Near-adiabatic parameter changes in correlated systems: Influence of the ramp protocol on the excitation energy

We study the excitation energy for slow changes of the hopping parameter in the Falicov-Kimball model with nonequilibrium dynamical mean-field theory. The excitation energy vanishes algebraically for long ramp times with an exponent that depends on whether the ramp takes place within the metallic phase, within the insulating phase, or across the Mott transition line. For ramps within metallic or insulating phase the exponents are in agreement with a perturbative analysis for small ramps. The perturbative expression quite generally shows that the exponent depends explicitly on the spectrum of the system in the initial state and on the smoothness of the ramp protocol. This explains the qualitatively different behavior of gapless (e.g., metallic) and gapped (e.g., Mott insulating) systems. For gapped systems the asymptotic behavior of the excitation energy depends only on the ramp protocol and its decay becomes faster for smoother ramps. For gapless systems and sufficiently smooth ramps the asymptotics are ramp-independent and depend only on the intrinsic spectrum of the system. However, the intrinsic behavior is unobservable if the ramp is not smooth enough. This is relevant for ramps to small interaction in the fermionic Hubbard model, where the intrinsic cubic fall-off of the excitation energy cannot be observed for a linear ramp due to its kinks at the beginning and the end.


Introduction
In equilibrium thermodynamics, adiabatic processes are defined as quasistatic processes without heat exchange with the environment.The entropy remains constant during an adiabatic process, while it always increases if the process takes place in a finite time and is therefore no longer quasistatic and reversible.These fundamental concepts are closely related to the adiabatic theorem of quantum mechanics [1][2][3] for an isolated system which evolves according to the Schrödinger equation with a time-dependent Hamiltonian H(t), i.e., a system that is subject to external fields or to changes of its parameters, but not coupled to heat or particle reservoirs.The adiabatic theorem states that a system that is initially in the ground state evolves to the new ground state during an infinitesimally slow change of the Hamiltonian, whereas it cannot follow a parameter change that takes place in a finite time, resulting in a nonzero excitation energy.The paradigm for this crossover from adiabatic to nonadiabatic behavior in a quantum system is the exactly solvable Landau-Zener model [4,5], i.e., a two-level system H LZ (t) = vtσ z + γσ x that is driven through an avoided level crossing with finite speed v > 0 (σ z and σ x are Pauli matrices).When the system is in the ground state |φ 0 (−∞) = (1, 0) + at time t = −∞, the probability to find the system in the excited state |φ 1 (∞) = (1, 0) + at time t → ∞ vanishes exponentially when the speed v is small compared to the scale γ 2 / set by the gap γ at the avoided crossing, The above Landau-Zener formula can be generalized to various multilevel cases [6][7][8][9], from which, e.g., the demagnetization probability for the transverse-field Ising model was obtained [10].However, for correlated systems in general the Landau-Zener results cannot be directly applied, because essentially all matrix elements of an interacting many-particle Hamiltonian change in a complicated way upon variation of one of its parameters.The investigation of slow changes of external parameters in correlated systems has recently received considerable attention due to its relevance for experiments with ultracold atomic gases in optical lattices [11], in which quantum-many body systems can be kept under well-controlled conditions.In those systems timedependent control of the parameters is not only of practical importance (as discussed below), but it also allows to test fundamental theoretical predictions.For example, the Landau-Zener result was indeed experimentally confirmed in a Bose-Einstein condensate loaded into an accelerated optical lattice [12].
Various slow parameter changes in many-body systems have recently been studied [13][14][15][16][17][18][19][20][21][22][23][24][25][26][27][28][29][30][31].For a general ramp the system is initially in the ground state and some parameter of the Hamiltonian is then changed to a new value within a time interval τ , either linearly or nonlinearly with time.To investigate the crossover from the extreme nonadiabatic limit τ = 0 (i.e., a sudden quench of the Hamiltonian) to possibly adiabatic behavior in the limit τ → ∞ a measure for the degree of nonadiabaticity is needed.A popular quantity for this purpose is the excitation energy ∆E(τ ) after the ramp, i.e., ∆E(τ ) = E(τ ) − E 0 (τ ), (2) where E 0 (τ ) is the ground-state energy of the Hamiltonian after the ramp.For initial states at non-zero temperature, the entropy increase provides a more natural measure of nonadiabaticity in general.However, entropy is uniquely defined only for thermal equilibrium, and thus it can only be computed after the ramp is complete and the system has thermalized.On the other hand, isolated many-body systems do not necessarily thermalize quickly after changes in the Hamiltonian [32][33][34][35][36][37][38][39][40][41][42][43][44][45], in particular for integrable systems, as demonstrated experimentally with ultracold gases [46].In contrast to the entropy the internal energy is always well-defined, regardless of whether the system passes through a series of thermal or nonthermal states in the limit of a quasistationary process.
