Reduction of one-dimensional non-Hermitian point-gap topology by correlations

Introduction–. Topological insulators and superconductors have been extensively analyzed in these 15 years Thouless et al. (1982); Hatsugai (1993); Kitaev (2001); Kane and Mele (2005a, b); Qi et al. (2008); Hasan and Kane (2010); Qi and Zhang (2011); Sato and Fujimoto (2016). In particular, considerable efforts have been devoted to understanding correlation effects on the non-trivial topology, which has revealed a variety of unique phenomena. For instance, correlation effects induce topological ordered phases Tsui et al. (1982); Laughlin (1983); Jain (1989); Wen (2004); Levin and Wen (2005); Kitaev (2003, 2006) which host anyons. In addition, it has turned out that correlation effects change $\mathbb{Z}$-classification of topological superconductors at the mean-field level to $\mathbb{Z}_{8}$-classification Fidkowski and Kitaev (2010). Such a reduction phenomenon of possible topological phases for a given symmetry class has been theoretically reported for arbitrary spatial dimensions Turner et al. (2011); Fidkowski and Kitaev (2011); Yao and Ryu (2013); Ryu and Zhang (2012); Qi (2013); Lu and Vishwanath (2013); Levin (2013); Isobe and Fu (2015); Yoshida et al. (2017); Fidkowski et al. (2013); Wang et al. (2014); Metlitski et al. (2014); Wang and Senthil (2014); You and Xu (2014); Morimoto et al. (2015). Furthermore, a theoretical work Jian and Xu (2018) has elucidated that the reduction can also occur in synthetic dimensions which are considered to be fabricated in cold atoms Boada et al. (2012); Celi et al. (2014); Nakajima et al. (2016); Lohse et al. (2016). These developments reveal the ubiquity of the reduction phenomena.

Along with the above significant progress, understanding of the non-Hermitian band topology has been rapidly developed in these years Hatano and Nelson (1996); Bender and Boettcher (1998); Hu and Hughes (2011); Esaki et al. (2011); Bergholtz et al. (2021); Ashida et al. (2020). Remarkably, it has been elucidated that the point-gap topology induces novel phenomena which do not have Hermitian counterparts Martinez Alvarez et al. (2018); Kunst et al. (2018); Yao and Wang (2018); Yao et al. (2018); Yokomizo and Murakami (2019); Edvardsson et al. (2019); Gong et al. (2018); Carlström and Bergholtz (2018); Carlström et al. (2019); Zhou and Lee (2019); Kawabata et al. (2019a). A prime example is the emergence of the exceptional points Katō (1966); Rotter (2009); Berry (2004); Heiss (2012); Shen et al. (2018) (and their symmetry-protected variants Budich et al. (2019); Yoshida et al. (2019a); Okugawa and Yokoyama (2019); Zhou et al. (2019); Kawabata et al. (2019b); Kimura et al. (2019); Yoshida et al. (2020a); Delplace et al. (2021); Mandal and Bergholtz (2021)) on which the point-gap topology induces band touching for both the real and the imaginary parts. Another remarkable phenomenon is a non-Hermitian skin effect which results in extreme sensitivity to the presence/absence of boundaries Yao and Wang (2018); Lee and Thomale (2019); Borgnia et al. (2020); Zhang et al. (2020a); Okuma et al. (2020); Yoshida et al. (2020b); Okugawa et al. (2020); Kawabata et al. (2020). So far, the non-Hermitian topological band theory has been applied to a wide range of systems from quantum Jin and Song (2009a); Lee (2016); San-Jose et al. (2016); Xu et al. (2017); Jin and Song (2009b); Kozii and Fu (2017); Zyuzin and Zyuzin (2018); Yoshida et al. (2018); Shen and Fu (2018); Papaj et al. (2019); Matsushita et al. (2019); Michishita and Peters (2020) to classical systems Guo et al. (2009); Rüter et al. (2010); Regensburger et al. (2012); Zhen et al. (2015); Hassan et al. (2017); Zhou et al. (2018); Takata and Notomi (2018); Ozawa et al. (2019); Xiao et al. (2020); Weidemann et al. (2020); Hofmann et al. (2020); Helbig et al. (2020); Yoshida and Hatsugai (2019); Ghatak et al. (2020); Scheibner et al. (2020).

