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

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

REVIEW 2 major objections 5 minor 100 references

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

T0 review · 2 major / 5 minor · reviewed 2026-08-08 · deepseek-v4-flash

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Pith's one-line read Dirac heat capacity, entropy, and orbital moment obey one scaling law

desk verdict A mathematically clean set of scaling identities for a real-spectrum non-Hermitian Dirac model, with the central weakness honestly flagged: the assumed Gibbs state is a convention, not a derived property of any open system. read the letter →

arxiv 2608.04369 v1 pith:FIUWQLE6 submitted 2026-08-05 quant-ph cond-mat.stat-mech

Francisco J. Peña , Bastian Castorene , Juan Pablo Esparza , Vladimir Juričić , Patricio Vargas This is my paper · ORCID

classification quant-phcond-mat.stat-mech MSC 81Q1282B30

keywords non-Hermitianthermodynamicsquasi-HermitianHamiltonianLandaulevelsDiracfermionsheatcapacityscalingorbitalmagneticresponsegrapheneadiabaticcooling

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper claims that for a non-Hermitian Dirac Hamiltonian with a real spectrum (the deformation parameter $\beta$ with $|\beta|<1$), every equilibrium thermodynamic observable is exactly the corresponding Hermitian observable evaluated at rescaled arguments. The load-bearing identity is the grand partition function $\Xi_\beta(T,\mu;B)=\Xi_0(T/\lambda,\mu/\lambda;B)$ with $\lambda=\sqrt{1-\beta^2}$, which follows from a similarity transformation $H_\beta=\lambda S H_0 S^{-1}$. Consequently, at fixed projected filling factor the chemical potential and canonical heat capacity follow the compressed Landau-level ladder, while at fixed chemical potential the heat capacity, entropy, and orbital magnetic moment shift to an effective field $B_{\rm eff}=\lambda^2 B$. The paper argues this framework makes the non-Hermitian problem's caloric and magnetic response computable from the Hermitian one, and so provides a starting point for extrapolating the same scaling to interactions and disorder.

What carries the argument

The central object is the similarity transformation $S=e^{\theta M/2}$ with $\theta=\operatorname{arctanh}\beta$, applied to the non-Hermitian Dirac Hamiltonian $H_\beta=(1+\beta M)H_0$ to obtain $H_\beta=\lambda S H_0 S^{-1}$. Because $M$ anticommutes with the Dirac matrices, this transformation rescales every eigenenergy by $\lambda=\sqrt{1-\beta^2}$ and transforms the eigenvectors nonunitarily. Its thermodynamic workhorse is the effective-field representation $B_{\rm eff}=\lambda^2 B$, which makes the reduced density of states identical to the Hermitian one, $D_\beta(E;B)=D_0(E;B_{\rm eff})$. Inserting this density of states into the standard Fermi–Dirac integrals for particle number, internal energy, entropy, and grand potential yields the entire set of scaling relations.

What would settle it

Measure the canonical heat capacity of a Landau-quantized graphene sample at fixed perpendicular field and fixed projected filling factor for two deformation strengths $\beta_1$ and $\beta_2$. The scaling predicts $C_{\beta_2}(T)=C_{\beta_1}(T\,\lambda_1/\lambda_2)$ at every temperature; any heat-capacity peak that fails to shift by the factor $\lambda_1/\lambda_2$ would falsify the central identity. Equivalently, at fixed $\mu$ the field positions of heat-capacity or magnetization oscillations must scale as $B_n(\beta)=B_n(0)/(1-\beta^2)$.

