The Time-Dependent Exchange-Correlation Functional for a Hubbard Dimer: Quantifying Non-Adiabatic Effects
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The time-dependent exchange-correlation functional for a Hubbard dimer: quantifying non-adiabatic effects Johanna I. Fuks∗,1, 2 Mehdi Farzanehpour∗,1 Ilya V. Tokatly,1, 3 Heiko Appel,4 Stefan Kurth,1, 3 and Angel Rubio1, 4 1Nano-Bio Spectroscopy group and ETSF, Dpto. F´ısica de Materiales, Universidad del Pa´ısVasco, Centro de F´ısica de Materiales CSIC-UPV/EHU-MPC and DIPC, Av. Tolosa 72, E-20018 San Sebasti´an,Spain 2Department of Physics and Astronomy, Hunter College and the Graduate Center of the City University of New York, New York City, United States 3IKERBASQUE, Basque Foundation for Science, E-48011 Bilbao, Spain 4Fritz-Haber-Institut der Max-Planck-Gesellschaft, Faradayweg 4-6, D-14195 Berlin, Germany (Dated: January 13, 2021) We address and quantify the role of non-adiabaticity ("memory effects”) in the exchange- correlation (xc) functional of time-dependent density functional theory (TDDFT) for describing non-linear dynamics of many-body systems. Time-dependent resonant processes are particularly challenging for available TDDFT approximations, due to their strong non-linear and non-adiabatic character. None of the known approximate density functionals are able to cope with this class of problems in a satisfactory manner. In this work we look at the prototypical example of the resonant processes by considering Rabi oscillations within the exactly soluble 2-site Hubbard model. We con- struct the exact adiabatic xc functional and show that (i) it does not reproduce correctly resonant Rabi dynamics, (ii) there is a sizable non-adiabatic contribution to the exact xc potential, which turns out to be small only at the beginning and at the end of the Rabi cycle when the ground state population is dominant. We then propose a "two-level" approximation for the time-dependent xc potential which can capture Rabi dynamics in the 2-site problem. It works well both for resonant and for detuned Rabi oscillations and becomes essentially exact in the linear response regime. This new, fully non-adiabatic and explicit density functional constitutes one of the main results of the present work. PACS numbers: 31.15.ee,42.65.-k,71.15.Mb I. INTRODUCTION uses the instantaneous density as input for an approxi- mate ground-state functional. Thus, this approximation completely neglects both the history and the initial-state Due to the favorable balance between efficiency dependence of the exact functional. and accuracy, time-dependent density functional theory (TDDFT) is becoming the theory of choice to describe The successes and failures of the adiabatic approxima- the interaction of many-electron systems with external tion to describe linear response phenomena have been ad- electromagnetic fields of arbitrary intensity, shape and dressed in many works [1, 2, 15{17]. However, much less time dependence. Within this theory the observables is known about the performance of adiabatic TDDFT for are expressible as functionals of the time-dependent den- general dynamics beyond linear response. In some of our sity. Similarly to static DFT, in TDDFT one can de- past studies [8, 11, 18, 19] on one-dimensional model sys- fine an auxiliary non-interactiong Kohn-Sham (KS) sys- tems we have shown that adiabatic xc functionals fail to tem which reproduces the exact time-dependent dynam- describe dynamical processes where the density changes ics of the density. It is the propagation of this aux- significantly in time (e. g. in photo-physical and chem- iliary system in an (unknown) effective local potential ical processes where valence electrons are promoted to which makes TDDFT computationally powerful. How- empty states). There are few cases where the exact time- ever, despite of the great success of the theory in de- dependent xc potential is known and can thus be used to scribing optical properties of a large variety of molecules test approximations [11, 20, 21]. In these works it has and nanostructures [1{3], the available approximations been shown numerically that novel dynamical steps ap- arXiv:1312.1667v1 [cond-mat.other] 5 Dec 2013 for the exchange-correlation (xc) potential exhibit seri- pear in the xc potential which are fundamental to capture ous deficiencies in the description of non-linear processes, the proper resonant versus non-resonant dynamics and long range charge transfer [4{6] and double excitations charge localization. While the construction of accurate [7{9], to mention a few. approximations to the exact universal xc functional of The theoretical challenge is to improve the available TDDFT for