Physical constraints on effective non-Hermitian systems

This paper demonstrates that incorporating the anti-Hermitian part of a Hamiltonian directly into Matsubara Green's functions is incompatible with interacting many-body systems, proposing instead a consistent physical description based on pseudo-Hermitian quantum mechanics and characterizing the resulting zero-temperature distribution functions and electromagnetic responses.

The Gist

Imagine you are trying to describe how a ball moves through a room. In the standard world of physics (Hermitian physics), the ball is a closed system; it bounces, it rolls, but the total amount of "energy" or "information" in the system is perfectly preserved. It's like a billiard game where no balls ever fall off the table.

However, in the real world, things are often "open." Balls lose energy to friction, or they might be part of a system where particles are constantly being created and destroyed. To describe these messy, open systems, physicists often use a mathematical tool called a Non-Hermitian Hamiltonian (NHH). Think of this as a "shortcut" or a "shadow" description that accounts for things leaking out or being absorbed, without having to track every single particle in the environment.

The paper by Aaron Kleger and Rufus Boyack is essentially a rulebook check. They are asking: "When we use these shortcut descriptions for complex, interacting systems, are we following the rules of the game?"

Here is the breakdown of their findings using simple analogies:

1. The "Shortcut" vs. The "Real Thing"

For a long time, many physicists have treated these Non-Hermitian systems by simply plugging the "leaky" numbers directly into their standard equations. It's like trying to drive a car with a flat tire by just turning up the engine louder.

The authors show that this common approach is broken when you have interactions (when particles talk to each other). If you just take the standard rules and add a "leak" term, you end up with a description that violates the fundamental laws of physics, specifically causality (the idea that the future can't affect the past) and gauge invariance (a fancy way of saying the laws of physics shouldn't change just because you change your coordinate system).

2. The "Magic Mirror" Solution (Pseudo-Hermitian Quantum Mechanics)

The paper proposes that if you want to use these Non-Hermitian shortcuts correctly, you can't just use the standard rules. You have to use a specific framework called Pseudo-Hermitian Quantum Mechanics (PHQM).

The Analogy:
Imagine you are looking at a reflection in a funhouse mirror. The reflection looks distorted (Non-Hermitian).

• The Old Way: People tried to measure the reflection directly using a ruler meant for flat surfaces. The measurements didn't add up.
• The New Way (PHQM): The authors say, "You need a special, flexible ruler (called a pseudo-metric operator) that bends to match the mirror's shape."

When you use this special ruler, the distorted reflection actually behaves exactly like a normal, healthy object. The "leakiness" isn't actually a loss of energy; it's just a different way of looking at a system that is actually perfectly stable and "unitary" (energy-conserving) underneath.

3. The "Sign" Problem

One of the most technical but crucial points they make involves a mathematical "sign" (a plus or minus) that appears in the equations.

• In standard physics: When you have a leaky system, the math requires a specific "sign" to flip depending on the direction of time or frequency. It's like a traffic light that must change color to keep traffic flowing safely.
• In the authors' framework: If you are using the "Magic Mirror" (PHQM), that sign flip doesn't happen for the main part of the system. The "leakiness" is actually just a reshaping of the system, not a loss.

They found that many previous studies were mixing these two worlds up. They were using the "Magic Mirror" math but applying the "Standard Leaky" rules, which creates a contradiction.

4. The "Tachyon" Test Drive

To prove their point, the authors took a specific model called the "Tachyon Dirac Model" (a theoretical particle that behaves like a wave in a 1D line) and ran it through three different "engines":

1. Standard Leaky Engine: Treats the system as losing energy to the environment.
2. Magic Mirror Engine (PHQM): Treats the system as a reshaped, stable system.
3. Post-Selection Engine: A method where you only count the outcomes where nothing "leaked" out (a specific experimental trick).

The Result:
They calculated how well these systems conduct electricity (optical conductivity). They found that:

• The Standard Leaky engine and the Magic Mirror engine gave different answers.
• The "leakiness" in the standard engine acts like friction, slowing things down.
• The "leakiness" in the Magic Mirror engine acts like a change in the particle's mass, changing how it moves without necessarily slowing it down in the same way.

The Bottom Line

The paper argues that you cannot treat all Non-Hermitian systems the same way.

• If your system is truly losing energy to the environment (dissipative), you must use the standard "leaky" rules, which include specific mathematical signs to keep physics consistent.
• If your system is being described by the "Magic Mirror" (PHQM), the "leakiness" is actually just a mathematical trick to describe a stable system. In this case, you must use a different set of rules (a different "ruler") to get the right physical predictions.

The authors conclude that many previous papers might have been using the wrong ruler for the job, leading to incorrect predictions about how these exotic systems behave. They provide the correct "rulebook" to ensure that when we study these strange, non-Hermitian worlds, our math actually matches physical reality.

Technical Summary

Technical Summary: Physical Constraints on Effective Non-Hermitian Systems

Problem Statement
Non-Hermitian (NH) systems have become a focal point for studying exotic phenomena such as skin effects, enhanced sensitivity, and unique topological phases. While standard quantum mechanics requires Hermitian Hamiltonians, interactions between particles and their environments often lead to effective non-Hermitian Hamiltonians (NHHs) in quasiparticle descriptions. A prevalent approach in the literature incorporates the anti-Hermitian part of the Hamiltonian directly into the Matsubara Green's function for many-body systems. However, this paper identifies a critical inconsistency: such an approach violates fundamental physical constraints required for systems with interactions, specifically the relationship between causal Green's functions and the requirements of gauge invariance and causality. The authors note a lack of consensus regarding how to compute expectation values and define equilibrium density matrices when utilizing frameworks like biorthogonal quantum mechanics (BQM), parity-time (PT) symmetry, or pseudo-Hermitian quantum mechanics (PHQM). The central problem is to determine the physical constraints on effective actions and Green's functions for NH systems and to distinguish between descriptions arising from dissipative interactions versus those arising from pseudo-Hermitian structures.

