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sha1_base64="Kb2JlA5x3IdX0EP9KZN4Xg9M=">AB/3icbVDLSsNAFJ3UV62vqAsXboJFqFBKIoIKIkU3LrqopbGFJoTJdNIOnUzCzEQoRt/xY0LFbf+hjv/xkmbhbYeuJfDOfcyc48fUyKkaX5rhaXldW14npY3Nre0f3XsQUcIRtlFEI971ocCUMGxLIinuxhzD0Ke49uM7/ziLkgEWvLcYzdEA4YCQiCUkmeflBpe43rteqOlUnTLzGVdZbJ5eNmvmFMYisXJSBjmanv7l9COUhJhJRKEQPcuMpZtCLgmieFJyEoFjiEZwgHuKMhi4abTAybGsVL6RhBxVUwaU/X3RgpDIcahryZDKIdi3svE/7xeIoMLNyUsTiRmaPZQkFBDRkaWhtEnHCNJx4pAxIn6q4GkEMkVWYlFYI1f/IisU9rlzXz/qxcv8nTKIJDcAQqwALnoA7uQBPYAIEJeAav4E170l60d+1jNlrQ8p198Afa5w+yEpS/</latexit> <latexit sha1_base64="Kb2JlA5x3IdX0EP9KZN4Xg9M=">AB/3icbVDLSsNAFJ3UV62vqAsXboJFqFBKIoIKIkU3LrqopbGFJoTJdNIOnUzCzEQoRt/xY0LFbf+hjv/xkmbhbYeuJfDOfcyc48fUyKkaX5rhaXldW14npY3Nre0f3XsQUcIRtlFEI971ocCUMGxLIinuxhzD0Ke49uM7/ziLkgEWvLcYzdEA4YCQiCUkmeflBpe43rteqOlUnTLzGVdZbJ5eNmvmFMYisXJSBjmanv7l9COUhJhJRKEQPcuMpZtCLgmieFJyEoFjiEZwgHuKMhi4abTAybGsVL6RhBxVUwaU/X3RgpDIcahryZDKIdi3svE/7xeIoMLNyUsTiRmaPZQkFBDRkaWhtEnHCNJx4pAxIn6q4GkEMkVWYlFYI1f/IisU9rlzXz/qxcv8nTKIJDcAQqwALnoA7uQBPYAIEJeAav4E170l60d+1jNlrQ8p198Afa5w+yEpS/</latexit> Two-terminal (thermoelectric) power production Right ( R ) Left ( L ) reservoir reservoir S T , T , L L R R P = [( µ R − µ L ) /e ] J e ( T L > T R , µ L < µ R ) The upper bound to efficiency is given by the Carnot efficiency (expected only at zero power; intuitively, finite currents entail dissipation): η C = 1 − T R T L
Scattering theory for two reservoirs Conserved currents: Heat currents: First law of thermodynamics:
<latexit sha1_base64="(nul)">(nul)</latexit> <latexit sha1_base64="(nul)">(nul)</latexit> <latexit sha1_base64="(nul)">(nul)</latexit> <latexit sha1_base64="(nul)">(nul)</latexit> Thermoelectric efficiency (power production) Charge current Heat current from reservoirs: � ∞ J q. α = 1 J h, α dE ( E − µ α ) τ ( E )[ f L ( E ) − f R ( E )] h −∞ Efficiency:
Delta-energy filtering and Carnot efficiency If transmission is possible only inside a tiny energy window around E=E ✶ then Carnot efficiency Carnot efficiency obtained in the limit of reversible transport (zero entropy production) and zero output power [Mahan and Sofo, PNAS 93, 7436 (1996); Humphrey et al., PRL 89, 116801 (2002)]
Example: single-level quantum dot Dot’s scattering matrix: The Green’s function is for a non-Hermitian effective Hamiltonian taking into account coupling to the dots operator coupling the single-level dot to reservoirs:
Short intermezzo: Cyclic thermal machines The upper bound to efficiency is given by the Carnot efficiency: Carnot efficiency obtained for quasi-static transformation (zero extracted power) The ideal Carnot engine is a reversible machine, since there is no dissipation (no entropy production)
Finite-time thermodynamics I: endoreversible cyclic engines Dissipation is due to finite thermal conductances between heat reservoirs and the ideal heat engine
Output power: Optimize power with respect to
The efficiency at maximum power (Curzon-Ahlborn efficiency) is independent of the heat conductances: [Yvon, 1955; Chambadal, 1957; Novikov, 1958; Curzon and Ahlborn, Am. J. Phys. 43, 22 (1975)] Within linear response:
Finite-time thermodynamics II: exoreversible cyclic engines Irreversibility only arises due to internal dissipative processes Stochastic thermodynamics [Seifert, Rep. Prog. Phys. 75, 126001 (2012)] Time-dependent trapping potential Time-dependent probability density
Fokker-Planck equation: is the mobility Gaussian distribution Exactly solvable model
Schmiedl-Seifert efficiency at maximum power: related to the ratio of entropy production during the hot and cold isothermal steps of the cycle for the symmetric case [Schmiedl and Seifert, EPL 81, 20003 (2008)] Within linear response:
Low-dissipation engines The entropy production vanishes in the limit of infinite-time cycles:
The CA limit is recovered for symmetric dissipation: dots: efficiencies of various thermal power plants [Esposito, Kawai, Lindenberg, Van den Broeck, PRL 105, 150603 (2010)]
Bekenstein-Pendry bound There is an purely quantum upper bound on the heat current through a single transverse mode [Bekenstein, PRL 46 , 923 (1981); Pendry, JPA 16 , 2161 (1983) ] For a reservoir coupled to another reservoir at T=0 through a -mode constriction which lets particle flow at all energies:
Maximum power of a heat engine Since the heat flow must be less than the Bekenstein- Pendry bound and the efficiency smaller than Carnot efficiency also the output power must be bounded Within scattering theory: [Whitney, PRL 112 , 130601 (2014); PRB 91 , 115425 (2015)]
Efficiency optimization (at a given power) Find the transmission function that optimizes the heat-engine efficiency for a given output power [Whitney, PRL 112 , 130601 (2014); PRB 91 , 115425 (2015)]
Trade-off between power and efficiency Carnot efficiency f o r b i d d e 1 n Efficiency Maximum 2 possible power, P max gen increase voltage power generated, P gen Result from (nonlinear) scattering theory [Whitney, PRL 112 , 130601 (2014); PRB 91 , 115425 (2015)]
Power-efficiency trade-off including phonons no phonons Efficiency weak phonons strong phonons 0 Power output, P [see Whitney, PRB 91 , 115425 (2015)]
Boxcar transmission in topological insulators Graphene nanoribbons with heavy adatoms and nanopores [Chang et al., Nanolett., 14, 3779 (2014)]
Linear response for coupled (particle and heat) flows Stochastic baths : ideal gases at fixed temperature and electrochemical potentia l ∆ µ = µ L − µ R Onsager relation (for time- reversal symmetric systems): ∆ T = T L − T R Positivity of entropy production: (we assume T L > T R , µ L < µ R )
Onsager and transport coefficients Note that the positivity of entropy production implies that the (isothermal) electric conductance G>0 and the thermal conductance K>0
Seebeck and Peltier coefficients Seebeck and Peltier coefficients are related a Onsager reciprocal relation (when time symmetry is not broken, we simply have )
Interpretation of the Peltier coefficient Entropy current: entropy transported by the electron flow each electron carries an entropy of advective term in thermal transport (reversible) open-circuit term in thermal transport (by electrons and phonons, irreversible)
Entropy production/ heat dissipation rate Joule heating heat lost by thermal resistance disappears for time-reversal To minimize symmetric systems dissipation large G and small K are needed
Linear response? (exhaust gases) (room temperature) [Vining, Nat. Mater. 8 , 83 (2009)] Linear response for small temperature and electrochemical potential differences (compared to the average temperature) on the scale of the relaxation length Exhaust pipe: temperature drop over a mm scale: temperature drop of 0.003 K on the relaxation length scale (of 10 nm)
<latexit sha1_base64="(nul)">(nul)</latexit> <latexit sha1_base64="(nul)">(nul)</latexit> <latexit sha1_base64="(nul)">(nul)</latexit> <latexit sha1_base64="(nul)">(nul)</latexit> Maximum efficiency Within linear response and for steady-state heat to work conversion: Find the maximum of η over , for fixed (i.e., over the applied voltage Δ V for fixed temperature difference Δ T ) ( T L ≈ T R ≈ T )
Thermoelectric figure of merit ZT ≡ L 2 det L = GS 2 eh T K
Conditions for Carnot efficiency ZT diverging implies that the Onsager matrix is ill- conditioned, that is, the condition number diverges: In such case the system is singular (tight coupling limit): J h ∝ J e (the ratio J h /J e is independent of the applied voltage and temperature gradients)
