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Role of electron-electron interaction in the Mpemba effect in quantum dots

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Pith's one-line read A single-level quantum dot shows the Mpemba effect whenever the initially hotter state carries less charge-mode amplitude, and attractive electron-electron interaction makes the effect large.

desk verdict Solid theory paper: the interaction-sign dependence of the quantum Mpemba effect in a quantum dot is new, the attractive-U regime is the right place to look, and the paper deserves peer review. read the letter →

arxiv 2412.18456 v3 pith:UOQOMDJO submitted 2024-12-24 cond-mat.mes-hall cond-mat.stat-mech

Juliane Graf , Janine Splettstoesser , Juliette Monsel This is my paper · ORCID
classification cond-mat.mes-hallcond-mat.stat-mech PACS 73.63.Kv73.23.-b
keywords electron-electroninteractionquantumMpembaeffectsingle-leveldotfermionicdualityrelaxationdecaymodesnonequilibriumfreeenergyattractive
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

The paper asks when a single-level quantum dot, tunnel-coupled to a thermal bath, shows the Mpemba effect: an initial state that starts farther from equilibrium relaxes back to equilibrium faster than one that starts closer. Using the relative entropy (equivalently the nonequilibrium free energy) and the internal dot energy as distance measures, it identifies the condition as $D(\varrho'_{\rm in})>D(\varrho_{\rm in})$ combined with $|c'_{\rm in}|<|c_{\rm in}|$, where $c$ is the amplitude of the slow charge decay mode. The sign and magnitude of the electron-electron interaction $U$ then decide how visible the effect is: strongly attractive interaction makes the charge mode nearly frozen ($\gamma_c \ll \gamma_p$), so the initially hotter state overtakes the colder one early and clearly. The paper also shows that the internal energy reveals the effect only when $U$ is finite. If the analysis is right, quantum dots with phonon-mediated attractive interaction are a practical platform for observing the quantum Mpemba effect.

What carries the argument

The machinery is the fermionic-duality eigenmode decomposition of the quantum-dot master equation. Fermionic duality is a dissipative symmetry of the evolution kernel under energy inversion, $\epsilon \to -\epsilon$, $U\to -U$, and it turns the non-Hermitian relaxation matrix into three physically labeled modes: charge, spin, and parity, with rates $\gamma_c$, $\gamma_s$, and $\gamma_p=2\Gamma$. The charge and parity modes carry the argument: the charge amplitude $c=(c'|\rho(0))$ and parity amplitude $p=(p'|\rho(0))$ are the overlaps that decide whether a Mpemba crossing occurs, and the rate ratio $\gamma_c/\gamma_p$ sets how early the crossing appears. The same decomposition, together with a Taylor expansion of the logarithm in the relative entropy, produces the two-exponential form of the energy decay and the asymptotic relative-entropy difference that yield conditions (23) and (25).

What would settle it

A time-resolved experiment on a single-level dot with attractive interaction ($\beta U \lesssim -5$) in the Coulomb-blockaded region could settle the claim: prepare two states by a temperature quench such that $D(\varrho'_{\rm in})>D(\varrho_{\rm in})$ and $|c'_{\rm in}|<|c_{\rm in}|$, then record the relative entropy or dot energy versus time. If no crossing appears before both curves decay at the slow charge rate, or if the measured ratio $\gamma_c/\gamma_p$ is not much smaller than 1, the central claim fails.

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Extended reading notes

Core claim

On its own terms, the paper's discovery is a mode-level criterion for anomalous relaxation in an interacting quantum dot. The relaxation of the dot populations is decomposed into a charge mode and a parity mode with rates $\gamma_c$ and $\gamma_p=2\Gamma$, and the Mpemba effect occurs for two initial states when the one that is thermodynamically farther from equilibrium carries the smaller charge-mode amplitude: $D(\varrho'_{\rm in})>D(\varrho_{\rm in})$ and $|c'_{\rm in}|<|c_{\rm in}|$. When the charge amplitude of the hotter state is exactly zero, the relaxation is exponentially faster and the strong Mpemba effect appears. The interaction determines how strong the effect is because $\gamma_c/\gamma_p$ depends on $U$ and the level position; for strong attractive interaction inside the Coulomb-blockaded region this ratio is very small, so the parity mode dominates and the crossing in the relative entropy happens early. The energy-based version is not equivalent: for finite $U$, the energy decays as a sum of two exponentials with rates $\gamma_c$ and $\gamma_p$, and a crossing occurs under a different condition on $c'_{\rm in}$ and $c_{\rm in}$; at $U=0$ the parity contribution vanishes and the energy cannot show the Mpemba effect.

