††thanks: $\\#$G. Yang and Y.-K. Li contributed equally.

# Complex semiclassical theory for non-Hermitian quantum systems

Guang Yang$\\#$ International Center for Quantum Materials, School of Physics,
Peking University, Beijing, 100871, China  Yong-Kang Li$\\#$ International
Center for Quantum Materials, School of Physics, Peking University, Beijing,
100871, China  Yongxu Fu  International Center for Quantum Materials, School
of Physics, Peking University, Beijing, 100871, China  Zhenduo Wang
International Center for Quantum Materials, School of Physics, Peking
University, Beijing, 100871, China  Yi Zhang<EMAIL_ADDRESS>International Center for Quantum Materials, School of Physics, Peking
University, Beijing, 100871, China

###### Abstract

Non-Hermitian quantum systems exhibit various novel characteristics, such as
exotic topological phenomena and non-Hermitian skin effect, yet their
exponential localization and non-orthogonal eigenstates stranglehold analysis
and calculations. The potent semiclassical theory, which offers physical
pictures and quantitative results for various Hermitian systems, remains
limited in non-Hermitian systems. Here, we establish a complex semiclassical
theory for non-Hermitian quantum systems as a straightforward analytical
continuation of their quantization conditions and physical variables, such as
momentum, position, time, and energy, with consistent physical interpretations
as wave packets with additional factors. Generally applicable to both
continuous and lattice models, such complex semiclassical theory yields
accurate complex energy spectra (e.g., Landau levels) and quantum states,
presenting a helpful bedrock and an alternative perspective towards important
physics such as nontrivial topology, skin effect, transport, dissipative
behaviors, etc.

_Introduction_ —The non-Hermitian skin effect (NHSE), where most eigenstates
localize at the open boundary, reveals that the physics of non-Hermitian
quantum systems may differ significantly from Hermitian systems [1, 2, 3, 4,
5, 6, 7, 8, 9]. Examples of non-Hermitian origins include dissipation in
quantum optics [10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20], open systems in
cold atom systems [21, 22, 23, 24, 25], and finite quasiparticle lifetime in
condensed matters with interactions or disorder [26, 27, 28, 29], etc.
Generalizing the recent successes of topological materials and theories to
non-Hermitian quantum systems, physicists have discovered a series of non-
Hermitian topological phases [30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41,
42, 43, 44, 45, 46]. These have brought non-Hermitian quantum systems to the
research frontier.

The semiclassical theory [47, 48, 49] is a conventional yet powerful framework
capable of offering quantitative property explorations and explicit physical
pictures for quantum materials and models, such as the quantum Hall effect in
two and three dimensions [50, 51, 52, 53, 54, 55, 56], the quantum anomalous
[57, 58, 59] and spin Hall effect [60, 61, 62, 63], and other topological
phenomena. Given the emerging difficulties in non-Hermitian quantum systems
due to their eigenstates’ non-orthogonality and exponential amplification [7,
64, 39], such a semiclassical theory would be helpful, especially on large
systems. Nevertheless, despite various semiclassical studies with Berry
curvature [65], external electric field [66], real spectra [67], etc., a
semiclassical theory for non-Hermitian quantum systems in generic
electromagnetic fields is still lacking.

In this letter, we present and study a complex semiclassical theory applicable
to non-Hermitian quantum systems. In particular, we generalize the
conventional equations of motion (EOMs) and the quantization conditions for
Hermitian quantum systems via an analytical continuation of the physical
variables, e.g., (the center of mass) momentum and position. At first glimpse,
it may seem odd to assign complex values to such quantities with classical
implications; however, we show a consistent physical interpretation in terms
of wave packets with augmented factors naturally connected to non-unitary
dynamics. Complex energy levels emerge wherever exists closed orbits with
satisfactory quantization conditions. We also establish a straightforward
strategy to search and establish closed orbits in the complex domain, as well
as approximations for the corresponding quantum eigenstates. We demonstrate
our conclusions with continuous and lattice model examples, where the complex
semiclassical theory yields consistent results with their quantum benchmarks
and breeds insights into non-Hermitian topology, skin effect, dissipative
physics, and more.

_Complex semiclassical theory_ —Conventionally, the semiclassical theory for a
Hermitian quantum system begins with the EOMs of a wave packet following its
quantum dynamics [49]:

$\displaystyle\dot{\bf r}$ $\displaystyle=$ $\displaystyle{\bf
v}=\partial\epsilon/\partial{\bf p},$ $\displaystyle\dot{\bf p}$
$\displaystyle=$ $\displaystyle{\bf F}=-\partial\epsilon/\partial{\bf r},$ (1)

where energy $\epsilon$ depends on the momentum $\bf p$ and position $\bf r$
of the wave packet’s center of mass (COM). Such a wave-packet description
remains a valid approximation as long as the length scale of energy variation
surpasses its extent, e.g., when the external electromagnetic fields are weak.
Once the resulting trajectory forms a closed orbit that satisfies the Bohn-
Sommerfeld quantization condition:

$\oint{\bf p}\cdot d{\bf r}=(n+\gamma)h,n\in\mathbb{Z},$ (2)

we have achieved a valid state at the corresponding energy level. For
simplicity, we neglect the Berry curvature [49, 65], and $\gamma=1/2$. As the
solutions and analysis of the EOMs are essentially classical, the
semiclassical theory offers excellent efficiency and transparency. For
instance, the cycles perpendicular to a magnetic field vividly depict various
topological phenomena, such as the quantum Hall effect [50, 51, 52, 53, 54,
55, 56].

Commonly, the variables in the EOMs undertake real values, as they are
classical quantities or expectation values of Hermitian observables.
Interestingly, to apply the semiclassical theory on non-Hermitian quantum
systems, we need to analytically continue the physical variables such as $\bf
r$, $\bf p$, $t$, and $\epsilon$ in Eqs. 1 and 2 to complex values. Hence the
name complex semiclassical theory. The WKB approximation offers a special case
with $\bf p\in\mathbb{C}$ and $\bf r\in\mathbb{R}$. We will discuss the
interpretation of complex variables, derivations, and various consequences
later.

Given the non-unitary nature of the time evolution in non-Hermitian quantum
systems, it is natural to encompass complex $\epsilon$ and $t$. Under a
specific initialization, we map out the trajectory following Eq. 1, where the
complex energy $\epsilon$ remains conserved. Once we have located a closed
orbit - a problem in differential equations known as the stable orbit [68, 69,
70, 71, 72, 73, 74, 75], we evaluate the complex integral $\oint{\bf p}\cdot
d{\bf r}$, and, according to Eq. 2, deem whether $\epsilon$ and the underlying
states are physical.

However, this is easier said than done: the complex nature of $t$ may render
the search for a closed orbit aimless and rather challenging, especially given
that the complex period $T$ is initially unknown and may differ between the
orbits. We propose the following strategy: (1) upon an open cycle that fails
to form a close orbit (yet reasonably approaches the initial point), we can
evaluate the differences $\Delta x$ and $\Delta p$ between initial and current
variables; (2) given $\dot{x}$ and $\dot{p}$ of the current step, we choose
the complex phase of $dt$, so that $dx$ and $dp$ reduce $\Delta x$ and $\Delta
p$; (3) the variables at the next step inch towards initialization, and so on
until we reach a closed orbit. As complex integrals do not depend on their
particular paths (but instead on their windings around certain “fixed
points”), such make-shift closed orbits provide equal bases for evaluating
vital quantities, such as the period $T$ and $\int p\cdot dx$ accumulated over
the cycle. This approach does not require fine-tuning in initialization or
perturbative treatment of the non-Hermitian terms. Nevertheless, once $T$
becomes available, there lay cycles along $t=\lambda T$, $\lambda\in[0,1]$,
which are commonly smoother and more aesthetic.

_Toy model examples_ —First, let’s consider a quadratic model in 1D 111The
linear and constant terms are trivial as we can absorb them in operator
definitions as displacements.:

$\hat{H}=\alpha\hat{x}^{2}+\beta\hat{p}^{2}+\eta(\hat{x}\hat{p}+\hat{p}\hat{x})/2,$
(3)

where $\alpha,\beta,\eta\in\mathbb{C}$, and $\hat{H}$ is non-Hermitian in
general. We set $\hbar=1$ here and afterward. Quantum mechanically, we can
obtain the energy spectrum $\epsilon_{n}=(n+1/2)\omega$,
$\omega=\sqrt{4\alpha\beta-\eta^{2}}$ is generally complex-valued,
$n\in\mathbb{Z}$, and the corresponding eigenstates $\hat{a}|0\rangle=0$,
$|n+1\rangle=\hat{b}^{\dagger}|n\rangle/\sqrt{n+1}$ with the modified ladder
operators $[\hat{a},\hat{b}^{\dagger}]=1$:

$\hat{a}=\frac{2\alpha\hat{x}+(\eta+i\omega)\hat{p}}{2\sqrt{\alpha\omega}},\hat{b}^{\dagger}=\frac{2\alpha\hat{x}+(\eta-i\omega)\hat{p}}{2\sqrt{\alpha\omega}},$
(4)

which we compare with the complex semiclassical theory.