Watch

Extended reading notes

Core claim

The central claim is that uniform spectral compression fully determines equilibrium thermodynamics. The non-Hermitian deformation multiplies every nonzero Landau level by $\lambda=\sqrt{1-\beta^2}$, preserves the zero mode and the level ordering, and therefore maps the density of states at field $B$ onto the Hermitian density of states at the effective field $B_{\rm eff}=\lambda^2 B$. Starting from this master identity, the paper derives the scaling relations of Eq. (42): $\mu_\beta(T,B;N_c)=\mu_0(T,B_{\rm eff};N_c)$, $C_{B,\beta}(T,B;N_c)=C_{B,0}(T,B_{\rm eff};N_c)$, $c_{\mu,\beta}(T,\mu;B)=c_{\mu,0}(T,\mu;B_{\rm eff})$, $M^{\rm phys}_{0,+,\beta}(T,\mu;B)=M^{\rm phys}_{0,+,0}(T,\mu;B_{\rm eff})$, and $m^{\rm phys}_{0,+,\beta}(T,\mu;B)=\lambda^2 m^{\rm phys}_{0,+,0}(T,\mu;B_{\rm eff})$. The difference between the total orbital moment and the per-particle moment arises because the physical orbital degeneracy $D_B=eBA/h$ is fixed by the applied field, not the effective field. The paper further shows that quasistatic tuning of $\beta$ at fixed filling yields isothermal entropy change and adiabatic temperature scaling $T_f/\lambda_f=T_i/\lambda_i$.

Load-bearing premise

The system must actually thermalize with respect to the quasi-Hermitian Hamiltonian, meaning a physical bath or reservoir that realizes the corresponding Gibbs ensemble; if no such reservoir exists, the scaling identities describe a formal mathematical model rather than measurable non-Hermitian matter.

Editorial extensions

If this is right

At fixed projected filling factor $N_c$, the canonical heat capacity at any $\beta$ collapses onto the Hermitian curve under the temperature rescaling $T\to T/\lambda$; the paper illustrates this collapse for $\beta=0.75$ and $\beta=0.9$.
At fixed chemical potential, the oscillatory heat capacity and entropy as functions of field are shifted by the inverse factor: their peaks occur at $B_n(\beta)=B_n(0)/\lambda^2$, so field sweeps directly reveal the spectral-compression factor.
The total orbital moment of the projected electron sector equals the Hermitian total moment evaluated at the effective field, while the moment per projected electron carries an additional factor $\lambda^2$; this distinction follows from the physical, field-dependent orbital degeneracy.
Quasistatic tuning of $\beta$ at fixed filling is a reversible thermodynamic process: it produces an isothermal entropy increase and an adiabatic temperature change $T_f/T_i=\lambda_f/\lambda_i$, governed entirely by the Hermitian canonical heat capacity.
Any departure from these scaling relations in a real sample would signal an additional energy scale that does not share the uniform $\lambda$ compression, such as interactions, disorder, Zeeman splitting, or finite single-particle linewidth.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

Editorial inference: the scaling laws offer a parameter-free experimental diagnostic for spectral compression: measure the field position of any resolvable Landau-level feature at two values of the deformation and extract $\lambda^2$ directly from the ratio of fields, without needing to know the microscopic origin of the deformation.
Editorial inference: the framework extends to interacting or disordered systems only if those perturbations respect the anticommutation structure that produces the uniform $\lambda$ factor; if they do not, their $\beta$-dependence will break the collapse and thereby fingerprint the perturbation's symmetry.
Editorial inference: because the mapping is purely spectral and leaves the orbital degeneracy untouched, experiments comparing extensive quantities (total particle number, total moment) with per-particle quantities must include the extra $\lambda^{-2}$ factor; the paper's total-versus-per-particle distinction makes that correction explicit.
Editorial inference: the adiabatic temperature scaling suggests a possible cooling protocol: quasistatically increasing $|\beta|$ at fixed filling lowers the temperature of the Dirac electron system, which could be explored in driven dissipative graphene analogues if the required Gibbs reservoir can be engineered.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

First thing to know: this is a careful, honest paper. It takes the known spectral compression H_beta = lambda S H0 S^{-1} from Ref. [76] and works out its thermodynamic consequences for Dirac fermions in a magnetic field. The new content is the ensemble bookkeeping: scaling identities for the chemical potential, entropy, heat capacity, and orbital moment under fixed projected filling factor versus fixed chemical potential, and the distinction between total orbital moment and per-particle moment. Those are not in Ref. [76] or in the companion paper [77], and they are derived cleanly. I checked Eq. (11), the effective-field identity Eq. (19), and the Appendix B derivatives. They are internally consistent. The authors also deserve credit for keeping the physical orbital degeneracy separate from the spectral compression, which is exactly where this kind of calculation tends to go wrong. The zero-field lattice benchmark in Appendix A is a sensible sanity check and grounds the continuum results.