In the present work we only consider systems that are initially in the ground state.If the excitation energy ∆E(τ ) vanishes in the limit of long ramp times, τ → ∞, the system is considered to behave adiabatic.It is expected that the excitation energy is still small for finite ramp times τ , just as in the Landau-Zener formula, when the ground state is protected by a gap for all parameters throughout the ramp [24].However, the excitation energy is not exponentially small (∆E(τ ) ∝ exp(const/τ )) in general.As we will show below, the asymptotic decrease of ∆E(τ ) for large τ can depend both on the intrinsic properties of the many-body system and on the ramp protocol.In particular for gapped systems the ramp protocol can be used to make ∆E(τ ) arbitrarily small, but in general it often vanishes only algebraically.This behavior is known from the Landau-Zener model, where the excitation is exponentially small only when the avoided level crossing is traversed from t = −∞ to t = +∞, whereas the excitation probability is proportional to 1/τ 2 and hence much larger if the evolution takes place from t = 0 to t = ∞, i.e., starting exactly at the center of the level crossing [18,47].
The situation is completely different for gapless systems, such as the exactly solvable one-dimensional transverse-field Ising model in which the gap vanishes at exactly one value of the transverse field.When the magnetic field is ramped across this critical point the excitation energy is [14][15][16] with a rational exponent η = 1 2 .Similar results were obtained for a number of other quantum critical systems, such as the Bose-Hubbard model [21] or the random field Ising model [20].However, the existence of a quantum critical point is not a necessary condition to obtain a nonanalytic relation ∆E(τ ) [23,24].Equation (3), with various values of the exponent η, holds for ramps within gapless phases of several gapless systems [24].For a continuous bath of harmonic oscillators, which model the lowenergy excitations of a large class of systems, the exponent η for a slow squeeze of the oscillator mass depends on the spatial dimension [24]: An analytic relation ∆E(τ ) ∼ τ −2 is found for all dimensions d ≥ 3, while η is noninteger for d = 2.For d = 1, the thermodynamic limit does not commute with the limit of large τ , i.e., the prefactor in Eq. (3) increases with system size [24], suggesting that adiabatic behavior is impossible for that class of one-dimensional systems.
The excitation energy during a nonadiabatic ramp, and its dependence on the ramp duration τ is not only a fundamental property of a quantum many-body system, but it is also of practical interest for experiments with cold atomic gases.Various ramping procedures are used in experiment to transform one phase into another, and the available time for the process cannot be too long in order to avoid extrinsic losses.On the other hand, whether theoretical predictions are actually observable in experiment can depend in a subtle way on the unavoidable excitation during the preparation of the state [17,[48][49][50][51][52][53].When the ramp duration is fixed to a given maximum value, it thus becomes important to find the optimal ramp through which a given point in parameter space can be reached through minimal excitation of the system [25].In general, it is plausible that any additional term in the Hamiltonian should be switched on slowly, so as to build up the correlations that it favors without incurring to high energy cost, and increasing the speed at later times.
In view of these issues the question arises to what extent the dependence of the excitation energy on the ramp duration τ is determined by intrinsic properties of the system, and to what extent it is influenced by the details of the ramp.In this paper we give a perturbative argument that holds in the limit of small ramp amplitudes and allows to separate an intrinsic contribution to the excitation energy and a ramp shape dependent contribution.In some cases the latter can mask the intrinsic contribution such that the behavior of the excitation energy in the limit of long ramp times τ is completely determined by the ramp shape.Furthermore, we present results for the excitation of the Falicov-Kimball model after various ramps.In this model, which can be solved exactly using nonequilibrium dynamical mean-field theory (DMFT), Eq. ( 3) is found to hold with an exponent η that is different for ramps across the metal-insulator transition, within the metallic phase, and within the insulating phase.Our numerical results for η in this model support the scenario obtained from the perturbative argument.
The paper is organized as follows.In Sec. 2 we show results for the excitation energy in the Falicov-Kimball model in nonequilibrium dynamical mean-field theory.In Sec. 3 we develop a perturbative argument for small ramp amplitudes and discuss the implications for gapped and gapless systems, such as the metallic and Mott insulating phases of the Falicov-Kimball and the fermionic Hubbard model.A conclusion in Sec. 4 closes the presentation.

