Non-Hermitian physics

Abstract

A review is given on the foundations and applications of non-Hermitian classical and quantum physics. First, key theorems and central concepts in non-Hermitian linear algebra, including Jordan normal form, biorthogonality, exceptional points, pseudo-Hermiticity, and parity-time symmetry, are delineated in a pedagogical and mathematically coherent manner. Building on these, we provide an overview of how diverse classical systems, ranging from photonics, mechanics, electrical circuits, and acoustics to active matter, can be used to simulate non-Hermitian wave physics. In particular, we discuss rich and unique phenomena found therein, such as unidirectional invisibility, enhanced sensitivity, topological energy transfer, coherent perfect absorption, single-mode lasing, and robust biological transport. We then explain in detail how non-Hermitian operators emerge as an effective description of open quantum systems on the basis of the Feshbach projection approach and the quantum trajectory approach. We discuss their applications to physical systems relevant to a variety of fields, including atomic, molecular and optical physics, mesoscopic physics, and nuclear physics with emphasis on prominent phenomena and subjects in quantum regimes, such as quantum resonances, superradiance, the continuous quantum Zeno effect, quantum critical phenomena, Dirac spectra in quantum chromodynamics, and nonunitary conformal field theories. Finally, we introduce the notion of band topology in complex spectra of non-Hermitian systems and present their classifications by providing the proof, first given by this review in a complete manner, as well as a number of instructive examples. Other topics related to non-Hermitian physics, including nonreciprocal transport, speed limits, nonunitary quantum walk, are also reviewed.

Keywords:

• Non-Hermitian systems
• nonunitary dynamics
• photonics
• mechanics
• acoustics
• electrical circuits
• open quantum systems
• quantum optics
• quantum many-body physics
• dissipation
• topology
• bulk-edge correspondence
• topological invariants
• edge modes
• nonreciprocal transport
• quantum walk

Acknowledgments

We are grateful to Kyosuke Adachi, Yohei Fuji, Takeshi Fukuhara, Hosho Katsura, Kyogo Kawaguchi, Flore K. Kunst, Naoto Nagaosa, Masaya Nakagawa, Shuta Nakajima, Daiki Nishiguchi, Takahiro Sagawa, Kazuki Sone, Kazuaki Takasan, Kazumasa A. Takeuchi, and Hidenori Tanaka for fruitful discussions.

We also thank Konstantin Bliokh, Jan Carl Budich, Johan Carlström, Aashish Clerk, Yaoming Chu, Joshua Feinberg, Yan V. Fyodorov, Ryo Hanai, Oleg Kirillov, Yannis Kominis, Lijun Lang, Tao Liu, Jamir Marino, Igor Mekhov, Benedetto Militello, Ali Mostafazadeh, Franco Nori, Roberto Onofrio, Lucas Sa, Sujit Sarkar, Masatoshi Sato, Mathias Scheurer, Henning Schomerus, Boris Shklovskii, Gal Shmuel, Luis Foa Torres, Constantinos Valagiannopoulos, Lorenza Viola, Bao Wang, Zhesen Yang, Qi-Bo Zeng, Yujun Zheng for useful email correspondence.

Y.A. acknowledges support from the Japan Society for the Promotion of Science through Grant No. JP16J03613, and Harvard University for hospitality. Z.G. was supported by MEXT. We acknowledge support through KAKENHI Grant No. JP18H01145 and a Grant-in-Aid for Scientific Research on Innovative Areas “Topological Materials Science” (KAKENHI Grant No. JP15H05855) from the Japan Society for the Promotion of Science.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Supplemental data

Supplemental data and all the Appendices for this article can be accessed http://dx.doi.org/10.1080/00018732.2021.1876991.

Notes

1 The condition number of an invertible matrix A measures the extent to which the error in the solution $x$ to a linear equation $Ax=b$ is amplified by the error in $b$ [59]. Denoting the latter error as $e$, the ratio of the relative error is upper bounded by the condition number: $(\parallel A^{-1}e\parallel /\parallel A^{-1}b\parallel )/(\parallel e\parallel /\parallel b\parallel )=(\parallel A^{-1}e\parallel /\parallel e\parallel )(\parallel A{A}^{-1}b\parallel /\parallel A^{-1}b\parallel )\le \parallel A^{-1}\parallel \parallel A\parallel =\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{d}\left(A\right)$.