The soft spot is the thermal state, and it is load-bearing. The paper assumes equilibrium with respect to H_beta and notes that a bath must support the corresponding Gibbs ensemble, but it never specifies the density operator and does not exhibit a reservoir dynamics that would realize the state. The stress-test makes a sharper version of this point: if you define the thermal state using the quasi-Hermitian metric eta = S^{-2}, so the partition function is Tr(eta e^{-beta H_beta}), the scaling identities do not survive. Everything the paper derives relies on the ordinary trace of the non-Hermitian Gibbs operator e^{-beta H_beta}, which is not a standard thermal density matrix. That is a convention, not a theorem about open non-Hermitian systems. The authors are transparent about the assumption, but they do not confront the metric-trace issue, and that makes the central claim conditional rather than established for measurable NH matter.

The overlap with prior work is real but not disqualifying. The spectral compression is from Ref. [76], and Ref. [77] already covers fixed-mu quantum capacitance. The genuinely new part is the thermodynamic scaling structure, and it is a useful organizing result for people working on NH thermodynamics.

Who should read this: researchers looking for a compact dictionary between non-Hermitian Dirac thermodynamics and Hermitian Dirac thermodynamics, especially in the graphene Landau-level context. It deserves a serious referee. My recommendation: send it to review, but ask the authors to state their density-operator convention explicitly and to discuss whether any metric-compatible thermal state would obey the same scaling. Without that, the paper is a rigorous statement about an isospectral Hermitian model, which is fine but more modest than the title claims.

Referee Report

2 major / 5 minor

Summary. The paper studies a specific quasi-Hermitian (non-Hermitian but real-spectrum) deformation of the two-dimensional Dirac Hamiltonian in a uniform magnetic field, H_beta = (1+beta M)H_0. Via a similarity transformation S with H_beta = lambda S H_0 S^{-1}, lambda = sqrt(1-beta^2), the authors show that the grand partition function obeys Xi_beta(T, mu; B) = Xi_0(T/lambda, mu/lambda; B) (Eq. (11)), and equivalently that the reduced density of states satisfies D_beta(E;B) = D_0(E;B_eff) with B_eff = lambda^2 B (Eq. (19)). From these master identities they derive scaling relations for the chemical potential, entropy, heat capacities, and orbital magnetic moment at fixed projected filling factor and at fixed chemical potential (Eq. (42)), as well as caloric responses to quasistatic changes of beta (Eqs. (54), (60)-(64)). The derivations are internally consistent and are complemented by a zero-field tight-binding benchmark in Appendix A and detailed effective-field calculations in Appendix B. The paper is explicit that the results assume thermalization with respect to the quasi-Hermitian Hamiltonian or its Hermitian representative.

Significance. If the assumed equilibrium state is physically realizable, the paper provides a complete and elegant reduction of the equilibrium thermodynamics of this non-Hermitian Dirac model to the Hermitian problem: every spectral response is determined by the Hermitian functions at rescaled temperature, chemical potential, or effective field. The paper's strengths are its transparent algebraic derivations, the careful distinction between spectral compression (B_eff) and the physical orbital degeneracy (D_B proportional to B), the explicit equations for Landau-level crossing fields (Eqs. (48)-(49)), and the self-contained appendices. The results are falsifiable: the predicted field positions and amplitudes of the caloric and magnetic oscillations can be checked in any system that realizes the assumed Gibbs state. However, the physical significance is conditional on the existence of a bath that supports the quasi-Hermitian Gibbs ensemble; the paper does not construct such a bath, so the results are currently a rigorous statement about an isospectral Hermitian model rather than a demonstrated property of measurable non-Hermitian matter.

