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Equilibrium Thermodynamics of Non-Hermitian Dirac Fermions: Caloric and Magnetic Responses

This paper establishes scaling relations for the equilibrium thermodynamics of real-spectrum non-Hermitian Dirac fermions in a magnetic field by demonstrating that a similarity transformation maps the system to a Hermitian model with reduced velocity and effective magnetic field, thereby deriving rescaled caloric and magnetic responses across different thermodynamic ensembles.

Original authors: Francisco J. Peña, Bastian Castorene, Juan Pablo Esparza, Vladimir Juričić, Patricio Vargas

Published 2026-08-06

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CC BY 4.0

Original authors: Francisco J. Peña, Bastian Castorene, Juan Pablo Esparza, Vladimir Juričić, Patricio Vargas

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine a world where the rules of physics aren't just about solid, unchanging objects, but about systems that can gain or lose energy like a leaky bucket or a glowing lightbulb. In the strange realm of quantum mechanics, scientists often study "Hermitian" systems—think of them as perfectly sealed, frictionless boxes where energy is conserved and everything behaves predictably. But the real world is messier. Systems interact with their environment, lose particles, or gain energy from outside. To describe these "non-Hermitian" systems, physicists use special math that usually leads to weird, imaginary numbers that don't make sense for things like temperature or pressure. However, there's a special, rare corner of this messy world where the math stays "real" and sensible, even though the system is open and interacting with its surroundings. This is the playground for a new study: figuring out how heat and magnetism behave in these tricky, real-spectrum quantum systems. Why does anyone care? Because materials like graphene (a super-thin, super-strong sheet of carbon) can host these strange particles, and understanding how they react to heat and magnetic fields could unlock new ways to build ultra-sensitive sensors or quantum computers.

Now, let's dive into what this paper actually does. The researchers are looking at a specific type of quantum particle called a "Dirac fermion" (which acts like a massless, super-fast electron) inside a magnetic field. Normally, when you put these particles in a magnetic field, their energy levels snap into a neat, stepped ladder called "Landau levels." The paper asks: What happens to this ladder if we tweak the system to be "non-Hermitian" but keep the energy levels real? They found that the non-Hermitian tweak doesn't scramble the ladder; instead, it acts like a magical compression machine. It squishes all the rungs of the energy ladder closer together by a specific amount, depending on how strong the "tweak" is.

Here is the cool part: The authors discovered that you don't need to invent new laws of physics to understand this squished ladder. You can simply pretend the system is a normal, standard quantum system, but with two tricks up your sleeve. First, you can pretend the particles are moving slower than usual. Second, and more importantly, you can pretend the magnetic field you are applying is actually weaker than it really is. This "effective magnetic field" is the key. The paper shows that every time you measure something like heat capacity (how much energy it takes to warm the system up) or magnetic response (how the system reacts to the magnet), the results for this weird, squished system are exactly the same as the results for a normal system, just shifted to a lower magnetic field.

The researchers explored two different ways of looking at this. In the first scenario, they kept the number of particles fixed (like a sealed jar). They found that as they increased the "squish" factor, the system's temperature and chemical potential (a measure of how eager particles are to move) had to adjust to follow the compressed ladder. It's like if you squeezed a spring, the whole spring would shift down, but the pattern of the coils would stay the same. In the second scenario, they kept the chemical potential fixed (like connecting the system to a giant energy reservoir). Here, the "squish" caused the energy levels to cross the reservoir's energy line at different magnetic field strengths. This created a wavy, oscillating pattern in the heat and magnetic responses. The paper shows that these waves are just the normal waves, but they appear at magnetic fields that are shifted by the squish factor.

One of the most fascinating findings is about the "orbital magnetic moment," which is basically how much the electrons want to spin around in the magnetic field. The paper reveals a subtle but important difference: the total magnetic moment of the whole system shifts just like the energy levels, but the magnetic moment per particle gets an extra boost from the squish factor. It's as if the whole crowd of electrons moves to a new spot, but each individual electron also gets a little stronger.

Finally, the team looked at what happens if you slowly change the "squish" factor itself (by tweaking the system's environment) while keeping the temperature or entropy constant. They found that changing this factor does work on the system, similar to compressing a gas. If you do this without letting heat escape (an adiabatic process), the temperature of the system changes in a predictable way. The paper provides a complete set of rules (scaling relations) that let scientists take the known behavior of normal quantum systems and instantly predict how these weird, non-Hermitian systems will behave, as long as the energy levels stay real and the "squish" isn't too extreme.

The study is based on theoretical calculations and simulations, not new experimental data, but it provides a solid framework for future experiments. The authors are careful to note that these rules work best when the energy levels are clearly separated; if the "squish" gets too strong or the temperature gets too high, the levels blur together, and the neat rules start to break down. This work doesn't solve every mystery of non-Hermitian physics, but it lays down a clear map for understanding how heat and magnetism dance in these strange, real-spectrum quantum worlds.

