The adiabatic theorem for non-Hermitian quantum systems with real eigenvalues and the complex geometric phase

This paper rigorously proves that the adiabatic theorem remains valid for diagonalizable non-Hermitian quantum systems with real eigenvalues by employing the complex geometric phase, biorthogonal functional calculus, and the Grönwall inequality, thereby justifying the definition of a complex Berry phase in such systems.

The Gist

Here is an explanation of the paper "The adiabatic theorem for non-Hermitian quantum systems with real eigenvalues and the complex geometric phase," translated into simple, everyday language with creative analogies.

The Big Picture: A Hiker in a Shifting Landscape

Imagine you are hiking up a mountain. The Adiabatic Theorem is a famous rule in physics that says: If you move slowly enough, you will stay on the path you started on.

In the quantum world, this means if a particle starts in a specific energy state (a "path") and the environment changes very slowly, the particle will stay in that same energy state, only picking up a little bit of "travel fatigue" (a phase shift) along the way.

For a long time, physicists thought this rule only worked for "perfect" systems (called Hermitian systems), where energy is conserved and everything behaves predictably, like a frictionless slide.

The Problem:
Recently, scientists have become very interested in Non-Hermitian systems. Think of these as "leaky" systems. They might lose energy to the environment, or gain energy from outside (like a laser or a system with friction). In these messy, leaky worlds, the old hiking rule often breaks down. If you try to walk slowly in a swamp, you might accidentally slip off the path entirely.

The Paper's Discovery:
Authors Minyi Huang and Ray-Kuang Lee asked: "Is there a special type of leaky system where the slow-walking rule still works?"

They found the answer: Yes! If the leaky system has real eigenvalues (meaning its energy levels are still "real" numbers, not imaginary weirdness) and can be diagonalized (meaning it can be untangled into simple, independent parts), then the Adiabatic Theorem holds true.

The Secret Weapon: The "Ghost" Compass

To prove this, the authors had to invent a new way to navigate.

1. The Old Map (Hermitian): In normal systems, you have a clear map. You know exactly where you are relative to your destination.
2. The New Map (Non-Hermitian): In leaky systems, the map is distorted. The "left" and "right" directions don't match up perfectly anymore.

The authors used a tool called Biorthogonal Systems.

• Analogy: Imagine trying to measure a room with a ruler that is slightly bent. If you use the ruler alone, your measurements are wrong. But, if you use a second, "mirror-image" ruler that is bent in the opposite way, and you combine the two measurements, the errors cancel out, and you get the true shape of the room.
• In the paper, they use a pair of vectors (one for the state, one for the "mirror" state) to keep the math honest.

The "Complex" Twist: The Berry Phase

When you hike a loop in the real world, you might end up facing a different direction than when you started, even if you walked in a circle. In quantum mechanics, this is called the Berry Phase.

• In Normal Systems: This phase is like a simple rotation (like turning a dial).
• In This Paper: Because the system is "leaky" (non-Hermitian), the phase becomes Complex.
  • Analogy: Imagine your compass needle doesn't just rotate; it also grows or shrinks as you walk. This "Complex Berry Phase" accounts for both the direction change and the gain/loss of energy in the system.

The paper proves that even with this weird, growing/shrinking compass, if you move slowly enough, you still stay on the right path.

Why Was This Hard to Prove? (The "Boundedness" Issue)

In normal physics, if you walk slowly, you never run off to infinity. But in these leaky systems, there was a fear that the math might explode (go to infinity) even if you moved slowly.

The authors had to use a mathematical "safety net" called the Grönwall Inequality.

• Analogy: Imagine you are trying to prove that a balloon won't pop even if you keep blowing into it slowly. You can't just guess; you need a strict mathematical rule that says, "As long as the air goes in at a certain rate, the balloon will stretch but never burst."
• They used this inequality to prove that the system's behavior stays "tame" (bounded) and doesn't go crazy, ensuring the theorem is valid.

The "Aha!" Moment

The authors also realized something clever in their discussion:
If a leaky system has real energy levels, it is mathematically "similar" to a perfect, non-leaky system.

• Analogy: Imagine a distorted photo of a mountain. If you stretch and squeeze the photo back into its original shape, it's the same mountain.
• However, in the quantum world, the "stretching" (the transformation) changes as you move. You can't just stretch the photo once and be done; the photo keeps changing shape as you walk. This is why they couldn't just copy the old proof; they had to build a new one that accounts for the photo changing while you walk.

Summary: What Does This Mean for Us?

1. The Rule Holds: Even in messy, energy-leaking quantum systems, if the energy levels are "real" and you move slowly, the system stays stable.
2. New Navigation: We now have a rigorous way to calculate the "phase" (the travel fatigue) in these systems, which includes a complex, growing/shrinking component.
3. Future Tech: This is huge for Quantum Computing and Lasers. Many modern quantum devices rely on non-Hermitian physics (like lasers that lose and gain light). This paper gives engineers the confidence to design these devices, knowing that if they control the speed of change correctly, the quantum states will behave predictably.

In a nutshell: The authors proved that even in a quantum world where things leak and change shape, if you move slowly enough and use the right "mirror" math, you can still trust the path you are on.

Technical Summary

Here is a detailed technical summary of the paper "The adiabatic theorem for non-Hermitian quantum systems with real eigenvalues and the complex geometric phase" by Minyi Huang and Ray-Kuang Lee.

1. Problem Statement

The adiabatic theorem is a cornerstone of quantum mechanics, stating that a system evolving slowly under a time-dependent Hamiltonian remains in its instantaneous eigenstate, acquiring only a phase factor. While rigorously proven for Hermitian systems (where eigenvalues are real and eigenvectors are orthogonal), the theorem's validity in non-Hermitian systems is not guaranteed.

