Thermodynamic signatures of non-Hermiticity in Dirac materials via quantum capacitance

Francisco J. Pe\~na; Juan Pablo Esparza; Patricio Vargas; Vladimir Juri\v{c}i\'c

arxiv: 2604.14150 · v1 · submitted 2026-04-15 · ❄️ cond-mat.mes-hall · quant-ph

Thermodynamic signatures of non-Hermiticity in Dirac materials via quantum capacitance

Juan Pablo Esparza , Francisco J. Pe\~na , Patricio Vargas , Vladimir Juri\v{c}i\'c This is my paper

Pith reviewed 2026-05-10 11:59 UTC · model grok-4.3

classification ❄️ cond-mat.mes-hall quant-ph

keywords non-HermitianDirac materialsquantum capacitanceexceptional pointsgraphenethermodynamic density of statesPetermann factornon-reciprocal hopping

0 comments

The pith

In Dirac materials modeled with non-reciprocal hopping, quantum capacitance and thermodynamic density of states diverge as the exceptional point is approached.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper examines a minimal model of graphene with an imbalance in hopping amplitudes, parameterized by β. As β approaches 1 from below, the effective Dirac velocity drops as sqrt(1 - β²), which causes the low-energy density of states, the thermodynamic density of states, and the quantum capacitance to all grow proportionally to 1/(1 - β²). At charge neutrality this keeps the capacitance linear in temperature, but with a prefactor that diverges; in a magnetic field the Landau-level spacings collapse. The same scaling appears in the Petermann factor of the biorthogonal Bloch states, isolating the effect of eigenvector non-orthogonality. These thermodynamic quantities therefore supply an equilibrium, bulk signature of non-Hermiticity that does not require dynamical or wave-based measurements.

Core claim

In the weakly non-Hermitian regime of the non-reciprocal Dirac model, the thermodynamic density of states and the quantum capacitance exhibit a universal approach to the exceptional point: both quantities, together with the low-energy density of states, scale as (1 - β²)^{-1} as |β| → 1^-. At charge neutrality the capacitance remains linear in temperature but with a diverging prefactor, while the inverse response softens linearly. In a magnetic field the Landau-level spacing collapses and thermally active levels crowd together. The biorthogonal states carry a Petermann factor K = (1 - β²)^{-1} that isolates the non-Hermitian contribution from eigenvector non-orthogonality.

What carries the argument

Hopping-imbalance parameter β in the non-reciprocal Dirac Hamiltonian, which reduces the Fermi velocity to v_F = v sqrt(1 - β²) and produces the shared (1 - β²)^{-1} scaling in the density of states, capacitance, and Petermann factor.

If this is right

Quantum capacitance stays linear in temperature at charge neutrality, but its prefactor grows without bound as the exceptional point is approached.
The inverse of the capacitance response softens linearly with (1 - β²) on approach to the exceptional point.
In an applied magnetic field the Landau-level spacing collapses proportionally to sqrt(1 - β²), crowding more levels into the thermally active window.
The Petermann factor of the biorthogonal Bloch states grows identically as (1 - β²)^{-1}, quantifying the non-orthogonality contribution.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

Capacitance measurements at charge neutrality could serve as a practical, equilibrium diagnostic for effective non-Hermiticity in engineered Dirac materials without needing time-resolved or scattering probes.
The same velocity-reduction mechanism may produce analogous thermodynamic divergences in other two-dimensional Dirac systems once non-reciprocal terms are introduced.
If the scaling survives disorder or interactions, it would link the approach to the exceptional point directly to measurable thermodynamic response functions.

Load-bearing premise

The minimal non-reciprocal graphene model with hopping imbalance β accurately captures the low-energy physics of real Dirac materials near the exceptional point.

What would settle it

Measure quantum capacitance at charge neutrality in a Dirac material whose hopping imbalance can be tuned toward β = 1; if the temperature-linear prefactor does not diverge as (1 - β²)^{-1}, the scaling claim is false.

Figures

Figures reproduced from arXiv: 2604.14150 by Francisco J. Pe\~na, Juan Pablo Esparza, Patricio Vargas, Vladimir Juri\v{c}i\'c.

**Figure 2.** Figure 2: FIG. 2. Thermodynamic density of states [PITH_FULL_IMAGE:figures/full_fig_p002_2.png] view at source ↗

**Figure 3.** Figure 3: FIG. 3. Thermodynamic density of states [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗

**Figure 4.** Figure 4: FIG. 4. Dimensionless thermodynamic density of states [PITH_FULL_IMAGE:figures/full_fig_p005_4.png] view at source ↗

read the original abstract

Non-Hermitian band descriptions capture how loss, gain, and environmental coupling reshape quantum matter, yet most experimental tests rely on wave-based or dynamical probes. Here we establish a new equilibrium route to exceptional physics in Dirac materials: in the weakly non-Hermitian regime, the thermodynamic density of states and the quantum capacitance exhibit a universal equilibrium approach to the exceptional point. In our minimal non-reciprocal graphene model, the hopping imbalance reduces the Dirac velocity as $v_F=v\sqrt{1-\beta^2}$, implying that the low-energy density of states, the thermodynamic density of states, and the quantum capacitance all scale as $(1-\beta^2)^{-1}$ as $|\beta|\to 1^-$. Consequently, at charge neutrality the quantum capacitance remains linear in temperature but with a diverging prefactor, while the inverse response softens linearly on approaching the exceptional point. In a magnetic field, this manifests as a collapse of the Landau-level spacing and a corresponding crowding of thermally active levels. Complementarily, the biorthogonal Bloch states exhibit a Petermann factor $K=(1-\beta^2)^{-1}$, which isolates the irreducibly non-Hermitian effect of eigenvector non-orthogonality. These results identify quantum capacitance as an experimentally accessible bulk equilibrium probe of effective non-Hermiticity in Dirac materials.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper ties quantum capacitance to velocity renormalization near exceptional points in a Dirac model, but the biorthogonal DOS definition looks like it may need an extra check.

