Abstract
The discrete-time quantum walk provides a versatile platform for exploring abundant topological phenomena due to its intrinsic spin-orbit coupling. In this work, we study the non-Hermitian second-order topology in a two-dimensional non-unitary coinless discrete-time quantum walk, which is realizable in the three-dimensional photonic waveguides. By adding the non-unitary gain-loss substep operators into the one-step operator of the coinless discrete-time quantum walk, we find the appearance of the four-degenerate zero-dimensional corner states at ReE = 0 when the gain-loss parameter of the system is larger than a critical value. This intriguing phenomenon originates from the nontrivial second-order topology of the system, which can be characterized by a second-order topological invariant of polarizations. Finally, we show that the exotic corner states can be observed experimentally through the probability distributions during the multistep non-unitary coinless discrete-time quantum walks. Our work potentially pave the way for exploring exotic non-Hermitian higher-order topological states of matter in coinless discrete-time quantum walks.
1 Introduction
Due to the unique bulk-boundary correspondence, topological phases of matter have attracted great attention in recent years [1–3]. The standard bulk-boundary correspondence generates the emergence of robust gapless eigenstates localized at the boundary of the nontrivial topological sample. However, in 2017, Benalcazar et al extend the concept of the topological phases of matter by introducing the higher-order topological insulators, which obey a generalized bulk-boundary correspondence [4, 5]. Specifically, for a d-dimensional nth nontrivial topological system, the robust gapless eigenstates localized at (d − n)-dimensional boundary of the system will appear [6–8]. Generally speaking, the nontrivial topological phases can be generated through engineering specific hoppings in lattice models [9, 10]. In addition, more exotic topological properties can arise due to other characteristics of the system, such as periodic driving [11–13], non-Hermiticity [14–18], and disorder [19–22], to mention a few.
As a typical time-periodic driving (Floquet) system, the discrete-time quantum walk (DTQW), which is a dynamical evolution process of particles (called walkers) in discrete position space at discrete points in time, exhibits abundant topological properties [23, 24] and has been realized experimentally in systems of cold atoms [25, 26], trapped ions [27, 28], photons [29–32], superconducting circuits [33], and nuclear magnetic resonance [34]. Although the intriguing first-order topological phenomena in unitary [35–49] and non-unitary [50–59] DTQWs have been widely studied both in theory and experiment, little attention has been paid to the connection between the DTQWs and the higher-order topology [60]. As the quantum counterpart of classical random walk, the walker’s internal degree of freedom (IDF) of the DTQW plays the role of a quantum coin [61]. Thus, the walker’s internal state is also called the coin state. According to the coin state, DTQW can be divided into the coined DTQW (IDF1) and the coinless DTQW (IDF = 1) [62]. Here we study the second-order topology in two-dimensional non-unitary coinless DTQWs. Compared with the coined DTQW, in which the walker’s direction of motion depends on its coin states, the research of the coinless one without coin states is still lack [63, 64], especially its topological properties [57, 60, 65, 66]. However, the coinless DTQW can be directed constructed from the static Hamiltonian and thus can easily simulate rich physical phenomena originate from the static Hamiltonian. Furthermore, novel phenomena beyond the static Hamiltonian can emerge in coinless DTQW, such as the emergence of the topological boundary states at energy π, which is unique for the Floquet systems.
In this paper, we construct a non-unitary one-step operator of a two-dimensional coinless DTQW, which can be realized using three-dimensional photonic waveguides. Through the quasi-energy spectrum and the collective distributions of the eigenstates, we observe four energy-degenerate corner-localized eigenstates induced solely by the gain-loss term in our proposed non-unitary coinless DTQW. The existence of such corner states originate from the nontrivial second-order topology of the system. To characterize the topological properties of the system, we calculate numerically a second-order topological invariant of polarizations through constructing the biorthogonal nested Wilson loops and give the topological phase diagram. Moreover, we numerically demonstrate that the corner states governed by the nontrivial second-order topology can be experimentally observed through the probability distributions in multistep coinless non-unitary DTQWs. Our work potentially pave the way for studying exotic non-Hermitian higher-order topological states of matter in coinless discrete-time quantum walks.
The structure of this paper is organized as follows. In Section 2, the one-step operator of a two-dimensional coinless non-unitary DTQW is constructed. In Section 3, we numerically calculate the quasienergy spectra and observe the second-order topological corner states. In Section 4, we show the topological phase diagram characterized by a topological invariant of polarizations. In Section 5, we illustrate how to observe the corner states in such system. Discussion and conclusion are finally drawn in Section 6.