Semiclassically, we begin with the energy relation $\epsilon=\alpha
x^{2}+\beta p^{2}+\eta xp$, and the resulting EOMs following Eq. 1 are exactly
solvable 222A heuristic derivation of such orbits from a quantum wave-packet
perspective is in the supplemental materials [78].:

$\displaystyle x(t)$ $\displaystyle=$ $\displaystyle z_{1}e^{i\omega
t}+z_{2}e^{-i\omega t},$ $\displaystyle p(t)$ $\displaystyle=$
$\displaystyle\frac{(i\omega-\eta)z_{1}}{2\beta}e^{i\omega
t}-\frac{(i\omega+\eta)z_{2}}{2\beta}e^{-i\omega t},$ (5)

where $z_{1},z_{2}\in\mathbb{C}$ are complex constants determined by the
initial conditions. These orbits are closed with a period of $2\pi/\omega$ for
$\omega t\in\mathbb{R}$ (Fig. 1a); for a generic $t$, however, $e^{\pm i\omega
t}$ introduces an exponential amplitude, and the trajectories end up as open
spirals (Fig. 1b). Fortunately, after the trajectory reaches a good spot, we
can achieve a closed orbit by choosing the complex phase of $dt$ so that $da$
is along $a(0)-a(t)$, $a=(x+ip)/\sqrt{2}$ for the following steps (Fig. 1c).

Figure 1: Following the EOMs of the model in Eq. 3, the semiclassical
trajectories form (a) a closed loop for $\omega t\in\mathbb{R}$, or (b) an
open spiral for $t\in\mathbb{R}$, where we may (c) close the loop by adjusting
$dt$’s phase. (d) The corresponding trajectories in the complex $t$ plane show
that the different closed orbits with identical winding (around its fixed
point $x=p=0$) are equivalent and eventually reach the same period $T$, which
is complex given the non-Hermicity and the complex nature of $\omega$.
$\alpha=1+0.5i$, $\beta=1$, and $\eta=0.7+0.3i$.

We can then evaluate the physical quantities over these closed orbits:
$\epsilon=z_{1}z_{2}\omega^{2}/\beta$ and $\oint p\cdot dx=2\pi
z_{1}z_{2}\omega/\beta$. The quantization condition requires
$z_{1}z_{2}=(n+1/2)\beta/\omega$, which in turn indicates the valid energy
levels are $\epsilon_{n}=(n+1/2)\omega$ and consistent with the quantum
benchmark. We include further examples, such as continuous models with higher-
order potential and kinetic terms, in the supplemental materials [78].

_Derivation and interpretation_ —Given a quantum state $|\Phi\rangle$ at $t=0$
and a non-Hermitian Hamiltonian $\hat{H}$, we define the time dependence of a
Hermitian observable $\hat{O}$’s expectation value as:

$\langle\hat{O}\rangle(t)=\langle\Phi|U^{-1}(t)\hat{O}U(t)|\Phi\rangle,$ (6)

where $U(t)=\exp(-i\hat{H}t)$. An alternative to Eq. 6 is
$\langle\Phi|U^{\dagger}(t)\hat{O}U(t)|\Phi\rangle$, which guarantees the
expectation values to be real and coincides with Eq. 6 for Hermitian
$\hat{H}$. In comparison, our choice of Eq. 6 has the following advantages:
(1) $\langle\Phi|U^{-1}(t)\cdot\mathbf{1}\cdot U(t)|\Phi\rangle=1$ remains
constant, thus normalization does not evolve in time. (2) Intimately related
to the quantum-classical crossover and the conservation laws, the Heisenberg
picture $\dot{\hat{O}}=-i[\hat{O},\hat{H}]/\hbar$ remains valid. (3) Eq. 6 is
consistent with the expectation value $\langle nL|\hat{O}|nR\rangle$ and the
time evolution operator $U(t)=\sum_{n}e^{-iE_{n}t}|nR\rangle\langle nL|$ under
the biorthogonal basis $\langle nL|mR\rangle=\delta_{nm}$ [79]. However, there
is a trade-off: given a non-Hermitian $\hat{H}$, $\dot{\hat{O}}$ is no longer
necessarily Hermitian for a Hermitian $\hat{O}$; therefore, the expectation
values $\langle\hat{O}\rangle$ corresponding to the physical variables cannot
remain real-valued and need analytical continuation to the complex domains. We
further discuss our conventions in Ref. [78].

The semiclassical theory originates from quantum states’ wave-packet
representation. We can describe a wave packet with center-of-mass (COM)
position $x_{0}$ and momentum $p_{0}$ as
$|x_{0},p_{0}\rangle=\hat{W}(x_{0},p_{0})|0\rangle$, where
$\hat{W}(x_{0},p_{0})=\exp[i(p_{0}\hat{x}-x_{0}\hat{p})]$ is a translation
operator and $|0\rangle$ is a wave packet with zero average momentum and
position. The generalization to $x_{0},p_{0}\in\mathbb{C}$ is valid and
straightforward:

$\displaystyle\langle x_{0},p_{0}|\hat{x}|x_{0},p_{0}\rangle$ $\displaystyle=$
$\displaystyle x_{0},$ $\displaystyle\langle
x_{0},p_{0}|\hat{p}|x_{0},p_{0}\rangle$ $\displaystyle=$ $\displaystyle
p_{0},$ (7) $\displaystyle\langle x_{0},p_{0}|x_{0},p_{0}\rangle$
$\displaystyle=$ $\displaystyle 1,$

establishing a mapping between a wave packet $|x_{0},p_{0}\rangle$ and its
complex-valued physical variables $(x_{0},p_{0})$.

What is the physical interpretation of such a wave packet? Defining
$x^{\prime}_{0}$ and $x^{\prime\prime}_{0}$ ($p^{\prime}_{0}$ and
$p^{\prime\prime}_{0}$) as the real and imaginary parts of $x_{0}$ ($p_{0}$),
we note:

$\displaystyle|x_{0},p_{0}\rangle$ $\displaystyle=$
$\displaystyle\exp(-\frac{x^{2}_{0}+p^{2}_{0}}{4})\sum_{n=0}\frac{(x_{0}+ip_{0})^{n}}{2^{n/2}\sqrt{n!}}|n\rangle$
(8) $\displaystyle=$ $\displaystyle A(x_{0},p_{0})|x_{r},p_{r}\rangle,$

is a wave packet with real-valued COM position
$x_{r}=x^{\prime}_{0}-p^{\prime\prime}_{0}$ and momentum
$p_{r}=p^{\prime}_{0}+x^{\prime\prime}_{0}$, but with an additional factor:

$\displaystyle A(x_{0},p_{0})$ $\displaystyle=$
$\displaystyle\exp(\frac{x_{r}^{2}+p_{r}^{2}-x_{0}^{2}-p_{0}^{2}}{4})\in\mathbb{C}.$
(9)

We can also convert the real-valued COM variables and the overall factor into
complex $x_{0}$ and $p_{0}$, as both parameterizations employ four real-valued
parameters. The variable modulus of $|x_{0},p_{0}\rangle$ is a direct
signature of non-conserved probability in non-Hermitian quantum systems.