read the letter

The new piece here is showing that quantum capacitance at charge neutrality stays linear in T but picks up a diverging prefactor (1-β²)^{-1} as the non-reciprocal hopping parameter β approaches the exceptional point, all coming from the renormalized Dirac velocity in their minimal model. They also keep the Petermann factor K separate as the pure non-Hermitian signature from eigenvector overlap. That gives a concrete equilibrium bulk probe that could be measured without driving the system out of equilibrium, which is a useful angle for Dirac materials work. The derivation from the Hamiltonian to the capacitance scaling is direct and the magnetic-field Landau-level crowding follows logically from the same velocity drop. The model is minimal but clean for illustration. The soft spot is exactly the one in the stress-test note. The thermodynamic DOS is written as the usual 2D Dirac form with only the velocity renormalization, yet the paper invokes the biorthogonal formalism for K. If the proper spectral density or integrated density of states in the left-right inner product picks up an extra K factor, the claimed scaling for capacitance would change. The abstract and the description do not show the explicit biorthogonal trace or Green's function step that would confirm the DOS stays free of that overlap. That is not a fatal gap but it is the load-bearing step for the central claim. The assumption that the minimal non-reciprocal model captures real materials near the point is standard for this kind of paper and not over-sold. This is for people working on non-Hermitian band structures or 2D Dirac systems who want new measurable signatures. A reader who already knows the Petermann factor literature will see the connection quickly. It is worth sending to a serious referee because the experimental suggestion is straightforward to test and the algebra is simple enough that any DOS issue can be fixed or ruled out in revision.

Referee Report

2 major / 2 minor

Summary. The paper analyzes a minimal non-reciprocal graphene model with hopping imbalance β, deriving that the Dirac velocity renormalizes to v_F = v √(1-β²). This implies that the low-energy density of states, thermodynamic density of states, and quantum capacitance all diverge as (1-β²)^{-1} approaching the exceptional point from the Hermitian side. At charge neutrality the quantum capacitance remains linear in temperature with a diverging prefactor; in a magnetic field the Landau level spacing collapses. The biorthogonal Petermann factor K=(1-β²)^{-1} is presented as a separate, irreducibly non-Hermitian correction arising from eigenvector non-orthogonality. The central claim is that quantum capacitance provides an experimentally accessible equilibrium thermodynamic signature of non-Hermiticity in Dirac materials.

Significance. If the thermodynamic DOS derivation is shown to be free of additional biorthogonal corrections, the work supplies a concrete, falsifiable prediction for capacitance measurements near exceptional points and cleanly separates velocity renormalization from the Petermann factor. This strengthens the case for equilibrium probes of non-Hermitian physics in condensed-matter systems and could guide experiments on engineered Dirac materials.

major comments (2)

[Thermodynamic density of states and quantum capacitance section] The derivation of the thermodynamic density of states (and hence quantum capacitance) invokes the standard Hermitian 2D Dirac formula DOS(E) ∝ |E|/v_F² with the renormalized velocity. In the biorthogonal formalism already used for the Petermann factor K=(1-β²)^{-1}, the proper spectral density involves the left-right eigenvector overlap; the manuscript does not explicitly demonstrate that this overlap contributes no extra factor of K or modified normalization to the thermodynamic DOS. This point is load-bearing for the claimed (1-β²)^{-1} scaling of the capacitance.
[Discussion of experimental implications] The claim that the minimal non-reciprocal model captures the low-energy physics of real Dirac materials near the exceptional point, and that quantum capacitance can be measured at charge neutrality without confounding effects, is stated but not supported by additional analysis of disorder, interactions, or finite-size effects. This assumption underpins the experimental relevance of the predicted divergence.

minor comments (2)

[Abstract] The abstract states that the scalings are 'universal'; this is accurate only within the minimal model and should be qualified to avoid overstatement.
[Model and definitions] Notation for the thermodynamic DOS versus the spectral DOS could be clarified with an explicit equation relating the two in the non-Hermitian case.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and valuable comments on our manuscript. We address each major comment below in detail and have revised the manuscript accordingly to improve clarity and rigor.

read point-by-point responses

Referee: The derivation of the thermodynamic density of states (and hence quantum capacitance) invokes the standard Hermitian 2D Dirac formula DOS(E) ∝ |E|/v_F² with the renormalized velocity. In the biorthogonal formalism already used for the Petermann factor K=(1-β²)^{-1}, the proper spectral density involves the left-right eigenvector overlap; the manuscript does not explicitly demonstrate that this overlap contributes no extra factor of K or modified normalization to the thermodynamic DOS. This point is load-bearing for the claimed (1-β²)^{-1} scaling of the capacitance.

Authors: We appreciate the referee's emphasis on this foundational point. In our minimal model the non-Hermitian Hamiltonian possesses a real spectrum for |β| < 1, and the thermodynamic density of states is obtained from the partition function (or equivalently the free energy), which depends only on the eigenvalues. The dispersion remains linear with the renormalized velocity v_F = v √(1-β²), yielding the standard 2D Dirac DOS form without additional factors. The left-right eigenvector overlap that produces the Petermann factor K affects biorthogonal normalization and dynamical response functions, but does not enter the spectral counting for equilibrium thermodynamics. We have added an explicit derivation in the 'Thermodynamic density of states and quantum capacitance' section, showing via the Green's function trace that the overlap contributes no multiplicative K to the DOS, thereby confirming that the (1-β²)^{-1} scaling originates solely from velocity renormalization. revision: yes
Referee: The claim that the minimal non-reciprocal model captures the low-energy physics of real Dirac materials near the exceptional point, and that quantum capacitance can be measured at charge neutrality without confounding effects, is stated but not supported by additional analysis of disorder, interactions, or finite-size effects. This assumption underpins the experimental relevance of the predicted divergence.