2 Non-Unitary Coinless Discrete-Time Quantum Walk
Based on the point that the coinless DTQW can be constructed by dividing the static Hamiltonian, we first introduce an extended Benalcazar-Bernevig-Hughes (BBH) Hamiltonian with on-site gain and lossThe first term in Eq. 1 is exactly the BBH Hamiltonian [4, 5].where is the creation (annihilation) operator of a spinless particle at the site (x, y), tx(y) = t + (−1)x(y)δt are the hopping amplitudes in the x (y) direction respectively, and H.c. is the Hermitian conjugate. Nx and Ny are the numbers of the lattice sites in the x and y directions, respectively. We can see that there are two types of hopping amplitudes t − δt and t + δt. To simplify the writing in the following paper, we relabel these two types of hopping amplitudes as t − δt = J1 and t + δt = J2. Furthermore, the on-site gain-loss Hamiltonian is introduced aswhere γ is the gain-loss parameter. has a period of four lattice sites (gain-loss-loss-gain) both in x and y directions. Since the previous work [67, 68] have demonstrated that such gain-loss-loss-gain typed non-Hermitian term occurring in one-dimensional lattice system can dramatically affect the topology of the system, its two-dimensional extension are also expected to bring some novel topological phenomena.
In order to construct a coinless discrete-time quantum walk, we first divide the Hamiltonian (1) into five partswhere and represent the sum of the hoppings with parameter J1 (J2) along the x and y directions, respectively. Thus, in the Hilbert space |x⟩ ⊗|y⟩ (or |x, y⟩) with x ∈ {1, Nx} and y ∈ {1, Ny}, an one-step operator of the coinless DTQW can be constructed asOther alternative one-step operators will also be discussed in the next section. For simplicity, we use the units ΔT = ℏ = 1 hereafter. By directly calculating the matrix exponential, we can write these four unitary substep operators aswith the coupling operators defined asThe operator denotes a Nx × Nx (Ny × Ny) identity matrix in the sub-Hilbert space |x⟩ (|y⟩).
Similarly, the non-unitary substep operator can be written in the following form
By applying the one-step operator in Eq. 5 many times, a multiple non-unitary coinless DTQW can be realized, as shown schematically in Figure 1. Furthermore, when the gain-loss parameter γ = 0, the substep operator is exactly a (Nx ⋅ Ny) × (Nx ⋅ Ny) identity matrix and the corresponding one-step operator will become unitary.
FIGURE 1
Based on recent experimental progress of quantum walks in waveguides [66, 69–73], the realization of Eq. 5 is accessible under the flexible control of the three-dimensional photonic waveguides. Specifically, the above discussed coinless DTQW except the gain-loss term can be realized by the directional coupling of two waveguides [60, 66, 70, 74]. And the alternative gain or loss can be introduced in a single waveguide [57, 75].
3 Spectra and Corner States
In order to illustrate the effect of the gain-loss term on the topological features of this non-unitary coinless DTQW, in this section we fix the coupling parameter J1/J2 = 1.1, which corresponds to a trivial phase when the gain-loss parameter γ is zero [60]. Since the one-step operator is non-unitary, its effective Hamiltion is non-Hermitian with complex energy. In Figure 2A, we plot the real part of the quasi-energy spectrum, obtained from diagonalizing , as a function of γ under the open boundary condition in both directions. We find that when the gain-loss parameter γ is large than a critical value γc, four energy-degenerate states will emerge at ReE = 0. The value of γc is related to the specific form of the one-step operator except the value of the coupling parameter J1/J2. The one-step operator of Eq. 5 contains four unitary operators, each of which is followed by a non-unitary gain-loss operator . However, the one-step operator can also be constructed as , which consists of only six substep operators. In Figure 2B, we show the real part of the quasi-energy spectrum of varying with γ. Similarly, we observe the existence of four energy-degenerate states at ReE’ = 0 when the gain-loss parameter γ is large than a critical value . Compared with the critical value γc for , the critical value for is larger since the latter contains less gain-loss operators . Since the numerical methods used to analyse the topological features of the systems generated by the above two one-step operators are similar, we only discuss the former in the following paper. We notice that no four energy-degenerate states will emerge at ReE = 0 in the real part of the quasi-energy varying with γ when the system is generated by another one-step operator . Furthermore, we do not show the imag part of the quasi-energy varying with γ since no more valuable information can be obtained.