Model
Below we present results for the excitation energy ∆E(τ ) for ramps of different types in the Falicov-Kimball model [54], with Hamiltonian Here c ( †) i and f ( †) i are annihilation (creation) operators for the itinerant and immobile electrons, respectively, and are their local densities.Hopping between sites i and j, with amplitude V ij (t) = V (t)t ij , is possible only for the mobile c particles.Note that although the f electrons are immobile, the equilibrium state of H Falicov-Kimball does not correspond to one quenched f configuration but rather to a state with annealed disorder, where each f state contributes according to the free energy of the c particles.
In the context of dynamical mean-field theory (DMFT) [55], which becomes exact in infinite dimensions [56], the Falicov-Kimball model has a long history because it can be mapped onto a solvable single-site problem [57][58][59][60].A Mott metal-insulator transition occurs at a critical interaction U c for half-filling (at density n c = n f = 1  2 ), as well as a transition to a charge-ordered state at sufficiently low temperatures.The physics of the Falicov-Kimball model thus partly resembles that of its parent, the fermionic Hubbard model, with two mobile spin species.DMFT can be applied to nonequilibrium situations [39,45,[61][62][63][64][65][66][67][68], in which case the effective single-site problem for the Falicov-Kimball model is still quadratic and can be solved using equations of motion [62].Here we extend the exact solution of the Falicov-Kimball model for an interaction quench [39] to a numerical solution that can be applied to arbitrary time dependencies in V (t) and U(t) (Appendix A).This allows us to study the excitation after ramps of the hopping or the interaction strength.We employ a set of hopping amplitudes for which the density of states has a semielliptic shape, where ǫ k are the eigenvalues of the hopping matrix t ij and L is the number of lattice sites.Furthermore, we consider only the homogeneous phase at half-filling, for which the chemical potential is fixed at µ = U/2 and the f -orbital energy at E f = 0.In this case the critical interaction for the equilibrium Mott transition is U c = 2V [58].

Linear ramp protocol
We consider linear ramps in the Falicov-Kimball model ( 4) in DMFT for the homogeneous paramagnetic phase at half-filling.We assume that the system is in the ground state for times t < 0. For 0 ≤ t ≤ τ the hopping parameter V is changed according to the ramp protocol where V i is the initial hopping amplitude, τ is the total ramp time, ∆V is the ramp amplitude, and r(x) is the ramp shape.The latter is a monotonously increasing function with r(0) = 0 and r(1) = 1.We set the energy scale by V i ≡ V ≡ 1, so that time is measured in units of 1/V .(From now on we set = 1.)The energy of the system per   7), r(x) = x] within the metallic phase (U = 1, V f = 2), within the insulating phase (U = 3, V f = 0.5), and across the metal-insulator transition (U = 1, V f = 0).The energy scale is set by V i ≡ V = 1.The curves become independent of τ in the quench regime τ 1/V .The solid black lines, with a slope 1/2, 1, and 2 (from top to bottom), correspond to the asymptotic behavior (9).Inset: Internal energy E(t) [Eq.( 8)] during ramps (7) of the hopping amplitude in the Falicov-Kimball model (r(x) = x, U = 1, and V f = 0), using various ramp durations τ .For t < 0 and t > τ , the energy is constant.The solid black line is the internal energy E(t) in the ground state at U = 1 and hopping V (t).
lattice site is given by The excitation after a ramp is then obtained from the difference (2), where E 0 (τ ) is the energy (8) of the ground state of the Hamiltonian after the ramp.The DMFT solution for ramps in V (t) (and also U(t)) is described in Appendix A.
The time evolution of the energy ( 8) is plotted in the inset of Fig. 1 during a ramp (7) with linear profile r(x) = x.For small ramp-durations (τ = 1), the energy rises linear with time.In this case, the system is essentially quenched, i.e., its state |ψ(t) remains unchanged during the ramp, and the energy is thus only determined by the ramp protocol, E(t) ≈ ψ(0)|H(t)|ψ(0) .In the opposite limit τ → ∞, the energy adiabatically follows the ground-state energy E 0 (t) for hopping parameter V (t) (solid line in inset of Fig. 1), in accordance with the adiabatic theorem.