Authors: We agree that a comprehensive treatment of disorder, interactions, and finite-size effects lies beyond the scope of the present minimal-model study. Our goal was to isolate the clean thermodynamic signature of non-Hermiticity. We have revised the 'Discussion of experimental implications' section to explicitly state the assumptions of the ideal model, to note that disorder and interactions will broaden or regularize the divergence in real samples, and to emphasize that the predicted scaling nevertheless provides a falsifiable benchmark for future experiments in sufficiently clean engineered Dirac systems. This addition clarifies the intended scope while preserving the central claim for the minimal case. revision: partial

Circularity Check

0 steps flagged

No significant circularity in derivation chain

full rationale

The claimed scaling of thermodynamic DOS and quantum capacitance as (1-β²)^{-1} follows algebraically from the model-defined velocity renormalization v_F = v √(1-β²) inserted into the standard 2D Dirac DOS formula DOS(E) ∝ |E|/v_F². This is a direct consequence of the non-reciprocal hopping model and is not a self-referential fit or redefinition of the target observable. The Petermann factor K=(1-β²)^{-1} is isolated as a separate biorthogonal effect. No load-bearing self-citations, ansatz smuggling, or uniqueness theorems from prior author work are invoked to force the result. The derivation remains self-contained against the external Hermitian Dirac benchmark and the explicit model Hamiltonian.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The central claim rests on the assumption that a minimal non-reciprocal tight-binding model for graphene captures the essential non-Hermitian physics, together with the definition of the control parameter β that tunes the system to the exceptional point.

free parameters (1)

β
Dimensionless non-reciprocity parameter that controls the hopping imbalance and is taken to approach 1 from below to reach the exceptional point.

axioms (1)

domain assumption The low-energy physics of the material is described by a minimal non-reciprocal graphene Hamiltonian with asymmetric nearest-neighbor hopping.
Invoked to derive the velocity renormalization v_F = v √(1-β²) and the subsequent density-of-states scaling.

pith-pipeline@v0.9.0 · 5557 in / 1497 out tokens · 45972 ms · 2026-05-10T11:59:51.891932+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

76 extracted references · 76 canonical work pages

[1]

Z. Gong, Y. Ashida, K. Kawabata, K. Takasan, S. Hi- gashikawa, and M. Ueda, Topological phases of non- hermitian systems, Phys. Rev. X8, 031079 (2018)

work page 2018
[2]

L. E. F. Foa Torres, Perspective on topological states of non-hermitian lattices, Journal of Physics: Materials3, 014002 (2019)

work page 2019
[3]

E. J. Bergholtz, J. C. Budich, and F. K. Kunst, Ex- ceptional topology of non-hermitian systems, Rev. Mod. Phys.93, 015005 (2021)

work page 2021
[4]

Jin, Topological phases and edge states in a non- hermitian trimerized optical lattice, Phys

L. Jin, Topological phases and edge states in a non- hermitian trimerized optical lattice, Phys. Rev. A96, 032103 (2017)

work page 2017
[5]

L. Li, C. H. Lee, and J. Gong, Topological switch for non-hermitian skin effect in cold-atom systems with loss, Phys. Rev. Lett.124, 250402 (2020)

work page 2020
[6]

Liang, D

Q. Liang, D. Xie, Z. Dong, H. Li, H. Li, B. Gadway, W. Yi, and B. Yan, Dynamic signatures of non-hermitian skin effect and topology in ultracold atoms, Phys. Rev. Lett.129, 070401 (2022)

work page 2022
[7]

L. Zhou, H. Li, W. Yi, and X. Cui, Engineering non- hermitian skin effect with band topology in ultracold gases, Communications Physics5, 252 (2022)

work page 2022
[8]

Z.-F. Cai, T. Liu, and Z. Yang, Non-hermitian skin effect in periodically driven dissipative ultracold atoms, Phys. Rev. A109, 063329 (2024)

work page 2024
[9]

E. Zhao, Z. Wang, C. He, T. F. J. Poon, K. K. Pak, Y.- J. Liu, P. Ren, X.-J. Liu, and G.-B. Jo, Two-dimensional 6 non-hermitian skin effect in an ultracold fermi gas, Na- ture637, 565 (2025)

work page 2025
[10]

Zhang, X

Y. Zhang, X. Dai, and Y. Xiang, Topological hierarchy in non-hermitianthree-dimensionalphotoniccrystals,Phys. Rev. B110, 104103 (2024)

work page 2024
[11]

X. Zhou, S. K. Gupta, Z. Huang, Z. Yan, P. Zhan, Z. Chen, M. Lu, and Z. Wang, Optical lattices with higher-order exceptional points by non-hermitian cou- pling, Applied Physics Letters113, 101108 (2018)

work page 2018
[12]

S. M. Rafi-Ul-Islam, H. Sahin, Z. B. Siu, and M. B. A. Jalil, Interfacial skin modes at a non-hermitian hetero- junction, Phys. Rev. Res.4, 043021 (2022)

work page 2022
[13]

Liu, Non-hermitian second-order topology induced by resistances in electric circuits, Phys

J.Wu, X.Huang, Y.Yang, W.Deng, J.Lu, W.Deng,and Z. Liu, Non-hermitian second-order topology induced by resistances in electric circuits, Phys. Rev. B105, 195127 (2022)

work page 2022
[14]

Ezawa, Experimental observation of non-hermitian higher-order skin interface states in topological electric circuits, Phys

B.Liu, Y.Li, B.Yang, X.Shen, Y.Yang, Z.H.Hang,and M. Ezawa, Experimental observation of non-hermitian higher-order skin interface states in topological electric circuits, Phys. Rev. Res.5, 043034 (2023)

work page 2023
[15]

Ochkan, R

K. Ochkan, R. Chaturvedi, V. Könye, L. Veyrat, R. Gi- raud, D. Mailly, A. Cavanna, U. Gennser, E. M. Han- kiewicz, B. Büchner,et al., Non-hermitian topology in a multi-terminal quantum hall device, Nature Physics20, 395 (2024)

work page 2024
[16]

Juričić and B

V. Juričić and B. Roy, Yukawa-lorentz symmetry in non- hermitian dirac materials, Communications Physics7, 169 (2024)

work page 2024
[17]

S. A. Murshed and B. Roy, Quantum electrodynamics of non-hermitian dirac fermions, Journal of High Energy Physics2024, 1 (2024)

work page 2024
[18]

X.-J. Yu, Z. Pan, L. Xu, and Z.-X. Li, Non-hermitian strongly interacting dirac fermions, Phys. Rev. Lett.132, 116503 (2024)

work page 2024
[19]

S. A. Murshed and B. Roy, Yukawa-lorentz symmetry of interacting non-hermitian birefringent dirac fermions, SciPost Phys.18, 073 (2025)

work page 2025
[20]

