Note: On the memory kernel and the reduced system propagator
NNote: On the memory kernel and the reduced system propagator
Lyran Kidon,
1, 2
Haobin Wang, Michael Thoss, and Eran Rabani
1, 2, 5 Department of Chemistry, University of California, Berkeley, California 94720USA Materials Science Division, Lawrence Berkeley National Laboratory, Berkeley, California 94720,USA Department of Chemistry, University of Colorado Denver, Denver, Colorado 80217-3364,USA Institute of Physics, University of Freiburg, Hermann-Herder-Strasse 3, 79104 Freiburg,Germany The Sackler Center for Computational Molecular and Materials Science, Tel Aviv University, Tel Aviv,Israel 69978 (Dated:)
The generalized quantum master equation (GQME)formalism has recently been proven highly successfulfor the calculation of the reduced dynamics of com-plex many-body systems driven away from equilibrium.Two approaches exist: the Nakajima–Zwanzig–Moritime-convolution (TC) approach, and the Tokuyama–Mori time-convolutionless (TCL) approach. In both ap-proaches, the complexity of solving the many-body quan-tum Liouville equation is replaced with the need to evalu-ate a time-dependent kernel/super-operator from whichthe dynamics of the reduced system can be inferred atall times. The formalism becomes advantageous whenthe characteristic decay time of the kernels is short com-pared to the approach to equilibrium or steady state, whereby brute-force numerically-converged solvers provide efficient schemes to obtain the kernels. Alterna-tively, the GQME is also an excellent starting point forapproximate schemes based on semiclassical and mixedquantum-classical approaches.
The TC memory kernel and the TCL kernel/generatorare computed by means of auxiliary super-operators. Forthe TC memory kernel, this involves the solution ofan integro-differential equation and requires the calcula-tion of a super-operator whose form depends explicitlyon the system-bath coupling, which often becomesan intractable task. For the TCL kernel, the calcula-tion is based on the so-called reduced system propaga-tor super-operator which requires only generic systemobservables, but requires an inversion of the reducedpropagator which can lead to numerical instabilities.Here, we outline a simple framework to obtain theTC memory kernel from the reduced system propaga-tor alone, circumventing the need to obtain its inverse(TCL) or calculate higher order system-bath observables(TC). The form of the reduced system propagator is uni-versal for any system-bath Hamiltonian, allowing for areduction in the complexity of obtaining the TC memorykernel. We demonstrate this on the nonequilibrium gen-eralized Anderson-Holstein impurity model. The frame-work also provides direct relations between the TC andTCL kernels in terms of the reduced system propagator.Consider an open quantum system coupled to an en-vironment (bath), described by H = H S + H B + H SB ,where H S , H B are the system and bath Hamiltonians, and H SB the coupling between the two. The exacttime evolution of the reduced density matrix (RDM), σ ( t ) = Tr B { ρ ( t ) } ( ρ ( t ) is the full density matrix),within the TCL approach, is given in terms of a time-localkernel, ∂∂t σ ( t ) = K ( t ) σ ( t ) , where we assumed a non-correlated initial state, namely that ρ (0) = σ (0) ⊗ ρ B (0) .A simple approach to obtain K ( t ) , is based on the re-duced system propagator, U S ( t ) = Tr B (cid:8) e L t ρ B (cid:9) (cid:54) = e L S t ,where L = − i (cid:126) [ H , · · · ] . Using U S ( t ) , the TCL generatoris given by: K ( t ) = ˙ U S ( t ) U − S ( t ) (1)The matrix elements of the super-operator U S ( t ) can beobtained directly from the reduced density matrix ele-ments, σ ( t ) evolved from different initial conditions ofthe system. Since the time evolution of the reduceddensity operator in matrix form reads σ ij ( t ) = (cid:88) kl U S,ij,kl ( t ) σ kl (0) , (2)it clearly follows that an initial state with σ mm (0) =1 and the remaining values of σ kl (0) = 0 , will give U S,ij,mm ( t ) = σ ij ( t ) . For more details see Ref. 10. Im-portantly, the relation between U S ( t ) and σ ( t ) holds forany system-bath Hamiltonian and thus, simplifies the cal-culation of K ( t ) for complex model systems. However,Eq. (1) is ill-defined when U S ( t ) is singular (for example,when two system states are degenerate and couple to thebath in the same way), limiting its applicability.An alternative to the TCL approach, describes thetime evolution of the reduced density matrix usinga non-local memory term, ∂∂t σ ( t ) = L S σ ( t ) + (cid:126) ´ t dτ κ ( τ ) σ ( t − τ ) . For this approach (TC), it is wellknown that one can rewrite the memory kernel κ ( t ) interms of a Volterra equation of the second kind: κ ( t ) = ∂ Φ ( t ) ∂t − Φ ( t ) L S − (cid:126) ˆ t dτ Φ ( t − τ ) κ ( τ ) , (3)where Φ ( t ) = (cid:126) Tr B (cid:8) L e L t ρ B (cid:9) is a super-operator thatcan be calculated by a variety of solvers. However,unlike the reduced propagator,