We now focus on the excitation ∆E(τ ) after the ramp, which is plotted in Fig. 1 for linear ramps (7) within the gapless metallic phase (U = 1, V f = 2), within the gapped insulating phase (U = 3, V f = 0.5), and across the metal-insulator transition (U = 1, V f = 0), which occurs in the equilibrium system at U = 1 and V = 0.5.From Fig. 1 one can estimate the crossover timescale τ quench , which separates the regime in which the state of the system cannot follow the parameter change (τ < τ quench ) from the adiabatic regime in which ∆E(τ ) decreases with increasing ramp-duration τ (τ > τ quench ).Independent of the ramp parameters, τ quench turns out to be of the order of few times the inverse bandwidth.The decrease of ∆E(τ ) for τ > τ quench can be fitted with a power law (3) for τ 10.The exponent turns out to be a rational number, which depends only on the phase in which the system is before the ramp (metallic phase for U < 2, insulating phase for U > 2) and after the ramp (metallic phase for U < 2V f , insulating phase for U > 2V f ).These results can be summarized as 2 linear ramp across the transition, τ −1 linear ramp in metallic phase, τ −2 linear ramp in insulating phase.(9) How do these exponents arise and how do they depend on the ramp shape?Further data show that the exponent η = 1 2 for the excitation across the metal-insulator transition is independent of the ramp shape r(x).At present we have now simple explanation of this exponent.It would be interesting to determine how this exponent is related to the critical behavior of equilibrium correlation functions, such as the density of states at the transition [59].On the other hand, the behavior for ramps within either the metallic or the insulating phase will be explained in the next section by a perturbative argument, which applies to small ramps of arbitrary shape in any quantum system.In particular we will see that the exponent η = 1 is a consequence of the non-Fermi-liquid behavior of the metallic phase in the Falicov-Kimball model, while the exponent η = 2 in the insulating phase is not an intrinsic property of the Falicov-Kimball model but is in fact due to the linear ramp shape.

Small ramps of arbitrary shape without traversing phase boundaries
Our numerical results for ramps of the hopping amplitude in the Falicov-Kimball model show that the exponent η in Eq. ( 9) does not depend on the precise values of the ramp parameters V i and V f , but only on the thermodynamic phase of the initial and final state.This finding suggests to study the excitation energy perturbatively in the limit of small ramp amplitudes, but for arbitrary ramp shapes and ramp durations.In the remainder of this section we will derive the excitation energy ∆E(τ ) up to second order in the ramp amplitude for an arbitrary Hamiltonian.In particular we discuss the asymptotic behavior of ∆E(τ ) in the limit τ → ∞ and how it may be influenced by the ramp shape, and illustrate these general results with data for the specific case of the Falicov-Kimball model.

Perturbative result for the excitation energy
We consider the general Hamiltonian where H 0 is the Hamiltonian before the ramp, W is the operator that is switched on, and κ(t) is the ramp function.As in Eq. ( 7), we characterize κ(t) by the ramp amplitude ∆κ, the ramp duration τ , and the ramp shape r(x), i.e., κ(t) = ∆κ r(t/τ ).In order to expand ∆E(τ ) for fixed ramp duration τ and ramp shape r(x) in powers of ∆κ, we decompose the quantum state |ψ(t) of the system in the instantaneous eigenbasis |φ n (t) of the Hamiltonian (10), which satisfies the condition at any instance of time.We assume the |φ n (t) to be nondegenerate for simplicity.
After fixing the phase of the eigenvectors in a convenient way we obtain the eigenstate decomposition of |ψ(t) as so that the Schrödinger equation implies using the notation ǫ nm (t) = ǫ n (t) − ǫ m (t).The matrix element on the right-hand side of Eq. ( 13) is given by where the first equality follows from Eq. ( 11) and the last from the explicit form of the Hamiltonian [Eq.(10)].
Because the system is assumed to be in the ground state |φ 0 (0) of H 0 for t ≤ 0, Eq. ( 13) must be solved with the initial condition a m (0) = δ m0 .Together Eq. ( 14) this implies that a n (t) = O(∆κ) for n = 0.In order to obtain the leading term in the expansion of a n (t) (for n = 0) we can thus restrict the sum in Eq. ( 13) to the single term m = 0, This expression was used before as a starting point for the discussion of ramps across a quantum critical point [14].Here we study ramps which do not cross a phase boundary, and we assume that the instantaneous eigenenergies ǫ n0 (t) and eigenfunctions |φ n (t) , which depend on time t only through the parameter κ, can be expanded around κ = 0.
