Ex) Article Title, Author, Keywords
Ex) Article Title, Author, Keywords
New Phys.: Sae Mulli 2023; 73: 1178-1182
Published online December 31, 2023 https://doi.org/10.3938/NPSM.73.1178
Copyright © New Physics: Sae Mulli.
Shu Hamanaka1*, Kazuki Yamamoto1,2†, Tsuneya Yoshida1‡
1Department of Physics, Kyoto University, Kyoto 606-8502, Japan
2Department of Physics, Tokyo Institute of Technology, Meguro, Tokyo 152-8551, Japan
Correspondence to:*hamanaka.shu.45p@st.kyoto-u.ac.jp
†yamamoto@phys.titech.ac.jp
‡yoshida.tsuneya.2z@kyoto-u.ac.jp
This is an Open Access article distributed under the terms of the Creative Commons Attribution Non-Commercial License(http://creativecommons.org/licenses/by-nc/3.0) which permits unrestricted non-commercial use, distribution, and reproduction in any medium, provided the original work is properly cited.
In the non-Hermitian system, the complex-valued spectrum leads to a unique topology known as the point-gap topology. Recent studies within a single-particle picture have shown that the nontrivial point-gap topology induces the non-Hermitian skin effect (NHSE), which denotes significant dependence of the eigenstates and eigenvalues on boundary conditions. However, most of the studies focus on the noninteracting case, and little is known about the effect of interactions on the point-gap topology. In this paper, we study the one-dimensional non-Hermitian fermionic system with a complex-valued interaction. By performing the numerical diagonalization, we demonstrate that the strong interaction induces the NHSE. Moreover, we show that the many-body topological invariant takes a nonzero value corresponding to the NHSE.
Keywords: Condensed Matter Physics, Non-Hermitian Physics, Topological Phase
Recently, non-Hermitian physics is attracting much attention both theoretically and experimentally[1]. The non-Hermiticity allows eigenvalues of the Hamiltonian to be complex, which leads to the two types of gap structures in non-Hermitian systems[1]. One is the line-gap and the other is the point-gap. Although the line-gap topology is the generalization of the Hermitian topology, the point-gap topology is unique to non-Hermitian systems[1,2]. In the single-particle picture, it has been elucidated that the nontrivial point-gap topology causes the non-Hermitian skin effect (NHSE)[4, 5], which exhibits the extreme sensitivity of eigenvalues and eigenstates to boundary conditions[6]. Specifically, under open boundary conditions (OBC), an extensive number of eigenstates are localized near the boundary. The NHSE has been experimentally observed in classical and quantum systems such as mechanical metamaterials[7] and ultracold atoms[8].
Although a lot of studies have been conducted to explore non-Hermitian topological phenomena, most of them have been restricted to noninteracting cases, and several studies have focused on the NHSE in interacting systems[9-12]. As representative examples, Refs.[9] and[10] treat the Hamiltonian with asymmetric hopping, which causes the NHSE even in single-particle systems. We note that although the Liouvillian skin effect induced by on-site interactions was proposed in Ref.[13], the eigenstates and eigenvalues of the non-Hermitian effective Hamiltonian were not analyzed in detail. Thus, the possibility of the NHSE induced by onsite interactions without asymmetric hopping remains unclear.
In this paper, we investigate the effect of onsite interactions on the point-gap topology. Specifically, we find that onsite interactions induce the NHSE in one-dimensional fermionic systems. Correspondingly, we observe that the topological invariant defined under the twisted boundary condition takes a nonzero value. The findings of this study suggest the crucial importance of interactions in the non-Hermitian topology.
In this section, we provide the model, which exhibits the NHSE in interacting systems. We study a Falicov-Kimball model[14] with a complex-valued interaction. The Hamiltonian is written by
Here,
where
Here,
In this section, we discuss how the many-body winding number has been treated in non-Hermitian interacting systems. Although the many-body winding number is fixed to zero under the inversion symmetry [see Eq. (7)], the winding number can be finite for our model because the first term of Eq. (4a) breaks such symmetry constraints.
First of all, when we compute the topological invariant, we have to block diagonalize the Hamiltonian. The non-Hermitian Hamiltonian
By making use of the relation given in Eq. 5, we can obtain the block-diagonalized Hamiltonian
Here,
Next, we discuss the symmetry constraint on the topological invariant defined in Eq. (6). When the Hamiltonian has the inversion symmetry
the winding number
Finally, we show that the Falicov-Kimball model given in Eqs. (1) and (2) breaks the inversion symmetry even for the noninteracting case (
This leads to the constraints
In this section, we numerically study the effects of interactions on the point-gap topology. In particular, we demonstrate that onsite interactions make the point-gap topology nontrivial and induce the NHSE. We note that when the system is noninteracting, i.e.
First, we show the case where the total number of particles in the up-spin state equals to one, i.e.
Second, we discuss another case where the total number of particles in the up-spin state equals to two. As shown in Fig. 2(a), the spectrum exhibits a complicated structure. We observe that the spectrum forms three clusters. Moreover, we find that the spectrum under OBC collapses inside the spectrum under PBC. This is the qualitatively similar behavior to that of the
In this paper, we have studied the effect of interactions on the point-gap topology. Specifically, by analyzing the fermionic system with a complex-valued interaction, we have demonstrated that onsite interactions induce the NHSE. Correspondingly, we have shown that the winding number takes a nonzero value. Our results shed light on the significant role of onsite interactions in non-Hermitian many-body systems.
It is worth noting that some symmetries and two or more spatial dimensions lead to various kinds of skin effects such as symmetry-protected skin effects and higher-dimensional skin effects in noninteracting systems[4]. Analyzing the effect of interactions on these novel skin effects is left for a future study. We also note that the generalization of our results to the Lindblad master equation is interesting.
We thank Masaki Tezuka for the useful discussion. S.H. was supported by SCES2019 Fund and WISE Program. K.Y. was supported by JSPS KAKENHI Grant-in-Aid for JSPS fellows Grant No. JP20J21318 and JP23K19031. K.Y. acknowledges the support by the research grant by Yamaguchi Educational and Scholarship Foundation. This work was supported by JSPS KAKENHI Grants No. JP22H05247 and No. JP21K13850.