Efficiency at maximum power Output power Find the maximum of P over , for fixed (over the applied voltage Δ V for fixed Δ T ) Maximum output power L 2 h = 1 P max = T eh F 2 4 S 2 G ( ∆ T ) 2 4 L ee Power factor
P quadratic function of , with maximum at half of the stopping force : Efficiency at maximum power ZT η ( ω max ) = η C ZT + 2 ≤ η CA ≡ η C 2 2 η CA Curzon-Ahlborn upper bound
η max η ( ω max ) P max
Efficiency versus power ⇒
Maximum refrigeration efficiency (heat extracted from the cold reservoir) Cooling power Coefficient of performance (COP) η ( r ) = J h P (can be >1) ZT is the figure of merit also for refrigeration
ZT is an intrinsic material property? For mesoscopic systems size-dependence for G,K,S can be expected In the diffusive transport regime Ohm’s and Fourier’s scaling laws hold:
Local equilibrium Under the assumption of local equilibrium we can write phenomenological equations with ∇ T and ∇ µ rather than Δ T and Δ µ charge and heat current densities In this case we connect Onsager coefficients to electric and thermal conductivity rather than to conductances � j e � j h � � , σ = κ = � V � T � T =0 j e =0
Crossover from endoreversible to exoreversible regimes Thermoelectric generator: internal dissipation (Joule heating, thermal conductance) and external dissipation (dissipative thermal coupling to reservoirs) [Apertet, Ouerdane, Goupil, Lecoeur, PRE 85, 031116 (2012)]
Linear response and Landauer formalism The Onsager coefficients are obtained from the linear response expansion of the charge and thermal currents L ee = e 2 TI 0 , L eh = L he = eTI 1 , L hh = TI 2
Wiedemann-Franz law Phenomenological law: the ratio of the thermal to the electrical conductivity is directly proportional to the temperature, with a universal proportionality factor. Lorenz number
Sommerfeld expansion The Wiedemann-Franz law can be derived for low- temperature non-interacting systems both within kinetic theory or Landauer approach In both cases it is substantiated by Sommerfeld expansion. Within Landauer approach we consider � ∞ J q. α = 1 dE ( E − µ α ) τ ( E )[ f L ( E ) − f R ( E )] h −∞ We assume smooth transmission functions τ (E) in the neighborhood of E=µ:
To leading order in k B T/E F with G = e 2 I 0 ≈ e 2 I 2 − I 2 ≈ π 2 k 2 K = 1 � � B T 1 h τ ( µ ) , τ ( µ ) 3 h T I 0 Neglected I 12 /I 0 with respect to I 2 , which in turn implies L ee L hh >>(L eh ) 2 and Wiedemann-Franz law: � 2 G ≈ π 2 � k B K T e 3
Wiedemann-Franz law and thermoelectric efficiency ZT = GS 2 T = S 2 K L Wiedemann-Franz law derived under the condition L ee L hh >>(L eh ) 2 and therefore Wiedemann-Franz law violated in - low-dimensional interacting systems that exhibit non- Fermi liquid behavior - (smll) systems where transmission can show significant energy dependence
(Violation of) Wiedemann-Franz law in small systems Consider a (basic) model of a molecular wire coupled to electrodes: Transmission: Green’s function: Level broadening functions: Self-energies:
Wide band limit: level widths energy independent: Take Transmission: Green’s function obtained by inverting
Mott’s formula for thermopower For non-interacting electrons (thermopower vanishes when there is particle-hole symmetry) � � � ∞ − ∂ f −∞ dE ( E − µ ) τ ( E ) S = 1 = 1 I 1 ∂ E � � � ∞ eT I 0 eT − ∂ f −∞ dE τ ( E ) ∂ E Consider smooth transmissions Electron and holes contribute with opposite signs: we want sharp, asymmetric transmission functions to have large S (ex: resonances, Anderson QPT, see Imry and Amir, 2010), violation of WF, possibly large ZT.