Since a n (t) is already of order O(∆κ), ǫ n0 (t) and |φ n (t) in Eq. ( 15) can be replaced by ǫ n0 ≡ ǫ n0 (0) and |φ n ≡ |φ n (0) , respectively.The excitation energy, ∆E(τ ) = , is then given by ∆E(τ ) = ∆κ 2 E(τ ) + O(∆κ 3 ) ( 16) Eqs. ( 16)-( 19) constitute the main result of this section.The correlation function R(ω), which can be interpreted as the spectral density of possible excitations induced by the operator W , is independent of the ramp shape r(x) and the ramp duration τ .Conversely, the ramp spectrum F (x) does not depend on the Hamiltonian but only on details of the ramp.For continuous ramp shapes r(x) it follows that F (x) → 0 for |x| → ±∞, such that F (ωτ ) becomes increasingly peaked around ω = 0 in the limit τ → ∞.In fact, making the replacement F (ωτ ) ∝ δ(ω)/τ is equivalent to Fermi's Golden Rule for |a n (t)| 2 , and the nonadiabatic excitation ( 17) is due to deviations of F (ωτ ) from δ(ω).The crossover scale τ quench that was discussed in Sec. 2 is thus given by the value of τ below which F (ωτ ) = F (0) + O(ω 2 τ 2 ) is approximately constant over the entire bandwidth Ω of R(ω), i.e., τ quench ≈ 1/Ω.In Sec.3.5 we confirm this estimate numerically for small interaction ramps in the metallic phase of the Falicov-Kimball model.
In the following we will analyze the asymptotic behavior of Eq. ( 17) in the adiabatic limit, τ → ∞.For this we need the behavior of the ramp spectrum F (x) at large values of x, which follows from Eq. (19) as where the exponent is given by α = 2n if the nth derivative of r(x) is discontinuous (i.e., the (n − 1)st derivative has a kink), but all lower derivatives are continuous; this behavior follows from the Riemann-Lebesgue lemma [69].For example, in case of a linear r(x) = x the first derivative r ′ (x) = Θ(x)Θ(1−x) is discontinuous at x = 0 and x = 1, so that the ramp spectrum F (x) decays like x −2 [cf.Eq. (29b) below].In general, when the ramp shape has a finite number of kinks, the large-x asymptotics of the ramp spectrum ( 19) is given by a finite sum of oscillating terms, By choosing a smooth ramp one can always increase the exponent α or even make F (x) decay exponentially for x → ∞.However, in practice ramp protocols often have kinks that lead to a power-law decay (20).
To estimate the magnitude of the integral (17) in the limit τ → ∞ we distinguish two cases, namely (i) the gapless case, in which R(ω) vanishes like a power law at ω = 0, and (ii), the case of a gapped excitation spectrum, in which R(ω) has a finite gap Ω gap above ω = 0.In both cases we assume that R(ω) is zero beyond some highfrequency scale Ω, although the argument remains valid if R(ω) vanishes exponentially for ω > Ω.Since ] we can also assume that any singularities of R(ω) are integrable.

Case (i): Gapless excitation spectrum
In this paragraph we discuss the case in which the excitation spectrum R(ω) is gapless and vanishes like a power law at ω = 0, First we assume α > ν, where α is the exponent that characterizes the ramp shape [Eq.( 20)].Writing R(ω) = ω ν R(ω), the integral (17) becomes, after a change of variables, Using the asympotic behavior (20) we find that the integral in this expression remains finite in the limit τ → ∞, so that in this case the leading contribution to the excitation energy is given by with It is important to note that the exponent does not depend on the ramp shape, but only on the density of possible excitations above ω = 0.Because the latter is an intrinsic property of the system we will refer to E intr (τ ) as the intrinsic contribution to the excitation energy in the following.In principle, a ramp between two parameter values can always be made so smooth that E intr (τ ) becomes the dominating contribution to the excitation energy (i.e., α > ν), as in Eq. (23).However, as we will see in the following paragraph, if the ramp is not smooth enough (i.e., if α ≤ ν), the intrinsic contribution will be masked by a nonuniversal contribution that is essentially determined by the ramp shape.
For the case of α ≤ ν we estimate the integral (17) as follows.For the moment we assume that the spectral density R(ω) has no singularities at finite frequencies and use the bounds with positive constants C 1 , C 2 , Ω 1 , Ω 2 .Together with Eqs. ( 20) and ( 17) we obtain for the excitation energy for τ → ∞, with positive constants C ′ 1 and C ′ 2 .The upper bound holds because f (x) = O(1) in Eq. (20).To obtain the lower bound it is sufficient to note that although f (x) can have infinitely many zeros, the moving average f (x) = x+h x dxf (x) over any small finite interval of given length h is larger than some positive constant.This property is satisfied in particular when the ramp shape has a finite number of kinks, as discussed below Eq. ( 20).Finally we note that Eqs. ( 25) and ( 26) hold also if R(ω) has integrable singularities, because a small frequency interval around each of them contributes to the integral (17) in the same way as the gapped spectrum [Sec.3.3], namely ∝ τ −α [Eq.(27)].
