Modeling heat transport through completely positive maps
Open accessAbstract
We investigate heat transport in a spin-1/2 Heisenberg chain, coupled locally to independent thermal baths of different temperature. The analysis is carried out within the framework of the theory of open systems by means of appropriate quantum master equations. The standard microscopic derivation of the weak-coupling Lindblad equation in the secular approximation is considered, and shown to be inadequate for the description of stationary nonequilibrium properties like a nonvanishing energy current. Furthermore, we derive an alternative master equation that is capable of describing a stationary energy current and, at the same time, leads to a completely positive dynamical map. This paves the way for efficient numerical investigations of heat transport in larger systems based on Monte Carlo wave function techniques.
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TL;DR
The paper shows that the standard weak-coupling Lindblad master equation in the secular approximation predicts a vanishing stationary energy current in a spin-1/2 Heisenberg chain subject to a temperature gradient, making it unsuitable for heat transport. It derives an alternative completely positive Lindblad master equation valid for weak internal couplings that supports a nonequilibrium steady-state energy current and enables Monte Carlo wave function simulation.
Key findings
- In the secular approximation, the stationary state is diagonal in the energy eigenbasis; for any plausible current operator satisfying \(\langle n|\hat J|n\rangle=0\), the stationary current vanishes, so no energy flows from the hot to the cold bath.
- The Redfield master equation can describe a finite current but is not generally in Lindblad form and may violate positivity; it cannot be treated with the standard Monte Carlo wave function technique.
- An alternative weak-internal-coupling Lindblad QME is derived by decomposing the coefficient matrix and neglecting a small non-positive contribution \(\gamma_B\); it preserves complete positivity and features a nonequilibrium steady state with an energy current.
- For \(n=3\) weakly coupled spins with \(\beta_L=0.41\), \(\beta_R=1.39\), \(\lambda=\kappa=0.01\), \(\Omega=1\), the Redfield equation, the weak-coupling Lindblad equation, and a stochastic wave function simulation with \(10^5\) realizations show excellent agreement.
Methods
- Model: a spin-1/2 Heisenberg chain of length \(n\), with local field \(\Omega\) and nearest-neighbor coupling \(\lambda\), coupled locally at both ends to independent bosonic Ohmic heat baths at inverse temperatures \(\beta_L\) and \(\beta_R\); the energy current operator is obtained from a discretized continuity equation.
- Derivation: Born-Markov Redfield master equation, secular approximation, weak internal coupling approximation replacing the interaction-picture system operator by free dynamics, and a Lindblad-form derivation using a Hilbert-Schmidt operator basis and coefficient-matrix decomposition.
- Numerical analysis: exact diagonalization in Liouville space for the Redfield and Lindblad equations, plus stochastic wave function simulation of the Lindblad equation with \(10^5\) realizations for \(n=3\).
Limitations
- Stated: the studies are restricted to one-dimensional spin chains; it remains open whether the secular approximation generally wipes out all local coupling aspects and whether the findings extend to larger classes of open systems and coupling models.
- Stated: the weak internal coupling assumption is fundamental for the model, and neglecting \(\gamma_B\) is valid for not too low temperatures; the Redfield equation may violate positivity, and larger-system numerical results are deferred to a forthcoming paper.
- Evident: the numerical comparison is shown only for \(n=3\), so finite-size scaling is not established in this paper.
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0 of 16 cited sources checked · 0 supported · 0 flagged
- Findingreasoning§ A. Secular Approximation · p. 5
For any plausible choice of a current operator satisfying that energy eigenstates carry no current, the stationary current in the secular-approximation steady state vanishes.
“by any plausible choice of a current operator, stationary currents will vanish by virtue of Eq. (44).”
- Also stated in: § IV. CONCLUSIONS · p. 7 · plain
“Therefore, no energy current is flowing from the hot towards the cold bath, a situation which is highly unphysical.”
Quote in the paperSubstantial claim- The authors state this as an assumption: The stationary solution of Eq. (39) is unique.