2 Conversely, we can also impose the normalization condition on the left eigenvectors, but again the right eigenvectors are not normalized in general.

3 This is always possible for a sufficiently small $\parallel E\parallel$ even if M is not diagonalizable, since the eigenvalues are continuous functions in terms of the parameters of a matrix [61], although their derivatives can be singular.

4 One can readily check the relation $N_j=(M-{\lambda }_{j}I)P_j$ from this relation.

5 The second inequality results from the fact that ${\mathrm{\Delta }}_{C_j}\le \left[\underset{z\in C_j}{max}\rho \right(\frac{1}{M-zI}){]}^{-1}\le (\underset{z\in C_j}{max}|z-{\lambda }_j{|}^{-1}{)}^{-1}=\underset{z\in C_j}{min}|z-{\lambda }_j|$, where we have used the inequality in Equation (39(39) $\rho \left(M\right)\le \underset{1\le j\le n}{max}{\sigma }_j,\rho \left(M\right)=\underset{k\to \mathrm{\infty }}{lim}\parallel M^k{\parallel }^{\frac{1}{k}}.$(39) ).

6 These are the points around which $\lambda \left(\kappa \right)$ behaves asymptotically as $(\kappa -{\kappa }_0{)}^{\alpha }$, with α being a non-integer rational number. Since the eigenvalues should be bounded, we have $\alpha >0$.

7 If the degree of degeneracy at ${\kappa }_0$ decreases due to the lift of degeneracy between ${\lambda }_1\left(\kappa \right)$ and ${\lambda }_2\left(\kappa \right)$, we have ${\lambda }_{12}\left(\kappa \right)\equiv {\lambda }_1\left(\kappa \right)-{\lambda }_2\left(\kappa \right)\ne 0$ at ${\kappa }_0$ but ${\lambda }_{12}\left(\kappa \right)=0$ near ${\kappa }_0$. This is impossible for continuous ${\lambda }_{12}\left(\kappa \right)$.

8 The characteristic polynomial (4(4) $p_M\left(\lambda \right)\equiv \mathrm{d}\mathrm{e}\mathrm{t}(\lambda I-M)=0,$(4) ) is given by $(x-\gamma {)}^3+\gamma (x-\gamma {)}^2+\gamma$, whose zeros are given by $x=y+\gamma$ with $y^3+\gamma y^2+\gamma =0$. When $|\gamma |$ is small, then both x and y are of the order of $O\left({\gamma }^{1/3}\right)$ due to the fact that γ and $\gamma y^2$ are higher-order small quantities.

9 Rigorously speaking, we have to assume that all the mass points share the same mass. Nevertheless, even if the diagonal mass matrix M is not proportional to the identity, by redefining $x$ as $\sqrt{M}x$, we can recover Equation (131(131) $\ddot{x}=-Dx+\mathrm{\Gamma }\dot{x},$(131) ).

10 These forces take the form $f=\sum _{j,k,l}symbol\mathrm{\Omega }\times \dot{\phantom{a}}x=\sum _{j,k,l}{ϵ}_{jkl}{\mathrm{\Omega }}_j{\dot{x}}_k{e}_l$, where $\sum _j{ϵ}_{jkl}{\mathrm{\Omega }}_j$ indeed reverses its sign upon the interchange of k and l.

11 After Fourier transformation, we obtain $\begin{array}{rl}\left[D\right(symbolk){]}_{\alpha \text{β}}& =\sum _{\gamma ,symbol{r}^{″},symbolr}{Q}_{symbolr-symbol{r}^{″},\alpha \gamma }{Q}_{symbol{r}^{″}-symbol{r}^{\prime },\gamma \text{β}}^{\mathrm{T}}{e}^{-isymbolk\cdot (symbolr-symbol{r}^{\prime })}=\sum _{\gamma ,symbol{r}^{″}}Q(symbolk{)}_{\alpha \gamma }{Q}_{symbol{r}^{\prime }-symbol{r}^{″},\text{β}\gamma }{e}^{isymbolk\cdot (symbol{r}^{\prime }-symbol{r}^{″})}\\ & =\sum _{\gamma }Q\left(symbolk{)}_{\alpha \gamma }Q\right(symbolk{)}_{\text{β}\gamma }^{\ast }=\left[Q\right(symbolk\left)Q\right(symbolk{)}^{†}{]}_{\alpha \text{β}}.\end{array}$

12 The crease angle refers to the angle formed by a crease (marked in red and blue in Figure 15b) and the vertical axis (dashed line in the same figure). Here we take ${\text{β}}_1={\text{β}}_2=\text{β}$ and ${\gamma }_1={\gamma }_2=\gamma$.