The result that is stated in Eqs. ( 25) and ( 26) has a simple interpretation: Kinks in the ramp shape increase the probability of excitations to high energy states, as expressed by the slowly decaying tail of the ramp spectrum F (x).When the ramp is not smooth enough the integral ( 17) is therefore dominated by the high-frequency part of R(ω), leading to a nonuniversal, ramp-shape dependent excitation energy.For the often-considered linear ramp (α = 2) any intrinsic contribution E intr (τ ) with ν ≥ 2 will therefore be unobservable in E(τ ).This is precisely what happens for weak-coupling interaction ramps in the Hubbard model, as discussed below in Sec.3.5.

Case (ii): Gapped excitation spectrum
We now turn to the case of an excitation density which has a gap Ω gap at ω = 0.The integral ( 17) then starts at the finite lower bound Ω gap , such that F (x) can be replaced by its asymptotic behavior (20) in the entire integration range.As a consequence we have The integral gives a finite constant in the limit τ → ∞ provided that R(ω) is not singular.Otherwise the integral may give a τ -dependent but bounded contribution, as shown in the next subsection for ramps in the insulating phase of the Falicov-Kimball model.The gapped case [Eq.(27)] is thus similar to the gapless case with α < ν [Eq.( 25)].In both cases the excitation energy is dominated the high-frequency behavior of F (x) and is therefore completely determined by the ramp shape, while the intrinsic contribution ( 23) is unobservable.Our analysis so far can be summarized as follows.The excitation energy after a ramp may be either dominated by the intrinsic contribution (23) or set by rampshape dependent terms [Eqs.(25), (26), and ( 27)], depending on the large-frequency asymptotics (20) of the ramp spectrum and the small-frequency behavior (21) of the excitation density.This fact will be illustrated in the following two subsections for ramps in the insulating and metallic phase of the Falicov-Kimball model and the Hubbard model.

Insulating phase
In previous subsection we have shown that the excitation energy after a ramp within a gapped phase behaves in a nonuniversal way because the intrinsic contribution ( 23 vanishes.A significant dependence of the nonadiabatic excitation energy on the ramp shape is therefore expected also for ramps with finite amplitude.In the following we will demonstrate this fact for ramps within the insulating phase of the Falicov-Kimball model, where it turns out that the asymptotic behavior for τ → ∞ is indeed correctly described by the analytic expression ( 27) that was obtained for small ramps.
For this purpose we focus on three particular ramp shapes, Here r n (x) is chosen in such a way that its nth derivative is discontinuous at x = 0 and x = 1 (Fig. 2a), i.e., r ′ n (x) ∝ sin n (πx) for 0 < x < 1.The corresponding ramp spectra [Eq.(19)] are These functions vanish like F n (x) ∼ x −2n for x → ∞ (Fig. 2b).We now perform ramps of the hopping amplitude as the energy scale.We consider only the paramagnetic insulating phase at half-filling, i.e., U > 2 = 2V i and U > 2V f .The excitation energy after such ramps is plotted as a function of the ramp duration τ in Fig. 3.The curves can be fitted with power laws (3) for large τ , with an exponent η = 2 for the linear ramp (28a) and η = 4 for the ramp (28b), respectively.For the ramp (28c) the results are consistent with an exponent η = 6, but the excitation energy is too small for a power-law fit in the accessible range.Hence the large-τ behavior of the nonadiabatic excitation in the case of ramps with finite amplitude turns out to be the same as in the limit of small ramps, i.e., a power law with an exponent that is determined by the singularities of the derivatives of the ramp shape [cf.Eq. ( 27)] rather than by intrinsic properties of the system.
To check Eq. ( 17) explicitly for arbitrary ramps we would have to compute the density of excitations R(ω), for which no general solution is available.Nevertheless one can derive an expression in the atomic limit and compare the resulting excitation energy to ramps deep in the insulating phase.For ramps of the hopping V (t), the operator W in Eq. ( 18) is given by the kinetic energy operator.In the case of half-filling for both mobile and immobile particles there is exactly one particle per site in the ground state for V = 0. Therefore each hopping process creates exactly one doubly-occupied site, and the function R(ω) consists of a single delta peak at i.e., the Fresnel oscillations in ramp spectra such as Eqs.(29a) to (29a) become visible in the dependence of the excitation energy on τ .This result should only be slightly modified for ramps deep in the insulating phase (U ≫ V ), assuming that the delta-peak is then only slightly broadened and shifted in position.In fact, as seen in Fig. 4 the tail oscillations of F 1 (x) [Eq.(29a), Fig. 2] are also apparent in the excitation energy for ramps between states with U ≫ V .In the limit τ → ∞ these oscillations are washed out because R(ω) has a finite bandwidth ∆Ω for V > 0, such that the integral (17) averages over many oscillations for τ ≫ 1/∆Ω.