While most of the studies have focused on the non-interacting cases so far, correlation effects on the non-Hermitian topology attract growing interests Yoshida et al. (2019b); Xi et al. (2021); Yoshida et al. (2020c); Guo et al. (2020a); Matsumoto et al. (2020); Zhang et al. (2020b); Guo et al. (2020b); Shackleton and Scheurer (2020); Yang et al. (2021); Zhang et al. (2020c); Liu et al. (2020); Xu and Chen (2020); Pan et al. (2020); Mu et al. (2020); Lee (2021); Zhang et al. (2022); Kawabata et al. (2022); Tsubota et al. (2021); Qin et al. (2022); Orito and Imura (2022) due to the potential presence of novel non-Hermitian phenomena. Such interest of correlation effects on the non-Hermitian topology is further enhanced by recent development of technology in cold atoms which allows us to experimentally tune both dissipation and two-body interactions Tomita et al. (2017); Takasu et al. (2020). Despite these efforts, current understanding of the point-gap topology in correlated systems is quite limited. In particular, the knowledge about the reduction of the point-gap topology is limited only to zero dimension Yoshida and Hatsugai (2021), which poses the following significant question: fate of the higher dimensional point-gap topology under correlations.

We hereby tackle this question with particular focus on the one-dimensional point-gap topology in both cases of synthetic and spatial dimensions. We start with the topology in one synthetic dimension. Our analysis reveals the reduction of $\mathbb{Z}\times\mathbb{Z}\to\mathbb{Z}$ for systems with charge $\mathrm{U}(1)$ symmetry and spin-parity symmetry. We end up this conclusion by analyzing a toy model, as well as by an argument in terms of topological invariants. Furthermore, we analyze an extended Hatano-Nelson chain where such reduction results in a striking phenomenon: fragility of a skin effect against correlations in one spatial dimension.

Topological invariants in one synthetic dimension–. Firstly, we provide a generic argument in terms of topological invariants. Consider a quantum dot whose many-body Hamiltonian reads

with $\hat{H}_{0}(\theta)=\sum_{\alpha\beta}\hat{\Psi}^{\dagger}_{\alpha}h_{\alpha\beta}(\theta)\hat{\Psi}_{\beta},$ and $\hat{\Psi}^{T}=(\hat{c}_{a\uparrow},\hat{c}_{a\downarrow},\hat{c}_{b\uparrow},\hat{c}_{b\downarrow},\ldots).$ The second term $\hat{H}_{\mathrm{int}}$ denotes two-body interactions of fermions. Here, one-body Hamiltonian $h(\theta)$ is non-Hermitian and satisfies $h(2\pi)=h(0)$. The synthetic dimension is parameterized by $\theta$ which corresponds to a tunable parameter in experiments (e.g., a hopping integral in cold atoms). The operator $\hat{c}^{\dagger}_{l\sigma}$ ($\hat{c}_{l\sigma}$) creates (annihilates) a fermion in orbital $l$ ($l=a,b,\ldots$) and spin state $\sigma$ ($\sigma=\uparrow,\downarrow$). The subscript $\alpha$ labels the set of $l$ and $\sigma$.

Throughout this paper, we suppose that the Hamiltonian (1) respects the charge $\mathrm{U}(1)$ symmetry and spin-parity symmetry. Namely, the zero-dimensional Hamiltonian satisfies

with $\hat{N}=\sum_{\alpha}\hat{\Psi}^{\dagger}_{\alpha}\hat{\Psi}_{\alpha}$ and $\hat{S}^{z}=\sum_{l=a,b,\ldots}(\hat{c}^{\dagger}_{l\uparrow}\hat{c}_{l\uparrow}-\hat{c}^{\dagger}_{l\downarrow}\hat{c}_{l\downarrow})/2$.

