Abstract
In the framework of quantum thermodynamics, we propose a method to quantitatively describe thermodynamic quantities for outofequilibrium interacting manybody systems. The method is articulated in various approximation protocols which allow to achieve increasing levels of accuracy, it is relatively simple to implement even for medium and large number of interactive particles, and uses tools and concepts from density functional theory. We test the method on the driven Hubbard dimer at half filling, and compare exact and approximate results. We show that the proposed method reproduces the average quantum work to high accuracy: for a very large region of parameter space (which cuts across all dynamical regimes) estimates are within 10% of the exact results.
Introduction
The last decade has seen an increasing interest in the study of outofequilibrium thermodynamics at the micro and nanoscale. Such interest is impelled by the development of quantum technology and experimental control methods at small scales. In this scenario energy fluctuations play an important role, thermodynamic quantities as work and entropy production are defined by their mean values, and the laws of thermodynamics still hold on average. The thermodynamic description of quantum manybody systems is significant for understanding the limits of the emerging quantum technology^{1,2,3,4,5,6}. Fluctuation theorems^{7,8,9,10,11,12,13,14} have been key tools to describe the unavoidable fluctuations in the nonequilibrium dynamics and related experiments have been performed for small, noninteracting systems^{15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30}, in both the classical and quantum domain. When we turn to the case of systems composed of interacting particles (‘manybody systems’), complex behavior arises as a consequence of the interactions. In general, dealing with a quantum manybody system entails a tremendous effort and it is usually necessary to employ approximations to tackle the problem. Recently, some efforts have been directed to study the outofequilibrium thermodynamics in many body systems like quantum harmonic oscillators’ chains and spin chains^{31,32,33,34,35,36,37,38,39,40}; however a method which can tackle a general system, and systems of increasing complexity is still lacking. In particular, a Ramseylike interferometry experimental protocol has been proposed in refs 41 and 42 to explore the statistics of energy fluctuations and work in quantum systems. This protocol has been so far implemented for single spin systems^{20, 22} while its application to a quantum manybody system implies, in fact, considerable experimental (and theoretical) difficulties. This protocol requires not only experimental control of each individual part of the system and of the interactions between its components, but also the ability to construct conditional quantum operations where a single ancilla should be used as a control qubit for the collective, coherent behavior of the whole quantum manybody system. This is experimentally extremely challenging and even the theoretical simulation of the dynamics generated by the circuit would become prohibitive as the number of particles in the interacting system increases.
In this paper, we propose a new method to accurately describe some thermodynamics quantities (such as the mean work) in outofequilibrium quantum systems which can be applied to small, medium and large interacting manybody systems. In addition this method could be used to make the Ramseylike interferometry experimental protocol accessible when consider manybody interacting systems. Our method is inspired by densityfunctional theory (DFT)^{43,44,45,46,47} and will use some of the tools developed within it. DFT is one of the most efficient nonperturbative methods for studying the properties of interacting quantum systems^{48}, and has produced tools widely used to describe diverse properties of manybody correlated systems, such as band insulators, metals, semiconductors, nanostructures, etc. refs 46 and 47. At the core of DFT there is the mapping between the interacting manybody system onto a noninteracting one, the KohnSham (KS) system^{49}, which is characterised by the same ground state density of the interacting system. In principle, from the density of such noninteracting system it is then possible to calculate exactly the groundstate (and even excited) properties of the interacting system, but in practice some approximations are required to obtain the so called exchangecorrelation potential, a crucial quantity in DFT. Despite, being a very popular and successful approach to quantum manybody physics, to our knowledge, DFT or related methods have not been applied, so far, to quantum thermodynamics. Here we will consider tools from a particular flavour of DFT, latticeDFT^{50}, but our method could be straightforwardly extended to different DFT flavours. LatticeDFT applies to model Hamiltonians such as the Hubbard or Heisenberg Hamiltonians, which are particularly important for quantum information processing and that describe well the type of spin systems of interest to the quantum thermodynamic community. As a testbed example we will study the outofequilibrium dynamics of the Hubbard dimer, which provides a variety of interesting regimes and behaviours to benchmark our method, since it can be solved exactly. We will compare exact and approximate results and show that our method provides very accurate estimates in most regimes of interest opening attractive further possibilities for applications.
