Non-Hermitian Subsystem Hamiltonians

• Non-Hermitian subsystem Hamiltonians are reduced operators that capture effective subsystem dynamics derived from larger Hermitian theories.
• They arise via various constructions such as operator-space reduction, bath elimination, and scattering approaches, leading to unique decay and stability properties.
• Their study uncovers rich symmetry, topological, and biorthogonal structures, enabling mappings to Hermitian counterparts through metric and Dyson transformations.

Non-Hermitian subsystem Hamiltonians are non-Hermitian operators attached not to an entire closed microscopic Hilbert space, but to a reduced sector, projected region, operator subspace, or enlarged auxiliary construction associated with part of a larger theory. In the literature, the term covers several inequivalent objects: an exact operator-space Bogoliubov-de Gennes generator on the linear span of bosonic creation and annihilation operators; an effective Hamiltonian for a designated subsystem obtained by eliminating a Hermitian bath or scattering leads; an exact non-Hermitian Hamiltonian on a decay-restricted subspace; and a non-Hermitian parent Hamiltonian on a doubled purification space for a mixed state. A recurring theme is that the full underlying theory may remain Hermitian or unitary even when the reduced description is non-Hermitian (Wakefield et al., 2023, Selim et al., 22 Jul 2025, Feiguin et al., 2013).

1. Conceptual scope and terminological distinctions

The most important conceptual distinction is that a non-Hermitian subsystem Hamiltonian is not a single canonical construction. In the quadratic-boson setting, the non-Hermitian object is the generator of Heisenberg evolution on the operator sector spanned by $a_j$ and $a_j^\dagger$, not a Hamiltonian acting on a reduced state Hilbert space. The full second-quantized Hamiltonian $H=H^\dagger$ acts unitarily on Fock space, while non-Hermiticity appears only after restricting the adjoint action $$i\partial_t \Box=[H,\Box]\equiv H_{\mathrm{eff}}\Box$$ to the finite-dimensional Nambu operator space (Wakefield et al., 2023).

A different notion appears in purification-based thermal constructions. There, the relevant non-Hermitian operator acts on a doubled Hilbert space $\mathcal H\otimes\widetilde{\mathcal H}$ and has the thermofield purification of $\rho(\beta)=e^{-\beta H}$ as its ground state. This is not a subsystem Hamiltonian obtained by tracing out degrees of freedom, and it is not the entanglement or modular Hamiltonian $K=-\log\rho_A$. It is a non-Hermitian parent Hamiltonian for a purified mixed state (Feiguin et al., 2013).

A third notion is the effective Hamiltonian of a designated subsystem embedded inside a larger Hermitian device. In photonic bath engineering, the subsystem can be the first one or two waveguides of a finite array, while the remaining waveguides form an engineered Hermitian bath whose elimination yields non-Hermitian subsystem response. In decaying open systems, the relevant reduced space can be $R=A\oplus B$, where decay from $R$ to lower-energy states $G$ generates an exact non-Hermitian Hamiltonian on $a_j^\dagger$0 itself (Selim et al., 22 Jul 2025, Militello et al., 2019).

These distinctions rule out a common misconception: non-Hermitian subsystem Hamiltonians are not always reduced density-matrix generators, and they are not always literal gain/loss Hamiltonians for an open subsystem. Depending on construction, they may instead be operator-space generators, scattering-center Hamiltonians, decay-restricted effective Hamiltonians, or parent Hamiltonians on doubled spaces.

2. Emergence from larger Hermitian systems

Several constructions show explicitly how non-Hermiticity can emerge from a larger Hermitian theory without abandoning global unitarity. In quadratic bosonic optics, the general $a_j^\dagger$1-mode Hamiltonian

$a_j^\dagger$2

induces the exact operator-space equation

$a_j^\dagger$3

Here the paper states explicitly that $a_j^\dagger$4 is Hermitian iff there are no single-mode or two-mode squeezing terms, and that $a_j^\dagger$5 is anti-Hermitian for pure amplification/absorption-type operator evolution. The non-unitary propagator $a_j^\dagger$6 is nevertheless symplectic, satisfying $a_j^\dagger$7, so the bosonic commutation relations are preserved (Wakefield et al., 2023).