Metal-insulator 3D Anderson transition x conductivity critical exponent [G.B., H. Ouerdane, C. Goupil, arXiv:1602.06590; Comptes Rendus Physique, in press]
Energy filtering For good thermoelectric we desire violation of WF law such that: No dispersion with delta-energy filtering: ZT diverges
Thermoelectricity in the Coulomb blockade regime, Kinetic equations. Quantum dot model
Multilevel interacting quantum dot Discrete energy levels: ideal to implement energy filtering Study the effects of Coulomb interaction between electrons [Erdmann, Mazza, Bosisio, G.B., Fazio, Taddei PRB 95 , 245432 (2017)]
Sequential (single-electron) tunnelling regime single-electron levels of the QC capacitance number of electrons in the dot electrostatic (Coulomb) interaction tunneling rate from level p to reservoir 𝛽 Weak coupling to the reservoirs: thermal energy , level spacing and charging energy much larger than the coupling energy between the QD and the reservoirs: charge quantized Electrostatic energy single-electron charging energy
Energy conservation Configuration determined by occupation numbers Non-equilibrium probability Energy conservation for tunnelling into or from reservoirs:
Kinetic equations One kinetic equation for each configuration: Stationary solution:
Steady-state currents Charge current: Energy current: Heat current:
Quantum limit Energy spacing and charging energy much bigger than Analytical results for equidistant levels: power factor (energy filtering)
Coulomb interaction may enhance the thermoelectric performance of a QD Compare interacting and non-interacting two-terminal QD with the same energy spacing T h e r m a l c o n d u c t a n c e suppressed by Coulomb interaction: ZT is greatly increased. For a single level K=0 (charge and heat current proportional). For at least two levels Coulomb blockade prevents a second electron to enter when one is already there (electrostatic energy to be paid).
Strongly interacting systems, Electronic Phase transitions, Power-efficiency trade-off, Power-efficiency-fluctuations trade-off, Carnot efficiency at finite power? Generality of Onsager reciprocal relations
Short intermezzo: a reason why interactions might be interesting for thermoelectricity thermal conductance at zero voltage If the ratio K’/K diverges, then the Carnot efficiency is achieved
Thermodynamic properties of the working fluid coupled equations:
Setting dN=0 in the coupled equations:
Thermodynamic cycle maximum efficiency (over d 𝜈 at fixed dT): thermodynamic figure of merit:
Analogy with a classical gas heat capacity at constant p or V
Power-efficiency trade-off: Is it possible to overcome the non-interacting bound? Noninteracting systems: for P/P max <<1, Bound not favorable for power-efficiency trade-off; due to the fact that delta-energy filtering is the only mechanism to achieve Carnot for noninteracting systems For interacting systems it is possible to achieve Carnot without delta-energy filtering
Interacting systems, Green-Kubo formula The Green-Kubo formula expresses linear response transport coefficients in terms of dynamic correlation functions of the corresponding current operators, cal- culated at thermodynamic equilibrium Non-zero generalized Drude weights signature of ballistic transport
