{"id":"297e38bd-212d-4319-ac59-8b04882ed2bb","arxiv_id":"2608.04369","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":3,"one_line_summary":"For real-spectrum non-Hermitian Dirac fermions, equilibrium thermodynamic responses collapse onto Hermitian responses at the effective field Beff=(1-beta^2)B, with additional amplitude factors for per-particle orbital moments.","lead":"This paper shows that a class of non-Hermitian Dirac fermions in a magnetic field has equilibrium heat capacity, entropy, and orbital magnetism identical to ordinary Dirac fermions in a rescaled effective magnetic field. The result could let experimenters predict thermodynamic signatures of gain-loss graphene-type systems using established Hermitian calculations.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The scaling identities are internally consistent, but they presuppose a Gibbs state for the non-Hermitian Hamiltonian; without a concrete bath realizing that state, Eq. (42) describes an isospectral Hermitian model rather than established equilibrium non-Hermitian matter.","rationale":"I read the paper in good faith and found no derivation error: the similarity transformation, the effective-field mapping, the fixed-PFF self-consistency, and the degeneracy bookkeeping in Appendix B are all internally consistent. The main numerical and analytic relations in Eq. (42) follow from the spectral equivalence Hβ = λ S H0 S^{-1}. The single load-bearing vulnerability is the physical status of the Gibbs state. The paper itself flags this requirement in the Introduction and near Eq. (68), and the reader's weakest assumption identifies the same point. My stress-test sharpens it: the exponential of the non-Hermitian Hamiltonian is not Hermitian, so one must choose between the isospectral Hermitian representative and a metric-dependent trace, and the paper does not supply a microscopic reservoir construction selecting either choice. This does not invalidate the mathematical content, but it does mean the claim about measurable non-Hermitian matter is conditional on an unproven dynamical assumption. Therefore I see no reason to change the reader's CONDITIONAL verdict.","tokens_in":18097,"tokens_out":17013,"duration_ms":167310,"concrete_test":"Construct the finite-lattice model of Appendix A with the magnetic field via Peierls substitution and couple it to a reservoir through local Lindblad operators c_{iσ} and c_{iσ}^† with a spectral density chosen to satisfy detailed balance with respect to the Hermitian representative h = λH0. Solve the Lindblad equation numerically for the stationary state and compute the projected-sector occupation, entropy, and heat capacity. If the steady-state occupations equal f(ε_n, μ, T) with ε_n = λ ε_n0 for all levels, then Eq. (42) can be realized by an open system; if the non-unitary similarity S forces a different fixed point (or no completely positive fixed point at all), the central claim reduces to a formal spectral identity and the CONDITIONAL verdict should be maintained.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim is Eq. (42): all equilibrium responses follow from the Hermitian Dirac problem via T/λ, μ/λ, or B_eff = λ^2B. I checked the algebra: Eq. (11) follows from Hβ = λ S H0 S^{-1}, and Eqs. (45)-(52) and (60)-(64) follow from the occupation formulas. The load-bearing step is not algebraic but physical: the paper never specifies the density operator of the equilibrium state. Equation (9) gives e^{-Hβ/kBT} = S e^{-λH0/kBT} S^{-1}, which is not Hermitian, hence not an admissible thermal state on the physical Hilbert space. One can instead use the Hermitian representative h = λH0, whose partition function is Tr e^{-λH0/kBT}; this makes the scaling identities true by construction but leaves open why the open non-Hermitian system should be described by h. Alternatively, if one adopts the quasi-Hermitian metric η = S^{-2} and defines the state by Z_η = Tr(η e^{-Hβ/kBT}), the identity changes to Z_η = Tr(e^{-λH0/kBT}η), which is not the Hermitian partition function and would not obey Eq. (11). The authors explicitly assume thermalization with respect to Hβ (Introduction; Discussion near Eq. (68)) and note that a bath must support the Gibbs ensemble, but no such bath is constructed. Until a completely positive reservoir dynamics with the required stationary state is exhibited, the scaling relations are a rigorous statement about an isospectral Hermitian Dirac model with velocity λv, not evidence about measurable equilibrium non-Hermitian matter.