In Hermitian-bath embeddings, the total Hamiltonian is partitioned as

$a_j^\dagger$8

and elimination of the bath yields the exact subsystem operator

$a_j^\dagger$9

Within the Wigner-Weisskopf/Markov design regime, this becomes $H=H^\dagger$0. For a single lossy site $H=H^\dagger$1, the effective term is $H=H^\dagger$2. The same framework realizes a PT-type dimer with

$H=H^\dagger$3

whose exceptional-point condition is $H=H^\dagger$4. In the reported implementation, the full array is conservative, while post-selection on subsystem outputs exposes the target non-Hermitian conditional evolution in both single- and two-photon regimes (Selim et al., 22 Jul 2025).

A scattering-theoretic construction replaces Hermitian leads by complex boundary self-energies. For a finite scattering center $H=H^\dagger$5 attached to two semi-infinite chains, the internal wavefunction of a fixed-energy scattering state satisfies the eigenvalue equation of the finite non-Hermitian Hamiltonian

$H=H^\dagger$6

The mapping is exact energy by energy: for a chosen incident plane wave with $H=H^\dagger$7, the wavefunction restricted to the center coincides with an eigenfunction of $H=H^\dagger$8 at the same energy. The imaginary parts of $H=H^\dagger$9 and $$i\partial_t \Box=[H,\Box]\equiv H_{\mathrm{eff}}\Box$$0 have opposite signs, so one contact acts as effective gain/source and the other as effective loss/sink, but the magnitudes need not be equal (Jin et al., 2011).

Construction	Reduced object	Source of non-Hermiticity
Bogoliubov operator reduction	Nambu/operator sector $$i\partial_t \Box=[H,\Box]\equiv H_{\mathrm{eff}}\Box$$1	Squeezing block $$i\partial_t \Box=[H,\Box]\equiv H_{\mathrm{eff}}\Box$$2
Hermitian bath embedding	Designated subsystem $$i\partial_t \Box=[H,\Box]\equiv H_{\mathrm{eff}}\Box$$3	Bath self-energy $$i\partial_t \Box=[H,\Box]\equiv H_{\mathrm{eff}}\Box$$4
Scattering-center reduction	Finite center $$i\partial_t \Box=[H,\Box]\equiv H_{\mathrm{eff}}\Box$$5	Lead-induced complex boundary potentials

Taken together, these constructions show that non-Hermiticity can originate from symplectic operator reduction, Hermitian dilation with bath elimination, or scattering self-energies, while the total microscopic dynamics remains Hermitian.

3. Symmetry, duality, and biorthogonal structure

For time-independent non-Hermitian Hamiltonians, $$i\partial_t \Box=[H,\Box]\equiv H_{\mathrm{eff}}\Box$$6 and $$i\partial_t \Box=[H,\Box]\equiv H_{\mathrm{eff}}\Box$$7 must be treated as distinct operators. Two symmetry relations are structurally central: $$i\partial_t \Box=[H,\Box]\equiv H_{\mathrm{eff}}\Box$$8 The second is $$i\partial_t \Box=[H,\Box]\equiv H_{\mathrm{eff}}\Box$$9-pseudohermiticity. In this framework, right and left eigenvectors are both required,

$\mathcal H\otimes\widetilde{\mathcal H}$0

and generalized invariants depend on whether one studies commuting symmetry or pseudohermitic intertwining. In particular, under $\mathcal H\otimes\widetilde{\mathcal H}$1, the product $\mathcal H\otimes\widetilde{\mathcal H}$2 is conserved rather than the normalized expectation value alone (Martínez et al., 2018).

This left-right structure is explicit in exactly solvable bosonic models. For a two-mode non-Hermitian bosonic Hamiltonian, complete biorthogonal families $\mathcal H\otimes\widetilde{\mathcal H}$3 and $\mathcal H\otimes\widetilde{\mathcal H}$4 are constructed, with

$\mathcal H\otimes\widetilde{\mathcal H}$5

The same work develops pseudo-bosonic ladder operators and a quasi-basis expansion on a dense subspace, which is precisely the kind of non-orthogonal mode structure encountered in effective non-Hermitian descriptions (Bebiano et al., 2020).