Metallic phase
As an application of Eq. ( 17) to ramps in a gapless phase we consider the turn-on of the interaction in the Falicov-Kimball model and the Hubbard model.For the following discussion it is convenient to write the Hamiltonian in momentum space Furthermore we change the notation with respect to Eq. ( 4) to allow for a unified description of the Hubbard model, where both spin species are mobile (ǫ k↑ = ǫ k↓ , µ ↑ = µ ↓ ) and the Falicov-Kimball model, where we take spin ↑ to be immobile ( We consider ramps at half filling (µ σ = U/2) in which the interaction is changed from zero to a finite value, U(t) = ∆Ur(t/τ ).For a ramp of the interaction strength in the Hubbard model and the Falicov-Kimball model, the operator W in Eq. ( 10) is given by the double occupation (32).As shown in Appendix B, the excitation density R(ω) at U = 0 can be expressed in terms of the second-order contribution to the self-energy q↓ (ω + ǫ q↓ + i0).
For comparison to our DMFT results we evaluate Eq. ( 33) in the limit of infinite dimensions [56] where the self-energy is independent of momentum q, [70] and the q-summation can be replaced by an integral over the density of states ρ ↓ (ǫ), For the Hubbard model, the second-order self-energy is given by [70] In accordance with Fermi liquid theory the imaginary part of the self-energy vanishes ∝ ω 2 , thus leading to well-defined quasiparticle excitations in the metallic phase of the Hubbard model.On the other hand, the imaginary part of the mobile electron self-energy in the Falicov-Kimball model remains finite at ω = 0 due to the scattering off fixed impurities.Its value can be obtained easily from the exact solution of the Falicov-Kimball model in DMFT [60], Here n f is the average density of localized particles, such that n f = 0.5 in case of halffilling.Eqs. ( 35) and ( 36) can then be inserted in Eq. (34), which in turn determined the intrinsic component (23) of the excitation energy, Hubbard: R(ω) Falicov-Kimball: R(ω) As discussed above, the intrinsic contribution can be masked completely by a rampshape dependent contribution if the ramp is not smooth enough, i.e., when the exponent in Eq. ( 20) satisfies α ≤ 1 or α ≤ 3 in case of the Falicov-Kimball and Hubbard model, respectively.However, the discussion below Eq. (20) shows that F (x) decays at least ∝ x −2 if the ramp is continuous, i.e., if it does not contain any abrupt finite changes.Hence we conclude that the intrinsic component (38) is always dominant for ramps that turn on a small interaction in the Falicov-Kimball model.This is consistent with our numerical results for ramps in the metallic phase of the Falicov-Kimball model which are not shown here, namely, that the 1/τ behavior for the metallic phase in (9) does not only hold for linear ramps (Fig. 1), but for all three ramps (28a) to (28c).
The situation is very different for ramps in the Hubbard model.Because the intrinsic contribution (37) vanishes ∝ τ −3 for τ → ∞ it is negligible with respect to the high-frequency contribution (25) for linear ramps, where the ramp spectrum decays as x −2 [Eq.(29a)].This is consistent with results of Möckel and Kehrein [31], who computed the excitation energy after a linear ramp of the interaction in the Hubbard model using Keldysh perturbation theory and found ∆E(τ ) ∼ τ −2 for τ → ∞.
we define by where c 1 is a normalization constant to satisfy the condition and c 2 is chosen such that r ′ (x) is sufficiently small at the boundary x = 0 and x = 1, i.e., the expression for F (x) holds up to terms which are exponentially small in c 2 .For U 1 numerical results for the excitation energy (scaled with the ramp amplitude ∆U 2 ) agree very well with the analytical expression (17), evaluated using Eqs.( 34), (36), and (40).This corroborates the validity argument of Sec. 3 and shows that it provides correct estimates for the nonadiabatic excitation energy after ramps which are not too large in amplitude.
Inset (b) of Fig. 5 illustrates the origin of the crossover from small to large ramp times τ .For fast ramps, e.g., τ = 1, the ramp spectrum F (ωτ ) averages over the entire bandwidth Ω of the excitation spectrum R(ω).As a consequence, the excitation energy becomes independent for ramp times smaller than the quench time scale τ 1/Ω = τ quench .Indeed we see in the numerical data that the weak-coupling quench time scale fits with the estimate 1/Ω ≈ 1/4.For larger quench times, e.g., τ 10 in inset (b) of Fig. 5, the ramp spectrum F (ωτ ) probes only the linear small-ω behavior of R(ω), leading to the universal power-law behavior ∆E(τ ) ∼ 1/τ for τ 10.