Load-bearing premise

The load-bearing premise is that the standard weak-coupling rate equation with a constant, spin-degenerate tunnel rate describes the dot faithfully for every interaction strength considered, including strongly attractive $U$; if higher-order tunneling, non-Markovian memory, or level broadening changes the relaxation at large $|\beta U|$, the predicted hierarchy $\gamma_c\ll\gamma_p$ and the Mpemba crossing could shift or disappear.

Editorial extensions

If this is right

  • For any interaction strength, a pair of initial states satisfying $D(\varrho'_{\rm in})>D(\varrho_{\rm in})$ and $|c'_{\rm in}|<|c_{\rm in}|$ will show a Mpemba crossing in the relative entropy, becoming strong when $c'_{\rm in}=0$.
  • A quantum dot with strong attractive interaction ($-\beta U \gg 1$) in the Coulomb-blockaded region has $\gamma_c \ll \gamma_p$, so the crossing occurs early and the effect is pronounced; repulsive interaction gives a late, weak crossing.
  • The internal dot energy can show the Mpemba effect only when $U\neq 0$; at $U=0$ the energy relaxes as a single exponential with rate $\gamma_c$ and no crossing is possible.
  • Initial states suitable for observing the effect can be prepared experimentally by a rapid gate-voltage switch or by a temperature quench, and combining both quenches gives access to states with exponential speedup.
  • The energy decay is not necessarily monotonic: when the charge and parity contributions have opposite signs, the dot energy can overshoot equilibrium before relaxing.

Reading between the lines

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

  • If the rate hierarchy holds, the same mode decomposition should predict where the Mpemba effect appears in other observables, such as heat currents or full counting statistics, even in devices where the steady state is a nonequilibrium one.
  • The criterion is tied to the relative-entropy distance; readers should not expect the same 'smaller charge amplitude' rule for other measures, since trace distance or entanglement asymmetry will weight the parity and charge modes differently.
  • A direct experimental test could use a phonon-mediated negative-$U$ dot, prepare two states by temperature quench, and check that the crossing time tracks $1/(\gamma_p-\gamma_c)$; a measured ratio near 1 would argue against the Markovian approximation.
  • Because the paper's preparation protocols realize initial states that are thermal at a different temperature or gate voltage, the effect could be probed without full state tomography, using only time-resolved energy or charge detection.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

0 major / 4 minor

Summary. The manuscript analyzes the Mpemba effect in a single-level quantum dot with local electron-electron interaction U, coupled to a thermal reservoir. Using a Markovian master equation in the weak-coupling limit and the fermionic duality symmetry, the authors decompose the relaxation into charge, spin, and parity modes. They derive sufficient conditions for the Mpemba effect based on the relative entropy (Eq. (23): the initially hotter state has a smaller charge-mode amplitude) and based on the internal dot energy (Eq. (25), which requires finite U). They show analytically and numerically that a strongly attractive interaction creates a large separation between the charge and parity decay rates, making the Mpemba effect particularly pronounced, and they propose experimentally relevant state-preparation protocols via gate-voltage switches, temperature quenches, interaction quenches, and two-reservoir initial states.

Significance. This is a solid theoretical contribution with a parameter-free, analytically derived criterion for the Mpemba effect in a realistic mesoscopic system. The use of fermionic duality to identify charge and parity modes gives clear physical insight into the relaxation mechanism, and the prediction that attractive-U quantum dots exhibit a pronounced Mpemba effect is falsifiable with existing experimental platforms (e.g., Refs. [31-34]). The proposed state-preparation protocols are concrete and experimentally relevant, and the appendices provide useful proofs of the physical-state triangle and the long-time behavior of the relative entropy. If the results hold, this paper establishes quantum dots as a promising platform for observing and controlling the Mpemba effect.

minor comments (4)
  1. [Sec. II C, around Eq. (10) and Fig. 1(d)] The statement "For 0 ≲ ϵ ≲ |U|, γc ≪ γp becomes almost zero" is too broad. For βU = -10, the ratio γc/γp is small only in a window around the particle-hole symmetry point ϵ ≈ -U/2 (e.g., βϵ ≈ 4.4 gives γc/γp ≈ 0.008), whereas at the edges ϵ = 0 and ϵ = |U| the ratio is approximately 0.25. Please correct this range to avoid overstating the region where the rate hierarchy holds.
  2. [Appendix B, Eqs. (B1)-(B3) and Eq. (18)] The Taylor expansion of log[Pn(t)] around Pn^eq converges only when |Pn(t) - Pn^eq| < Pn^eq for all n. Since the paper uses Eq. (18) to extract the long-time limit, please state explicitly that Eq. (18) is an asymptotic expansion valid for sufficiently long times (or under the stated convergence condition), and that the Mpemba conditions (23) are derived from the leading long-time term. This does not affect the validity of the conclusions, but it clarifies the mathematical status of the expansion.
  3. [Sec. II B, around Eq. (7)] It would be helpful to add a brief sentence clarifying why the sequential-tunneling master equation remains valid for strongly attractive U near the particle-hole symmetry point, where |0⟩ and |2⟩ are nearly degenerate. The authors could note that the relevant higher-order (cotunneling) corrections are suppressed by a factor of order Γ/|U| in the weak-coupling regime, so they do not alter the hierarchy γc ≪ γp.
  4. [Eq. (19a)] Please ensure that Eq. (19a) is typeset unambiguously; the coefficient should read A_k = (-1)^k / [k(k-1)].