C. A. Leong and B. Roy, Amplified magnetic catalysis in non-hermitian euclidean and hyperbolic dirac liquids (2025), arXiv:2510.02304 [cond-mat.str-el]

work page arXiv 2025
[21]

Roy, Zero modes and index theorems for non- hermitian dirac fermions, Phys

B. Roy, Zero modes and index theorems for non- hermitian dirac fermions, Phys. Rev. D112, 125016 (2025)

work page 2025
[22]

J. P. Esparza and V. Juričić, Exceptional magic angles in non-hermitian twisted bilayer graphene, Phys. Rev. Lett. 134, 226602 (2025)

work page 2025
[23]

Rotter, A non-hermitian hamilton operator and the physics of open quantum systems, Journal of Physics A: Mathematical and Theoretical42, 153001 (2009)

I. Rotter, A non-hermitian hamilton operator and the physics of open quantum systems, Journal of Physics A: Mathematical and Theoretical42, 153001 (2009)

work page 2009
[24]

E. J. Bergholtz and J. C. Budich, Non-hermitian weyl physics in topological insulator ferromagnet junctions, Phys. Rev. Res.1, 012003 (2019)

work page 2019
[25]

Kozii and L

V. Kozii and L. Fu, Non-hermitian topological theory of finite-lifetime quasiparticles: Prediction of bulk fermi arc due to exceptional point, Phys. Rev. B109, 235139 (2024)

work page 2024
[26]

Hamanaka, T

S. Hamanaka, T. Yoshida, and K. Kawabata, Non- hermitian topology in hermitian topological matter, Phys. Rev. Lett.133, 266604 (2024)

work page 2024
[27]

Lindblad, On the generators of quantum dynamical semigroups, Commun

G. Lindblad, On the generators of quantum dynamical semigroups, Commun. Math. Phys.48, 119 (1976)

work page 1976
[28]

Spohn, Entropy production for quantum dynamical semigroups, J

H. Spohn, Entropy production for quantum dynamical semigroups, J. Math. Phys.19, 1227 (1978)

work page 1978
[29]

Alicki, The quantum open system as a model of the heat engine, J

R. Alicki, The quantum open system as a model of the heat engine, J. Phys. A: Math. Gen.12, L103 (1979)

work page 1979
[30]

Lett.67, 565 (2005)

A.E.Allahverdyan, R.Balian,andT.M.Nieuwenhuizen, Maximal work extraction from finite quantum systems, Europhys. Lett.67, 565 (2005)

work page 2005
[31]

Talkner, E

P. Talkner, E. Lutz, and P. Hänggi, Fluctuation theo- rems: Workisnotanobservable,Phys.Rev.E75,050102 (2007)

work page 2007
[32]

Esposito, U

M. Esposito, U. Harbola, and S. Mukamel, Nonequilib- rium fluctuations, fluctuation theorems, and counting statistics in quantum systems, Rev. Mod. Phys.81, 1665 (2009)

work page 2009
[33]

Campisi, P

M. Campisi, P. Hänggi, and P. Talkner, Colloquium: Quantum fluctuation relations: Foundations and appli- cations, Rev. Mod. Phys.83, 771 (2011)

work page 2011
[34]

Deffner and E

S. Deffner and E. Lutz, Nonequilibrium entropy produc- tion for open quantum systems, Phys. Rev. Lett.107, 140404 (2011)

work page 2011
[35]

Kosloff, Quantum thermodynamics: A dynamical viewpoint, Entropy15, 2100 (2013)

R. Kosloff, Quantum thermodynamics: A dynamical viewpoint, Entropy15, 2100 (2013)

work page 2013
[36]

Deffner and E

S. Deffner and E. Lutz, Quantum speed limit for non- markovian dynamics, Phys. Rev. Lett.111, 010402 (2013)

work page 2013
[37]

Kosloff and A

R. Kosloff and A. Levy, Quantum heat engines and re- frigerators: Continuous devices, Annu. Rev. Phys. Chem. 65, 365 (2014)

work page 2014
[38]

Gelbwaser-Klimovsky, W

D. Gelbwaser-Klimovsky, W. Niedenzu, and G. Kurizki, Thermodynamics of quantum systems under dynamical control, Adv. At. Mol. Opt. Phys.64, 329 (2015)

work page 2015
[39]

Goold, M

J. Goold, M. Huber, A. Riera, L. del Rio, and P. Skrzypczyk, The role of quantum information in thermodynamics—a topical review, J. Phys. A: Math. Theor.49, 143001 (2016)

work page 2016
[40]

Vinjanampathy and J

S. Vinjanampathy and J. Anders, Quantum thermody- namics, Contemp. Phys.57, 545 (2016)

work page 2016
[41]

Strasberg, G

P. Strasberg, G. Schaller, T. Brandes, and M. Esposito, Quantum and information thermodynamics: A unifying framework based on repeated interactions, Phys. Rev. X 7, 021003 (2017)

work page 2017
[42]

Gardas, S

B. Gardas, S. Deffner, and A. Saxena, Non-hermitian quantum thermodynamics, Scientific reports6, 23408 (2016)

work page 2016
[43]

Bebiano, J

N. Bebiano, J. da Providência, and J. P. da Providên- cia, Toward non-hermitian quantum statistical thermo- dynamics, Journal of Mathematical Physics61, 022102 (2020)

work page 2020
[44]

F. C. E. Lima, A. R. P. Moreira, and C. A. S. Almeida, Information and thermodynamic properties of a non- hermitian particle ensemble, International Journal of Quantum Chemistry121, e26645 (2021)

work page 2021
[45]

Cao and S.-P

K. Cao and S.-P. Kou, Statistical mechanics for non- hermitian quantum systems, Phys. Rev. Res.5, 033196 (2023)

work page 2023
[46]

K. Cao, Q. Du, and S.-P. Kou, Information thermody- namics of non-hermitian quantum systems, Phys. Rev. A112, 062227 (2025)

work page 2025
[47]

Luryi, Quantum capacitance devices, Applied Physics Letters52, 501 (1988)

S. Luryi, Quantum capacitance devices, Applied Physics Letters52, 501 (1988)

work page 1988
[48]