major comments (2)

[II.A, VI (Eqs. (9), (11), (68))] The equilibrium state is never specified as a positive Hermitian density operator. Equation (9) gives e^{-H_beta/k_B T} = S e^{-lambda H_0/k_B T} S^{-1}, which is not Hermitian for beta != 0 and is therefore not an admissible thermal state on the physical Hilbert space. If the intended state is the Gibbs state of the Hermitian representative h = lambda H_0, the scaling identities in Eq. (42) follow by construction, but the physical mechanism by which an open non-Hermitian system reaches this state is not demonstrated. If, alternatively, the quasi-Hermitian metric eta = S^{-2} is used to define the thermal trace, the partition function becomes Tr(eta e^{-H_beta/k_B T}) != Tr(e^{-lambda H_0/k_B T}), so Eq. (11) would not hold. The paper acknowledges the need for a compatible bath at Eq. (68), but this is a load-bearing assumption rather than a limitation of presentation. Please provide a concrete reservoir construction (or a clear statement that the results are formal isospectral relations), and specify the density operator used in all thermodynamic averages.
[Abstract and VI] The abstract and conclusions state that the paper 'establish[es] a thermodynamic framework for real-spectrum non-Hermitian quantum matter' and the title refers to 'Equilibrium Thermodynamics of Non-Hermitian Dirac Fermions.' Given the unproven bath assumption, these claims overstate the physical scope. The scaling relations are exact properties of the spectral mapping, but without a physical realization of the Gibbs state they do not demonstrate that a non-Hermitian system in equilibrium exhibits these responses. Please either add a discussion of possible physical reservoirs (e.g., a Lindblad master equation whose stationary state is the quasi-Hermitian Gibbs state) or explicitly rephrase the conclusions to say that the results apply to the Hermitian representative, with the non-Hermitian deformation serving as a spectral mapping.

minor comments (5)

[II.A] The statement that the one-particle similarity transformation 'lifts directly to the fermionic Fock space' is imprecise because S is not unitary; the trace identity still holds by cyclicity, but the transformation is a non-unitary similarity rather than a canonical transformation. Please clarify.
[Eq. (11)] The notation Xi_0(T/lambda, mu/lambda; B) uses ratios as arguments; defining T' = T/lambda and mu' = mu/lambda would improve readability and avoid confusion with a temperature scale T/lambda.
[Ref. [83]] Reference [83] ('J. High Energy Phys. 01, 143') is missing the publication year; please complete the citation.
[Fig. 4] The caption's phrase 'Negative values of -Delta S/(N_c k_B)' is awkward; the plotted quantity is already -Delta S/(N_c k_B), so it would be clearer to say 'The plotted quantity -Delta S/(N_c k_B) is negative when spectral compression increases the entropy.'
[Appendix B, Eq. (B2)] Equation (B2) states the scaling D_beta(E;B) = (1/lambda) D_0(E/lambda; B) for the reduced DOS, but Eq. (19) uses D without specifying normalization. Please state explicitly in both places that D denotes the degeneracy-normalized (reduced) density of states, so the reader can distinguish it from the physical DOS.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: the scaling identities follow algebraically from the similarity transformation; the only load-bearing assumption is an explicitly stated thermalization postulate, not a fit or a self-citation.