Conclusion
We presented a general perturbative analysis of the excitation energy due to slow ramps of a parameter in a quantum system without crossing of phase boundaries, motivated by our numerical results for the Falicov-Kimball model obtained with nonequilibrium dynamical mean-field theory.We demonstrated that the excitation energy vanishes algebraically for large ramp duration τ [Eq.( 3)] under rather general circumstances.The exponent η can depend on one hand on the spectrum of the correlation function of the operator that is switched on, and on the other hand on the differentiability of the ramp function.Which of these influences dominates in η depends on the low-energy behavior of the excitation spectrum compared to the spectrum of the ramp protocol.In practice, any experimental ramp protocol can always be considered as differentiable on a short enough timescale.Our conditions on the degree of differentiability have to be interpreted in the sense that a ramp protocol must be considered as not differentiable if the slope or any higher derivative changes on a timescale shorter than the inverse bandwidth of the system.
For ramps in gapped systems the asymptotic behavior of the excitation energy depends only on the ramp protocol and can be made as small as desired by use of increasingly smooth ramp shapes.By contrast, for ramps in gapless systems the lowenergy excitation spectrum has no effect on η if the ramp is not smooth enough.Only if the ramp is sufficiently smooth does η become ramp-independent and reflects the low-energy excitation spectrum of the system.For the fermionic Hubbard model this implies that a linear ramp from U = 0 to a small value of U leads to an unnecessarily large excitation energy with η = 2, which can be reduced to the intrinsic exponent η = 3 if the ramp shape has at least two continuous derivatives.
Our results also indicate that in the Falicov-Kimball model the exact expression for the excitation energy in the limit of small ramps provides a good estimate up to quite large ramp amplitudes.This suggests to use the perturbative expression, which is valid for arbitrary systems, as a guide for finding ramp protocols that connect fixed parameters of the Hamiltonian and minimize the excitation energy for a given ramp time, thereby improving the preparation of states in experiments with ultracold atomic gases.To calculate R, the expectation value (B.1) is factorized using Wick's theorem and transformed to bosonic Matsubara frequencies.It turns out that the only nonvanishing contractions for η n = 0 is given by 3) where g 0 qσ (iω m ) = 1/(iω m − ǫ qσ ) is the noninteracting Green function at momentum q, and iω m are fermionic Matsubara frequencies.The expression has a simple diagrammatic representation (Fig. A1a).The diagram is split into one Green function line g 0 q↓ (iω m ) and the remainder, which we identify as the second-order contribution Σ One can now transform the Matsubara summation into a frequency integral, where it must be taken into account that the self-energy Σ(z) has a branch cut along the real axis with Σ(ω ± i0) ≡ ∓Im Σ(ω) (Fig. A1b).The result is R(iη n ) = q f (ǫ q↑ )Σ q↓ (ω + ǫ q↓ ).(B.6) Taking the limit of zero initial temperature then yields Eq. ( 33).

Figure 1 .
Figure 1.Excitation energy(2) after linear ramps of the hopping parameter [Eq.(7), r(x) = x] within the metallic phase (U = 1, V f = 2), within the insulating phase (U = 3, V f = 0.5), and across the metal-insulator transition (U = 1, V f = 0).The energy scale is set by V i ≡ V = 1.The curves become independent of τ in the quench regime τ 1/V .The solid black lines, with a slope 1/2, 1, and 2 (from top to bottom), correspond to the asymptotic behavior(9).Inset: Internal energy E(t) [Eq.(8)] during ramps(7) of the hopping amplitude in the Falicov-Kimball model (r(x) = x, U = 1, and V f = 0), using various ramp durations τ .For t < 0 and t > τ , the energy is constant.The solid black line is the internal energy E(t) in the ground state at U = 1 and hopping V (t).

Figure A1 .
Figure A1.Left panel: Diagrammatic representation of Eq. (B.3).Lines represent the noninteracting momentum-resolved Green function g 0 qσ (iω m ) = 1/(iω m − ǫ qσ ) for σ =↑ (solid lines) and σ =↓ (dashed lines).Momentum is conserved at the vertices, frequency iη n enters at the left vertex.Right panel: Transformation of the Matsubara sum (B.5) to the real frequency interval (B.6), using the usual expression iωm w(iω m ) = (iβ/2π) C1 dzf (z)w(z), where f (z) is the Fermi function and w(z) is some analytic integrand.The integrand in Eq. (B.5) has a branch cut at z = −iη n due to the branch cut of Σ(z) along the real axis, and a pole at ǫ q↑ .Then the contour C 1 is transformed into C 2 , which yields (B.5), using that the Fermi function is periodic under shift with bosonic Matsubara frequencies, f (ω − iη n ) = f (ω).