently long time. The non-equilibrium stationary state driven by $H$ is intimately associated with the phase diagram of the ground state (Fig. 1(a)). We focus on the time-dependent order parameters for the evolved state $\left|\psi(t)\right\rangle$. Based on the fact that $\left|\psi(t)\right\rangle$ represents the eigenstates of the operator $T$, the following explicit forms can be obtained:

and

where the key function is

Figs. 3(b1) and 3(b2) present $\zeta(t)$ and $\eta(t)$ as the function of time for finite-size systems with several typical system parameters $\left(\mu,\Delta\right)$. As expected, both quantities become steady after a long period. $\zeta(t)$ and $\eta(t)$ behave as damping oscillations around a certain equilibrium position. The quasi frequencies and decay rates can be evaluated through a saddle point integration in the continuum limit AS ; AAM ; SA . We rewrite $\eta\left(t\right)$ in the form

where

The results depends on the location $k_{0}$ of the extrema $\varepsilon_{k}$. (i) In this case with $k_{0}\neq 0,\pi$, we have quasi frequency $\omega\approx 2\varepsilon_{k_{0}}$ and $\delta\eta\left(t\right)\sim t^{-0.5}$. (ii) In this case with $k_{0}=0,\pi$, we have quasi frequency $\omega\approx 2\varepsilon_{k_{0}=0}$ and $\delta\eta\left(t\right)\sim t^{-1}$. Similar analysis can be applied to $\zeta\left(t\right)$. These results accord with the plots in Fig. 4.

On the other hand, such behaviors can also be understood by a simple picture, when the spectrum contains higher degeneracy, which relates to the peak of the density of state (DOS) of the system. The DOS at energy level $E$ is generally defined as

where d$N\left(E\right)$ indicates the number of energy levels that appear in the interval $\left[E,E+dE\right],$ with the normalization factor $\lambda=\frac{1}{N}$. Figs. 3(c1)-3(c4), depict the energy levels and DOS as the functions of momentum $k$ and $E$, respectively. When the peaks of $D\left(E\right)$ appear, the quasi frequency is related to the energy band gap $\delta E$, which is the distance between two peaks.

No matter what mechanism of the decay, $\zeta(t)$ and $\eta(t)$ approach to their stable values, which can be obtained from

and

It is two types of summation for the same function $F(k)$. In the large $N$ limit, $\overline{\eta}$ and $\overline{\zeta}$ can be expressed as the integral equations:

and

Straightforward derivations lead to the development of analytical expressions

where $\Lambda=\mu\Delta\pi^{-1}\left(1-\Delta^{2}\right)^{-1}$ at $\mu^{2}+\Delta^{2}=1$, and

when $\mu^{2}+\Delta^{2}\neq 1$. Conversely,

the time-averaged analytical expressions for $\eta$ and $\zeta$ obtained from Eqs. (33) and (35) are shown graphically in Fig. 3(a).

Similar to the static order parameters, analytical behavior of $\overline{\zeta}$ and $\overline{\eta}$ as function of $\left(\Delta,\mu\right)$ can still identify the phase diagram (see Appendix B). Fig. 5 shows the average order parameters $\overline{\zeta}$ and $\overline{\eta}$ obtained from Eqs. (29) and (30) as the function of $\mu$ and $\Delta$ for the case $N=4000$ in the first quadrant of the $\mu-\Delta$ plane. Fig. 6 presents $\partial_{\mu}\overline{\eta},$ $\partial_{\mu}\overline{\zeta}$, $\partial_{\Delta}\overline{\eta},$ and $\partial_{\Delta}\overline{\zeta}$, the partial dervatives of $\overline{\zeta}$ and $\overline{\eta}$ with respect to $\mu$ and $\Delta$, for finite system. Fig. 7(a1)-7(a3) show the profiles of $\overline{\zeta}\left(\mu,\Delta\right)$, $\overline{\eta}\left(\mu,\Delta\right)$, $\partial_{\mu}\overline{\eta}$ and $\partial_{\mu}\overline{\zeta}$ at the line $\Delta=0.6.$ Fig. 7(b1)-7(b3) present the profiles of $\overline{\zeta}\left(\mu,\Delta\right)$, $\overline{\eta}\left(\mu,\Delta\right)$, $\partial_{\Delta}\overline{\eta}$ and $\partial_{\Delta}\overline{\zeta}$ at the line $\mu=0.6$, indicating that four quantities, $\partial_{\mu}\overline{\eta},$ $\partial_{\mu}\overline{\zeta}$, $\partial_{\Delta}\overline{\eta},$ and $\partial_{\Delta}\overline{\zeta}$, can identify the phase boundary at $\mu=1$ and $\Delta=0$ $(\mu<1)$. Compared with $\eta_{\mathrm{g}}$ and $\zeta_{\mathrm{g}},$ which are always identical in all regions, $\overline{\zeta}$ and $\overline{\eta}$ vary in non-trivial regions, providing quantitative description of the pairing mechanism for a non-equilibrium state. Moreover, the non-equilibrium state is also a superconducting state because the values of $\overline{\zeta}$ and $\overline{\eta}$ have the same order as those of $\zeta_{\mathrm{g}}$ and $\eta_{\mathrm{g}}$. The fact that $\overline{\zeta}>\overline{\eta}$ indicates that BCS-like pairing in momentum space is favored. Before ending this work, we would like to point out that the obtained results depend on the initial state. For example, consider an initial state in the form

with $\left\{k_{a}\right\}\oplus\left\{k_{b}\right\}\in\left(0,\pi\right)$, which is the eigenstate of $H$ with arbitrary parameters $\left(J,\Delta,\mu\right)$ due to the fact

from the Appendix A. We find that

which result in $\overline{\eta}=\overline{\zeta}=0$, indicating that quantities $\overline{\eta}$ and $\overline{\zeta}\$cannot indicate the phase diagram.

In summary, we have studied the quench dynamics of a Kitaev chain by introducing two types of order parameters that characterize two different pairing mechanisms, namely local pairing in real space and BCS-like pairing in momentum space. We have shown exactly that the ground states over the entire parameter region support the identical values of them, indicating a balance between the two types of pairing channels; This is an unprecedented feature of the ground state. The balance is disturbed in a non-equilibrium state when the post-quench Hamiltonian is in a topologically non-trivial phase, whereas it is maintained in a topologically trivial phase. Furthermore, non-equilibrium superconducting states favor the BCS-like pairing. —Our findings present an alternative approach to generate a superconducting state from a trivial empty state and illuminate the pairing mechanism.

We acknowledge the support of NSFC (Grants No. 11874225).