full rationale

The central claim, Eq. (42), is derived from the spectral identity D_beta(E;B)=D_0(E;B_eff), which in turn follows from the similarity transformation H_beta = lambda S H_0 S^{-1} in Eq. (5), the trace identity in Eq. (10), and the Landau-level mapping in Eqs. (17)-(19). No parameter is fitted to any target observable: beta is a model control parameter, and the reduction to the Hermitian problem is obtained by direct change of variables in the Fermi-Dirac integrals, as shown in Appendix B. The paper explicitly states the physical assumption on which the whole framework rests: 'Throughout this work, equilibrium means that the system thermalizes with respect to the quasi-Hermitian Hamiltonian, or equivalently its Hermitian representative.' It also flags the corresponding limitation near Eq. (68): 'The equilibrium formulation assumes a bath or reservoir that realizes the Gibbs ensemble associated with the quasi-Hermitian Hamiltonian.' This is an applicability assumption, not a circular definition: even given that postulate, the scaling relations are nontrivial consequences of the isospectral compression of the Landau ladder. The self-citations are not load-bearing: Ref. [76] is cited for the model, but the quasi-Hermitian mapping is rederived in Eqs. (3)-(5) and in Appendix A, and the companion paper [77] is presented as a complementary study of quantum capacitance rather than as evidence for the scaling relations here. The effective-field parametrization B_eff = lambda^2 B is explicitly a representation of the spectral compression, not an independent empirical pattern being relabeled as a prediction. The numerical curves in Figs. 1-5 are evaluations of the derived identities, not fits. Therefore the derivation is self-contained modulo the stated thermalization assumption, and there is no circular step to flag.

Assumptions & free parameters 3 free parameters · 3 assumptions · 0 invented entities

There are no fitted parameters and no new physical entities. The only model parameter is the deformation strength beta, which is an external control parameter from prior work. The load-bearing assumptions are the quasi-Hermitian thermalization postulate and the regularization scheme for the projected electron sector; both are stated in the text and both restrict the domain of applicability of the scaling identities.

free parameters (3)

beta (spectral deformation strength)
Control parameter of the model inherited from Ref. [76]; not fitted. All scaling results hold for any |beta|<1, but the physical value for a concrete realization is not determined in this paper.
N_c (fixed projected filling factor) = 75
Illustrative numerical choice for the fixed-PFF figures; the analytic scaling does not depend on this value.
mu (fixed chemical potential) = 38 meV
Illustrative numerical choice for the grand-canonical figures; the analytic scaling does not depend on this value.

assumptions (3)

domain assumption The system thermalizes with respect to the quasi-Hermitian Hamiltonian H_beta, equivalently its Hermitian representative.
Invoked at Eq. (9)-(11) and repeated in the Discussion. Without a bath that realizes this Gibbs ensemble, the scaling identities describe a formal model rather than a measurable open system.
domain assumption The grand partition function is computed in a finite-band lattice regularization or after normal ordering and projection relative to the filled valence background.
Stated after Eq. (11) and used for the orbital magnetic response, where the regularized negative-energy Dirac sea is omitted.
standard math Fermi-Dirac statistics with real single-particle eigenvalues describes the equilibrium occupations.
Standard noninteracting fermion thermodynamics used in Sec. III A; valid once the Gibbs ensemble is granted.

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Cite this review

Pith. "Pith review of Equilibrium Thermodynamics of Non-Hermitian Dirac Fermions: Caloric and Magnetic Responses." pith.science (2026). https://pith.science/paper/FIUWQLE6

@misc{pith2026260804369,
  author       = {Pith},
  title        = {Pith review of: Equilibrium Thermodynamics of Non-Hermitian Dirac Fermions: Caloric and Magnetic Responses},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/FIUWQLE6}},
  note         = {Machine review of arXiv:2608.04369}
}

read the original abstract

We establish scaling relations governing the equilibrium thermodynamics of real-spectrum non-Hermitian Dirac fermions in a magnetic field. Assuming thermalization with respect to the quasi-Hermitian Hamiltonian, a similarity transformation maps the system at the same applied field onto a Hermitian Dirac model with reduced velocity, while the Landau-level spectrum also admits a representation in terms of a reduced effective magnetic field. This structure yields scaling relations for the chemical potential, entropy, heat capacities, and orbital magnetic response in different thermodynamic ensembles. At fixed projected filling factor (PFF), the self-consistent chemical potential follows the compressed ladder of Landau levels, and the canonical thermodynamic functions are rescaled Hermitian responses. At fixed chemical potential, Landau-level crossings generate oscillatory caloric and magnetic responses governed by the same spectral compression. Quasistatic non-Hermitian deformation at fixed PFF further yields adiabatic temperature scaling. More broadly, these results establish a thermodynamic framework for real-spectrum non-Hermitian quantum matter and provide a starting point for incorporating the effects of interactions and disorder within the same formalism.