The Hubbard Model
The Hubbard model was conceived in 1963 separately by Gutzwiller^{51}, Kanamori^{52}, and Hubbard^{53} to describe the interaction of electrons in solids. This model gives a microscopic understanding of the transition between the Mottinsulator and conductor systems^{54}, and allows for tunnelling of particles between adjacent sites of a lattice and interactions between particles occupying the same site. Here we consider the timedependent onedimensional Hubbard Hamiltonian^{55} described by
where ħ = 1 (atomic units), L is the number of sites, \({\hat{c}}_{i,\sigma }^{\dagger }\) (\({\hat{c}}_{i,\sigma }\)) are creation (annihilation) operators for fermions of spin σ, \({\hat{n}}_{i,\sigma }={\hat{c}}_{i,\sigma }^{\dagger }{\hat{c}}_{i,\sigma }\) is the site i number operator for the σ spin component, \({\hat{n}}_{i}={\hat{n}}_{i,\uparrow }+{\hat{n}}_{i,\downarrow }\), J is the hopping parameter, Δ_{ i }(t) is the timedependent amplitude of the spinindependent external potential at site i, and U is the onsite particleparticle interaction parameter. Analytical or numerically exact solutions for static manybody problems are rare. For the Hubbard model, there exist an exact solution to the homogeneous onedimensional case, however numerically exact solutions to the nonuniform case becomes quickly problematic as the number of particles and sites increases. Solutions to timedependent manybody problems are even more difficult to achieve, and the Hubbard model is no exception. However quantum work due to an external driving of a manybody dynamics is one of those problems in which including time dependence and the effects of manybody interactions is essential to capture – even qualitatively – the system behaviour. In the following, we propose a relatively simple approach to this problem which allows to include the key features stemming from the manybody interactions and dynamics, and yet maintaining the simplicity of simulating a noninteracting dynamics. We will show that, for a wide range of parameters, this simple approach allows to reproduce quantitatively the exact results.
DFTinspired estimate of quantum thermodynamic quantities
The KohnSham system
The KS system is a fictitious, noninteracting, quantum system defined as having the same ground state particle density as the original interacting physical system^{49}. The two systems have then the same number of particles. For each physical system, and a given manybody interaction, the KS system is uniquely defined and, in the limit in which the physical system is non interacting, the HohenbergKohn theorem^{48} ensures that the KS and the physical system coincide. Exceptions are generally pathological cases: for LatticeDFT, see^{56}, where the extension to latticeDFT of the onetoone correspondence between external potential and groundstate particle density is demonstrated and related exceptions are discussed.
Given an interacting system of Hamiltonian \(\hat{H}=\hat{K}+\hat{V}+{\hat{V}}_{ee}\), where \(\hat{K}\) is the kinetic energy operator, \(\hat{V}\) is the onebody external potential, and \({\hat{V}}_{ee}\) is the electron–electron repulsion, the corresponding KS system is described by the noninteracting Hamiltonian \({\hat{H}}_{KS}=\hat{K}+{\hat{V}}_{s}\), where \({\hat{V}}_{s}\) is the onebody potential \({\hat{V}}_{s}=\hat{V}+{\hat{V}}_{H}+{\hat{V}}_{xc}\). Here \({\hat{V}}_{H}\) is the Hartree potential corresponding to the classical electrostatic interaction and \({\hat{V}}_{xc}\) is the exchangecorrelation potential, the functional derivative of the exchangecorrelation energy E _{ xc }, which contains additional contributions from the manybody interactions as well as the manybody contributions to the kinetic energy. Usually, the exchangecorrelation potential is defined as a sum of exchange and correlation contributions, \({V}_{xc}={V}_{x}+{V}_{c}\), and, due to its unknown functional dependence on the groundstate particle density, approximations have to be used for calculating it.