Next, we analyze the dynamics of such a wave packet. After evolution under a
non-Hermitian Hamiltonian $\hat{H}(t)$ for an infinitesimal time
$\tau\rightarrow 0$, the wave packet’s COM becomes:

$\displaystyle\langle\hat{x}\rangle(t+\tau)$ $\displaystyle=$
$\displaystyle\langle
x(t),p(t)|e^{i\hat{H}\tau}\hat{x}e^{-i\hat{H}\tau}|x(t),p(t)\rangle$ (10)
$\displaystyle\approx$ $\displaystyle x(t)-i\tau\langle
x(t),p(t)|[\hat{x},\hat{H}]|x(t),p(t)\rangle$ $\displaystyle=$ $\displaystyle
x(t)+\tau\langle\frac{\partial\hat{H}}{\partial\hat{p}}\rangle(t),$
$\displaystyle\langle\hat{p}\rangle(t+\tau)$ $\displaystyle=$ $\displaystyle
p(t)-\tau\langle\frac{\partial\hat{H}}{\partial\hat{x}}\rangle(t),$

which characterizes the wave packet $|x(t+\tau),p(t+\tau)\rangle$, if it
survives to $t+\tau$. Therefore, we establish the evolution of the complex
variables:

$\displaystyle\dot{x}(t)$ $\displaystyle=$
$\displaystyle\left[x(t+\epsilon)-x(t)\right]/\tau$ (11) $\displaystyle=$
$\displaystyle\langle
x(t),p(t)|\left({\partial\hat{H}}/{\partial\hat{p}}\right)|x(t),p(t)\rangle$
$\displaystyle=$ $\displaystyle\langle
0|\frac{\partial\hat{H}}{\partial\hat{p}}\left(\hat{x}+x(t),\hat{p}+p(t)\right)|0\rangle\approx\partial
E/\partial p,$ $\displaystyle\dot{p}(t)$ $\displaystyle=$
$\displaystyle\langle
x(t),p(t)|\left(-{\partial\hat{H}}/{\partial\hat{x}}\right)|x(t),p(t)\rangle$
$\displaystyle=$ $\displaystyle-\langle
0|\frac{\partial\hat{H}}{\partial\hat{x}}\left(\hat{x}+x(t),\hat{p}+p(t)\right)|0\rangle\approx-\partial
E/\partial x,$

which leads to the EOMs in Eq. 1 with complex variables. In the last steps, we
have neglected the contributions from $\langle
0|\hat{x}^{n}\hat{p}^{m}|0\rangle$ in the expansions of
$\partial\hat{H}/\partial\hat{p}$ and $\partial\hat{H}/\partial\hat{x}$, which
is valid when the extent of the orbit, such as $x(t)$ and $p(t)$,
predominantly surpasses that of the wave packet, such as $\langle
0|\hat{x}^{2}|0\rangle$ \- the standard semiclassical prerequisites. The
approximation becomes exact for a quadratic $\hat{H}$, since
$\partial\hat{H}/\partial\hat{p}$ and $\partial\hat{H}/\partial\hat{x}$ are
linear in $\hat{x}$ and $\hat{p}$, where Eq. 7 applies.

$n$ | Quantum | Semiclassical | Outcome
---|---|---|---
0 | $|0\rangle_{0}$ | $|0\rangle_{0}$ | $|0\rangle_{0}$
1 | $(\hat{a}^{\dagger}_{0}+\kappa\hat{a}_{0})|0\rangle_{0}$ | $\hat{a}^{\dagger}_{0}|0\rangle_{0}$ | $|1\rangle_{0}$
2 | $(\hat{a}^{\dagger}_{0}+\kappa\hat{a}_{0})^{2}|0\rangle_{0}$ | $[(\hat{a}^{\dagger}_{0})^{2}+\kappa]|0\rangle_{0}$ | $\sqrt{2}|2\rangle_{0}+\kappa|0\rangle_{0}$
3 | $(\hat{a}_{0}^{\dagger}+\kappa\hat{a}_{0})^{3}|0\rangle_{0}$ | $[(\hat{a}^{\dagger}_{0})^{3}+3\kappa\hat{a}_{0}^{\dagger}]|0\rangle_{0}$ | $\sqrt{6}|3\rangle_{0}+3\kappa|1\rangle_{0}$
Table 1: The lowest eigenstates of the non-Hermitian quantum system in Eq. 3
obtained via quantum and semiclassical approaches compare consistently. These
eigenstates are non-orthogonal due to the non-Hermicity. We have employed the
orthonormal basis $|m\rangle_{0}$,
$a^{\dagger}_{0}a_{0}|m\rangle_{0}=m|m\rangle_{0}$:
$\hat{a}_{0}=(\hat{x}+i\hat{p})/\sqrt{2}$.
$\kappa=\eta/i\omega=(\alpha-\beta)/(\alpha+\beta)$.

The complex-variable description $|x_{0},p_{0}\rangle$ of the wave packet is
more redundant in terms of the choices of $x_{0},p_{0}\in\mathbb{C}$ for a
wave packet at $x_{r},p_{r}\in\mathbb{R}$, just with different $A$ factors.
Nevertheless, the orbits following different conventions describe physics
consistently. For example, we demonstrate the universal evolution of
$x_{r},p_{r}$ and $A$ for various initializations that differ merely in their
conventions in the supplemental materials [78]. Likewise, the orbits in Eq. 24
that differ by a conformal mapping $z_{1}e^{i\omega t}\rightarrow
z^{\prime}_{1}e^{i\omega t}$ are equivalent to each other. Note that
$z_{2}e^{-i\omega t}\rightarrow z^{\prime}_{2}e^{-i\omega t}$ under the
constraint of the quantization condition
$z_{1}z_{2}=z_{1}^{\prime}z_{2}^{\prime}=(n+1/2)\beta/\omega$.

The complex semiclassical theory also allows us to approximate the quantum
states from the corresponding closed orbits. For instance, let’s consider the
orbits in Eq. 24, and set $2\alpha=\omega-i\eta$ for simplicity: see the
second column of Table 1 for the resulting quantum eigenstates.
Semiclassically, we can take the limit $z_{2}\rightarrow\infty$ without loss
of generality, then safely neglect $z_{1}\rightarrow 0$ for an isotropic orbit
and wave packet: $x(t)=z_{2}e^{-i\omega t}$, $p(t)=z_{2}e^{-i\omega
t}\alpha/i\beta$, and the following COM location and $A$ factor:

$\displaystyle\frac{x(t)+ip(t)}{\sqrt{2}}$ $\displaystyle=$
$\displaystyle\frac{x(t)}{\sqrt{2}}(1+\frac{\alpha}{\beta}),$
$\displaystyle-\frac{x^{2}(t)+p^{2}(t)}{4}$ $\displaystyle=$
$\displaystyle\frac{x^{2}(t)}{4}(\frac{\alpha^{2}}{\beta^{2}}-1).$ (12)

Averaging the wave packets $|x(t),p(t)\rangle$ with the geometric phase
$\exp(in\omega t)$ over the closed orbit, we obtain 333Note that
$x^{2}_{r}+p_{r}^{2}=|z_{2}(1+\alpha/\beta)|$ is a trivial constant.:

$\displaystyle|n\rangle$ $\displaystyle\propto$
$\displaystyle\int_{0}^{T}dt\cdot\exp(in\omega
t)\exp[\frac{x^{2}}{4}(\frac{\alpha^{2}}{\beta^{2}}-1)+\frac{x}{\sqrt{2}}(1+\frac{\alpha}{\beta})\hat{a}^{\dagger}_{0}]|0\rangle_{0}$
(13) $\displaystyle\propto$ $\displaystyle\oint dx\cdot x^{-n-1}e^{\kappa
x^{2}/2+x\hat{a}^{\dagger}_{0}}|0\rangle_{0},$

where $\kappa=\eta/i\omega=(\alpha-\beta)/(\alpha+\beta)$, and $|0\rangle_{0}$
is an isotropic wave packet following $\hat{a}_{0}|0\rangle_{0}=0$,
$\hat{a}_{0}=(\hat{x}+i\hat{p})/\sqrt{2}$. We have also renormalized
$(1+\alpha/\beta)x/\sqrt{2}\rightarrow x$ in the second line. Following the
residual theorem, the contour integral returns the $x^{n}$’s coefficient in
the expansion of $\exp(x\hat{a}^{\dagger}+\kappa x^{2}/2)$, which compare
consistently with the quantum benchmarks as the examples in Table 1. However,
an analytical solution for the target quantum state may not be readily
available and generally need numerical calculations, e.g., integration under
the real-space basis for the wave function $\psi_{n}(x)$, and approximations
on the wave-packet shapes; see examples in the supplemental materials [78].