FIGURE 2
A unique phenomenon in non-Hermitian systems is the appearance of the skin effects [14], which means that all of the eigenstates will be localized near the boundary under the open boundary conditions. However, the skin effects will not emerge in all of the non-Hermitian systems [16]. A major consequence of non-Hermitian systems with the skin effects is that the bulk bands of the system under the open boundary conditions are considerably different from those of the system under the periodic boundary conditions. Thus, we show the complete quasi-energy spectrum under the open and periodic boundary conditions in Figures 2C,D, respectively. We find that the bulk bands under the different boundary conditions are consistent except the emergence of the four energy-degenerate states at ReE = 0 when we consider the open boundary conditions. Figures 2C,D strongly demonstrate that our system does not suffer from the skin effects and therefore it does not matter whether the right and/or left eigenstates are used to calculate the density distribution. In Figure 2E, we show the collective density distributions of these four energy-degenerate gapless states at ReE = 0, which are localized at the four corners of the lattice. And the remaining two energy-degenerate gapped states at ReE ≠ 0 are indeed extended in the bulk of the lattice, as shown in Figure 2F.
4 Topological Phase Diagram
The emergence of the exotic four energy-degenerate corner states at ReE = 0 can be attributed to the second-order bulk topology, which corresponds a kind of topological phase supporting lower-dimensional corner or hinge states, induced by the gain-loss term in Eq. 12. The nontrivial second-order bulk topology in non-Hermitian systems can be characterized by introducing the non-Bloch winding numbers or the biorthogonal nested Wilson loops [76–81]. In addition, due to the intrinsic 2π period of quasi-energy, a pair of topological invariants are required to predict the appearance of zero-energy and π-energy corner states in the Floquet second-order topological systems [82–87]. Specifically for the second-order topological characterisation of our model, only one invariant of polarizations (the quadrupole moments) constructed by the biorthogonal nested Wilson loops is enough since the C4 symmetry and the absence of the π-energy corner states.
The second-order topological invariant of polarizations are constructed in momentum space. Thus, we first need to renumber the lattice sites in terms of the unit cell, each of which contains 16 sublattices, as shown in Figure 3. Using the Fourier transformation, the one-step operator can be written in momentum space as , where and is a 16 × 16 matrix in the basis of . Then, we consider the right and left eigenstates of the effective Hamiltonian ,which satisfies the biorthogonal normalization with the band indices α and β. Alternatively, one can write the matrix of the effective Hamiltonian as Heff = VDV−1, where D is a diagonal matrix of quasi-energies and the columns of the matrixes V and are corresponding right and left eigenstates, respectively. Because of the absence of the non-Hermitian skin effects, the bulk-boundary correspondence based on the ordinary Bloch band theory is valid here. Thus, we can determine the topological phase transition points with the gapless real quasi-energy spectrum ReE(k), see Figures 4A–C. When the system is in the topological trivial or nontrivial phases, the real quasi-energy spectrum ReE(k) is all gapped.
FIGURE 3
FIGURE 4
To characterize the gapped phases using polarizations, we consider the case of half filling and define the biorthogonal Wilson loop operator along y direction aswhere is a 8 × 8 matrix with elements , ey is the unit vector in the y direction, and Δky = 2π/Ny. The two-dimensional Brillouin zone is discretized by using the interval (2π/Nx, 2π/Ny), such that there are (Nx + 1) (Ny + 1) k-points in total. Due to the non-Hermiticity of the effective Hmiltonian , the constructed operator is a non-unitary operator and corresponds to a non-Hermitian Wannier Hamiltonian . With the periodic boundary conditions, and , we can obtain the right and left Wannier states by diagonalizing Eq. 14 aswhere j is the Wannier band index and . These eight Wannier bands vy,j (j = 1, 2, … 8) can be divided into three Wannier sectors (labeled by ς = 0, ±) with finite gaps, see Figures 4D–F. Especially, all Wannier bands will tend to be flat when γ is pretty large. Each Wannier sector can carry their own biorthogonal topological invariants, which can be evaluated by constructing the biorthogonal nested Wilson loops. Utilizing eigenstates and , we can construct the biorthogonal Wannier states aswhere denotes the αth element of the 8-component state vector and . For each Wannier sector ς, with the periodic boundary conditions, and , the elements of the constructed nested Wilson loop operator along x direction arewhere ex is the unit vector in the x direction and Δkx = 2π/Nx, the indices l, l′ ∈ 1 … NW with NW the number of the Wanner bands in sector ς. In Eq. 17, summation is implied over repeated indices r, … , s ∈ 1 … NW over all Wannier bands in sector ς. After that, we can obtain the polarizations along x direction asIn a similar way, we can directly obtain the Wannier bands vx,j and polarizations along y direction by constructing the biorthoganal Wilson and nested Wilson loop operator along x and y directions, respectively. Due to the C4 symmetry, we have vy,j ≡ vx,j and .