13 This result can be derived from the spherical law of cosines. For the specific origami structure in Figure 15(b), we have $\begin{array}{rl}& -\cos (\alpha +\gamma )\cos (\alpha +\text{β})+\sin (\alpha +\gamma )\sin (\alpha +\text{β})\cos f_1\left(j\right)\\ & =-\cos (\alpha -\gamma )\cos (\alpha -\text{β})+\sin (\alpha -\gamma )\sin (\alpha -\text{β})\cos f_2\left(j\right)\end{array}$ and $\begin{array}{rl}& -\cos (\alpha +\gamma )\cos (\alpha +\text{β})+\sin (\alpha +\gamma )\sin (\alpha +\text{β})\cos f_2\left(j\right)\\ & =-\cos (\alpha -\gamma )\cos (\alpha -\text{β})+\sin (\alpha -\gamma )\sin (\alpha -\text{β})\cos f_1(j+1).\end{array}$

14 While Girko made remarkable progress on generalizing Ginibre's specific result for Gaussian distributions [418], his original proof was found to be flawed [419]. This conjecture was finally proved by Terence Tao and Van H. Vu in 2010 [420].

15 This Hamiltonian should be interpreted as $H_{PP}$ according to the notation of the previous subsection; for the sake of simplicity, we here denote it as H.

16 To emphasize this point, we introduce the tilde to denote a resonant mode in Equation (212(212) $|〈{\tilde{\mathrm{\Psi }}}_r|e^{-iHt}|{\tilde{\mathrm{\Psi }}}_r〉{|}^2=e^{-\mathrm{\Gamma }t}(t>0).$(212) ) and distinguish it from a vector in the Hilbert space.

17 The phase lapse refers to an abrupt change in the phase of the transmission wave.

18 We note that this model can be realized as a special case of Equation (260(260) ${\hat{H}}_{\mathrm{e}\mathrm{f}\mathrm{f}}=\sum _{mn}\left(J_{mn}^{\mathrm{r}}-i{J}_{mn}^{\mathrm{i}}\right)\left({\hat{b}}_m^{†}{\hat{b}}_n+\mathrm{H}.\mathrm{c}.\right)+\frac{U_{\mathrm{r}}-i{U}_{\mathrm{i}}}{2}\sum _m{\hat{b}}_m^{†}{\hat{b}}_m^{†}{\hat{b}}_m{\hat{b}}_m,$(260) ), including the position-dependent one-body loss. To see this, one can use mappings ${\sigma }^{-}\to b$ and ${\sigma }^z\to b^{†}b-1/2$ with b ($b^{†}$) being the annihilation (creation) operator of hard-core bosons ($b^{†2}=0$). A concrete experimental realization in ultracold atoms can be found in  [124].

19 This fact can be inferred from the facts that the master equation (257(257) $\frac{\mathrm{d}\mathbb{E}\left[{\hat{\rho }}_S\right]}{\mathrm{d}t}=-i\left({\hat{H}}_{\mathrm{e}\mathrm{f}\mathrm{f}}\mathbb{E}\left[{\hat{\rho }}_S\right]-\mathbb{E}\left[{\hat{\rho }}_S\right]{\hat{H}}_{\mathrm{e}\mathrm{f}\mathrm{f}}^{†}\right)+\sum _{m=1}^M{\hat{L}}_m\mathbb{E}\left[{\hat{\rho }}_S\right]{\hat{L}}_m^{†}.$(257) ) permits the solution $\rho \propto 1$ (identity matrix) when $L_j$ is Hermitian (or normal) for all $j=1,2,\dots ,M$ and that, in most cases, this is the unique solution due to the Perron-Frobenius theorem (see discussions below Equation 170(170) $\frac{\mathrm{d}p}{\mathrm{d}t}=Wp,$(170) ).