_Lattice model examples_ —The complex semiclassical theory also applies to
lattice models, for which we consider the following 2D non-Hermitian model for
illustration:

$\hat{H}=\sum_{\bf r}\eta c^{\dagger}_{{\bf r}-\hat{x}+\hat{y}}c_{\bf
r}-c^{\dagger}_{{\bf r}+\hat{x}}c_{\bf r}-c^{\dagger}_{{\bf r}-\hat{x}}c_{\bf
r}-Vc^{\dagger}_{{\bf r}+\hat{y}}c_{\bf r}-Vc^{\dagger}_{{\bf
r}-\hat{y}}c_{\bf r},$ (14)

with an external magnetic field of $Q/2\pi$ flux quantum per lattice
plaquette. Using the Landau gauge, we can re-express the model as a non-
Hermitian 1D lattice model:

$\displaystyle\hat{H}$ $\displaystyle=$ $\displaystyle\sum_{x}(\eta
e^{iQx}-1)c^{\dagger}_{x}c_{x+1}-c^{\dagger}_{x+1}c_{x}$ (15)
$\displaystyle-2V\cos(Qx)c^{\dagger}_{x}c_{x},$

which we diagonalize for its quantum spectrum and eigenstates and compare with
the complex semiclassical theory. We note that the diagonalization of non-
Hermitian Hamiltonians may suffer accumulated numerical errors [7, 64, 39],
especially for relatively large systems, e.g., sizable magnetic unit cells in
a small magnetic field, where the semiclassical theory excels. Also, these
model examples are beyond the framework of non-Bloch band theories [1, 3, 5].

Figure 2: For each complex energy $\epsilon$, the resulting (left) real and
(right) imaginary parts of $\oint p\cdot dx$, over its closed orbits, if any,
determine the satisfiability of the quantization condition. The translucent
planes denote $(2n+1)\pi$, $n\in\mathbb{Z}$ for the real part and $0$ for the
imaginary part, which pinpoints the physical $\epsilon_{n}$ in excellent
consistency with the quantum spectrum (red dots). The vertical values of the
red dots are set for straight comparisons. $V=1.0$, $\eta=0.1i$, and $Q=0.02$.

Semiclassically, the energy relation and EOMs are:

$\displaystyle\epsilon$ $\displaystyle=$ $\displaystyle-2\cos p-2V\cos Qx+\eta
e^{i(Qx+p)},$ $\displaystyle\dot{x}$ $\displaystyle=$ $\displaystyle 2\sin
p+i\eta e^{i(Qx+p)},$ $\displaystyle\dot{p}$ $\displaystyle=$
$\displaystyle-Q[2V\sin Qx+i\eta e^{i(Qx+p)}].$ (16)

We track the complex $(x,p)$ space trajectories via finite-time steps and
evaluate $\oint p\cdot dx$ over the closed orbits [69, 70, 71, 72, 73, 74,
75], generally obtainable using our strategy mentioned earlier. In Fig. 2, we
summarize $\oint p\cdot dx$ versus the complex energy $\epsilon$ over
nontrivial closed orbits. The real and imaginary parts of the quantization
condition $\oint p\cdot dx=(n+1/2)h$ offer two constraints and a discrete
series of energy levels $\epsilon_{n}$ in the complex $\epsilon$ plane, which
compare consistently with the quantum benchmark. We can also obtain
approximate quantum eigenstates corresponding to the qualified orbits. Further
details and additional lattice-model examples are in the supplemental
materials [78].

The complex semiclassical theory also offers straightforward recipes for
studying phases and phase transitions in non-Hermitian quantum systems. For
example, the NHSE emerges naturally from the complex semiclassical theory with
a soft potential $V(x)$ characterizing boundaries, as illustrated in the
supplemental materials [78]. Further, the interplay between non-Hermitian
topological physics and NHSE has received much attention recently[67, 81].
From a complex-semiclassical-theory perspective, these two scenarios are
separated by a Lifshitz transition: the former cases possess nontrivial closed
loops $p(T)=p(0)$ accompanied by discrete Landau levels and localized quantum
Hall states; in contrast, the latter cases defy a periodic closure, but rather
settle with open trajectories $p(T)=p(0)+{\bf G}$, where ${\bf G}=2\pi m\neq
0$ is a reciprocal lattice vector - while the lattice momentum nominally reset
to initialization, the wave packet keeps traversing across the system in the
real space.

Figure 3: The complex semiclassical theory characterizes the phases (bright
blue and pink regions) and phase boundary (black curve) for the non-Hermitian
lattice model in Eq. 14 based on the existence or absence of nontrivial closed
orbits. The dots and their color scales denote the quantum energy spectrum
$\epsilon_{n}$ and each corresponding eigenstate’s ratio of IPRs upon doubling
the system size. The insets illustrate the schematic trajectories of the two
phases. $V=0.3$, $\eta=0.1i$.

Navigating over the complex energy $\epsilon$, we summarize in Fig. 3 the
phase diagram of the non-Hermitian lattice model in Eq. 14. For benchmarking,
we also examine the quantum eigenstates and spectrum of the model. In
particular, we calculate the inverse participation ratio (IPR):
$(\sum_{x}|\psi(x)|^{4})^{-1}$ [82] for each eigenstate and characterize its
localization properties with the ratio of IPRs as we enlarge the system. The
results’ consistency is satisfactory: the quantum eigenstate retains a robust
IPR, thus its localization 444The calculations of the eigenstates and their
IPRs may be unstable due to accumulated numerical errors for non-Hermitian
Hamiltonians and sensitivity to boundary conditions in the delocalized region.
Despite unreliable values, the (loss of) IPRs’ robustness signals
delocalization., in and only in the quantum Hall region determined by the
complex semiclassical theory; more implicitly, sharp kinks appear in the
complex spectrum near the predicted boundary. Interestingly, we can further
generalize the models to complex-valued magnetic field $Q$, whose physics is
also nicely captured by the complex semiclassical theory (Fig. 3 and Ref.
[78]).

_Discussions_ —We have established a systematic complex semiclassical theory
with an analytical continuation of physical variables applicable to non-
Hermitian quantum systems. Previous work has also studied EOMs with complex-
valued position and momentum [67], yet is restricted to real-valued time and
energy levels. We have also provided a physical interpretation for such
complex variables in terms of non-unitary evolution, and a strategy for closed
orbits and approximate states, and demonstrated applicability and
effectiveness on continuous and lattice models. The complex semiclassical
theory may also apply beyond the framework of non-Hermitian Hamiltonians. For
example, it is straightforward to incorporate into the EOMs friction, such as
${\bf F_{\mu}}=-\mu{\bf p}$, which forgoes closed orbits conventionally in
real time; the complex semiclassical theory circumvents such difficulties; see
examples in Ref. [78]. Therefore, it may potentially provide alternative
approaches and insightful pictures toward dissipative or driven quantum
systems.

The generalization to multi-band scenario is also a meaningful direction. On
the one hand, it introduces Berry phases, which may possess complex values and
exotic properties, into the problems. On the other hand, it allows more forms
of complex energy relations so that the EOMs may involve multi-valued
functions and nontrivial complex geometries such as branch cuts. While we have
only considered models with relatively simple complex spaces and closed
orbits’ winding around certain fixed points, the possibilities of emergent
physics in a more general complex semiclassical theory seem fascinating.

_Acknowledgments_ —We acknowledge helpful discussions with Ryuichi Shindou,
Junren Shi, and Haoshu Li. We also acknowledge support from the National Key
R&D Program of China (No.2022YFA1403700) and the National Natural Science
Foundation of China (No.12174008 & No.92270102).

##  I Appendix A: More Continuous model examples

###  I.1 Closed orbits of non-Hermitian harmonic oscillators from a quantum
perspective

In this subsection, we study the quantum evolution of the wave packets under
the Hamiltonian in Eq. 3 in the main text, which offers an alternative,
quantum perspective of the orbits in Eq. 5 in the main text. As we have shown
in the main text, we can re-express the Hamiltonian:

$\displaystyle\hat{H}$ $\displaystyle=$
$\displaystyle\omega\left(b^{\dagger}a+1/2\right),$ $\displaystyle\hat{a}$
$\displaystyle=$
$\displaystyle\frac{2\alpha\hat{x}+(\eta+i\omega)\hat{p}}{2\sqrt{\alpha\omega}},$
$\displaystyle\hat{b}^{\dagger}$ $\displaystyle=$
$\displaystyle\frac{2\alpha\hat{x}+(\eta-i\omega)\hat{p}}{2\sqrt{\alpha\omega}},$
(17)

where $\omega=\sqrt{4\alpha\beta-\eta^{2}}$ is generally complex. Although
$\hat{b}^{\dagger}\neq\hat{a}^{\dagger}$, we still have the commutation
relation $[\hat{a},\hat{b}^{\dagger}]=1$, which leads to the ladder algebra:

$\displaystyle\hat{b}^{\dagger}|n\rangle$ $\displaystyle=$
$\displaystyle\sqrt{n+1}|n+1\rangle,$ $\displaystyle\hat{a}|n\rangle$
$\displaystyle=$ $\displaystyle\sqrt{n}|n-1\rangle,$
$\displaystyle\hat{b}^{\dagger}\hat{a}|n\rangle$ $\displaystyle=$
$\displaystyle n|n\rangle,$

where $\hat{a}|0\rangle=0$. Therefore, $|n\rangle$ and
$\epsilon_{n}=(n+1/2)\omega$ are the eigenstates and energy eigenvalues of
$\hat{H}$, respectively, $n\in\mathbb{Z}$.