Using the above procedure, we numerically calculate the polarizations and find that the polarizations and . In Figure 5A, we show the polarizations varying with γ, which equal to 0.5 (0) for topological (trivial) phases with (without) corner states. It means strongly that the polarizations here are good candidates for characterizing the second-order topology of this system. Thus, we define the topological invariant with the polarizations and ,The topological invariant W has two possible quantized values: 0 and 1, which corresponds to the trivial and topological phases, respectively. In Figure 5B, we show the topological phase diagram of the topological invariant P versus the coupling parameter J1 and the gain-loss parameter γ. When J1 ∈ (0.5, 1), the system with γ = 0 is in the topological phase and will still remain topological as γ is incremented from zero. However, the system with γ = 0 is in the trivial phase with J1 ∈ (1, 1.5) and will become topological when γ > γc. Furthermore, the value of γc will increase sharply as we increase J1.
FIGURE 5
5 Observation of Corner States
Experimentally, the exotic corner states can be observed through the localization of probability distributions in multi-step non-unitary coinless DTQWs. Without the existence of the local states, such as the above discussed corner states, the typical transfer behavior of the coinless DTQW is ballistic [63]. In this section, we demonstrate the existence of the corner states by showing the numerical results of probability distributions of multistep non-unitary coinless DTQWs with different gain-loss parameters and initial states. We fix the coupling parameter at J1 = 1.1 and tune the gain-loss parameter γ.
First, we tune the gain-loss parameter at γ = 3.5, which corresponds to a topological phase with the emergence of localized corner states, and initialize the walker at one corner (x, y) = (1, 16) of the lattice. As shown in Figure 6A, since the initial state has a large overlap with the corner state, the most part of the walker’s wave packet remains localized near the same corner as increasing the step of the quantum walk. Then, we tune the gain-loss parameter at γ = 0.5, which corresponds to a trivial phase. Since the absence of the localized corner states, the probability distributions of the walker spread ballistically into the bulk with increasing the step of the quantum walk, see Figure 6B. When the initial states are prepared at other three corners of the lattice, the numerical results are similar and thus are not shown here. Finally, we retune the gain-loss parameter at γ = 3.5 and initial the walker at the bulk (x, y) = (8, 8) of the lattice. Similiar to the second case, the walker’s wave packet extends into the bulk as increasing the step of the quantum walk, which further confirms the absence of the non-Hermitian skin effects with the nonlocalization of the bulk states, see Figure 6C.
FIGURE 6
6 Discussion and Conclusion
We first give a more detailed illustration of the topological phase diagram. In Section 4, we only show part of the complete topological phase diagram for simplicity. Actually, the complete phase diagram has a period of π in the J1 direction, see Figure 7. Moreover, when the coupling parameter J1 converges to qπ/2 (q is an integer), the value of γc will go to infinity. Especially, when the coupling parameter is exactly fixed at J1 = qπ/2, a 100%-coupling is present for each unitary sub-evolutionary process governed by (i = 1, 2, 3, 4). In such a case, a single walker does not feel the gain or loss after one step of the eight-step non-unitary coinless DTQW. Thus, the system with 100%-coupling is always trivial without corner states no matter the value of the gain-loss parameter γ.
FIGURE 7
In summary, we have constructed a two-dimensional non-unitary coinless DTQW which exhibits nontrivial second-order non-Hermitian topology. We have shown second-order non-Hermitian topological phase diagram characterized by polarizations. Finally, we have shown that the corner states can be observed through the probability distributions. Our work suggests that the coinless DTQW is a potential platform to explore novel non-Hermitian higher-order topological quantum phases, and may shed light on the ongoing exploration of topologically protected quantum information processing.
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The original contributions presented in the study are included in the article/Supplementary Material, further inquiries can be directed to the corresponding author.
Author contributions
YM: theory and writing.
Conflict of interest
The author declares that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.
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Summary
Keywords
coinless discrete-time quantum walks, non-Hermitian, higher-order topology, corner states, photonic waveguides
Citation
Meng Y (2022) Topological Corner States in Non-Unitary Coinless Discrete-Time Quantum Walks. Front. Phys. 10:861125. doi: 10.3389/fphy.2022.861125
Received
24 January 2022
Accepted
31 March 2022
Published
03 May 2022
Volume
10 - 2022
Edited by
Dong-Hui Xu, Chongqing University, China
Reviewed by
Jianming Wen, Kennesaw State University, United States
Pragya Shukla, Indian Institute of Technology Kharagpur, India
Updates
Copyright
© 2022 Meng.
This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.
*Correspondence: Ya Meng, mengya418@163.com
This article was submitted to Condensed Matter Physics, a section of the journal Frontiers in Physics
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