A closely related passive-wave example arises in transfer-matrix evolution. For quasi-one-dimensional flux-conserving, time-reversal-invariant waveguides, the generator $\mathcal H\otimes\widetilde{\mathcal H}$6 of transfer evolution is generally non-Hermitian but satisfies

$\mathcal H\otimes\widetilde{\mathcal H}$7

Thus the generator is simultaneously pseudo-Hermitian and anti-PT symmetric. Its allowed block form,

$\mathcal H\otimes\widetilde{\mathcal H}$8

shows that non-Hermiticity can emerge from an indefinite flux metric rather than gain/loss or openness (Chen et al., 2016).

4. Metrics, Dyson maps, and Hermitian companions

A large part of the theory of non-Hermitian subsystem Hamiltonians concerns the possibility of replacing the original inner product by a metric under which the effective Hamiltonian becomes self-adjoint. In pseudo-Hermitian formulations one seeks a positive-definite $\mathcal H\otimes\widetilde{\mathcal H}$9 satisfying

$\rho(\beta)=e^{-\beta H}$0

and then defines the modified adjoint

$\rho(\beta)=e^{-\beta H}$1

In the explicit two-dimensional oscillator with imaginary linear coordinate and momentum terms, the metric can be written as $\rho(\beta)=e^{-\beta H}$2, and the non-Hermitian Hamiltonian is mapped to a Hermitian oscillator by similarity transformation (Li et al., 2011).

A Heisenberg-picture variant starts from $\rho(\beta)=e^{-\beta H}$3-pseudo-Hermiticity,

$\rho(\beta)=e^{-\beta H}$4

defines $\rho(\beta)=e^{-\beta H}$5, and keeps the usual formal Heisenberg equation

$\rho(\beta)=e^{-\beta H}$6

while evaluating expectation values in the $\rho(\beta)=e^{-\beta H}$7-inner product. In the examples constructed there, the first-order Heisenberg equations are complex but the second-order equations become real, which then permits construction of a Hermitian counterpart $\rho(\beta)=e^{-\beta H}$8 with the same spectrum (Miao et al., 2012).

For quadratic $\rho(\beta)=e^{-\beta H}$9 Hamiltonians with anti-linear symmetry, the same question becomes more delicate. The paper develops linear and quadratic Dyson-map ansätze and shows that, whereas for linear $K=-\log\rho_A$0 Hamiltonians every anti-linearly symmetric model can be mapped to a Hermitian counterpart by an exponential of a linear combination of generators, the quadratic case is more complicated. Only a subclass is covered by a linear-exponent map, and more elaborate similarity transformations, including quadratic exponents, may be required (Assis, 2010).

Time dependence introduces a further distinction between an evolution generator and an observable Hamiltonian. Under a time-dependent Dyson map $K=-\log\rho_A$1, one has

$K=-\log\rho_A$2

and the observable associated with the non-Hermitian representation is

$K=-\log\rho_A$3

In the generic case, $K=-\log\rho_A$4 is not itself observable and belongs to an infinite gauge-linked chain of non-observable time-dependent non-Hermitian Hamiltonians; only under special choices making the metric time-independent does this chain collapse to a single observable quasi-Hermitian Hamiltonian (Luiz et al., 2017).

Hermitianization can also be purely computational. For a general non-Hermitian operator $K=-\log\rho_A$5, the companion Hermitian Hamiltonian

$K=-\log\rho_A$6

is positive semidefinite, and $K=-\log\rho_A$7 iff $K=-\log\rho_A$8 is an eigenvalue of $K=-\log\rho_A$9 with right eigenvector $R=A\oplus B$0. This device underlies variational MPS algorithms for non-Hermitian many-body systems, although the companion operator should be regarded as an auxiliary computational object rather than the physical subsystem Hamiltonian itself (Guo et al., 2022).