Then, we can define a wave-packet state
$|z_{0}\rangle=e^{-|z_{0}|^{2}/2}e^{z_{0}\hat{b}^{\dagger}}|0\rangle$,
$z_{0}\in\mathbb{C}$, whose time evolution under $\hat{H}$ is straightforward
to obtain: $e^{-i\omega t/2}|z_{t}\rangle$, where $z_{t}=e^{-i\omega t}z_{0}$.
Consequently, the wave packet’s center of mass (COM) shows the following time
dependence:

$\displaystyle\langle\hat{x}\rangle(t)$ $\displaystyle=$ $\displaystyle\langle
0|e^{-z_{t}\hat{b}^{\dagger}}e^{|z_{t}|^{2}/2}\hat{x}e^{-|z_{t}|^{2}/2}e^{z_{t}\hat{b}^{\dagger}}|0\rangle$
(18) $\displaystyle=$ $\displaystyle\langle
0|\hat{x}-z_{t}[\hat{b}^{\dagger},\hat{x}]|0\rangle$ $\displaystyle\approx$
$\displaystyle\frac{(\omega+i\eta)z_{t}}{2\sqrt{\alpha\omega}}=z_{2}e^{-i\omega
t},$ $\displaystyle\langle\hat{p}\rangle(t)$ $\displaystyle=$
$\displaystyle\langle 0|\hat{x}-z_{t}[\hat{b}^{\dagger},\hat{p}]|0\rangle$
$\displaystyle\approx$ $\displaystyle\frac{-2i\alpha
z_{t}}{2\sqrt{\alpha\omega}}=-\frac{(i\omega+\eta)z_{2}}{2\beta}e^{-i\omega
t},$

where $z_{2}=(\omega+i\eta)z_{0}/2\sqrt{\alpha\omega}$. We have neglected
$\langle 0|\hat{x}|0\rangle$ and $\langle 0|\hat{p}|0\rangle$ as the scale and
shape of the wave packet are much less significant than those of the orbit in
a semiclassical approximation. These results are fully consistent with the
$z_{2}$ terms in Eq. 5 in the main text obtained via semiclassical equations
of motion (EOMs). Note that we have employed inverse, instead of Hermitian
conjugate, in the expectation values $\langle\hat{x}\rangle(t)$ and
$\langle\hat{p}\rangle(t)$, which can now take on complex values.

To obtain the $z_{1}$ terms, on the other hand, we may start with a different
basis of ladder states: $\hat{b}^{\dagger}|0\rangle^{\prime}=0$,
$\hat{a}|n\rangle^{\prime}=\sqrt{n+1}|n+1\rangle^{\prime}$. The corresponding
wave-packet state evolves with an $e^{i\omega t}$ factor independently from
the $z_{2}$ terms. As we show in the main text and later in the next
subsection, the parameterizations are redundant with both $z_{1}$ and $z_{2}$,
and orbits that differ by a conformal mapping $z_{1}e^{i\omega t}\rightarrow
z^{\prime}_{1}e^{i\omega t}$, $z_{2}e^{-i\omega t}\rightarrow
z^{\prime}_{2}e^{-i\omega t}$ are equivalent to each other.

###  I.2 Redundancy and consistency of complex variables

In Eqs. 8 and 9 of the main text, we have established the connection between
the complex variables $x_{0},p_{0}\in\mathbb{C}$ and the wave packet
$|x_{r},p_{r}\rangle$ with real variables $x_{r},p_{r}\in\mathbb{R}$ and an
additional factor $A\in\mathbb{C}$. Therefore, there exists redundancy in
$x_{0},p_{0}$ with the same $x_{r},p_{r}\in\mathbb{R}$. They follow the same
equality $x_{r}+ip_{r}=x_{0}+ip_{0}$ and differ by their $A$ factors, which
can be attributed to quantum states’ normalization and phase conventions. In
this subsection, we demonstrate that the multiple orbits under different
conventions describe the same physics consistently.

Figure 4: We initialize two semiclassical orbits with $x=2,p=1$ and
$x^{\prime}=2.3+0.3i,p^{\prime}=0.7+0.3i$, which correspond to the same
$x_{r},p_{r}\in\mathbb{R}$, and (left) their wave packets’ COMs evolve
consistently, while (right) the ratio between their $A$ factors carries over
as a constant, therefore constitute merely conventions.

For example, we consider the continuous model in Eq. 3 in the main text and
map out the trajectories following various initializations, which correspond
to identical $x_{r},p_{r}\in\mathbb{R}$. Their subsequent evolution is
universal and consistent with each other, differing merely in conventions -
the difference in their $A$ factors carries through the entire orbits as
constant ratios; see Fig. 4. The results and conclusions generalize to other
models and conventions straightforwardly. Thus, the redundancy over complex
variables does not lead to self-contradiction.

###  I.3 Model examples with higher-order terms

In this subsection, we examine another continuous model - a non-Hermitian
quantum model with higher-order $\hat{x}$:

$\hat{H}=\beta\hat{x}^{4}+\gamma\hat{p}^{2}+\alpha\hat{a}^{2},$ (19)

where $\alpha,\beta,\gamma\in\mathbb{C}$. With the ladder operators
$\hat{x}=\frac{1}{\sqrt{2}}\left(\hat{a}+\hat{a}^{\dagger}\right)$ and
$\quad\hat{p}=\frac{i}{\sqrt{2}}\left(\hat{a}^{\dagger}-\hat{a}\right)$, we
can re-express the Hamiltonian:

$\displaystyle\hat{H}$ $\displaystyle=$
$\displaystyle\frac{\beta}{4}[\hat{a}^{4}+\left(\hat{a}^{\dagger}\right)^{4}+4\hat{a}^{\dagger}\hat{a}^{3}+4(\hat{a}^{\dagger})^{3}\hat{a}+6(\hat{a}^{\dagger})^{2}+6\hat{a}^{2}$
(20) $\displaystyle+$ $\displaystyle
6n^{2}+6n+3]-\frac{\gamma}{2}[(\hat{a}^{\dagger})^{2}+\hat{a}^{2}-2n-1]+\alpha\hat{a}^{2},$

which we diagonalize in the occupation number basis
$\hat{n}|n\rangle=n|n\rangle$ numerically (with a truncation at sufficiently
large $n$) for the quantum benchmarks.

Figure 5: Following the EOMs, the semiclassical trajectories of the model in
Eq. 19 (a) form closed loops along the $t/T\in\mathbb{R}$ direction and (b)
end up in open spirals along the $t\in\mathbb{R}$ direction, where we may
still form closed loops by adjusting $dt$, and the resulting trajectory is in
(c). (d) The corresponding trajectories in the complex $t$ plane show the
equivalence between different closed orbits reaching the complex period $T$.
$\alpha=0.3+0.5i$, $\beta=1.0+0.1i$, $\gamma=1.0-0.1i$, and $\oint p\cdot
dx=5\pi$ indicating the complex energy $\epsilon=7.04+1.24i$ qualifies for a
quantum level.

Semiclassically, we begin with the classical energy relation $\epsilon=\beta
x^{4}+\gamma p^{2}+\alpha(x+ip)^{2}/2$ and the resulting EOMs:

$\displaystyle\dot{x}$ $\displaystyle=$ $\displaystyle i\alpha
x+\left(2\gamma-\alpha\right)p,$ $\displaystyle\dot{p}$ $\displaystyle=$
$\displaystyle-4\beta x^{3}-\alpha x-i\alpha p,$ (21)

which we employ to search for the closed orbits. In the processes, the
strategy of $dt$ phase adjustment toward the initialization we mentioned in
the main text is especially helpful; see illustrations in Fig. 5. In
particular, the complex energies with qualified quantization conditions are
entirely consistent with the quantum benchmarks; see examples in Table. 2.

$n$ | 1 | 2 | 5 | 6.39+0.89i
---|---|---|---|---
$Re[\epsilon_{n}]$ | 3.622 | 7.037 | 19.861 | 10
$Im[\epsilon_{n}]$ | -0.558 | -1.240 | -3.936 | 0
$Re[\oint p\cdot dx]$ | 9.518 | 15.774 | 34.588 | 20.075
$Im[\oint p\cdot dx]$ | 0.005 | -0.001 | 0.001 | 2.781
Quantum | Yes | Yes | Yes | No
Table 2: The real and imaginary parts of the energies and geometric phases of
closed orbits following the complex semiclassical EOMs are consistent with the
quantization condition $\oint p\cdot dx=(2n+1)\pi$ for quantized, complex
energy eigenvalues for the model in Eq. 19. The parameters are
$\alpha=0.3+0.5i$, $\beta=1.0+0.1i$, and $\gamma=1.0-0.1i$.