Technical Summary: Equilibrium Thermodynamics of Non-Hermitian Dirac Fermions

Problem Statement
The paper addresses the equilibrium thermodynamics of two-dimensional Dirac fermions (specifically in graphene) subjected to a non-Hermitian (NH) deformation that preserves a real energy spectrum. While much of NH physics focuses on dynamical or non-equilibrium phenomena, this work investigates stationary deformations where the Hamiltonian is quasi-Hermitian. The central challenge is to establish a consistent thermodynamic framework for such systems, particularly under a perpendicular magnetic field where Landau quantization occurs. The authors aim to determine how spectral compression induced by NH parameters affects caloric (entropy, heat capacity) and magnetic (orbital response) observables in different thermodynamic ensembles.

Methodology
The authors employ a minimal NH deformation of the Dirac Hamiltonian, $H_\beta$ , characterized by a parameter $\beta$ ( $|\beta| < 1$ ) that introduces gain-loss imbalance or nonreciprocal coupling. The methodology relies on the following steps:

Quasi-Hermitian Mapping: The non-Hermitian Hamiltonian $H_\beta$ is mapped to a Hermitian representative $H_0$ via a similarity transformation $S$ . This transformation reveals that the spectrum of $H_\beta$ is uniformly compressed by a factor $\lambda = \sqrt{1-\beta^2}$ relative to the Hermitian case, such that $H_\beta = \lambda S H_0 S^{-1}$ .
Effective Field Representation: The authors demonstrate that this spectral compression can be equivalently represented by an effective magnetic field $B_{\text{eff}} = \lambda^2 B$ . This mapping preserves the Landau-level ordering and the zero mode while compressing the energy spacing of nonzero levels.
Thermodynamic Ensembles: The study analyzes two distinct protocols:
- Fixed Projected Filling Factor (PFF): The number of electrons per flux quantum ( $N_c$ ) is held constant, requiring the chemical potential $\mu$ to adjust self-consistently as the field or deformation changes.
- Fixed Chemical Potential: The system is coupled to a reservoir with a fixed $\mu$ , allowing the particle number to vary.
Scaling Derivation: Using the similarity transformation and the properties of the grand partition function, the authors derive scaling relations that express NH thermodynamic quantities in terms of their Hermitian counterparts evaluated at either rescaled temperatures/chemical potentials or the effective magnetic field $B_{\text{eff}}$ .

Key Contributions and Results

Scaling Relations: The primary result is a set of scaling relations (Eq. 42) that map the thermodynamics of the NH system to the Hermitian system.
- At fixed PFF, the self-consistent chemical potential and canonical heat capacity follow the compressed Landau spectrum. Specifically, $\mu_\beta(T, B) = \lambda \mu_0(T/\lambda, B)$ and $C_{B,\beta}(T, B) = C_{B,0}(T/\lambda, B)$ . This implies that spectral compression shifts thermally activated features to lower physical temperatures.
- At fixed chemical potential, the deformation shifts the positions of Landau-level crossings. The oscillatory caloric and magnetic responses are governed by the effective field $B_{\text{eff}} = (1-\beta^2)B$ , meaning the oscillation frequency in $1/B$ scales as $\lambda^{-2}$ .
Orbital Magnetic Response: The paper distinguishes between the total orbital magnetic moment and the moment per projected electron.
- The total orbital moment of the projected electron sector scales with the effective field alone: $M^{\text{phys}}_{0,+,\beta}(B) = M^{\text{phys}}_{0+,0}(B_{\text{eff}})$ .
- The moment per projected electron acquires an additional amplitude factor: $m^{\text{phys}}_{0,+,\beta}(B) = \lambda^2 m^{\text{phys}}_{0+,0}(B_{\text{eff}})$ . This distinction arises because the physical orbital degeneracy is determined by the applied field $B$ , not $B_{\text{eff}}$ .
Caloric Response to Deformation: The authors analyze quasistatic tuning of the deformation parameter $\beta$ $β$ .
- Changing $\beta$ performs reversible work associated with altering the Landau-level spacing.
- Under fixed PFF, an isentropic variation of $\beta$ leads to a predictable adiabatic temperature scaling: $T_f = T_i (\lambda_f / \lambda_i)$ . Increasing $|\beta|$ (increasing compression) lowers the temperature along an isentrope.
- Under fixed chemical potential, the entropy change oscillates in sign as $\beta$ varies, reflecting the difference in the Hermitian entropy evaluated at $B$ versus $B_{\text{eff}}$ .

Significance and Claims
The paper claims to establish a rigorous thermodynamic framework for real-spectrum non-Hermitian quantum matter. Its significance lies in demonstrating that equilibrium properties of these systems are not fundamentally new but are rescaled versions of Hermitian physics, provided the system thermalizes with respect to the quasi-Hermitian Hamiltonian (or its Hermitian representative).

The authors emphasize that these results provide a baseline for understanding NH effects. They note that the scaling relations serve as a reference for isolating thermodynamic effects arising from additional energy scales (such as Zeeman splitting, interactions, or disorder), which would introduce dependencies distinct from the uniform compression described here. The work also highlights the physical limits of this framework: as $|\beta| \to 1$ , the metric becomes singular, the spectrum collapses, and finite temperature or linewidth effects will eventually suppress the observable scaling before the singular threshold is reached. The paper does not propose specific experimental setups but suggests that the predicted scaling relations could be tested in graphene systems where Landau levels are spectrally resolved.

Technical Summary: Equilibrium Thermodynamics of Non-Hermitian Dirac Fermions

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