- Also stated in: § IV. CONCLUSIONS · p. 7 · plain
- Findingreasoning§ B. Weak Internal Coupling Approach · p. 6
The non-zero elements of γ_B rapidly vanish as temperature and N go to infinity.
“We see that the non-zero elements of γ B rapidly vanish as the temperature and, likewise, N goes to infinity”
Quote in the paperSubstantial claim - Findingreasoning§ A. Secular Approximation · p. 5
The QME in the secular approximation features a steady state that is diagonal in the energy representation.
“If we further assume that the stationary solution of Eq. (39) is unique, then this implies that the off-diagonal elements of ˆ ρ S decay and the stationary state ˆ ρ st S is diagonal in the energy representation.”
- Also stated in: § A. Secular Approximation · p. 5 · plain
“Summarizing, the QME in the SA [cf. Eq. (39)] features a steady state solution ˆ ρ st S that is diagonal in the energy representation, and by any plausible choice of a current operator, stationary currents will vanish by virtue of Eq. (44).”
- Also stated in: § IV. CONCLUSIONS · p. 7 · boosted
“Stress was laid on the fact that the secular approximation (SA) performed on a quantum master equation for a non-degenerate, one-dimensional system in a split bath scenario with temperature gradient (see Fig. 1) leads to a stationary state which is diagonal in the energy eigenbasis of the system.”
Quote in the paperSubstantial claim- The authors state this as an assumption: The stationary solution of Eq. (39) is unique.
- Also stated in: § A. Secular Approximation · p. 5 · plain
- Findingreasoning§ A. Secular Approximation · p. 5
The current operator (7) satisfies the condition that energy eigenstates carry no energy currents.
“It can be shown in a straight forward manner that our particular choice of the current operator (7) satisfies the condition in Eq. (43).”
Quote in the paperSubstantial claim - Findingreasoning§ I. INTRODUCTION · p. 2
The secular approximation, applied to the quantum master equation for the spin chain, yields a master equation that predicts a vanishing energy current in the presence of a finite temperature gradient.
“We observe that, once the SA has been performed, one is left with a QME that predicts a vanishing energy current in the presence of a finite temperature gradient.”
- Also stated in: § A. Secular Approximation · p. 5 · plain
“Summarizing, the QME in the SA [cf. Eq. (39)] features a steady state solution ˆ ρ st S that is diagonal in the energy representation, and by any plausible choice of a current operator, stationary currents will vanish by virtue of Eq. (44).”
Quote in the paperSubstantial claim- The authors state this as an assumption: The stationary solution of Eq. (39) is unique.
- Also stated in: § A. Secular Approximation · p. 5 · plain
- Findingreasoning§ B. Weak Internal Coupling Approach · p. 6
The final QME in the weak internal coupling limit (Eq. 62) is in Lindblad form.
“ending up, finally, with a master equation in Lindblad form.”
- Also stated in: § B. Weak Internal Coupling Approach · p. 6 · plain
“Thus, the Eq. (62) contains both vital properties: On the one hand it is in Lindblad form, preserving all properties of the density matrix and being, furthermore, well suited for all types of stochastical methods.”
Quote in the paperSubstantial claim - Also stated in: § B. Weak Internal Coupling Approach · p. 6 · plain
- Findingown data§ B. Weak Internal Coupling Approach · p. 6
The predictions of the Redfield master equation (12) and those of the Lindblad form (62) match excellently.
“Figure 2 demonstrates the excellent match between the predictions of the Redfield master equation (12) and those of the Lindblad form (62) that has been derived in this Section.”
Quote in the paperSubstantial claim - Interpretationreasoning§ A. Secular Approximation · p. 5
Therefore, in a nonequilibrium scenario, the differences of Bohr frequencies will eventually be too small to justify the application of the secular approximation.
“Thus, the differences of the Bohr frequencies will eventually be too small to justify the application of the SA in a nonequilibrium scenario.”
Quote in the paperSubstantial claim - Premisereasoning§ B. Weak Internal Coupling Approach · p. 5
The weak coupling assumption must be considered fundamental for the very model by virtue of Eqs. (7) and (26).
“We would like to stress that the weak coupling assumption must be considered fundamental for the very model by virtue of Eqs. (7) and (26).”