More generally, the applicability of complex semiclassical theory applies to
non-Hermitian systems with both potential and kinetic terms beyond quadratic
order. For instance, we consider the following continuous model:

$\hat{H}=\beta\hat{p}^{4}+\gamma\hat{x}^{4}+\alpha\hat{a}^{2},$ (22)

where $\alpha,\beta,\gamma\in\mathbb{C}$. In terms of the ladder operators,
the Hamiltonian becomes:

$\displaystyle\hat{H}$ $\displaystyle=$
$\displaystyle\frac{\beta}{4}[\hat{a}^{4}+\left(\hat{a}^{\dagger}\right)^{4}+4\hat{a}^{\dagger}\hat{a}^{3}+4(\hat{a}^{\dagger})^{3}\hat{a}+6(\hat{a}^{\dagger})^{2}+6\hat{a}^{2}$
(23) $\displaystyle+$ $\displaystyle
6n^{2}+6n+3]+\frac{\gamma}{4}[\hat{a}^{4}+\left(\hat{a}^{\dagger}\right)^{4}-4\hat{a}^{\dagger}\hat{a}^{3}-4(\hat{a}^{\dagger})^{3}\hat{a}$
$\displaystyle-$ $\displaystyle
6(\hat{a}^{\dagger})^{2}-6\hat{a}^{2}+6n^{2}+6n+3]+\alpha\hat{a}^{2},$

which we also diagonalize numerically for the quantum benchmarks.

Semiclassically, the EOMs following classical energy relation are:

$\displaystyle\dot{x}$ $\displaystyle=$ $\displaystyle 4\gamma p^{3}-\alpha
p+i\alpha x,$ $\displaystyle\dot{p}$ $\displaystyle=$ $\displaystyle-4\beta
x^{3}-\alpha x-i\alpha p,$ (24)

which we use to establish closed orbits (Fig. 6). We also evaluate the
integral $\oint pdx$ of these closed orbits, which determines their
qualifications as per the quantization conditions. We demonstrate several
examples in Table. 3. Interestingly, though relatively small, there are
noticeable departures of the qualified orbits’ $\oint pdx$ from the exact
half-integer values of $2\pi$, especially at small $n$: 9.518 versus 9.425 for
$n=1$, 15.774 versus 15.708 for $n=2$. Such differences contrast sharply with
previous continuous model examples, where we have at least four or five digits
of accuracy. Physically, the model in Eq. 22 deviates drastically from
quadratic forms; thus, the semiclassical approximation suffers at small $n$,
where the wave packets’ scale and shape become relatively influential. Indeed,
such deviations vanish asymptotically at larger $n$, e.g., $n\geq 5$.

Figure 6: Following the EOMs, the semiclassical trajectories of the quartic model in Eq. 22 (a) form closed loops along the $t/T\in\mathbb{R}$ direction and (b) end up in open spirals along the $t\in\mathbb{R}$ direction, where we may still form closed loops by adjusting $dt$, and the resulting trajectory is in (c). (d) The corresponding trajectories in the complex $t$ plane show the equivalence between different closed orbits reaching the complex period $T$. $\alpha=0.5+0.3i$, $\beta=1.0+0.1i$, and $\gamma=1.0+0.1i$. The trajectories’ energy is $\epsilon=18.65+1.87i$ with a geometric phase $\oint p\cdot dx=16.0101+0.0002i\approx 5\pi$. $n$ | 1 | 5 | 10 | 3.72-0.19i
---|---|---|---|---
$Re[\epsilon_{n}]$ | 7.137 | 87.540 | 317.226 | 10
$Im[\epsilon_{n}]$ | 0.721 | 8.760 | 31.729 | 0
$Re[\oint p\cdot dx]$ | 9.903 | 34.694 | 66.047 | 11.680
$Im[\oint p\cdot dx]$ | 0.001 | 1e-04 | 3e-05 | -0.586
Quantum | Yes | Yes | Yes | No
Table 3: The real and imaginary parts of the energies and geometric phases of
closed orbits following the complex semiclassical EOMs are nearly consistent
with the quantization condition $\oint p\cdot dx=(2n+1)\pi$ for quantized,
complex energy eigenvalues for the model in Eq. 22, especially for relatively
larger values of $n$. The parameters are $\alpha=0.5+0.3i$, $\beta=1.0+0.1i$,
and $\gamma=1.0+0.1i$.

These results show that our complex semiclassical theory generally applies to
continuous models of non-Hermitian quantum systems.

##  II Appendix B: Lattice model examples

###  II.1 Approximate quantum states

In this subsection, we establish approximate quantum states of non-Hermitian
quantum systems from the complex semiclassical theory. For example, we
consider the lattice model in Eq. 15 in the main text:

$\displaystyle\hat{H}$ $\displaystyle=$ $\displaystyle\sum_{x}(\eta
e^{iQx}-1)c^{\dagger}_{x}c_{x+1}-c^{\dagger}_{x+1}c_{x}-2V\cos(Qx)c^{\dagger}_{x}c_{x},$

which we diagonalize for its quantum energy eigenvalues and eigenstates. We
set $Q=0.005$, $V=0.3$, and without loss of generality, we choose its $n=3$
eigenstate as a benchmark for comparison with the complex semiclassical
theory. This eigenstate is sufficiently localized that the non-Hermitian error
accumulation does not pose a severe numerical issue.

Semiclassically, we begin with a closed orbit with the target quantization
condition $\oint p\cdot dx=7\pi$. The orbit shapes in the complex $x$ and $p$
spaces are in the inset of Fig. 7. Its energy $\epsilon_{3}=-2.5905+0.0959i$
is consistent with the corresponding quantum energy eigenvalue.

Figure 7: The complex semiclassical theory’s approximate state $\psi_{3}(x)$
compares qualitatively well with the corresponding exact quantum eigenstate.
We consider the $n=3$ eigenstate of the non-Hermitian model in Eq.
LABEL:eq:appstateham with $Q=0.005$ and $V=0.3$. For simplicity, we only show
the amplitude of the real-space wave function. The insets show the orbit
shapes in the complex $x$ and $p$ spaces. The corresponding energy eigenvalue
is $\epsilon_{3}=-2.5905+0.0959i$, consistent between the exact
diagonalization and the complex semiclassical theory.

Further, we can extract an approximation of the quantum state as:

$\displaystyle\psi_{3}(x)$ $\displaystyle\propto$
$\displaystyle\int_{0}^{T}dt\cdot\exp(6\pi it/T)\cdot
A_{r}\cdot\phi_{x_{t}+ip_{t}}(x),$ (26)

where $A_{r}=\exp\left[(|\kappa x_{t}+ip_{t}|^{2}/\kappa-\kappa
x_{t}^{2}-p_{t}^{2}/\kappa)/4\right]$, $\kappa=Q\sqrt{V}$, and
$\phi_{x_{t}+ip_{t}}(x)$ is a wave packet with momentum $Im(\kappa
x_{t}+ip_{t})$ and position $Re(x_{t}+ip_{t}/\kappa)$, obtainable via
translating $\phi_{0}(x)$ \- a wave packet with zero momentum at the origin.
Generally speaking, the shape of $\phi_{0}(x)$ is unknown, and we infer
$\phi_{0}(x)$ and its anisotropy $\kappa$ from the low-energy expansion of Eq.
LABEL:eq:appstateham as an approximation. Unlike the exactly solvable Eq. 13
in the main text, here we solve the integral numerically for a real-space wave
function $\psi_{3}(x)$, which compares qualitatively consistent with the
quantum benchmark (Fig. 7). Given the corresponding closed orbit in the
complex semiclassical theory, such an approach for the target quantum
eigenstate is straightforwardly generalizable.