20 Here we note that τ in general depends on N. This condition is to be contrasted with the weak non-Hermiticity regime $0<\underset{N\to \mathrm{\infty }}{lim}N(1-\tau )<\mathrm{\infty }$, which implies the presence of a scaling relation $\tau \sim 1-C/N$ for a large N, where C>0 is a constant independent of N.

21 Rigorously speaking, the full conformal-symmetry algebra consists of two copies of Virasoro algebra, which correspond to both holomorphic and anti-holomorphic infinitesimal transformations. What we discuss here is actually the diagonal CFT, whose representations of the two Virasoro algebras coincide with each other.

22 Assuming $\parallel 〈x+r|W_{x\alpha }〉\parallel \le c{e}^{-\kappa |r|}$ $\mathrm{\forall }\alpha$ with $c,\kappa >0$, we have $\parallel 〈x_1|P|x_2〉\parallel \le \sum _{x\in \mathbb{Z}}\sum _{\alpha \in \mathrm{o}\mathrm{c}\mathrm{c}.}{c}^2{e}^{-\kappa (|x-x_1|+|x-x_2|)}=n_{\mathrm{o}\mathrm{c}\mathrm{c}}{c}^2[|x_1-x_2|+\tanh (2\kappa \left)\right]e^{-\kappa |x_1-x_2|}$, where $n_{\mathrm{o}\mathrm{c}\mathrm{c}}$ is the number of occupied bands. Here we use $\parallel \cdot \parallel$ instead of $|\cdot |$ to emphasize the presence of internal states.

23 We remark, however, that the Lieb–Robinson bound can in general be violated in nonunitary dynamics conditioned on measurement outcomes, including non-Hermitian evolutions. A crucial difference is that the state vector should be normalized during the quench dynamics [79,201] (see Section 4.2.2 and Figure 27b).

24 Here $|\cdot )$ refers to a state in the virtual Hilbert space, which should be distinguished from a physical state denoted by $|\cdot 〉$.

25 This is a typical situation in the literature but only provides a sufficient condition. We may loose the condition to be no slower than a power-law decaying long-range hopping $H_{symbolra,symbol{r}^{\prime }{a}^{\prime }}\sim |symbolr-symbol{r}^{\prime }{|}^{-\alpha }$ with $\alpha >d$, where d is the spatial dimension.

26 By this, we mean that we do not consider Hopf insulators [991], which are not stable against additional bands.

27 Here “≃” means the Morita equivalence, which is the isomorphism, up to tensoring ${\mathrm{M}}_n\left(\mathbb{C}\right)$ or ${\mathrm{M}}_n\left(\mathbb{R}\right)$, i.e. up to an $n\times n$ matrix space with complex or real entries. For example, $\mathrm{C}{\ell }_{p+2}\left(\mathbb{C}\right)={\mathrm{M}}_2\left(\mathbb{C}\right)\otimes \mathrm{C}{\ell }_p\left(\mathbb{C}\right)$ and $\mathrm{C}{\ell }_{p+n,q+n}\left(\mathbb{R}\right)={\mathrm{M}}_{2^n}\left(\mathbb{R}\right)\otimes \mathrm{C}{\ell }_{p,q}\left(\mathbb{R}\right)$, where we provisionally use the symbol “=” for the usual isomorphism.

28 We emphasize that this is a statement for a general situation but does not rule out the possibility of introducing well-defined ground states in several setups, such as non-Hermitian systems with real spectra or long-lived states.

29 In the special case of diagonalizable matrices, the statement is shown in  [239] (see Section 5.5.2 for further details).

30 The construction of $V\left(symbolk\right)$ is not unique since it can be replaced by $V\left(symbolk\right)W\left(symbolk\right)$, where $W\left(symbolk\right)$ is invertible and continuous in $symbolk$ and satisfies $\left[W\right(symbolk),P(symbolk\left)\right]=0$. However, there is a natural construction in terms of a given continuous path $P_{\lambda }\left(symbolk\right)$ ($\lambda \in [0,1]$): $V\left(symbolk\right)=\hat{\mathrm{\Lambda }}{e}^{{\int }_0^{1}d\lambda [1-2P_{\lambda }(symbolk\left)\right]{\mathrm{\partial }}_{\lambda }{P}_{\lambda }\left(symbolk\right)}$, where $\hat{\mathrm{\Lambda }}$ indicates the ordering of λ, such that $P_1\left(symbolk\right)=V\left(symbolk\right)P_0\left(symbolk\right)V(symbolk{)}^{-1}$ and the smoothness in $symbolk$ follows that of $P_{\lambda }\left(symbolk\right)$.