Quote in the paperSubstantial claim - Interpretationreasoning§ B. Weak Internal Coupling Approach · p. 7
The dynamics will not differ in the type of damping.
“However, the dynamics will not differ in the type of damping.”
Quote in the paperSubstantial claim - Premisenone given§ A. Secular Approximation · p. 5
In chainlike systems, differences of Bohr frequencies between energy levels within bands go to zero as chain length goes to infinity, even in the limit of strong couplings.
“In chainlike systems, however, differences of Bohr frequencies between energy levels within the bands will go to zero with chainlengths going to infinity (even in the limit of strong couplings).”
Quote in the paperNot a substantial claim - Interpretationown data§ Modeling heat transport through completely positive maps · p. 1
The standard microscopic derivation of the weak-coupling Lindblad equation in the secular approximation is inadequate for describing stationary nonequilibrium properties such as a non-vanishing energy current.
“The standard microscopic derivation of the weak-coupling Lindblad equation in the secular approximation is considered, and shown to be inadequate for the description of stationary nonequilibrium properties like a non-vanishing energy current.”
- Also stated in: § IV. CONCLUSIONS · p. 7 · plain
“dard microscopic derivation of a Lindblad quantum master equation (QME) fails, in the sense that it is inappropriate to model nonequilibrium steady states when the system is subject to a temperature gradient induced by baths of different temperature.”
Quote in the paperSubstantial claim - Also stated in: § IV. CONCLUSIONS · p. 7 · plain
- Interpretationreasoning§ B. Weak Internal Coupling Approach · p. 7
The neglect of the off-diagonal entries in Eq. (52) directly yields a Lindblad QME of the same type previously used for investigations on heat transport in small quantum systems [14], which hitherto lacked a microscopic derivation.
“Moreover we would like to point out that the neglect of the off-diagonal entries in Eq. (52) directly yields a Lindblad QME which is of the same type that has previously served for investigations on heat transport in small quantum systems [14] but hitherto lacked a microscopic derivation.”
Quote in the paperNot a substantial claim0 of 1 sources checked- cites [14] M. Michel, M. Hartmann, J. Gemmer, and G. Mahler, Euro. Phys. J. B 34 , 325 (2003). source
Not checked in the free run.
- Interpretationreasoning§ I. INTRODUCTION · p. 2
The secular approximation is inappropriate for describing heat transport and nonequilibrium scenarios with a temperature gradient.
“The standard weak-coupling Lindblad master equation that is obtained on the basis of the SA is therefore inappropriate to describe heat transport.”
- Also stated in: § IV. CONCLUSIONS · p. 7 · plain
“Hence, the use of a QME in the SA is inappropriate for the description of this type of nonequilibrium scenarios.”
Quote in the paperSubstantial claim- Rests on an assumption: The stationary state of the spin chain coupled to baths of different temperature features a non-vanishing energy flux from the hotter to the colder bath.
- Also stated in: § IV. CONCLUSIONS · p. 7 · plain
- Interpretationreasoning§ B. Weak Internal Coupling Approach · p. 7
All effects that arise from different internal coupling schemes will only show up in the coherent part of the QME.
“All effects that arise from different internal coupling schemes will only show up in the coherent part of the QME.”
Quote in the paperSubstantial claim - Premisecitation§ I. INTRODUCTION · p. 2
The secular approximation is justified only when the time scale set by differences of Bohr frequencies of the open system is short compared to the relaxation time.
“The SA is justified as long as the time scale set by the differences of the Bohr frequencies of the open system is short compared to the relaxation time.”
Quote in the paperSubstantial claim0 of 2 sources checked - Interpretationnone given§ B. Weak Internal Coupling Approach · p. 7
The neglect of the interaction between spins in Eq. (45) leads to damping that concerns only the hypothetically uncoupled system.
“The neglect of the interaction between spins in Eq. (45) leads to the effect that the damping concerns the hypothetically uncoupled system only.”
Quote in the paperSubstantial claim - Premisecitation§ C. Local Environment Coupling · p. 4
The Redfield master equation may violate preservation of positivity of the reduced density operator.