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the Mpemba conditions are derived from the explicit rate kernel, not fitted or assumed.

full rationale

The derivation is self-contained. The central input is the explicit rate kernel W in Eq. (7), obtained from Fermi's golden rule, and the decay rates gamma_c = Gamma(f_epsilon^+ + f_U^-), gamma_s = Gamma(f_epsilon^- + f_U^+), and gamma_p = 2Gamma follow by direct diagonalization of that matrix, as stated in the paper: "By diagonalizing Eq. (7), we obtain W = ..." No parameter is fitted to any target observable. The mode decomposition in Eq. (16), together with the Taylor expansion in Appendix B, produces the relative-entropy series (Eq. 18) and its long-time limit (Eq. 20); the Mpemba sufficient condition (23), D(rho'_in) > D(rho_in) with |c'_in| < |c_in|, is a mathematical consequence of that long-time limit, not a restatement of the definition (3). Likewise, the energy-based condition (25) follows from the two-exponential expression (21). The fermionic duality results of Refs. [36,38] are used for compact analytical mode expressions and for the physical interpretation of charge and parity modes, but W is given explicitly and the eigenvalue/eigenmode structure is independently exhibited, so no load-bearing reduction to those self-citations occurs. No uniqueness theorem is invoked to rule out alternative approaches. The weak-coupling, sequential-tunneling assumption in Sec. II B is a stated physical approximation, not a circular input, and the predicted rate hierarchy gamma_c << gamma_p for attractive U and the associated crossing times are computed consequences of the model.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

No free parameters are fitted to data; U, epsilon, T, and Gamma are physical model parameters. The fictitious 'dual' equilibrium state rho_bar_eq from fermionic duality is a mathematical auxiliary, not a new physical entity. The central claim rests on standard Markovian weak-coupling assumptions and an accepted duality identity.

assumptions (5)
  • domain assumption Weak-coupling Markovian master equation with Fermi golden rule rates (Eq. 7) is valid.
    Stated in Sec. II B: hbar*Gamma << kBT and Gamma*tau_c << 1; coherences decouple from populations. This is the standard Born-Markov secular approximation.
  • domain assumption Wideband and spin-degenerate tunneling rates, Gamma_sigma(E) = Gamma.
    Used after Eq. (5) to write constant rates; spin symmetry makes the spin mode inactive in the dynamics.
  • domain assumption The steady state of the master equation is the Gibbs state of the dot at reservoir temperature T and chemical potential mu = 0 (Eq. 8).
    The single reservoir enforces thermal equilibrium; this is standard for this class of rate equations.
  • standard math Fermionic duality identities from Refs. [36,38] supply the eigenmode decomposition used in Eqs. (9)-(15).
    The paper uses this duality as an accepted formal tool without rederiving it; the Mpemba criteria are new consequences built on it.
  • domain assumption Initial states prepared by instantaneous quenches are equilibrium states at the pre-quench parameters (Sec. IV).
    The protocols assume gate-voltage or temperature switches are fast compared to the dot relaxation time, so the initial state is the thermal state before the switch.

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Pith. "Pith review of Role of electron-electron interaction in the Mpemba effect in quantum dots." pith.science (2026). https://pith.science/paper/UOQOMDJO

@misc{pith2026241218456,
  author       = {Pith},
  title        = {Pith review of: Role of electron-electron interaction in the Mpemba effect in quantum dots},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/UOQOMDJO}},
  note         = {Machine review of arXiv:2412.18456}
}
read the original abstract

The Mpemba effect has initially been noticed in macroscopic systems -- namely that hot water can freeze faster than cold water -- but recently its extension to open quantum systems has attracted significant attention. This phenomenon can be explained in the context of nonequilibrium thermodynamics of Markovian systems, relying on the amplitudes of different decay modes of the system dynamics. Here, we study the Mpemba effect in a single-level quantum dot coupled to a thermal bath, highlighting the role of the sign and magnitude of the electron-electron interaction in the occurrence of the Mpemba effect. We gain physical insights into the decay modes from a dissipative symmetry of this system called fermionic duality. Based on this analysis of the relaxation to equilibrium of the dot, we derive criteria for the occurrence of the Mpemba effect using two thermodynamically relevant measures of the distance to equilibrium, the nonequilibrium free energy and the dot energy. We furthermore compare this effect to a possible exponential speedup of the relaxation. Finally, we propose experimentally relevant schemes for the state preparation and explore different ways of observing the Mpemba effect in quantum dots in experiments.