Coulomb systems remains a challenge, sim- functionals in order to capture the nonlocality both in ple model Hamiltonians constitute a convenient frame- space and time of the exact xc functional which depends work to gain insights into the properties of the exact on the entire history of the density, the initial (interact- TDDFT functional. ing) many-body state and the initial KS state [10{14]. In the present work we exploit the possibilities of a We note that almost all TDDFT calculations today use solvable lattice model { the 2-site Hubbard model [22{ an adiabatic approximation for the xc potential, which 24] { to address the impact of non-locality in time in 2 the exchange correlation functional of TDDFT. Specif- oscillations in interacting systems as well as the main ically, we study resonant Rabi oscillations, a prototyp- difficulties of describing Rabi dynamics within TDDFT. ical example of non-linear external field driven dynam- The many-body time-dependent Schr¨odingerequation, ics where the population of states changes dramatically in time. We first derive here the exact ground-state i@tj (t)i = H(t)j (t)i; (2) Hartree-exchange-correlation (Hxc) functional for the 2- describes the evolution of the system from a given initial site model using the Levy-Lieb constrained search[25{27]. state j i. Since the Hamiltonian (1) is independent of This functional, when used in a TDDFT context with 0 spin, the spin structure of the wave function j (t)i is the instantaneous time-dependent density as input, con- fixed by the initial state. In the following we study the stitutes the exact adiabatic approximation which can be evolution from the ground state of the Hubbard dimer used as a reference to quantify the role of memory effects. and therefore it is sufficient to consider only the singlet By carefully studying and quantifying the dynamics pro- sector of our model. duced by TDDFT with the adiabatic Hxc potential we In the absence of an external potential, v = 0, the demonstrate that it fails both quantitatively and qualita- 1;2 stationary singlet eigenstates of the Hamiltonian (1) take tively to describe Rabi oscillations. In the second part of the form this work we apply an analytic density-potential map for lattice systems [10, 28] to derive an explicit, fully non- y y y y y y y y jgi = Ng c^1"c^1# +c ^2"c^2# + β+ c^1"c^2# − c^1#c^2" j0i(3a); adiabatic xc density functional which correctly captures p y y y y all features of Rabi dynamics in the Hubbard dimer. This je1i = 1= 2 c^1"c^1# +c ^2"c^2# j0i; (3b) functional is one of the main results of this paper. je i = N c^y c^y +c ^y c^y + β c^y c^y − c^y c^y j0i; The paper is organized as follows: in Sec. II we in- 2 e2 1" 1# 2" 2# − 1" 2# 1# 2" troduce the physics of the Rabi effect for the Hubbard (3c) dimer, showing how the dipole moment and state occu- pations evolve with time during the course of resonant Rabi oscillations. In Sec. III we address the same prob- Here j0i is the vacuum state, jgi is the ground state, and lem from a TDDFT perspective. In particular we use je1;2i are two excited singlet states. The Ng=e2 = (2 + the exact adiabatic xc functional as a reference to quan- 2 −1=2 2β±) are normalization factors and the coefficients tify memory effects. In the Sec. IV we consider the exact β± are defined as interacting system in a two-level approximation which al- p 2 2 lows us to derive a new approximate Hxc potential as an β± = (U ± 16T + U )=4T: (4) explicit functional of the time-dependent density. The excellent performance of this approximation is demon- The energy eigenvalues corresponding to the eigenstates strated and explained. We end the paper with our con- (3) are clusions in Sec. V. In the Appendix we derive the exact E = 2T β ; (5a) ground state xc potential for the Hubbard dimer using g − the Levy-Lieb constrained search. Ee1 = U; (5b) Ee2 = 2T β+ : (5c) II. RABI OSCILLATIONS FOR TWO-SITE To simplify notations, we rewrite the external potential HUBBARD MODEL part in Eq. (1) in the form X ∆v (v n^ + v n^ ) = (^n −n^ )+C(t)(^n +n ^ ) (6) We consider the dynamics of two electrons on a Hub- 1 1σ 2 2σ 2 1 2 1 2 bard dimer, that is, a two-site interacting Hubbard model σ with on-site repulsion U and hopping parameter T . The P wheren ^i = n^iσ is the operator of the number of Hamiltonian of the system reads σ particles on site i, ∆v = v1 −v2 is the difference of on-site potentials, and C(t) = (v (t)+v (t))=2. The last term in ^ X y y 1 2 H = − T c^1σc^2σ +c ^2σc^1σ + U (^n1"n^1# +n ^2"n^2#) Eq. (6) corresponds to a spatially uniform potential. This σ term can be trivially gauged away and will be ignored X + (v1(t)^n1σ + v2(t)^n2σ) ; (1) in the following without loss of generality.