“regardless of the well-known disadvantage, that these forms may violate the preservation of positivity of the reduced density operator.”
Quote in the paperSubstantial claim0 of 2 sources checked - Premisecitation§ A. Secular Approximation · p. 5
For systems with a non-degenerate spectrum, the secular approximation decouples the equations of motion for populations and coherences in the eigenbasis of the system Hamiltonian, even with two baths at different temperatures.
“For systems with a non-degenerate spectrum like the weakly coupled spin-chains treated in this work, the SA decouples the equations of motion for the populations and coherences, i.e., for the diagonal and for the offdiagonal elements of ˆ ρ I S in the eigenbasis of the system Hamiltonian ˆ H S [9, 27, 28, 29], even and especially in the case of two baths with different temperatures.”
Quote in the paperSubstantial claim0 of 4 sources checked- cites [9] H.-P. Breuer and F. Petruccione, The Theory of Open Quantum Systems (Oxford University Press, 2002).
Not checked in the free run.
- cites [27] R. Alicki and K. Lendi, Quantum Dynamical Semigroups and Applications (Springer Verlag, 1986).
Not checked in the free run.
- cites [28] M. Stollsteimer, Ph.D. thesis, Universit¨ at Stuttgart (2006). source
Not checked in the free run.
- cites [29] R. Bl¨ umel, A. Buchleitner, R. Graham, L. Sirko, U. Smilansky, and H. Walther, Phys. Rev. A 44 , 4521 (1991). source
Not checked in the free run.
- Premisecitation§ C. Local Environment Coupling · p. 4
The canonical Gibbs state is a stationary solution of the master equation (26) in an equilibrium scenario (β_L = β_R ≡ β).
“is a stationary solution of the master equation (26) [25].”
Quote in the paperNot a substantial claim0 of 1 sources checked- cites [25] R. Kubo, M. Toda, and N. Hashitsume, Statistical Physics II: Nonequilibrium Statistical Mechanics , no. 31 in Solid-State Sciences (Springer, Berlin, Heidelberg, New-York, 1991), 2nd ed.
Not checked in the free run.
- May combine two claims. 9
- Classified as probably not a substantial claim. 5
May combine two claims.
Findingreasoning§ B. Weak Internal Coupling Approach · p. 6The final QME (Eq. 62) features a nonequilibrium steady state with an energy current flowing through the system.
“On the other hand it features a nonequilibrium steady state with an energy current flowing through the system.”
- Also stated in: § IV. CONCLUSIONS · p. 7 · plain
“we have proposed an alternative derivation that does yield a Lindblad QME and can properly describe energy currents in the stationary state of weakly coupled spin1 2 Heisenberg chains.”
Quote in the paperSubstantial claim- Also stated in: § IV. CONCLUSIONS · p. 7 · plain
May combine two claims.
Findingreasoning§ Modeling heat transport through completely positive maps · p. 1The paper derives an alternative master equation that is capable of describing a stationary energy current and leads to a completely positive dynamical map.
“Furthermore, we derive an alternative master equation that is capable to describe a stationary energy current and, at the same time, leads to a completely positive dynamical map.”
- Also stated in: § IV. CONCLUSIONS · p. 7 · plain
“we have proposed an alternative derivation that does yield a Lindblad QME and can properly describe energy currents in the stationary state of weakly coupled spin1 2 Heisenberg chains.”
Quote in the paperSubstantial claim- Also stated in: § IV. CONCLUSIONS · p. 7 · plain
May combine two claims.
Findingreasoning§ B. Weak Internal Coupling Approach · p. 6The final QME (Eq. 62) preserves all properties of the density matrix and is well suited for stochastic methods.
“On the one hand it is in Lindblad form, preserving all properties of the density matrix and being, furthermore, well suited for all types of stochastical methods.”
Quote in the paperSubstantial claimMay combine two claims.
Findingreasoning§ B. Weak Internal Coupling Approach · p. 6The coefficient matrix γ in Eq. (52) is not positive, so the dissipator derived from the weak internal coupling approach is not in Lindblad form.