Conservation laws and thermoelectric efficiency Suzuki’s formula (which generalizes Mazur’s inequality) for finite-size Drude weights Q m relevant (i.e., non-orthogonal to charge and thermal currents), mutually orthogonal conserved quantities Assuming commutativity of the two limits,
Momentum-conserving systems Consider systems with a single relevant constant of motion, notably momentum conservation Ballistic contribution to vanishes since D ee D hh − D 2 eh = 0 ZT = σ S 2 T ∝ Λ 1 − α → ∞ when Λ → ∞ κ ( α < 1) (G.B., G. Casati, J. Wang, PRL 110, 070604 (2013))
For systems with more than a single relevant constant of motion, for instance for integrable systems, due to the Schwarz inequality eh = || x e || 2 || x h || 2 � � x e , x h � � 0 D ee D hh � D 2 � � x i = ( x i 1 , ..., x iM ) = 1 � J i Q 1 � , ..., � J i Q M � � � 2 Λ � Q 2 � Q 2 1 � M � M � � x e , x h � = x ek x hk k =1 Equality arises only in the exceptional case when the two vectors are parallel; in general det L ∝ L 2 , κ ∝ Λ , ZT ∝ Λ 0 ∝ Λ 2
Example: 1D interacting classical gas Consider a one dimensional gas of elastically colliding particles with unequal masses: m, M (integrable model) ZT = 1 (at µ = 0) ZT depends on the system size
Quantum mechanics needed: Relation between density and electrochemical potential Reservoirs modeled as ideal (1D) gases Maxwell-Bolzmann distribution of velocities injection rates grand partition function density de Broglie thermal wave length
Non-decaying correlation functions
Carnot efficiency at the thermodynamic limit ZT = σ S 2 Anomalous T thermal transport k [R. Luo, G. B., G. Casati, J. Wang, PRL 121 , 080602 (2018)]
Delta-energy filtering mechanism? A mechanism for achieving Carnot different from delta-energy filtering is needed
Validity of linear response The agreement with linear response improves with N
Non-interacting classical bound (but quantum mechanics needed) charge current heat current Maxwell-Boltzmann distribution (in 1D)
1.0 0.8 0.6 Η ê Η C 0.4 0.2 0.0 0.0 0.2 0.4 0.6 0.8 1.0 P ê P max
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sha1_base64="8lf/Y1auXT8/vunbgNUdnWycSs=">ACA3icbVA9T8MwEHX4LOUrwNjFokJiqhKEBGwVLIxFIrRSG1WOe2tOnZkO4gq6sDCX2FhAMTKn2Dj3+C0GaDlSc9vXdn370o4Uwbz/t2lpZXVtfWSxvlza3tnV13b/9Oy1RCKjkUrUioEzAYFhkMrUDiEMzGl3lfvMelGZS3JpxAmFMBoL1GSXGSl230hlG8iFrR2OcP0EUVqATKTSEk65b9WreFHiR+AWpogKNrvV6UmaxiAM5UTrtu8lJsyIMoxymJQ7qYaE0BEZQNtSQWLQYTY9YoKPrNLDfalsCYOn6u+JjMRaj+PIdsbEDPW8l4v/e3U9M/DjIkNSDo7KN+yrGROE8E95gCavjYEkIVs7tiOiSKUGNzK9sQ/PmTF0lwUruoeTen1fplkUYJVdAhOkY+OkN1dI0aKEAUPaJn9IrenCfnxXl3PmatS04xc4D+wPn8AagdmE4=</latexit> <latexit sha1_base64="8lf/Y1auXT8/vunbgNUdnWycSs=">ACA3icbVA9T8MwEHX4LOUrwNjFokJiqhKEBGwVLIxFIrRSG1WOe2tOnZkO4gq6sDCX2FhAMTKn2Dj3+C0GaDlSc9vXdn370o4Uwbz/t2lpZXVtfWSxvlza3tnV13b/9Oy1RCKjkUrUioEzAYFhkMrUDiEMzGl3lfvMelGZS3JpxAmFMBoL1GSXGSl230hlG8iFrR2OcP0EUVqATKTSEk65b9WreFHiR+AWpogKNrvV6UmaxiAM5UTrtu8lJsyIMoxymJQ7qYaE0BEZQNtSQWLQYTY9YoKPrNLDfalsCYOn6u+JjMRaj+PIdsbEDPW8l4v/e3U9M/DjIkNSDo7KN+yrGROE8E95gCavjYEkIVs7tiOiSKUGNzK9sQ/PmTF0lwUruoeTen1fplkUYJVdAhOkY+OkN1dI0aKEAUPaJn9IrenCfnxXl3PmatS04xc4D+wPn8AagdmE4=</latexit> <latexit