Methodology
The authors employ a field-theoretical approach to compare standard interacting quantum mechanics with pseudo-Hermitian quantum mechanics (PHQM). Key methodological steps include:

1. Green's Function Analysis: The paper analyzes the structure of retarded ($G^R$), advanced ($G^A$), and Matsubara Green's functions. It contrasts the standard interacting theory, where $G^A = (G^R)^\dagger$, with PHQM, where the inner product is modified by a pseudo-metric operator $\eta$.
2. Isospectral Mapping: The authors utilize the isospectral equivalence between a NHH ($H$) and a Hermitian Hamiltonian ($h$) via the transformation $h = \eta^{1/2}H\eta^{-1/2}$. This allows for the mapping of observables and time evolution between the two frameworks.
3. Model Application: To test these theoretical distinctions, the authors apply their framework to a specific (1+1)-dimensional NH Dirac model, referred to as the "Tachyon" model, characterized by a real mass $\Delta$ and an imaginary mass $m$.
4. Linear Response Theory: The authors generalize the Kubo formula for optical conductivity to NH systems described by PHQM. They compute the DC conductivity and optical sum rules under three distinct physical descriptions:
   • Standard dissipative dynamics (where the anti-Hermitian part arises from a self-energy $\Sigma''_R$).
   • PHQM with the current operator derived from the NHH ($J$).
   • PHQM with the current operator derived from the isospectral Hermitian Hamiltonian ($\tilde{j}$).
5. Distribution Functions: The paper derives zero-temperature distribution functions and equilibrium expectation values for these different frameworks, highlighting how the choice of inner product affects the definition of the ground state and thermal averages.

Key Contributions and Results

1. Incompatibility of Direct Anti-Hermitian Incorporation: The authors demonstrate that incorporating the anti-Hermitian part of the Hamiltonian directly into the Matsubara Green's function without a signum function ($\text{sgn}(\omega_n)$) is incompatible with standard interacting theory. In standard dissipative systems, the anti-Hermitian part of the self-energy must be accompanied by $\text{sgn}(\omega_n)$ in the Matsubara Green's function to satisfy causality ($G^A = (G^R)^\dagger$).
2. PHQM as a Consistent Framework: The paper establishes that NH systems where the Matsubara Green's function lacks the $\text{sgn}(\omega_n)$ factor are consistent with PHQM. In PHQM, the causal Green's functions satisfy a generalized relation: $G^R = \eta^{-1}(G^A)^\dagger \eta$. This framework describes unitary time evolution with respect to a modified inner product, distinguishing it from the non-unitary dynamics of dissipative open quantum systems.
3. Distinction in Physical Origins: The authors clarify that in PHQM, the non-Hermiticity of the NHH does not arise from a non-Hermitian self-energy (dissipation) but rather from the Hermitian renormalization of an isospectral Hermitian system. Consequently, the anti-Hermitian part of $H$ in PHQM does not carry the $\text{sgn}(\omega_n)$ factor, whereas the dissipative part (uniform decay rate $\gamma$) does.
4. Electromagnetic Response Differences: Applying these concepts to the Tachyon model, the authors calculate the DC conductivity ($\sigma_{DC}$) and optical sum rules. They find that the results differ significantly depending on the physical description:
   • Standard Dissipative: The imaginary mass $m$ acts as a relative inverse lifetime, renormalizing the scattering rate.
   • PHQM (Current $J$): The imaginary mass renormalizes the real mass gap ($\Delta \to \sqrt{\Delta^2 - m^2}$).
   • PHQM (Current $\tilde{j}$): The response matches the isospectral Hermitian system but involves a non-trivial renormalization of the velocity operator.
     The optical sum rules also differ, with PHQM yielding unconventional sum rules compared to the standard $e^2 v_F$ result, particularly in the dissipationless limit.
5. Equilibrium Expectation Values: The paper provides explicit formulas for zero-temperature expectation values. It shows that for standard dissipative systems, both right-left and left-right expectation values contribute, whereas for PHQM and postselected dynamics, the appropriate forms are right-right or left-right, respectively, depending on the specific dynamics and the nature of the eigenvalues.

Significance
The paper claims to resolve ambiguities in the formulation of many-body NH systems by rigorously distinguishing between effective theories arising from dissipative interactions and those described by PHQM. The authors argue that the choice of framework dictates the form of the Green's functions, the definition of current operators, and the resulting physical observables like conductivity. By identifying that approaches lacking the $\text{sgn}(\omega_n)$ factor in Matsubara Green's functions are consistent with PHQM (unitary dynamics) rather than standard dissipative physics, the work provides a consistent physical description for such systems. The differences in calculated transport properties and sum rules serve as potential experimental signatures to distinguish between these competing physical interpretations of non-Hermiticity. The authors emphasize that their work addresses the ambiguity of ensemble expectation values and provides a unified field-theoretical footing for comparing these approaches.