“However, the matrix given by Eq. (52) is not positive because γ 11 > 0 and”
- Also stated in: § B. Weak Internal Coupling Approach · p. 6 · boosted
“Hence, we conclude that the dissipator derived above is not in Lindblad form.”
Quote in the paperSubstantial claim- Also stated in: § B. Weak Internal Coupling Approach · p. 6 · boosted
Probably not a substantial claim: may not pass S3 (load-bearing). May combine two claims.
Premisecitation§ I. INTRODUCTION · p. 2The secular approximation is equivalent to the rotating wave approximation in many cases, but there exist counterexamples.
“In many cases it is equivalent to the rotating wave approximation, but there exist counterexamples [20].”
Quote in the paperNot a substantial claim0 of 1 sources checked- cites [20] S. Maniscalco, F. Intravaia, J. Piilo, and A. Messina, J. Opt. B: Quantum Semiclass. Opt. 6 , 98 (2004).
Not checked in the free run.
Classified as probably not a substantial claim.
Interpretationnone given§ IV. CONCLUSIONS · p. 7The usefulness of a QME of Lindblad form is the key link to efficient numerical investigations by standard stochastic wave function methods.
“The usefulness of a QME of Lindblad-form can not be overemphasized, because it is the key link to efficient numerical investigations by means of standard stochastic wave function methods.”
Quote in the paperSubstantial claimProbably not a substantial claim: may not pass S3 (load-bearing). May combine two claims.
Premisecitation§ C. Local Environment Coupling · p. 4The violation of positivity is usually observed only at the very early stage of the relaxation process and may possibly be cured by a slippage of initial conditions.
“This violation is usually observed at the very early stage of the relaxation process only and may possibly be cured by a slippage of initial conditions [23, 24].”
Quote in the paperNot a substantial claim0 of 2 sources checked- cites [23] S. Gnutzmann and F. Haake, Zeitschrift f¨ ur Physik B 101 , 263 (1996).
Not checked in the free run.
- cites [24] A. Suarez, R. Silbey, and I. Oppenheim, The Journal of Chemical Physics 97 , 5101 (1992).
Not checked in the free run.
Probably not a substantial claim: may not pass S3 (load-bearing).
Premisecitation§ III. COMPLETE POSITIVITY VS. NONEQUILIBRIUM STEADY STATES · p. 4The dynamics of any QME with a Lindblad generator can be represented through a stochastic process in Hilbert space.
“In fact, the dynamics of any QME with a generator of the form of Eq. (28) can be represented through a stochastic process in Hilbert space.”
Quote in the paperNot a substantial claim0 of 2 sources checked- cites [9] H.-P. Breuer and F. Petruccione, The Theory of Open Quantum Systems (Oxford University Press, 2002).
Not checked in the free run.
- cites [17] M. B. Plenio and P. L. Knight, Rev. Mod. Phys. 70 , 101 (1998). source
Not checked in the free run.
May combine two claims.
Interpretationreasoning§ A. Secular Approximation · p. 5If the stationary solution of Eq. (39) is unique, then the off-diagonal elements of ρ_S decay and the stationary state is diagonal in the energy representation.
“If we further assume that the stationary solution of Eq. (39) is unique, then this implies that the off-diagonal elements of ˆ ρ S decay and the stationary state ˆ ρ st S is diagonal in the energy representation.”
Quote in the paperSubstantial claim- The authors state this as an assumption: The stationary solution of Eq. (39) is unique.
May combine two claims.
Interpretationreasoning§ B. Weak Internal Coupling Approach · p. 6For not too low temperatures, the quantity A is small, and the contribution from γ_B to the dissipator may be neglected.
“For not too low temperatures the quantity A is small, and we may neglect the contribution from γ B to the dissipator, ending up, finally, with a master equation in Lindblad form.”
Quote in the paperSubstantial claimProbably not a substantial claim: may not pass S2 (checkable), S3 (load-bearing).
Interpretationreasoning§ B. Weak Internal Coupling Approach · p. 6Investigations on larger chain models now become feasible.
“Investigations on larger chain models now become feasible and more detailed numerical results will be presented in a forthcoming paper.”