• The Challenge: In general non-Hermitian systems, the adiabatic theorem can fail due to the lack of unitary evolution and the potential for complex eigenvalues or non-diagonalizable Hamiltonians (Jordan blocks).
• The Specific Gap: There has been a lack of rigorous mathematical proof confirming whether the adiabatic theorem holds for diagonalizable non-Hermitian systems with real eigenvalues. Furthermore, the definition and behavior of the geometric (Berry) phase in such systems require clarification, particularly regarding the use of biorthogonal bases.

2. Methodology

The authors employ a rigorous mathematical framework combining functional calculus, biorthogonal systems, and differential inequalities to prove the theorem.

• Biorthogonal Formalism: Unlike Hermitian systems where left and right eigenvectors are identical, non-Hermitian systems require a biorthogonal basis. The authors define right eigenvectors $v_i(s)$ and left eigenvectors $\xi_i(s)$ such that $\xi_i^\dagger(s) v_j(s) = \delta_{ij}$.
• Spectral Decomposition: They express the Hamiltonian $H(s)$ and the identity operator $I$ using spectral projections $P_k(s)$ derived from the biorthogonal basis:
  $$H(s) = \sum_k \lambda_k(s) P_k(s), \quad I = \sum_k P_k(s)$$
• Functional Calculus & Resolvent: They construct a matrix operator $S(s)$ (analogous to the reduced resolvent in Hermitian theory) to handle the off-diagonal terms:
  $$S(s) = \sum_{k \neq j} \frac{1}{\lambda_k(s) - \lambda_j(s)} P_k(s)$$
  This operator satisfies $(H(s) - \lambda_j(s)I)S(s) = I - P_j(s)$.
• Complex Geometric Phase: The proof explicitly incorporates a complex geometric phase factor derived from the biorthogonal inner product $\langle \xi_j | \dot{v}_j \rangle$.
• Grönwall Inequality: A critical methodological step involves proving that the evolution operator $U_T(s)$ and its inverse are uniformly bounded with respect to the adiabatic parameter $T$ (where $T \to \infty$). Since non-Hermitian evolution is not unitary, this boundedness is not trivial. The authors use the Grönwall inequality to bound the growth of the solution to the differential equations governing the evolution.

3. Key Contributions

1. Rigorous Proof for Real Eigenvalues: The paper provides a rigorous proof that the adiabatic theorem holds for diagonalizable non-Hermitian Hamiltonians with real eigenvalues.
2. Justification of Complex Berry Phase: The derivation naturally leads to the definition of a complex geometric phase (Berry phase) for non-Hermitian systems. The authors demonstrate that this phase is well-defined and physically meaningful even when eigenvectors are not normalized, provided a biorthogonal system is used.
3. Generalization of Kato's Techniques: The authors successfully generalize T. Kato's functional calculus techniques (originally developed for Hermitian systems) to the biorthogonal non-Hermitian context.
4. Addressing Non-Unitarity: By proving the uniform boundedness of $U_T(s)$ and $U_T^{-1}(s)$ using the Grönwall inequality, the paper overcomes the primary mathematical obstacle preventing the direct application of Hermitian proofs to non-Hermitian systems.

4. Main Results

The central result is the convergence of the time-evolution operator acting on an initial eigenstate. For a system starting in the $j$-th eigenstate $v_j(0)$ and evolving slowly ($T \to \infty$), the final state is:

$$U_T(s) v_j(0) \to e^{-iT \int_0^s \lambda_j(\sigma) d\sigma} e^{-\int_0^s \langle \xi_j(\sigma) | \dot{v}_j(\sigma) \rangle d\sigma} v_j(s)$$

• Dynamic Phase: The term $e^{-iT \int \lambda_j d\sigma}$ represents the standard dynamic phase (with real $\lambda_j$).
• Complex Geometric Phase: The term $e^{-\int \langle \xi_j | \dot{v}_j \rangle d\sigma}$ represents the geometric phase. In the non-Hermitian case, the inner product $\langle \xi_j | \dot{v}_j \rangle$ is generally complex, leading to a complex phase factor. This factor accounts for both a phase rotation and a change in the norm of the state vector.
• Subspace Stability: In the degenerate case, the system remains within the subspace spanned by the degenerate eigenvectors, implying a non-abelian generalization of the geometric phase.

5. Significance and Implications

• Theoretical Foundation: This work resolves a long-standing ambiguity regarding the applicability of the adiabatic theorem in non-Hermitian quantum mechanics. It establishes that as long as the system is diagonalizable and possesses real eigenvalues, adiabatic following is preserved.
• Complex Berry Phase: The paper validates the use of complex Berry phases in non-Hermitian physics. This is crucial for understanding topological phenomena in open quantum systems, PT-symmetric systems, and systems with gain and loss.
• Robustness against Basis Choice: The authors show that while individual eigenvectors are not uniquely defined (due to arbitrary scaling), the final adiabatically evolved state is independent of the specific choice of normalization for the biorthogonal basis.
• Applications: The results support the theoretical underpinnings of recent experimental and theoretical advances in non-Hermitian physics, such as the non-Hermitian skin effect, topological phase transitions in open systems, and adiabatic quantum computing protocols utilizing non-Hermitian Hamiltonians.

In conclusion, Huang and Lee successfully extend the adiabatic theorem to a broad class of non-Hermitian systems, providing the necessary mathematical rigor to justify the use of complex geometric phases in these contexts.