sha1_base64="8lf/Y1auXT8/vunbgNUdnWycSs=">ACA3icbVA9T8MwEHX4LOUrwNjFokJiqhKEBGwVLIxFIrRSG1WOe2tOnZkO4gq6sDCX2FhAMTKn2Dj3+C0GaDlSc9vXdn370o4Uwbz/t2lpZXVtfWSxvlza3tnV13b/9Oy1RCKjkUrUioEzAYFhkMrUDiEMzGl3lfvMelGZS3JpxAmFMBoL1GSXGSl230hlG8iFrR2OcP0EUVqATKTSEk65b9WreFHiR+AWpogKNrvV6UmaxiAM5UTrtu8lJsyIMoxymJQ7qYaE0BEZQNtSQWLQYTY9YoKPrNLDfalsCYOn6u+JjMRaj+PIdsbEDPW8l4v/e3U9M/DjIkNSDo7KN+yrGROE8E95gCavjYEkIVs7tiOiSKUGNzK9sQ/PmTF0lwUruoeTen1fplkUYJVdAhOkY+OkN1dI0aKEAUPaJn9IrenCfnxXl3PmatS04xc4D+wPn8AagdmE4=</latexit> <latexit sha1_base64="8lf/Y1auXT8/vunbgNUdnWycSs=">ACA3icbVA9T8MwEHX4LOUrwNjFokJiqhKEBGwVLIxFIrRSG1WOe2tOnZkO4gq6sDCX2FhAMTKn2Dj3+C0GaDlSc9vXdn370o4Uwbz/t2lpZXVtfWSxvlza3tnV13b/9Oy1RCKjkUrUioEzAYFhkMrUDiEMzGl3lfvMelGZS3JpxAmFMBoL1GSXGSl230hlG8iFrR2OcP0EUVqATKTSEk65b9WreFHiR+AWpogKNrvV6UmaxiAM5UTrtu8lJsyIMoxymJQ7qYaE0BEZQNtSQWLQYTY9YoKPrNLDfalsCYOn6u+JjMRaj+PIdsbEDPW8l4v/e3U9M/DjIkNSDo7KN+yrGROE8E95gCavjYEkIVs7tiOiSKUGNzK9sQ/PmTF0lwUruoeTen1fplkUYJVdAhOkY+OkN1dI0aKEAUPaJn9IrenCfnxXl3PmatS04xc4D+wPn8AagdmE4=</latexit> <latexit sha1_base64="8lf/Y1auXT8/vunbgNUdnWycSs=">ACA3icbVA9T8MwEHX4LOUrwNjFokJiqhKEBGwVLIxFIrRSG1WOe2tOnZkO4gq6sDCX2FhAMTKn2Dj3+C0GaDlSc9vXdn370o4Uwbz/t2lpZXVtfWSxvlza3tnV13b/9Oy1RCKjkUrUioEzAYFhkMrUDiEMzGl3lfvMelGZS3JpxAmFMBoL1GSXGSl230hlG8iFrR2OcP0EUVqATKTSEk65b9WreFHiR+AWpogKNrvV6UmaxiAM5UTrtu8lJsyIMoxymJQ7qYaE0BEZQNtSQWLQYTY9YoKPrNLDfalsCYOn6u+JjMRaj+PIdsbEDPW8l4v/e3U9M/DjIkNSDo7KN+yrGROE8E95gCavjYEkIVs7tiOiSKUGNzK9sQ/PmTF0lwUruoeTen1fplkUYJVdAhOkY+OkN1dI0aKEAUPaJn9IrenCfnxXl3PmatS04xc4D+wPn8AagdmE4=</latexit> Overcoming the non-interacting bound Non-interacting bound [by linear response]
Multiparticle collision dynamics (Kapral model) in 2D Streaming step: free propagation during a time τ Collision step: random rotations of the velocities of the particles in cells of linear size a with respect to the center of mass velocity: Momentum is conserved
Overcoming the (2D) non-interacting bound
Results can be extended to cooling linear response numerical data
Applications for cold atoms?
Power-efficiency trade-off at the verge of phase transitions For heat engines described as Markov processes: [N. Shiraishi, K. Saito, H. Tasaki, PRL 117 , 190601 (2016)] For a working substance at a critical point: [M. Campisi, R. Fazio, Nature Comm. 7 , 11895 (2016); see also Allahverdyan et al., PRL 111 , 050601 (2013)] Results compatible only with diverging amplitude A when approaching the Carnot efficiency
Power-efficiency-fluctuations trade-off For classical Markovian dynamics on a finite set of states and overdamped Langevin dynamics, trade-off between power, efficiency, and constancy for steady- state engines: [P. Pietzonka, U. Seifert, PRL 120 , 190602 (2018)] Bound violated in quantum mechanics, e.g. for resonant tunnelling transport (noninteracting system), but not close to Carnot efficiency. The problem for interacting systems is open. [J. Liu. D. Segal, PRE 99 , 062141 (2019)]
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