###  II.2 Non-Hermitian skin effect and generalized WKB approximation

In this subsection, we apply the complex semiclassical theory to another non-
Hermitian lattice model - the well-known Hatano-Nelson model:

$\displaystyle\hat{H}$ $\displaystyle=$
$\displaystyle\sum_{x}(t+\gamma)c^{\dagger}_{x+1}c_{x}+\sum_{x}(t-\gamma)c^{\dagger}_{x}c_{x+1}$
(27) $\displaystyle=$
$\displaystyle\sum_{k}\left[2t\cos(p)-2i\gamma\sin(p)\right]c^{\dagger}_{k}c_{k},$

where we set $t=1$ and $\gamma=\pm 0.1$ without loss of generality. In
particular, as a soft boundary condition, we introduce the following confining
potential:

$V(x)=U\left[e^{\alpha(x-L/2)}+e^{-\alpha(x+L/2)}\right],$ (28)

whose profile with $L=60$, $U=0.2$, and $\alpha=0.1$ is in Fig. 8a. Note that
instead of the common open boundary conditions that are hard walls, we have
employed smoother boundaries for better suitability of a semiclassical theory.

Figure 8: (a) The profile of the confining potential in Eq. 28 introduces two
soft boundaries. The red dashed lines show the setting for $L=60$, beyond
which the potential increases more rapidly. (b) The closed orbit following
EOMs in Eq. 29 retains a real-valued $x$ and a constant ${\rm Im}(p)=\mp
0.1003$ for $\gamma=\pm 0.1$, respectively, consistent with Eq. 30. The arrows
mark the direction of the cyclotron motion. (c) The additional factor and
amplitude modification to the wave packet due to the complex momentum offers
an alternative picture of the NHSE. Note the log scale along the vertical
direction.

We start with the following energy relation and EOMs in the complex
semiclassical theory:

$\displaystyle\epsilon$ $\displaystyle=$ $\displaystyle
2t\cos(p)+2i\gamma\sin(p)+V(x),$ $\displaystyle\dot{x}$ $\displaystyle=$
$\displaystyle\partial\epsilon/\partial p=-2t\sin(p)-2i\gamma\cos(p),$
$\displaystyle\dot{p}$ $\displaystyle=$
$\displaystyle-\partial\epsilon/\partial x=\alpha
U\left[e^{-\alpha(x+L/2)}-e^{\alpha(x-L/2)}\right].$ (29)

Interestingly, we can keep $x,t\in\mathbb{R}$ with an initialization that
satisfies:

${\rm Im}(x)=0,{\rm Im}(p)=\frac{1}{2}\ln\frac{t-\gamma}{t+\gamma}.$ (30)

Then, following the EOMs in Eq. 29, $\dot{p}$ becomes purely real, which keeps
${\rm Im}(p)$ constant with $t\in\mathbb{R}$, and in turn, $\dot{x}$ remains
purely real, Eq. 30 further holds, and so on so forth. Straightforwardly, we
can obtain a closed orbit; see Fig. 8b for example.

The complex semiclassical theory is not restricted to such real-valued $x$;
however, such a closed orbit is equally valid, analytically convenient, and,
at the same time, intimately connected with the WKB approximation.
Conventionally, while the position $x$ remains real in the WKB approximation,
the momentum $p$ retains a complex description as it becomes purely real or
imaginary depending on whether $E>V(x)$ or $E<V(x)$, respectively. The
semiclassical orbits such as Fig. 8b are just one step more general, as they
concern a fully complex $p$, instead of just real or imaginary. The
quantization condition is also consistent with that of the WKB approximation.

Notably, the non-Hermitian skin effect (NHSE) emerges naturally from the
complex semiclassical theory and the generalized WKB theory. Although the
closed orbit ventures across the entire length of the system (Fig. 8b), the
contribution of the wave packets to the target quantum state receives an
additional factor $\exp[-\int{\rm Im}(p)dx]$ contributed by the complex-valued
geometrical phase - an exponential weight amplification (attenuation) to the
right (left) - the NHSE (Fig. 8c). The tendency reverses with a sign change of
$\gamma$. Indeed, ${\rm Im}(p)$ given by Eq. 30 contributes a universal
factor:

$\exp(-0.5x\ln\frac{t-\gamma}{t+\gamma})=(\frac{t+\gamma}{t-\gamma})^{x/2},$
(31)

at position $x$, which is indeed consistent with the similarity-transformation
embodiment of the NHSE [46]. In the complex semiclassical theory, the factor
$A$ in Eq. 8 of the main text contributes a similar amplitude factor.

##  III Appendix C: Expectation-value conventions in non-Hermitian quantum
systems

We can diagonalize a non-Hermitian Hamiltonian as
$\hat{H}=\sum_{n}\epsilon_{n}\ket{nR}\bra{nL}$, where $\ket{nR}$ ($\bra{nL}$)
is the right (left) eigenstate with respect to the energy eigenvalue
$\epsilon_{n}$. Correspondingly, the time evolution operator is:

$U\left(t\right)=\exp{(-i\hat{H}t)}=\sum_{n}\exp{\left(-i\epsilon_{n}t\right)}\ket{nR}\bra{nL}.$
(32)

Alternatively, we can write the diagonalized $\hat{H}$ in more compact matrix
form as $\Lambda=V^{-1}\hat{H}V$, where the $n^{th}$ column (row) of $V$
($V^{-1}$) is exactly $\ket{nR}$ ($\bra{nL}$). $\epsilon_{n}$, on the diagonal
$\Lambda$, is the biorthogonal expectation value of $\hat{H}$:
$\epsilon_{n}=\braket{nL}{\hat{H}}{nR}$. Consistently, it is natural to deduce
that the expectation value of an operator $\hat{O}$ is the biorthogonal
counterpart, $\braket{\hat{O}}_{n}=\braket{nL}{\hat{O}}{nR}$, or more
generally, $\braket{\hat{O}}_{nm}=\braket{nL}{\hat{O}}{mR}$.

Given an initial quantum state $\ket{\Phi(0)}$, we can decompose it in the
biorthogonal basis $\ket{\Phi(0)}=\sum_{n}\ket{nR}\braket{nL}{\Phi(0)}$. It
evolves in time as:

$\displaystyle\ket{\Phi\left(t\right)}$
$\displaystyle=U\left(t\right)\ket{\Phi(0)}$
$\displaystyle=\sum_{nm}\exp{\left(-iE_{n}t\right)}\ket{nR}\braket{nL}{mR}\braket{mL}{\Phi(0)}$
$\displaystyle=\sum_{n}\exp{\left(-iE_{n}t\right)}\ket{nR}\braket{nL}{\Phi(0)},$
(33)

where we have used the property $\braket{nL}{mR}=\delta_{nm}$ of the
biorthogonal basis. Similarly, we have its bra $\langle\Phi(0)|$’s time
evolution as:

$\displaystyle\bra{\Phi(t)}$ $\displaystyle=\bra{\Phi(0)}U^{-1}(t)$
$\displaystyle=\sum_{n}\exp{\left(iE_{n}t\right)}\braket{\Phi(0)}{nR}\bra{nL},$
(34)

whereas the Hermitian conjugation of $\ket{\Phi(t)}$ is different:

$\displaystyle\bra{\Phi(t)}^{\prime}=\sum_{n}\exp{\left(iE_{n}^{*}t\right)}\braket{\Phi(0)}{nL}\bra{nR}.$
(35)

Thus, the amplitude of $\braket{\Phi(t)}{\Phi(t)}$ is conserved, while that of
$\braket{\Phi(t)}{{}^{\prime}}{\Phi(t)}$ grows or decays exponentially with
respect to $t$:

$\displaystyle\braket{\Phi(t)}{\Phi(t)}$
$\displaystyle=\braket{\Phi(0)}{U^{-1}(t)U(t)}{\Phi(0)}=1,$
$\displaystyle\braket{\Phi(t)}{{}^{\prime}}{\Phi(t)}$
$\displaystyle=\braket{\Phi(0)}{U^{\dagger}(t)U(t)}{\Phi(0)}$
$\displaystyle=\sum_{mn}\exp{\left[i(E_{m}^{*}-E_{n})t\right]}\braket{\Phi(0)}{mL}$
$\displaystyle\qquad\quad\times\braket{mR}{nR}\braket{nL}{\Phi(0)}.$ (36)

Therefore, the former convention is more convenient in non-Hermitian quantum
systems.