Figures

Figures reproduced from arXiv: 2608.04369 by the authors.

**Figure 3.** FIG. 3. Orbital magnetic moment per electron at [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗

**Figure 4.** FIG. 4. Isothermal entropy change at fixed [PITH_FULL_IMAGE:figures/full_fig_p008_4.png] view at source ↗

Figures from the paper (3 more)

**Figure 5.** Figure 5: FIG. 5. Normalized fixed- [PITH_FULL_IMAGE:figures/full_fig_p009_5.png]

**Figure 6.** Figure 6: The full-Brillouin-zone DOS includes both valleys and is displayed per spin, whereas the continuum expression below includes the spin and valley multiplicities explicitly through g = 4. 2. Heat capacity and spectral compression We next use the full-band DOS to evaluate…

**Figure 7.** Figure 7: FIG. 7. Dimensionless electronic heat capacity of pristine [PITH_FULL_IMAGE:figures/full_fig_p011_7.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

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Tight-binding spectrum and density of states We consider the nearest-neighbor tight-binding Hamil- tonian of graphene, HTB,0(k) =−t 0f(k) f ∗(k) 0 , f(k) = 3X j=1 eik·δj , (A1) wheret≃3 eV,a= 1.42 ˚A, and δ1 =a(1,0),δ 2,3 =a − 1 2 ,± √ 3 2 ! .(A2) Fort 2 = 0, the model has sublattice symmetry, {σz, HTB,0(k)}= 0.(A3) The NH deformation is introduced as HTB...
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Particle–hole symmetry pins the chemical potential at zero

Heat capacity and spectral compression We next use the full-band DOS to evaluate the zero- field thermal response att 2 = 0 andµ= 0. Particle–hole symmetry pins the chemical potential at zero. 1.0 0.5 0.0 0.5 1.0 = E/t 0.0 0.4 0.8 1.2 1.6 2.0 2.4D( ) = tD(E) = 0 = 0.75 = 0.90 FIG. 6. Dimensionless density of states eD(ε) =tD(E), with ε=E/t, for representa...
[3]

WithB eff =λ 2Bfrom Eq

Effective magnetic field For the relativistic Landau levels, the energy rescaling also has a representation in terms of an effective magnetic field. WithB eff =λ 2Bfrom Eq. (17), Dβ(E;B) =D 0(E;B eff ) = 1 λ D0 E λ ;B .(B4) It follows directly that, at fixedTandµ, Nβ(T, µ;B) =N 0(T, µ;Beff ),(B5) Uβ(T, µ;B) =U 0(T, µ;Beff ),(B6) Sβ(T, µ;B) =S 0(T, µ;Beff ...
[4]

Fixed chemical potential Changing variables fromEtox=E/λgives Nβ(T, µ;B) =N 0 T λ , µ λ ;B ,(B12) Uβ(T, µ;B) =λU 0 T λ , µ λ ;B ,(B13) Sβ(T, µ;B) =S 0 T λ , µ λ ;B ,(B14) Ωβ(T, µ;B) =λΩ 0 T λ , µ λ ;B .(B15) The grand-canonical heat capacity Cµ,β =T ∂Sβ ∂T µ,B (B16) therefore obeys Cµ,β(T, µ;B) =C µ,0 T λ , µ λ ;B .(B17) Since the occupation transforms un...
[5]

Using Eq

Fixed projected filling factor Letµ β(T, B;Nc) be the solution of Nβ(T, µβ;B) =N c.(B19) 12 ForT >0 and a prescribed occupation inside the finite spectral range,N β is strictly increasing withµ, so the solution is unique; theT→0 result is understood as the corresponding limit. Using Eq. (B12), this gives µβ(T, B;Nc) =λµ 0 T λ , B;Nc .(B20) Substitution al...
[6]