González, F

J. González, F. Guinea, and M. Vozmediano, The elec- tronic spectrum of fullerenes from the dirac equation, Nuclear Physics B406, 771 (1993)

work page 1993
[49]

N. M. R. Peres, F. Guinea, and A. H. Castro Neto, Elec- tronic properties of disordered two-dimensional carbon, Phys. Rev. B73, 125411 (2006). 7

work page 2006
[50]

A. H. Castro Neto, F. Guinea, N. M. R. Peres, K. S. Novoselov, and A. K. Geim, The electronic properties of graphene, Rev. Mod. Phys.81, 109 (2009)

work page 2009
[51]

Petermann, Calculated spontaneous emission factor for double-heterostructure injection lasers with gain- induced waveguiding, IEEE Journal of Quantum Elec- tronics15, 566 (1979)

K. Petermann, Calculated spontaneous emission factor for double-heterostructure injection lasers with gain- induced waveguiding, IEEE Journal of Quantum Elec- tronics15, 566 (1979)

work page 1979
[52]

A. E. Siegman, Excess spontaneous emission in non- hermitian optical systems. i. laser amplifiers, Physical Review A39, 1253 (1989)

work page 1989
[53]

A. E. Siegman, Excess spontaneous emission in non- hermitian optical systems. ii. laser oscillators, Physical Review A39, 1264 (1989)

work page 1989
[54]

Cheng, C

Y.-J. Cheng, C. G. Fanning, and A. E. Siegman, Experi- mental observation of a large excess quantum noise factor inthelinewidthofalaseroscillatorhavingnonorthogonal modes, Physical Review Letters77, 627 (1996)

work page 1996
[55]

Wang, Y.-H

H. Wang, Y.-H. Lai, Z. Yuan, M.-G. Suh, and K. Va- hala, Petermann-factor sensitivity limit near an excep- tional point in a brillouin ring laser gyroscope, Nature Communications11, 1610 (2020)

work page 2020
[56]

A. Das, S. Pisana, B. Chakraborty, S. Piscanec, S. K. Saha, U. V. Waghmare, K. S. Novoselov, H. R. Krish- namurthy, A. K. Geim, A. C. Ferrari,et al., Monitoring dopants by raman scattering in an electrochemically top- gated graphene transistor, Nature nanotechnology3, 210 (2008)

work page 2008
[57]

J. Xia, F. Chen, J. Li, and N. Tao, Measurement of the quantum capacitance of graphene, Nature nanotechnol- ogy4, 505 (2009)

work page 2009
[58]

Mišković and N

Z. Mišković and N. Upadhyaya, Modeling electrolytically top-gated graphene, Nanoscale research letters5, 505 (2010)

work page 2010
[59]

C. Zhan, J. Neal, J. Wu, and D.-e. Jiang, Quantum ef- fects on the capacitance of graphene-based electrodes, The Journal of Physical Chemistry C119, 22297 (2015)

work page 2015
[60]

S.Ilani, L.A.Donev, M.Kindermann,andP.L.McEuen, Measurement of the quantum capacitance of interacting electrons in carbon nanotubes, Nature Physics2, 687 (2006)

work page 2006
[61]

G. L. Yu, R. Jalil, B. Belle, A. S. Mayorov, P. Blake, F. Schedin, S. V. Morozov, L. A. Ponomarenko, F. Chi- appini, S. Wiedmann, U. Zeitler, M. I. Katsnelson, A. K. Geim, K. S. Novoselov, and D. C. Elias, Interaction phe- nomena in graphene seen through quantum capacitance, Proceedings of the National Academy of Sciences110, 3282 (2013)

work page 2013
[62]

J. W. McClure, Diamagnetism of graphite, Phys. Rev. 104, 666 (1956)

work page 1956
[63]

Zheng and T

Y. Zheng and T. Ando, Hall conductivity of a two- dimensional graphite system, Phys. Rev. B65, 245420 (2002)

work page 2002
[64]

Lukose, R

V. Lukose, R. Shankar, and G. Baskaran, Novel electric field effects on landau levels in graphene, Phys. Rev. Lett. 98, 116802 (2007)

work page 2007
[65]

N. M. R. Peres and E. V. Castro, Algebraic solution of a graphene layer in transverse electric and perpendicular magnetic fields, Journal of Physics: Condensed Matter 19, 406231 (2007)

work page 2007
[66]

Jiang, E

Z. Jiang, E. A. Henriksen, L. C. Tung, Y.-J. Wang, M. E. Schwartz, M. Y. Han, P. Kim, and H. L. Stormer, In- frared spectroscopy of landau levels of graphene, Phys. Rev. Lett.98, 197403 (2007)

work page 2007
[67]

Li and E

G. Li and E. Y. Andrei, Observation of landau levels of dirac fermions in graphite, Nature physics3, 623 (2007)

work page 2007
[68]

Koshino and E

M. Koshino and E. McCann, Landau level spectra and the quantum hall effect of multilayer graphene, Phys. Rev. B83, 165443 (2011)

work page 2011
[69]

Luican, G

A. Luican, G. Li, and E. Y. Andrei, Quantized landau level spectrum and its density dependence in graphene, Phys. Rev. B83, 041405 (2011)

work page 2011
[70]

G. G. Pyrialakos, H. Ren, P. S. Jung, M. Khajavikhan, and D. N. Christodoulides, Thermalization dynamics of nonlinear non-hermitian optical lattices, Phys. Rev. Lett. 128, 213901 (2022)

work page 2022
[71]

D. F. Munoz-Arboleda, R. Arouca, and C. M. Smith, Thermodynamics and entanglement entropy of the non- hermitian su-schrieffer-heeger model, Phys. Rev. B110, 115135 (2024)

work page 2024
[72]

D. M. Pino, Y. Meir, and R. Aguado, Thermodynam- ics of non-hermitian josephson junctions with exceptional points, Phys. Rev. B111, L140503 (2025)

work page 2025
[73]

A.Felski, A.Beygi,andS.P.Klevansky,Thermodynamic properties of non-hermitian nambu–jona-lasinio models, Phys. Rev. D107, 016015 (2023)

work page 2023
[74]

Fan, G.-Y

A. Fan, G.-Y. Huang, and S.-D. Liang, Complex berry curvature pair and quantum hall admittance in non- hermitian systems, Journal of Physics Communications 4, 115006 (2020)

work page 2020
[75]