31 While $f^{\prime }\left(z\right)$ in Equation (339(339) $\begin{array}{rl}\mathrm{\Lambda }\left(L\right)& =\left\{f\right(z):|z|=1,z\in \mathbb{C}\},\\ \mathrm{\Lambda }\left(T\right)& =\mathrm{\Lambda }\left(L\right)\cup \{\lambda \in \mathbb{C}\mathrm{\setminus }\mathrm{\Lambda }(L):w(\lambda )\ne 0\},w\left(\lambda \right)\equiv {\oint }_{|z|=1}\frac{\mathrm{d}z}{2\pi i}\frac{f^{\prime }\left(z\right)}{f\left(z\right)-\lambda },\\ {\mathrm{\Lambda }}_{ϵ}\left(T\right)& =\underset{L\to \mathrm{\infty }}{lim}{\mathrm{\Lambda }}_{ϵ}\left(T_L\right)\Rightarrow \mathrm{\Lambda }\left(T\right)=\underset{ϵ\to 0}{lim}\underset{L\to \mathrm{\infty }}{lim}{\mathrm{\Lambda }}_{ϵ}\left(T_L\right),\end{array}$(339) ) might diverge at some z under this condition, this does affect the well-definedness of the integral since it simply counts the phase winding of $f\left(z\right)-\lambda$, which is well-defined (i.e. bounded) along the path.

32 Since $u_{\mu }$ has D elements and there are totally $2R$ $u_{\mu }$'s, the total number of rows is given by 2RD = 2M.

33 We can assume $U_X{U}_X^{\ast }=e^{i\varphi }I\iff U_X^{\ast }{U}_X=e^{-i\varphi }I$ from the unitarity. Then $U_X^{\ast }=e^{i\varphi }{U}^{†}=e^{-i\varphi }{U}^{†}$, leading to $e^{i\varphi }=\pm 1$.

34 If a matrix H is Hermitian and has a unique polar decomposition H = UP, then we have $H=H^{†}=P{U}^{†}=U^{†}\left(UP{U}^{†}\right)$. From the uniqueness, we have $U=U^{†}$ and thus $U^2=U{U}^{†}=I$, implying that the spectrum of U consists of $\pm 1$.

35 Regarding the case with exceptional points, it is stated in  [239] that “they can be pair-annihilated without closing a line gap”. However, how this can be achieved, especially under various symmetry constraints, has remained to be clarified.

36 See the Supplemental Material of [1024] for further details.

37 To confirm the Hermiticity, we have only to use the fact that $〈n_t|{\mathrm{\partial }}_t{n}_t〉$ is purely imaginary and ${\mathrm{\partial }}_{t}I={\mathrm{\partial }}_t(\sum _n|n_t〉〈n_t|)=\sum _n|{\mathrm{\partial }}_t{n}_t〉〈n_t|+\sum _n|n_t〉〈{\mathrm{\partial }}_t{n}_t|=0$ (I: identity).

38 A coin operator refers to an operator acting nontrivially only on the internal (spin) degree of freedom. Here $\mathbb{I}$ denotes the identity acting on the spatial degree of freedom.

39 Precisely speaking, what is realized in the loss-only photonic quantum walk experiment [811] is a passive PT symmetry, in the sense that Equation (398(398) $U\left(k\right)={\sigma }^z\left[U\right(k{)}^{-1}{]}^{\ast }{\sigma }^z.$(398) ) is valid only after a background loss in $U\left(k\right)$ is removed. The experimental data shown in Figure 48(c) and (d) are corrected in this manner.

Additional information

Funding

This work was supported by Japan Society for the Promotion of Science [JP16J03613] and Japan Society for the Promotion of Science [JP18H01145] and Japan Society for the Promotion of Science [JP15H05855].