Here, let us discuss the point-gap topology of the above system. In terms of the one-body Hamiltonian, we can introduce two distinct $\mathbb{Z}$-invariants. Because the one-body Hamiltonian $h(\theta)$ is periodic in $\theta$, we can introduce the winding number $w$

with the reference energy $\epsilon_{\mathrm{ref}}\in\mathbb{C}$. The derivative with respect to $\theta$ is denoted by $\partial_{\theta}$. The symbol “$\mathrm{tr}$” denotes the trace of a matrix (i.e., $\mathrm{tr}h=\sum_{\alpha}h_{\alpha\alpha}$).

In addition, we can introduce spin winding number $w_{\mathrm{s}}$

with $(s^{z})_{\alpha\beta}=\mathrm{sgn}(\sigma)\delta_{\alpha\beta}$. Here, $\delta_{\alpha\beta}$ takes $1$ ($0$) for $\alpha=\beta$ ($\alpha\neq\beta$), and $\mathrm{sgn}(\sigma)$ takes $1$ ($-1$) for $\sigma=\uparrow$ $(\downarrow)$. For the spin winding number the spin-parity symmetry is essential; the one-body Hamiltonian satisfies $[s^{z},h(\theta)]=0$ in the presence of the spin-parity symmetry.

The above results indicates that in the presence of the $\mathrm{U}(1)$ symmetry and the spin-parity symmetry, the point-gap topology of $h(\theta)$ is characterized by two distinct $\mathbb{Z}$-invariants.

Now, let us discuss the point-gap topology of the many-body Hamiltonian. In the presence of the spin-parity symmetry, the Hamiltonian $\hat{H}$ can be block-diagonalized with $\hat{N}$ and $\hat{P}:=(-1)^{\hat{N}_{\uparrow}}=e^{i\frac{\pi}{2}\hat{N}}e^{i\pi\hat{S}^{z}}$. Here, $\hat{N}_{\uparrow}$ denotes the operator of total number of fermions in the up-spin state. Therefore, for each Fock space, the following many-body winding number $W_{(N,P)}$ can be introduced sec ;

where $N$ and $P$ are eigenvalues of $\hat{N}$ and $\hat{P}$, respectively. The reference energy is denoted by $E_{\mathrm{ref}}\in\mathbb{C}$. By $\hat{H}_{(N,P)}$, we denote the many-body Hamiltonian for the subsector with $(N,P)$. The symbol “$\mathrm{Tr}$” denotes the trace over the subsector of the Fock space.

In the absence of interactions, eigenvalues of the many-body Hamiltonian $\hat{H}_{(N,P)}$ for each Fock space is computed from the eigenvalues of the one-body Hamiltonian $h(\theta)$ whose point-gap topology is characterized by $w$ and $w_{\mathrm{s}}$.

The above results indicate that the point-gap topology of the one-body Hamiltonian $h(\theta)$ is characterized by a set of two $\mathbb{Z}$-invariants $(w,w_{\mathrm{s}})$ while the topology of the many-body Hamiltonian $\hat{H}$ is characterized by the $\mathbb{Z}$-invariant $W_{(N,P)}$ for each sector of the Fock space. This fact implies that the non-trivial topology characterized by $(w,w_{\mathrm{s}})=(0,1)$ is trivialized by introducing the interactions, which we see below.

Two orbital quantum dot: non-interacting case–. As a specific case of Eq. (1), let us consider a two-orbital quantum dot ($l=a,b$) with a diagonal matrix $h(\theta)$ [$h_{\alpha\beta}(\theta)=h_{\alpha}(\theta)\delta_{\alpha\beta}$] whose diagonal elements are written as

Here, $\lambda$ and $\epsilon_{l\sigma}$ ($l=a,b$ and $\sigma=\uparrow,\downarrow$) are real numbers. At the non-interacting level, couplings between orbitals are absent. The one-body Hamiltonian of orbital $a$ corresponds to the small cycle limit of an extended Hatano-Nelson chain under the twisted boundary condition [see Eq. (13)].

The topology of $h(\theta)$ is characterized as $(w,w_{\mathrm{s}})=(0,1)$ for $\epsilon_{\mathrm{ref}}=0$ and $|\epsilon_{a\sigma}|<\lambda$ ($\sigma=\uparrow,\downarrow$).