P. R. Bueno and J. J. Davis, Measuring quantum ca- pacitance in energetically addressable molecular layers, Analytical Chemistry86, 1337 (2014)

work page 2014
[76]

T. Cea, N. R. Walet, and F. Guinea, Twists and the electronic structure of graphitic materials, Nano Letters 19, 8683 (2019)

work page 2019

[1] [1]

Z. Gong, Y. Ashida, K. Kawabata, K. Takasan, S. Hi- gashikawa, and M. Ueda, Topological phases of non- hermitian systems, Phys. Rev. X8, 031079 (2018)

work page 2018

[2] [2]

L. E. F. Foa Torres, Perspective on topological states of non-hermitian lattices, Journal of Physics: Materials3, 014002 (2019)

work page 2019

[3] [3]

E. J. Bergholtz, J. C. Budich, and F. K. Kunst, Ex- ceptional topology of non-hermitian systems, Rev. Mod. Phys.93, 015005 (2021)

work page 2021

[4] [4]

Jin, Topological phases and edge states in a non- hermitian trimerized optical lattice, Phys

L. Jin, Topological phases and edge states in a non- hermitian trimerized optical lattice, Phys. Rev. A96, 032103 (2017)

work page 2017

[5] [5]

L. Li, C. H. Lee, and J. Gong, Topological switch for non-hermitian skin effect in cold-atom systems with loss, Phys. Rev. Lett.124, 250402 (2020)

work page 2020

[6] [6]

Liang, D

Q. Liang, D. Xie, Z. Dong, H. Li, H. Li, B. Gadway, W. Yi, and B. Yan, Dynamic signatures of non-hermitian skin effect and topology in ultracold atoms, Phys. Rev. Lett.129, 070401 (2022)

work page 2022

[7] [7]

L. Zhou, H. Li, W. Yi, and X. Cui, Engineering non- hermitian skin effect with band topology in ultracold gases, Communications Physics5, 252 (2022)

work page 2022

[8] [8]

Z.-F. Cai, T. Liu, and Z. Yang, Non-hermitian skin effect in periodically driven dissipative ultracold atoms, Phys. Rev. A109, 063329 (2024)

work page 2024

[9] [9]

E. Zhao, Z. Wang, C. He, T. F. J. Poon, K. K. Pak, Y.- J. Liu, P. Ren, X.-J. Liu, and G.-B. Jo, Two-dimensional 6 non-hermitian skin effect in an ultracold fermi gas, Na- ture637, 565 (2025)

work page 2025

[10] [10]

Zhang, X

Y. Zhang, X. Dai, and Y. Xiang, Topological hierarchy in non-hermitianthree-dimensionalphotoniccrystals,Phys. Rev. B110, 104103 (2024)

work page 2024

[11] [11]

X. Zhou, S. K. Gupta, Z. Huang, Z. Yan, P. Zhan, Z. Chen, M. Lu, and Z. Wang, Optical lattices with higher-order exceptional points by non-hermitian cou- pling, Applied Physics Letters113, 101108 (2018)

work page 2018

[12] [12]

S. M. Rafi-Ul-Islam, H. Sahin, Z. B. Siu, and M. B. A. Jalil, Interfacial skin modes at a non-hermitian hetero- junction, Phys. Rev. Res.4, 043021 (2022)

work page 2022

[13] [13]

Liu, Non-hermitian second-order topology induced by resistances in electric circuits, Phys

J.Wu, X.Huang, Y.Yang, W.Deng, J.Lu, W.Deng,and Z. Liu, Non-hermitian second-order topology induced by resistances in electric circuits, Phys. Rev. B105, 195127 (2022)

work page 2022

[14] [14]

Ezawa, Experimental observation of non-hermitian higher-order skin interface states in topological electric circuits, Phys

B.Liu, Y.Li, B.Yang, X.Shen, Y.Yang, Z.H.Hang,and M. Ezawa, Experimental observation of non-hermitian higher-order skin interface states in topological electric circuits, Phys. Rev. Res.5, 043034 (2023)

work page 2023

[15] [15]

Ochkan, R

K. Ochkan, R. Chaturvedi, V. Könye, L. Veyrat, R. Gi- raud, D. Mailly, A. Cavanna, U. Gennser, E. M. Han- kiewicz, B. Büchner,et al., Non-hermitian topology in a multi-terminal quantum hall device, Nature Physics20, 395 (2024)

work page 2024

[16] [16]

Juričić and B

V. Juričić and B. Roy, Yukawa-lorentz symmetry in non- hermitian dirac materials, Communications Physics7, 169 (2024)

work page 2024

[17] [17]

S. A. Murshed and B. Roy, Quantum electrodynamics of non-hermitian dirac fermions, Journal of High Energy Physics2024, 1 (2024)

work page 2024

[18] [18]

X.-J. Yu, Z. Pan, L. Xu, and Z.-X. Li, Non-hermitian strongly interacting dirac fermions, Phys. Rev. Lett.132, 116503 (2024)

work page 2024

[19] [19]

S. A. Murshed and B. Roy, Yukawa-lorentz symmetry of interacting non-hermitian birefringent dirac fermions, SciPost Phys.18, 073 (2025)

work page 2025

[20] [20]

C. A. Leong and B. Roy, Amplified magnetic catalysis in non-hermitian euclidean and hyperbolic dirac liquids (2025), arXiv:2510.02304 [cond-mat.str-el]

work page arXiv 2025

[21] [21]

Roy, Zero modes and index theorems for non- hermitian dirac fermions, Phys

B. Roy, Zero modes and index theorems for non- hermitian dirac fermions, Phys. Rev. D112, 125016 (2025)

work page 2025

[22] [22]

J. P. Esparza and V. Juričić, Exceptional magic angles in non-hermitian twisted bilayer graphene, Phys. Rev. Lett. 134, 226602 (2025)

work page 2025

[23] [23]

Rotter, A non-hermitian hamilton operator and the physics of open quantum systems, Journal of Physics A: Mathematical and Theoretical42, 153001 (2009)