Putting both the expectation-value and time-evolution conventions together, we
define and evaluate the time evolution of the expectation value of observable
$\hat{O}$ as:

$\displaystyle\braket{\hat{O}(t)}=\braket{\Phi(t)}{\hat{O}}{\Phi(t)}=\braket{\Phi(0)}{U^{-1}\hat{O}U(t)}{\Phi(0)},$
(37)

which converts back to the common definition in Hermitian systems:

$\displaystyle\braket{\hat{O}(t)}$
$\displaystyle=\braket{\Phi(0)}{U^{-1}\hat{O}U(t)}{\Phi(0)}$
$\displaystyle=\braket{\Phi(0)}{U^{\dagger}\hat{O}U(t)}{\Phi(0)}=\braket{\Phi(t)}{\hat{O}}{\Phi(t)}.$
(38)

Whether the expectation value in non-Hermitian quantum systems should be
defined and calculated within the biorthogonal basis or merely the right
eigenstates remains a subtle matter. When we speak of the NHSE, the focus is
solely on (the amplitude of) the right eigenstates:
$\braket{\psi(x)}{{}^{\prime}}{\psi(x)}$; in contrast, however, the NHSE
disappears when we employ the biorthogonal basis: $\braket{\psi(x)}{\psi(x)}$.
Even when it comes to many-body non-Hermitian quantum systems, these two views
are both present in the literature: the spatial density of an $n$-fermion
state $\ket{\psi_{\mu}(x_{1},\ldots,x_{n})}$ is calculated via
$\braket{\psi_{\mu}}{{}^{\prime}\hat{n}_{x}}{\psi_{\mu}}$ with the right
eigenstates in Ref. [84], while the non-Hermitian many-body bulk polarization
takes the biorthogonal form $P^{LR}$ [85].

##  IV Appendix D: Physical picture of complex magnetic field $Q$

The complex semiclassical theory allows us to locate interesting quantum
physics and phenomena from a novel perspective.

For instance, we can re-express the EOMs in Eq. 16 in the main text with three
complex variables $p$, $Qx$, and $Qt$:

$\displaystyle\partial Qx/\partial Qt$ $\displaystyle=$ $\displaystyle 2\sin
p+i\eta e^{i(Qx+p)},$ $\displaystyle\partial p/\partial Qt$ $\displaystyle=$
$\displaystyle-[2V\sin Qx+i\eta e^{i(Qx+p)}].$ (39)

As in the main text, we determine the resulting phases with distinctive
properties via the presence or absence of close orbits. Eq. 39 indicates that
such physics does not depend on the exact value $Q$ of the magnetic field or
even whether $Q$ is real or complex. Indeed, it is natural to analytically
continue $Q$ to the complex domain, especially considering that $p$, $x$, and
$t$ are already complex variables.

Inserting complex $Q$ into the lattice model in Eq. 15 in the main text, we
obtain the energy spectra and eigenstates via exact diagonalization. Further,
we determine the localization properties and phases via the inverse
participation ratios (IPRs), which remain consistent with the complex
semiclassical theory, as shown in Fig. 3 in the main text as well as Fig. 9.

The physics of such a complex magnetic field deserves further study. While a
regular, real-valued magnetic field introduces a geometric phase for any path
encircling its magnetic flux, a complex magnetic field allows amplification
and attenuation - a “geometric amplitude” effect. Even by itself, this effect
is non-unitarity, offering a new source for non-Hermitian quantum physics,
potentially realizable in circuit-model simulators with directional dampers
and amplifiers [86, 87, 88].

Figure 9: The semiclassical trajectories (insets) characterize the phases of
the non-Hermitian lattice model in Eqs. 14 and 15 in the main text. The dots
denote the quantum spectrum, where $\oint p\cdot dx=2\pi(n+1/2)$ according to
the quantization condition. $V=0.3$, $\eta=0.1i$. Conventionally, given a
real-valued magnetic field $Q$, the changes to the geometric phase $\oint
p\cdot dx$ are real-valued, altering the spectrum along the curve it sits in
the complex $\epsilon$ space. A complex-valued $Q$ alters the quantization
conditions in a nontrivial way that completely shifts the $\oint p\cdot
dx\in\mathbb{R}$ curve and thus the whole spectrum.

Such a geometric effect also poses direct and nontrivial consequences on the
quantization conditions; see the arrows in Fig. 9. When we vary a real-valued
magnetic field $Q$, it alters $\oint p\cdot x\in\mathbb{R}$, thus the spacings
between and locations of the quantum levels along the original curve in the
complex $\epsilon$ space. In comparison, however, a complex $Q\in\mathbb{C}$
can shift the entire curve $\oint p\cdot x\in\mathbb{R}$ elsewhere, thus
completely recasting the spectrum. Such impacts are explicit in the current
model example, where $\oint p\cdot x=Q^{-1}\oint p\cdot d(Qx)\propto Q^{-1}$.

##  V Appendix E: Complex semiclassical theory beyond Hamiltonians - EOMs with
friction

In this subsection, we test the complex semiclassical theory on a damped
harmonic oscillator:

$\displaystyle\dot{x}$ $\displaystyle=$ $\displaystyle p,$
$\displaystyle\dot{p}$ $\displaystyle=$ $\displaystyle-x-\mu p,$ (40)

where the coefficient $\mu$ characterizes a friction $F_{\mu}=-\mu p$. The
quantum theory of such a dissipative system has been in continual
investigation and controversy for decades and commonly requires external
reservoirs to properly handle the dissipation of energy [89].

The EOMs in Eq. 40 are exactly solvable:

$\displaystyle x$ $\displaystyle=$ $\displaystyle
z_{1}e^{-c_{1}t}+z_{2}e^{-c_{2}t},$ $\displaystyle p$ $\displaystyle=$
$\displaystyle-z_{1}c_{1}e^{-c_{1}t}-z_{2}c_{2}e^{-c_{2}t},$ (41)

where $c_{1/2}=(\mu\pm\sqrt{\mu^{2}-4})/2$. When $\mu<2$, $c_{1/2}$ are
partially imaginary, the oscillatory behavior partially remains, and the
system is underdamped; when $\mu>2$, $c_{1/2}$ is completely real, and the
system becomes overdamped. $z_{1}$ and $z_{2}$ are constants determined by the
initial conditions.

Assuming $\mu<2$ and setting $z_{1}=z_{2}=x_{0}/2$, we obtain the familiar
form of a classical mechanical damped oscillator:

$\displaystyle x(t)$ $\displaystyle=$ $\displaystyle x_{0}e^{-\mu
t/2}\cos(\omega t),$ $\displaystyle p(t)$ $\displaystyle=$ $\displaystyle-
x_{0}e^{-\mu t/2}\left[\omega\sin(\omega t)+\mu\cos(\omega t)/2\right],$ (42)

where $\omega=\sqrt{1-\mu^{2}/4}$. Due to the damping factor $e^{-\mu t/2}$,
the absence of closed orbit in real $t\in\mathbb{R}$ is apparent; see Fig.
10b.

Figure 10: Following the EOMs, the semiclassical trajectories of the model in
Eq. 40 with a friction coefficient $\mu=0.2$ (a) form closed loops along the
$t/T\in\mathbb{R}$ direction and (b) end up in open spirals along the
$t\in\mathbb{R}$ direction, where we may still form closed loops by adjusting
$dt$, and the resulting trajectory is in (c). Here in (a) and (c), we
characterize the orbit closure with $a(T)=a(0)$, $a=x+ip$. (d) The
corresponding trajectories in the complex $t$ plane show the equivalence
between different closed orbits reaching the complex period $T$. $\mu=0.2$ and
$T=6.28+0.63i\approx 2\pi i/[(\mu+\sqrt{\mu^{2}-4})/2]$.

In the complex semiclassical theory, on the other hand, the variables $z_{1}$,
$z_{2}$, $x$, $p$, and, importantly, $t$ may take on complex values.
Consequently, we can establish closed orbits straightforwardly in complex time
$t\in\mathbb{C}$, e.g., $ic_{1}t\in\mathbb{R}$ in Eq. 41. More generally, we
can obtain closed orbits numerically via finite-time steps of the EOMs in Eq.
40, followed by the strategy we propose in the main text; see Fig. 10c. We can
also establish closed orbits along $t\propto T$; see Fig. 10a. As long as the
winding is identical, different closed orbits are equivalent, e.g., with the
same period $T\approx 2\pi i/c_{1}$.

Unlike the aforementioned dissipation-less models in the main text and the
supplemental materials, the EOMs in Eq. 40 face energy losses, and in general,
we cannot return both $x$ and $p$ to their original complex values. Instead,
we require $a(T)=a(0)$, $a=x+ip=x_{r}+ip_{r}$ return the real-valued COM
position and momentum of the wave packet $x_{r},p_{r}\in\mathbb{R}$ to their
initial conditions. Nevertheless, the additional factor $A$ augmenting the
wave packet, as in Eq. 9 in the main text, generally differs. Accordingly, we
may need to modify the quantization conditions so that the geometric phase
$\oint p\cdot dx$ compensates for such differences to establish wave-packet
consistency upon cycling around the orbits. Such studies will be fascinating
yet beyond the scope of the current work, and we lack a comparable quantum-
theory counterpart as our benchmark.

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