Quote in the paperNot a substantial claimMay combine two claims.
Premisecitation§ C. Local Environment Coupling · p. 4Extended stochastic schemes exist for solving general QMEs such as the Redfield equation, but they are generally less efficient than the standard approach for a Lindblad QME.
“Although extended stochastic schemes exist for the solution of general QME's as the Redfield equation [26], these methods turn out to be less efficient, in general, than the standard approach for a Lindblad QME”
Quote in the paperSubstantial claim0 of 1 sources checked- cites [26] H.-P. Breuer, B. Kappler, and F. Petruccione, Physical Review A 59 , 1633 (1999).
Not checked in the free run.
The authors state this as an assumption.
The bath is in a thermal equilibrium state at inverse temperature β.
“To this end, we invoke the Born-Markov approximation and assume that the bath is in a thermal equilibrium state at inverse temperature β .”
The authors state this as an assumption.
For weak couplings of neighboring spins, X(-τ) is approximately given by the free dynamics.
“To this end, we assume that for weak couplings of neighboring spins ˆ X ( -τ ) [cf. Eq. (14)] is approximately given by the free dynamics (see [30]), i.e.,”
The authors state this as an assumption.
The stationary solution of Eq. (39) is unique.
“If we further assume that the stationary solution of Eq. (39) is unique, then this implies that the off-diagonal elements of ˆ ρ S decay and the stationary state ˆ ρ st S is diagonal in the energy representation.”
The paper relies on this without support.
The stationary state of the spin chain coupled to baths of different temperature features a non-vanishing energy flux from the hotter to the colder bath.
“The stationary state features a non-vanishing energy flux through the chain of spins from the hotter towards the colder heat bath.”
Claims other papers make about this one
Each is checked against this paper’s own text: does it say what the citing paper says it does?
This result is consistent with [15], which discusses the absence of any currents as expressed through system observables in the global approach.
“This is consistent with the results of [15], which discusses the absence of any currents as expressed through system observables in the global approach.”
From Markovian master equations for quantum thermal machines: Local versus global approach (2017) · cites this paper as [15]
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References 32 · Cited by 224
- Authors
- Wichterich, H., Henrich, M. J., Breuer, H.-P., Gemmer, J., & Michel, M.
- Affiliations
- Osnabrück University (DE)
- University of Stuttgart (DE)
- University of Freiburg (DE)
- ORCID
- Breuer, Heinz-Peter
- Venue
- Physical Review E
- Publisher
- American Physical Society (APS)
- Published
- 2007-09-12
- Volume
- 76
- Issue
- 3
- Pages
- 031115-031115
- ISSN
- 1539-3755 (ISSN-L)
- 1550-2376
- OpenAlex
- W2060287714
- Cited by
- 224
- References
- 32
- Citations by year
- 2026: 7 · 2025: 17 · 2024: 11 · 2023: 12 · 2022: 19 · 2021: 16 · 2020: 22 · 2019: 16 · 2018: 9 · 2017: 15 · 2016: 10 · 2015: 13 · 2014: 9 · 2013: 6 · 2012: 8
- FWCI
- 5.65
- Type
- journal article
- Language
- en
- Open access
- green
- arxiv.org
- Locations
- Physical Review E (publishedVersion)
- PubMed (publishedVersion)
- arXiv (Cornell University) (submittedVersion)
- FreiDok plus (Universitätsbibliothek Freiburg) (submittedVersion)
- arXiv (Cornell University)
- Topics
- Physical Sciences › Physics and Astronomy › Statistical and Nonlinear Physics › Advanced Thermodynamics and Statistical Mechanics
- Physical Sciences › Physics and Astronomy › Atomic and Molecular Physics, and Optics › Quantum many-body systems
- Physical Sciences › Physics and Astronomy › Atomic and Molecular Physics, and Optics › Quantum and electron transport phenomena
- Keywords
- heat transport, spin-1/2 Heisenberg chain, completely positive maps, Lindblad master equation, quantum master equation, open quantum systems, energy current, thermal baths, weak-coupling approximation, secular approximation
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