I. Rotter, A non-hermitian hamilton operator and the physics of open quantum systems, Journal of Physics A: Mathematical and Theoretical42, 153001 (2009)

work page 2009

[24] [24]

E. J. Bergholtz and J. C. Budich, Non-hermitian weyl physics in topological insulator ferromagnet junctions, Phys. Rev. Res.1, 012003 (2019)

work page 2019

[25] [25]

Kozii and L

V. Kozii and L. Fu, Non-hermitian topological theory of finite-lifetime quasiparticles: Prediction of bulk fermi arc due to exceptional point, Phys. Rev. B109, 235139 (2024)

work page 2024

[26] [26]

Hamanaka, T

S. Hamanaka, T. Yoshida, and K. Kawabata, Non- hermitian topology in hermitian topological matter, Phys. Rev. Lett.133, 266604 (2024)

work page 2024

[27] [27]

Lindblad, On the generators of quantum dynamical semigroups, Commun

G. Lindblad, On the generators of quantum dynamical semigroups, Commun. Math. Phys.48, 119 (1976)

work page 1976

[28] [28]

Spohn, Entropy production for quantum dynamical semigroups, J

H. Spohn, Entropy production for quantum dynamical semigroups, J. Math. Phys.19, 1227 (1978)

work page 1978

[29] [29]

Alicki, The quantum open system as a model of the heat engine, J

R. Alicki, The quantum open system as a model of the heat engine, J. Phys. A: Math. Gen.12, L103 (1979)

work page 1979

[30] [30]

Lett.67, 565 (2005)

A.E.Allahverdyan, R.Balian,andT.M.Nieuwenhuizen, Maximal work extraction from finite quantum systems, Europhys. Lett.67, 565 (2005)

work page 2005

[31] [31]

Talkner, E

P. Talkner, E. Lutz, and P. Hänggi, Fluctuation theo- rems: Workisnotanobservable,Phys.Rev.E75,050102 (2007)

work page 2007

[32] [32]

Esposito, U

M. Esposito, U. Harbola, and S. Mukamel, Nonequilib- rium fluctuations, fluctuation theorems, and counting statistics in quantum systems, Rev. Mod. Phys.81, 1665 (2009)

work page 2009

[33] [33]

Campisi, P

M. Campisi, P. Hänggi, and P. Talkner, Colloquium: Quantum fluctuation relations: Foundations and appli- cations, Rev. Mod. Phys.83, 771 (2011)

work page 2011

[34] [34]

Deffner and E

S. Deffner and E. Lutz, Nonequilibrium entropy produc- tion for open quantum systems, Phys. Rev. Lett.107, 140404 (2011)

work page 2011

[35] [35]

Kosloff, Quantum thermodynamics: A dynamical viewpoint, Entropy15, 2100 (2013)

R. Kosloff, Quantum thermodynamics: A dynamical viewpoint, Entropy15, 2100 (2013)

work page 2013

[36] [36]

Deffner and E

S. Deffner and E. Lutz, Quantum speed limit for non- markovian dynamics, Phys. Rev. Lett.111, 010402 (2013)

work page 2013

[37] [37]

Kosloff and A

R. Kosloff and A. Levy, Quantum heat engines and re- frigerators: Continuous devices, Annu. Rev. Phys. Chem. 65, 365 (2014)

work page 2014

[38] [38]

Gelbwaser-Klimovsky, W

D. Gelbwaser-Klimovsky, W. Niedenzu, and G. Kurizki, Thermodynamics of quantum systems under dynamical control, Adv. At. Mol. Opt. Phys.64, 329 (2015)

work page 2015

[39] [39]

Goold, M

J. Goold, M. Huber, A. Riera, L. del Rio, and P. Skrzypczyk, The role of quantum information in thermodynamics—a topical review, J. Phys. A: Math. Theor.49, 143001 (2016)

work page 2016

[40] [40]

Vinjanampathy and J

S. Vinjanampathy and J. Anders, Quantum thermody- namics, Contemp. Phys.57, 545 (2016)

work page 2016

[41] [41]

Strasberg, G

P. Strasberg, G. Schaller, T. Brandes, and M. Esposito, Quantum and information thermodynamics: A unifying framework based on repeated interactions, Phys. Rev. X 7, 021003 (2017)

work page 2017

[42] [42]

Gardas, S

B. Gardas, S. Deffner, and A. Saxena, Non-hermitian quantum thermodynamics, Scientific reports6, 23408 (2016)

work page 2016

[43] [43]

Bebiano, J

N. Bebiano, J. da Providência, and J. P. da Providên- cia, Toward non-hermitian quantum statistical thermo- dynamics, Journal of Mathematical Physics61, 022102 (2020)

work page 2020

[44] [44]

F. C. E. Lima, A. R. P. Moreira, and C. A. S. Almeida, Information and thermodynamic properties of a non- hermitian particle ensemble, International Journal of Quantum Chemistry121, e26645 (2021)

work page 2021

[45] [45]

Cao and S.-P

K. Cao and S.-P. Kou, Statistical mechanics for non- hermitian quantum systems, Phys. Rev. Res.5, 033196 (2023)

work page 2023

[46] [46]

K. Cao, Q. Du, and S.-P. Kou, Information thermody- namics of non-hermitian quantum systems, Phys. Rev. A112, 062227 (2025)

work page 2025

[47] [47]

Luryi, Quantum capacitance devices, Applied Physics Letters52, 501 (1988)

S. Luryi, Quantum capacitance devices, Applied Physics Letters52, 501 (1988)

work page 1988

[48] [48]

González, F

J. González, F. Guinea, and M. Vozmediano, The elec- tronic spectrum of fullerenes from the dirac equation, Nuclear Physics B406, 771 (1993)

work page 1993

[49] [49]

N. M. R. Peres, F. Guinea, and A. H. Castro Neto, Elec- tronic properties of disordered two-dimensional carbon, Phys. Rev. B73, 125411 (2006). 7

work page 2006

[50] [50]

A. H. Castro Neto, F. Guinea, N. M. R. Peres, K. S. Novoselov, and A. K. Geim, The electronic properties of graphene, Rev. Mod. Phys.81, 109 (2009)

work page 2009

[51] [51]