To be concrete, we plot a spectral flow of the one-body Hamiltonian in Fig. 1 for $(\lambda,\epsilon_{a\uparrow},\epsilon_{a\downarrow},\epsilon_{b\uparrow},\epsilon_{b\downarrow})=(1,0.2,-0.1,0.35,-0.25)$. This figure indicate that increasing $\theta$ from $0$ to $2\pi$, an eigenvalue winds around the origin in the clockwise (counter-clockwise) direction for the subsector $\sigma=\uparrow$ ($\sigma=\downarrow$). The above numerical data support that the topology of $h(\theta)$ is characterized as $(w,w_{\mathrm{s}})=(0,1)$.

Figure 2(a) displays a spectral flow of the many-body Hamiltonian $\hat{H}_{0}$ for the subsector with $(N,P)=(2,1)$ of the Fock space. We can observe the loop structure of the spectral flow due to the topology of the one-body Hamiltonian $h(\theta)$. However, this figure indicates that $\hat{H}_{(2,1)}$ is topologically trivial [i.e., $W_{(2,1)}=0$] for $E_{\mathrm{ref}}=0$ because an eigenvalue winds around the origin in the clockwise direction, and the other eigenvalue winds around the origin in the opposite direction.

Two orbital quantum dot: interacting case–. Now, let us introduce the following two-body interaction

with real numbers $J$ and $V$. Here, “$\mathrm{h.c.}$” denotes the Hermitian conjugate of the corresponding operator [e.g., $iJ(\hat{S}^{+}_{a}\hat{S}^{-}_{b}+\mathrm{h.c.})=iJ(\hat{S}^{+}_{a}\hat{S}^{-}_{b}+\hat{S}^{-}_{a}\hat{S}^{+}_{b})$]. The spin operator $\hat{S}^{\pm}_{l}$ is defined as $\hat{S}^{\pm}_{l}=\hat{S}^{x}_{l}\pm i\hat{S}^{y}_{l}$ with $\hat{S}^{x(y)}_{l}$ being the $x$ ($y$) component of the spin operator for orbital $l$. The above two-body interactions respect charge $\mathrm{U}(1)$ symmetry and spin-parity symmetry; applying the operator $e^{i\pi\hat{S}^{z}}$ transforms the spin operators as $e^{i\pi\hat{S}^{z}}\hat{S}^{\pm}_{l}e^{-i\pi\hat{S}^{z}}=-\hat{S}^{\pm}_{l}$, meaning that the interactions respect spin-parity symmetry.

For the sake of simplicity, we focus on the subsector with $(N,P)=(2,1)$. The results for the subsector $(N,P)=(2,-1)$ are provided in Sec. S1 of Supplemental Material sup . Figure 2(b) displays the spectral flow for $V=J=1$. Remarkably, this figure indicate that the interactions open a imaginary gap; interactions split the loops which wind the origin at the non-interacting level [see Fig. 2(a)].

This fact indicates that interactions [Eq. (7)] allow a smooth deformation of the spectral flow for $\lambda=1$ to that for $\lambda=0$ without closing the point-gap at $E_{\mathrm{ref}}=0$ the latter of which is obviously trivial.

Indeed, the following deformation smoothly connects the Hamiltonian $\hat{H}(\theta)$ for $\lambda=1$ and that for $\lambda=0$: (i) Increasing $V$ from $0$ to $1$ for $\lambda=1$ and $J=V$ [see Fig. 2(c)]; (ii) Decreasing $\lambda$ from $1$ to $0$ for $J=V=\sqrt{\lambda}$ [see Fig. 2(d)]. This deformation demonstrates that the many-body Hamiltonian $\hat{H}_{(2,1)}(\theta)$ is topologically trivial.

We note that difference of the symmetry constraint of the spin-parity symmetry is essential for the imaginary gap at $\mathrm{Im}E=0$ in Fig. 2(b). As discussed above, the symmetry constraints (2), which results in $[s^{z},h(\theta)]=0$, forbids hybridization terms between two distinct subsectors with $(N,S^{z})$. In contrast, the symmetry constraint allows such hybridization terms of two-body interactions $\hat{H}_{\mathrm{int}}$. Therefore, the two-body interactions can destroy the loop structure arising from the non-trivial topology of the one-body Hamiltonian [see Figs. 2(a) and 2(b)].