Φ ( t ) depends explicitlyon the form of the system-bath coupling via the full Liou-villian L = L S + L B + L SB , and requires the calculation of1 a r X i v : . [ c ond - m a t . o t h e r] J u l igh–order observables in the system and bath degrees offreedom, not required for U S ( t ) . Here we show, the needto calculate higher–order observables can be removed bynoting a simple connection between Φ ( t ) and U S ( t ) : Φ ( t ) (cid:126) = Tr B (cid:8) L e L t ρ B (cid:9) = ∂∂t Tr B (cid:8) e L t ρ B (cid:9) = ∂ U S ( t ) ∂t . (4)The equation for the TC memory kernel may now berewritten in terms of the reduced system propagator andthe system Liouvillian alone: κ ( t ) = (cid:126) ¨ U S ( t ) − (cid:126) ˙ U S ( t ) L S − ˆ t dτ ˙ U S ( t − τ ) κ ( τ ) . (5)Eq. (5) is the main result of this note. It pro-vides a scheme to calculate the memory kernel for ageneral form of the coupling Hamiltonian without theneed to calculate any terms involving the bath opera-tors, as long as the time derivatives of U S ( t ) are ob-tained numerically. We demonstrate this for the gener-alized Anderson-Holstein model describing an impuritywith on-site electron-electron (e-e) interactions, coupledto three baths: a phonon bath and two fermionic baths(leads) at different chemical potentials (full descriptiongiven in Ref. 23). We deployed the multilayer mul-ticonfiguration time-dependent Hartree (ML-MCTDH)method to numerically compute the RDM at shorttimes from independent initial system states. The re-duced propagator was obtained from the ML-MCTDHresults according to Eq. (2), and its time derivatives per-formed numerically ( points finite difference). Finally,the memory kernel was computed according to Eq. (5). t [ h _ / Γ ] -0.2-0.100.10.20.3 κ nn , mm ( t ) κ κ κ κ κ κ κ t [ h _ / Γ ]U = 0 Γ σ nn ( t ) U = 2 Γ Figure 1. The RDM and memory kernel for two values ofthe interaction energy U . Upper panels: The RDM ele-ments propagated from an initially empty dot using the ML-MCTDH method (squares) and the TC GQME approach(solid lines). σ nn are the probabilities of the dot being empty(black), occupied by one electron (red), and occupied by twoelectrons (green). Lower panels: The seven distinct nonzeromemory-kernel elements. All quantities are shown in units ofthe system-leads coupling strength Γ . In Fig. 1 we shown the results for two values of theon-site e-e repulsion, U . The elements of the memory computed from the reduced system propagator are shownin the lower panels and the resulting elements of the re-duced density are shown in the upper panels. The pop-ulations obtained by solving the TC GQME with thememory kernel given by Eq. (5) are in excellent agree-ment with the numerical results obtained directly fromthe ML-MCTDH method (upper panels), reassuring thenumerical procedure to obtain κ ( t ) from U S ( t ) .In summary, we have related the memory kernel κ ( t ) in the Nakajima–Zwanzig–Mori TC formalism to the re-duced system propagator U S ( t ) , which can be obtainedat short times from an impurity solver. Compared to pre-vious formulations our approach provides a robust andsimpler framework, circumventing the need to computehigh-order system-bath observables. Moreover, unlikethe Tokuyama–Mori TCL approach, the current formal-ism does not rely on the inversion of a super-operatorwhich can be singular. We illustrated the correctnessof the proposed approach for a model system describ-ing both electron-electron and electron-phonon correla-tions and find excellent agreement between the accurateML-MCTDH results and the generalized quantum mas-ter equation. REFERENCES S. Nakajima, Prog. Theor. Phys. , 948 (1958). R. Zwanzig, J. Chem. Phys. , 1338 (1960). H. Mori, Prog. Theor. Phys. , 423 (1965). M. Tokuyama and H. Mori, Prog. Theor. Phys. , 1073 (1976). Q. Shi and E. Geva, J. Chem. Phys. , 12063 (2003). G. Cohen and E. Rabani, Phys. Rev. B , 075150 (2011). G. Cohen, E. Gull, D. R. Reichman, A. J. Millis, and E. Rabani,Phys. Rev. B , 195108 (2013). E. Y. Wilner, H. Wang, G. Cohen, M. Thoss, and E. Rabani,Phys. Rev. B , 045137 (2013). E. Y. Wilner, H. Wang, M. Thoss, and E. Rabani, Phys. Rev.B , 205129 (2014). L. Kidon, E. Y. Wilner, and E. Rabani, J. Chem. Phys. ,234110 (2015). N. Makri and D. E. Makarov, J. Chem. Phys. , 4600 (1995). L. Mühlbacher and E. Rabani, Phys. Rev. Lett. , 176403(2008). S. Weiss, J. Eckel, M. Thorwart, and R. Egger, Phys. Rev. B , 195316 (2008). H. Wang and M. Thoss, J. Chem. Phys. , 024114 (2009). E. Gull, D. R. Reichman, and A. J. Millis, Phys. Rev. B ,075109 (2010). D. Segal, A. J. Millis, and D. R. Reichman, Phys. Rev. B ,205323 (2010). G. Cohen, E. Gull, D. R. Reichman, and A. J. Millis, Phys.Rev. Lett. , 266802 (2015). M.-L. Zhang, B. J. Ka, and E. Geva, J. Chem. Phys. ,044106 (2006). T. C. Berkelbach, T. E. Markland, and D. R. Reichman, J.Chem. Phys. , 084104 (2012). A. Kelly, A. Montoya-Castillo, L. Wang, and T. E. Markland,J. Chem. Phys. , 184105 (2016). A. Montoya-Castillo and D. R. Reichman, J. Chem. Phys. ,184104 (2016). B. B. Laird, J. Budimir, and J. L. Skinner, J. Chem. Phys. ,4391 (1991). H. Wang and M. Thoss, J. Chem. Phys. , 134704 (2013). H. Wang and M. Thoss, J. Chem. Phys. , 1289 (2003)., 1289 (2003).