Petermann, Calculated spontaneous emission factor for double-heterostructure injection lasers with gain- induced waveguiding, IEEE Journal of Quantum Elec- tronics15, 566 (1979)

K. Petermann, Calculated spontaneous emission factor for double-heterostructure injection lasers with gain- induced waveguiding, IEEE Journal of Quantum Elec- tronics15, 566 (1979)

work page 1979

[52] [52]

A. E. Siegman, Excess spontaneous emission in non- hermitian optical systems. i. laser amplifiers, Physical Review A39, 1253 (1989)

work page 1989

[53] [53]

A. E. Siegman, Excess spontaneous emission in non- hermitian optical systems. ii. laser oscillators, Physical Review A39, 1264 (1989)

work page 1989

[54] [54]

Cheng, C

Y.-J. Cheng, C. G. Fanning, and A. E. Siegman, Experi- mental observation of a large excess quantum noise factor inthelinewidthofalaseroscillatorhavingnonorthogonal modes, Physical Review Letters77, 627 (1996)

work page 1996

[55] [55]

Wang, Y.-H

H. Wang, Y.-H. Lai, Z. Yuan, M.-G. Suh, and K. Va- hala, Petermann-factor sensitivity limit near an excep- tional point in a brillouin ring laser gyroscope, Nature Communications11, 1610 (2020)

work page 2020

[56] [56]

A. Das, S. Pisana, B. Chakraborty, S. Piscanec, S. K. Saha, U. V. Waghmare, K. S. Novoselov, H. R. Krish- namurthy, A. K. Geim, A. C. Ferrari,et al., Monitoring dopants by raman scattering in an electrochemically top- gated graphene transistor, Nature nanotechnology3, 210 (2008)

work page 2008

[57] [57]

J. Xia, F. Chen, J. Li, and N. Tao, Measurement of the quantum capacitance of graphene, Nature nanotechnol- ogy4, 505 (2009)

work page 2009

[58] [58]

Mišković and N

Z. Mišković and N. Upadhyaya, Modeling electrolytically top-gated graphene, Nanoscale research letters5, 505 (2010)

work page 2010

[59] [59]

C. Zhan, J. Neal, J. Wu, and D.-e. Jiang, Quantum ef- fects on the capacitance of graphene-based electrodes, The Journal of Physical Chemistry C119, 22297 (2015)

work page 2015

[60] [60]

S.Ilani, L.A.Donev, M.Kindermann,andP.L.McEuen, Measurement of the quantum capacitance of interacting electrons in carbon nanotubes, Nature Physics2, 687 (2006)

work page 2006

[61] [61]

G. L. Yu, R. Jalil, B. Belle, A. S. Mayorov, P. Blake, F. Schedin, S. V. Morozov, L. A. Ponomarenko, F. Chi- appini, S. Wiedmann, U. Zeitler, M. I. Katsnelson, A. K. Geim, K. S. Novoselov, and D. C. Elias, Interaction phe- nomena in graphene seen through quantum capacitance, Proceedings of the National Academy of Sciences110, 3282 (2013)

work page 2013

[62] [62]

J. W. McClure, Diamagnetism of graphite, Phys. Rev. 104, 666 (1956)

work page 1956

[63] [63]

Zheng and T

Y. Zheng and T. Ando, Hall conductivity of a two- dimensional graphite system, Phys. Rev. B65, 245420 (2002)

work page 2002

[64] [64]

Lukose, R

V. Lukose, R. Shankar, and G. Baskaran, Novel electric field effects on landau levels in graphene, Phys. Rev. Lett. 98, 116802 (2007)

work page 2007

[65] [65]

N. M. R. Peres and E. V. Castro, Algebraic solution of a graphene layer in transverse electric and perpendicular magnetic fields, Journal of Physics: Condensed Matter 19, 406231 (2007)

work page 2007

[66] [66]

Jiang, E

Z. Jiang, E. A. Henriksen, L. C. Tung, Y.-J. Wang, M. E. Schwartz, M. Y. Han, P. Kim, and H. L. Stormer, In- frared spectroscopy of landau levels of graphene, Phys. Rev. Lett.98, 197403 (2007)

work page 2007

[67] [67]

Li and E

G. Li and E. Y. Andrei, Observation of landau levels of dirac fermions in graphite, Nature physics3, 623 (2007)

work page 2007

[68] [68]

Koshino and E

M. Koshino and E. McCann, Landau level spectra and the quantum hall effect of multilayer graphene, Phys. Rev. B83, 165443 (2011)

work page 2011

[69] [69]

Luican, G

A. Luican, G. Li, and E. Y. Andrei, Quantized landau level spectrum and its density dependence in graphene, Phys. Rev. B83, 041405 (2011)

work page 2011

[70] [70]

G. G. Pyrialakos, H. Ren, P. S. Jung, M. Khajavikhan, and D. N. Christodoulides, Thermalization dynamics of nonlinear non-hermitian optical lattices, Phys. Rev. Lett. 128, 213901 (2022)

work page 2022

[71] [71]

D. F. Munoz-Arboleda, R. Arouca, and C. M. Smith, Thermodynamics and entanglement entropy of the non- hermitian su-schrieffer-heeger model, Phys. Rev. B110, 115135 (2024)

work page 2024

[72] [72]

D. M. Pino, Y. Meir, and R. Aguado, Thermodynam- ics of non-hermitian josephson junctions with exceptional points, Phys. Rev. B111, L140503 (2025)

work page 2025

[73] [73]

A.Felski, A.Beygi,andS.P.Klevansky,Thermodynamic properties of non-hermitian nambu–jona-lasinio models, Phys. Rev. D107, 016015 (2023)

work page 2023

[74] [74]

Fan, G.-Y

A. Fan, G.-Y. Huang, and S.-D. Liang, Complex berry curvature pair and quantum hall admittance in non- hermitian systems, Journal of Physics Communications 4, 115006 (2020)

work page 2020

[75] [75]

P. R. Bueno and J. J. Davis, Measuring quantum ca- pacitance in energetically addressable molecular layers, Analytical Chemistry86, 1337 (2014)

work page 2014

[76] [76]

T. Cea, N. R. Walet, and F. Guinea, Twists and the electronic structure of graphitic materials, Nano Letters 19, 8683 (2019)

work page 2019