5. Effective dynamics, stability, and confinement

When a non-Hermitian subsystem Hamiltonian is interpreted as an open-system generator, its anti-Hermitian part directly controls purity, decay, and dynamical stability. A standard decomposition is

$R=A\oplus B$1

where $R=A\oplus B$2 is the Hermitian subsystem Hamiltonian and $R=A\oplus B$3 is the environment-induced decay operator. For the non-normalized density operator $R=A\oplus B$4,

$R=A\oplus B$5

while the normalized density operator $R=A\oplus B$6 obeys the nonlinear equation

$R=A\oplus B$7

In this framework, pure states remain fixed points of the purity equation but can be locally stable or unstable against mixing fluctuations depending on the structure and sign of $R=A\oplus B$8 (Zloshchastiev, 2015).

Hilbert-space partitioning gives a more geometric reduced-dynamics picture. For a decaying system with $R=A\oplus B$9, restriction to $R$0 yields the exact non-Hermitian Hamiltonian

$R$1

Writing

$R$2

the regime

$R$3

induces dynamical confinement in $R$4. To second order, the effective eigenvalues in $R$5 are

$R$6

so the protected sector itself acquires non-Hermitian corrections. The interpretation is clearest in the pure-decay limit $R$7, where the confinement is Zeno-like, but the paper emphasizes that generic phases mix Zeno behavior with off-resonant detuning (Militello et al., 2019).

A more formal unification is provided by the Hamiltonian reformulation of linear non-Hermitian systems. For a diagonalizable non-Hermitian generator $R$8,

$R$9

one can introduce a conjugate left state $G$0 and define the canonical Hamiltonian function

$G$1

The non-Hermitian right-state evolution is then exactly equivalent to Hamilton’s canonical equations on the enlarged phase space $G$2. In this formulation the conserved quantity is the biorthogonal overlap $G$3, not the ordinary norm (Zhang, 2023).

6. Topology, frequency dependence, and scattering structure

Non-Hermitian subsystem Hamiltonians often retain a band or scattering interpretation, but the relevant topology and spectral singularities differ sharply from the Hermitian case. For periodic non-Hermitian Bloch Hamiltonians $G$4, the paper distinguishes separable bands, defined by $G$5 for all $G$6 and all $G$7, from isolated bands, whose entire complex-energy images do not overlap. In two dimensions, a separable band carries a biorthogonal Chern number built from left and right eigenvectors; in one dimension, the basic invariant is the vorticity

$G$8

so topology is controlled by spectral winding of complex energies rather than by eigenstates alone. The same framework identifies exceptional points as generic defective degeneracies with vorticity $G$9 (Shen et al., 2017).

For effective subsystem Hamiltonians derived from Green’s functions, frequency dependence becomes intrinsic. The retarded Green’s function is written as

$a_j^\dagger$00

so momentum- and frequency-dependent non-Hermitian effective Hamiltonians are in one-to-one correspondence with single-particle Green’s functions of systems that may be interacting or open. Their classification yields 54 symmetry classes and the K-theoretic decomposition

$a_j^\dagger$01

which separates a momentum contribution, a pure-frequency contribution, and a mixed momentum-frequency contribution. This shows that reduced effective Hamiltonians can carry topology absent from any static $a_j^\dagger$02 description (Kotz et al., 2023).

Scattering supplies a complementary spectral language. For one-dimensional non-Hermitian scattering Hamiltonians with Hermiticity, parity pseudohermiticity, time-reversal symmetry, or $a_j^\dagger$03 symmetry, the poles of the eigenvalues of the $a_j^\dagger$04-matrix satisfy

$a_j^\dagger$05

so the associated energies $a_j^\dagger$06 are either real or occur in complex-conjugate pairs. The result applies not only to bound states but also to antibound states, resonances, and antiresonances. For effective subsystem Hamiltonians arising from leads, reservoirs, or scattering reduction, this pole symmetry is often more informative than a discrete-spectrum analysis alone (Simón et al., 2018).

Within this broader topological and scattering context, non-Hermitian subsystem Hamiltonians are best viewed as effective spectral objects whose geometry may live in operator space, subsystem space, momentum space, frequency space, or the analytic continuation of scattering amplitudes. Their defining feature is not merely $a_j^\dagger$07, but the fact that the non-Hermitian operator captures a reduced description of a larger structure while preserving enough algebraic, dynamical, or topological information to remain calculable and physically interpretable.