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":18381,"tokens_out":18131,"duration_ms":162441,"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":[{"comment":"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.","section":"II.A, VI (Eqs. (9), (11), (68))"},{"comment":"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.","section":"Abstract and VI"}],"minor_comments":[{"comment":"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.","section":"II.A"},{"comment":"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.","section":"Eq. (11)"},{"comment":"Reference [83] ('J. High Energy Phys. 01, 143') is missing the publication year; please complete the citation.","section":"Ref. [83]"},{"comment":"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.'","section":"Fig. 4"},{"comment":"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.","section":"Appendix B, Eq. (B2)"}],"recommendation":"major_revision","confidential_remarks":"The main technical issue is the unspecified density operator and physical bath. The authors are transparent about the assumption, but it is load-bearing for the title's claim of non-Hermitian equilibrium thermodynamics. If the authors can supply a concrete reservoir model (even in a simplified setting) or restrict their conclusions to the isospectral Hermitian problem, the paper could be a solid contribution. The overlap with the companion paper [77] appears complementary rather than duplicative."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","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.\n\nThe 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.\n\nThe 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.\n\nWho 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.","headline":"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.","tokens_in":19002,"tokens_out":4522,"would_cite":false,"duration_ms":44655,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["81Q12","82B30"],"pacs":[],"model":"deepseek-v4-flash","headline":"Dirac heat capacity, entropy, and orbital moment obey one scaling law","keywords":["non-Hermitian thermodynamics","quasi-Hermitian Hamiltonian","Landau levels","Dirac fermions","heat capacity scaling","orbital magnetic response","graphene","adiabatic cooling"],"falsifier":"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 $B_n(\\beta)=B_n(0)/(1-\\beta^2)$.","tokens_in":17845,"feed_emoji":"🧲","tokens_out":11418,"duration_ms":92180,"temperature":0.7,"pith_summary":"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.","feed_headline":"Dirac heat capacity, entropy, and orbital moment obey one scaling law","feed_subtitle":"A single factor λ=√(1−β²) shifts temperature and field for real-spectrum non-Hermitian Dirac fermions.","key_machinery":"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.","core_discovery":"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 $T_f/\\lambda_f=T_i/\\lambda_i$.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Supplies the minimal non-Hermitian Dirac Hamiltonian whose real-spectrum deformation renormalizes the Dirac velocity to $v\\sqrt{1-\\beta^2}$.","marker":"[76]"},{"why":"Establishes pseudo-Hermiticity as the condition for a real spectrum and introduces the metric operator used in the equilibrium construction.","marker":"[56]"},{"why":"Provides the quasi-Hermitian representation and the similarity transformation that maps the deformed Hamiltonian to a Hermitian representative.","marker":"[57]"},{"why":"Defines quasi-Hermitian operators and the physical inner product that legitimizes the Gibbs-ensemble treatment of the deformed Hamiltonian.","marker":"[87]"},{"why":"Establishes the non-Hermitian quantum thermodynamics framework whose Gibbs ensemble the paper adopts for the quasi-Hermitian Hamiltonian.","marker":"[58]"},{"why":"Gives an independent stability-based equilibrium criterion for real-spectrum non-Hermitian systems, supporting the thermalization assumption.","marker":"[60]"}],"fun_headline_variants":["One factor λ maps non-Hermitian Dirac thermodynamics to Hermitian","λ-scaling unifies thermodynamic response of non-Hermitian Dirac matter","Spectral compression controls heat capacity and orbital moment","Non-Hermitian Dirac fermions: caloric and magnetic scaling laws unified"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["One factor λ maps non-Hermitian Dirac thermodynamics to Hermitian","λ-scaling unifies thermodynamic response of non-Hermitian Dirac matter","Spectral compression controls heat capacity and orbital moment","Non-Hermitian Dirac fermions: caloric and magnetic scaling laws unified"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.00077,"raw_usage":{"total_tokens":3473,"prompt_tokens":1069,"completion_tokens":2404,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":685,"completion_tokens_details":{"reasoning_tokens":2330}},"tokens_in":685,"tokens_out":2404,"duration_ms":41384,"temperature":1.0,"reasoning_tokens":2330,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-08T18:51:46.126418+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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 $B_n(\\beta)=B_n(0)/(1-\\beta^2)$.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the minimal non-Hermitian Dirac Hamiltonian whose real-spectrum deformation renormalizes the Dirac velocity to $v\\sqrt{1-\\beta^2}$."},{"cited_title":"Roy, Zero modes and index theorems for non- Hermitian Dirac fermions, Phys","cited_arxiv_id":null,"evidence_quote":"Defines quasi-Hermitian operators and the physical inner product that legitimizes the Gibbs-ensemble treatment of the deformed Hamiltonian."}],"review_version":1}