For instance, in the subsector with $(N,P)=(2,1)$, the Hamiltonian is written as

Here, we have chosen the following basis vectors spanning the subsector of the Fock space $\left(\hat{c}^{\dagger}_{a\uparrow}\hat{c}^{\dagger}_{b\uparrow}|0\rangle,\ \hat{c}^{\dagger}_{a\downarrow}\hat{c}^{\dagger}_{b\downarrow}|0\rangle\right)$. The vacuum state is denoted by $|0\rangle$ (i.e., $\hat{c}_{l\sigma}|0\rangle=0$ for arbitrary $l$ and $\sigma$).

Diagonalizing the above Hamiltonian, we obtain

with $2\delta_{0}=\epsilon_{a\uparrow}+\epsilon_{b\uparrow}+\epsilon_{a\downarrow}+\epsilon_{b\downarrow}$ and $2\delta_{3}=\epsilon_{a\uparrow}+\epsilon_{b\uparrow}-\epsilon_{a\downarrow}-\epsilon_{b\downarrow}$. Equation (11) elucidates that spin-parity symmetry allows the hybridization term between states with $(N,S^{z})=(2,1)$ and $(N,S^{z})=(2,-1)$ which opens the line-gap $\mathrm{Im}[E_{+}(\theta)-E_{-}(\theta)]>0$ [see Fig. 2(b)]. In contrast, spin-parity symmetry forbids such hybridization terms for the quadratic Hamiltonian $\hat{H}_{0}$.

The above numerical results supports that the many-body Hamiltonian $\hat{H}_{(2,1)}(\theta)$ is topologically trivial despite the loop structure due to the topology of the one-body Hamiltonian with $(w,w_{\mathrm{s}})=(0,1)$.

Putting the argument in terms of the topological invariants and the above results of the toy model together, we end up with the reduction of the point-gap topology $\mathbb{Z}\times\mathbb{Z}\to\mathbb{Z}$.

Topology in one spatial dimension and fragility of a skin effect–. By analyzing an extended Hatano-Nelson chain [see Fig. 3(a)], we elucidate that correlations reduce the point-gap topology in one spatial dimension as is the case in one synthetic dimension. Remarkably, this reduction phenomenon results in fragility of a skin effect against interactions.

Let us consider an extended Hatano-Nelson chain [see Fig. 3(a)] whose Hamiltonian reads

with a diagonal matrix $h(k,\theta)$ [$h_{\alpha\beta}(k,\theta)=h_{\alpha}(k,\theta)\delta_{\alpha\beta}$, $(kL/2\pi=0,1,\ldots,L-1)$] whose diagonal elements are

Here, we have imposed the twisted boundary condition in order to compute the winding numbers (for more details, see Sec. S2.1 of Supplemental Material sup ). The operator $\hat{\Psi}_{k\alpha}$ is the Fourier transformed annihilation operator $\Psi_{k\alpha}:=\frac{1}{\sqrt{L}}\sum_{j=0,\ldots,L-1}e^{ikj}\Psi_{j\alpha}$ with $\Psi_{j}^{T}=(\hat{c}_{ja\uparrow},\hat{c}_{ja\downarrow},\hat{c}_{jb\uparrow},\hat{c}_{jb\downarrow})$. The two-body term $\hat{H}_{\mathrm{int}}$ describes the interaction between fermions in orbital $a$ and localized fermions in orbital $b$. This model also preserves charge $\mathrm{U(1)}$ and spin-parity symmetry, meaning that $\hat{H}_{\mathrm{eHN}}(\theta)$ can be block-diagonalized with $\hat{N}$ and $\hat{P}=(-1)^{\hat{N}_{a\uparrow}+\hat{N}_{b\uparrow}}$ where $\hat{N}_{l\sigma}$ and $\hat{N}$ are defined as $\hat{N}_{l\sigma}=\sum_{j}\hat{c}^{\dagger}_{jl\sigma}\hat{c}_{jl\sigma}$ and $\hat{N}=\sum_{l\sigma}\hat{N}_{l\sigma}$, respectively. The Hamiltonian $\hat{H}_{\mathrm{eHN}}(\theta)$ also commutes with $\hat{n}_{jb}=\sum_{\sigma}\hat{c}^{\dagger}_{jb\sigma}\hat{c}_{jb\sigma}$ for $j=0,\ L-1$, and thus, we suppose that orbital $b$ is occupied at both edges ($j=0,\ L-1$).