Figures

Figures reproduced from arXiv: 2412.18456 by the authors.

Figure 1
Figure 1. FIG. 1 [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2 [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. Time evolution of the energy distance to equilibrium, [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗
Figures from the paper (3 more)
Figure 4
Figure 4. Figure 4: FIG. 4 [PITH_FULL_IMAGE:figures/full_fig_p009_4.png]
Figure 5
Figure 5. Figure 5: FIG. 5. Accessible initial states by simultaneous gate-voltage and [PITH_FULL_IMAGE:figures/full_fig_p013_5.png]
Figure 6
Figure 6. Figure 6: FIG. 6 [PITH_FULL_IMAGE:figures/full_fig_p014_6.png]

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  1. The quantum Mpemba effects

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    A review of the quantum Mpemba effect covering open and isolated quantum systems, key theories, experiments, and open questions.

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    |ρeq) = |0), which occurs in the limit βϵ, β(ϵ + U ) ≫ 1, or |ρeq) = |2), in the limit βϵ, β(ϵ + U ) ≪ −1, then the only physical value of p when c = 0 is p = 0

    Exponential speedup and pure equilibrium state If the equilibrium state is a pure state, i.e. |ρeq) = |0), which occurs in the limit βϵ, β(ϵ + U ) ≫ 1, or |ρeq) = |2), in the limit βϵ, β(ϵ + U ) ≪ −1, then the only physical value of p when c = 0 is p = 0. Indeed, putting c0 = p0 = c = 0 [|ρeq) = |0)] in inequalities (A1a) and (A1c) gives c1p ≥ 0 and −c2p ...

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    From the master equation (6) and the kernel eigenmode decomposition [Eq

    Time-evolution of the relative entropy First, we compute D[ρ(t)] for any time t > 0 for a relax- ation from an arbitrary physical initial state ρ(0). From the master equation (6) and the kernel eigenmode decomposition [Eq. (9)], we can write the state of the dot at any time using Eq. (16). Then, the relative entropy can be expressed as in Eq. (17). We now...

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    Exponential speedup From Eq. (18), we can now study the time scales of the re- laxation toward equilibrium. We always haveγc < γp, though γc → γp when ϵ → −U/2 and U → +∞. The slowest decay rate is then γ2,0 = 2γc, and in the long time limit γ2,0t ≫ 1, D[ρ(t)] ∼ γct→∞ B2,0 2 c(0)2e−2γct, (B3) provided that the initial state has a finite overlap with the c...

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    Then, at time t, the difference in relative entropy of the two states is given by, using Eq

    Mpemba effect Let us consider two initial states |ϱin) and |ϱ′ in) such that D(ϱ′ in) > D(ϱin). Then, at time t, the difference in relative entropy of the two states is given by, using Eq. (18), D[ϱ′(t)] − D[ϱ(t)] = X k>1 Ak kX j=0 Bk,j (p′j in c′k−j in − pj inck−j in )e−γk,j t, (B7) where cin = (c′|ϱin), pin = (p′|ϱin), and c′ in = (c′|ϱ′ in), p′ in = (p...

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    IV C, one can wonder to what extent we can widen the range of accessible initial states by applying simultaneously a gate-voltage switch and a temperature quench

    Simultaneous gate-voltage switch and temperature quench Following the discussion in Sec. IV C, one can wonder to what extent we can widen the range of accessible initial states by applying simultaneously a gate-voltage switch and a temperature quench. In that case, the dot would be ini- tially at equilibrium for energy ϵin and temperature Tin, such (a) βU...

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    The dot is initially at equilibrium for interaction en- ergy Uin, then at time t = 0 the interaction energy is instan- taneously set to a new value U

    Interaction quench For completeness, we consider here an interaction-energy quench. The dot is initially at equilibrium for interaction en- ergy Uin, then at time t = 0 the interaction energy is instan- taneously set to a new value U. This means that the prepared initial state is ρ(0) = ρeq(Uin) = exp( −βHUin )/ZUin, where HUin is the dot Hamiltonian as g...

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    For simplicity, we investigate either the case of a potential bias ∆µ or the case of a temperature bias ∆T = TL − TR

    Nonequilibrium steady state We prepare the initial state by coupling, for t < 0, two reservoirs, that we will call left and right, with temperatures TL/R and electrochemical potentials µL/R, such that |ρ(0)) is the nonequilibrium steady state |ρss) corresponding to this two-terminal setup [38]. For simplicity, we investigate either the case of a potential...

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