Magnetic-field observables From Eq. (B8) and∂B eff /∂B=λ 2, the reduced spec- tral orbital response satisfies Msp,β(T, µ;B) =λ 2Msp,0(T, µ;Beff ).(B29) The extensive projected grand potential instead obeys Ωphys β (T, µ;B) =λ −2Ωphys 0 (T, µ;Beff ),(B30) becauseD Beff =λ 2DB. Differentiation with respect to the physical field therefore gives M phys 0,+,β(...

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Pith tools

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[3] [1]

Tight-binding spectrum and density of states We consider the nearest-neighbor tight-binding Hamil- tonian of graphene, HTB,0(k) =−t 0f(k) f ∗(k) 0 , f(k) = 3X j=1 eik·δj , (A1) wheret≃3 eV,a= 1.42 ˚A, and δ1 =a(1,0),δ 2,3 =a − 1 2 ,± √ 3 2 ! .(A2) Fort 2 = 0, the model has sublattice symmetry, {σz, HTB,0(k)}= 0.(A3) The NH deformation is introduced as HTB...

[4] [2]

Particle–hole symmetry pins the chemical potential at zero

Heat capacity and spectral compression We next use the full-band DOS to evaluate the zero- field thermal response att 2 = 0 andµ= 0. Particle–hole symmetry pins the chemical potential at zero. 1.0 0.5 0.0 0.5 1.0 = E/t 0.0 0.4 0.8 1.2 1.6 2.0 2.4D( ) = tD(E) = 0 = 0.75 = 0.90 FIG. 6. Dimensionless density of states eD(ε) =tD(E), with ε=E/t, for representa...

[5] [3]

WithB eff =λ 2Bfrom Eq

Effective magnetic field For the relativistic Landau levels, the energy rescaling also has a representation in terms of an effective magnetic field. WithB eff =λ 2Bfrom Eq. (17), Dβ(E;B) =D 0(E;B eff ) = 1 λ D0 E λ ;B .(B4) It follows directly that, at fixedTandµ, Nβ(T, µ;B) =N 0(T, µ;Beff ),(B5) Uβ(T, µ;B) =U 0(T, µ;Beff ),(B6) Sβ(T, µ;B) =S 0(T, µ;Beff ...

[6] [4]

Fixed chemical potential Changing variables fromEtox=E/λgives Nβ(T, µ;B) =N 0 T λ , µ λ ;B ,(B12) Uβ(T, µ;B) =λU 0 T λ , µ λ ;B ,(B13) Sβ(T, µ;B) =S 0 T λ , µ λ ;B ,(B14) Ωβ(T, µ;B) =λΩ 0 T λ , µ λ ;B .(B15) The grand-canonical heat capacity Cµ,β =T ∂Sβ ∂T µ,B (B16) therefore obeys Cµ,β(T, µ;B) =C µ,0 T λ , µ λ ;B .(B17) Since the occupation transforms un...

[7] [5]

Using Eq

Fixed projected filling factor Letµ β(T, B;Nc) be the solution of Nβ(T, µβ;B) =N c.(B19) 12 ForT >0 and a prescribed occupation inside the finite spectral range,N β is strictly increasing withµ, so the solution is unique; theT→0 result is understood as the corresponding limit. Using Eq. (B12), this gives µβ(T, B;Nc) =λµ 0 T λ , B;Nc .(B20) Substitution al...

[8] [6]

Magnetic-field observables From Eq. (B8) and∂B eff /∂B=λ 2, the reduced spec- tral orbital response satisfies Msp,β(T, µ;B) =λ 2Msp,0(T, µ;Beff ).(B29) The extensive projected grand potential instead obeys Ωphys β (T, µ;B) =λ −2Ωphys 0 (T, µ;Beff ),(B30) becauseD Beff =λ 2DB. Differentiation with respect to the physical field therefore gives M phys 0,+,β(...