Now, we demonstrate that for $N_{a}=1$ (i.e., $N=3$), a skin effect observed at the non-interacting level is fragile against the two-body interactions due to trivial topology of the many-body Hamiltonian. Let us start with the non-interacting level. Under the twisted boundary condition, the spectral flow shows a loop structure [see Fig. 3(b)] due to the point-gap topology of one-body Hamiltonian $h(\theta):=\bigoplus_{k}h(k,\theta)$ characterized by $(w,w_{\mathrm{s}})=(0,1)$ for $\epsilon_{\mathrm{ref}}=0$. This non-trivial topology of $h$ induces the skin effect at the non-interacting level. In the presence of the boundaries, all of the eigenvalues [$E_{n}$ ($n=0,1,\ldots$)] become zero in contrast to the eigenvalues in the absence of the boundaries [see Fig. 3(b)]. In addition, a fermion in the up- (down-) spin state is localized around the right (left) edge under the open boundary condition [see Fig. 3(c)].

However, interactions destroy the above skin effect, which is due to the trivial topology of the many-body Hamiltonian $W_{(3,-1)}=0$ for $E_{\mathrm{ref}}=0$ (for computation of the many-body winding number, see Fig. S2 of Supplemental Material sup ). Because of the trivial topology, we can observe that interactions destroy the loop structure of the spectral flow and open a line gap for $J=V=1$ [see Fig. 3(d)], which is also confirmed by analysis based on the perturbation theory (see Sec. S2.2 of Supplemental Material sup ). This result verifies the reduction of the point-gap topology $\mathbb{Z}\times\mathbb{Z}\to\mathbb{Z}$ for the subsector with $(N,P)=(3,-1)$. Correspondingly, the interactions destroy the extreme sensitivity of the spectrum to the presence/absence of boundaries [see Fig. 3(d)]. Furthermore, in the presence of interactions fermions extend to the bulk even under the open boundary condition [see Fig. 3(e)]. This result is also intuitively understood as follows: while the one-body term $\hat{H}_{0}(\theta)$ localizes the fermions in orbital $a$ and the up- (down-) spin state around the right (left) edge, the two-body interactions $\hat{H}_{\mathrm{int}}$ flip their spins at edges, which suppresses the effects of boundaries. The above results indicate that the skin effect observed at the non-interacting level is fragile against the two-body interactions. Our numerical calculation indicate that such fragility of the skin effect is also observed for the case of many fermions in orbital $a$ [see Fig. 4]. More detailed data are provided in Sec. S2.3 of Supplemental Material sup .

Summary and discussion–. We have analyzed correlation effects on the one-dimensional point-gap topology in both cases of synthetic and spatial dimensions. Our analysis has elucidated that the reduction $\mathbb{Z}\times\mathbb{Z}\to\mathbb{Z}$ occurs for systems of synthetic one dimension with charge $\mathrm{U(1)}$ symmetry and spin-parity symmetry. This conclusion is obtained by the argument of topological invariants as well as by explicit analysis of the toy model. Furthermore, we have also analyzed the extended Hatano-Nelson chain which exhibits striking correlation effects: correlations reduce the point-gap topology and destroy the skin effect at the non-interacting level.

The above discoveries shed new light on non-Hermitian correlated systems and open up a new directions of researches on non-Hermitian topological physics. For instance, the above results imply the possibility of similar reduction phenomena for other cases of symmetry and dimensions. As well as the above theoretical open question, experimental observation of the reduction is also a significant issue to be addressed. We expect that cold atoms are promising candidate where interactions and non-Hermiticity can be tuned in experiments.

Acknowledgements–. This work is also supported by JSPS KAKENHI Grant No. JP21K13850 and also by JST CREST, Grant No. JPMJCR19T1, Japan.

Supplemental Materials:
Reduction of one-dimensional non-Hermitian point-gap topology by correlations