Work in a quantum system
In a nonequilibrium driven quantum system work is defined as the mean value of a work probability distribution \(\langle W\rangle =\int WP(W)dW\) that takes into account energylevel transitions (system histories) that can occur in a quantum dynamics^{57}. In this scenario an external agent is performing (extracting) work on (from) a quantum system and the concept of work takes into account both the intrinsic nondeterministic nature of quantum mechanics and the effects of nonequilibrium fluctuations. The work probability distribution P(W) contains all the information about the possible transitions in the Hamiltonian energy spectrum produced by an external potential (field) that drives the system outofequilibrium between the initial time t = 0 and the final time t = τ of a protocol. This distribution is defined as \(P(W)={\sum }_{n,m}\,{p}_{n}{p}_{mn}\delta (W{\rm{\Delta }}{\varepsilon }_{m,n})\), where p _{ n } is the probability to find the system in the eigenstate n〉 (with eigenenergy \({\varepsilon }_{n}^{0}\)) of the initial Hamiltonian \(\hat{H}(t=\mathrm{0)}\), p _{ mn } is the transition probability to evolve the system to the eigenstate m〉 (with eigenenergy \({\varepsilon }_{m}^{\tau }\)) of the final Hamiltonian \(\hat{H}(t=\tau )\) given the initial state n〉, and \({\varepsilon }_{m,n}={\varepsilon }_{m}^{\tau }{\varepsilon }_{n}^{0}\) is the energy difference associated with the transition probability p _{ mn }. We will consider the work done on a system which starts in the thermal equilibrium state \({\hat{\rho }}_{0}=\exp [\beta \hat{H}(t=\mathrm{0)}]/{Z}_{0}\), where \(\beta =\mathrm{1/}{K}_{B}T\) is the inverse temperature, K _{ B } is the Boltzmann constant, T is the absolute temperature, and with \({Z}_{t}={\rm{Tr}}{e}^{\beta \hat{H}(t)}\) being the partition function for the instantaneous Hamiltonian \(\hat{H}(t)\) at time t. Then the system evolves up to time τ according to some driven protocol described by the time evolution operator generated by \(\hat{H}(t)\) at the constant inverse temperature β. The final state of the system will not be, in general, an equilibrium state. This describes the nonequilibrium dynamics that we are interested in and it is the typical scenario explored in quantum thermodynamic protocols where fluctuations theorems can be applied.
DFTinspired methods
We consider the Hubbard system Eq. (1) and propose methods to accurately, quantitatively estimate the average quantum work 〈W〉 produced by its driven dynamics. The key idea will be to use the KS Hamiltonian \({\hat{H}}_{KS}\) as a zerothorder Hamiltonian in a perturbation expansion scheme which converges to the exact manybody Schrödinger equation^{58,59,60}. Using this expansion at its zerothorder, gives a simple method of including interactions within a formally noninteracting scheme through the DFT exchangecorrelation (\({\hat{V}}_{XC}\)) and Hartree (\({\hat{V}}_{H}\)) terms. For the static case a similar scheme has been proven very effective to largely improve results over a wide range of parameters for the description of entanglement with respect to standard perturbation schemes^{60}.
The formulation of DFT for the Hubbard model that we employ is the siteoccupation functional theory (SOFT)^{61, 62}, where the traditional density in real space \(n({\bf{r}})=\langle {\sum }_{\sigma }\,{\hat{\psi }}_{\sigma }^{\dagger }({\bf{r}}){\hat{\psi }}_{\sigma }({\bf{r}})\rangle \) is replaced by the site occupation \({n}_{i}=\langle {\sum }_{\sigma }\,{\hat{c}}_{i,\sigma }^{\dagger }{\hat{c}}_{i,\sigma }\rangle \). Therefore, using the SOFT we can write the KS Hamiltonian for the Hubbard model as
where
and the Hartree potential is \({V}_{H,i}=U{n}_{i}\mathrm{/2}\). The timedependent potential Δ_{ i }(t) defines the driven protocol and the exchangecorrelation potential V _{ xc,i } reduces, for the Hubbard model, to the correlation potential \({V}_{c,i}=\delta {E}_{c}/\delta {n}_{i}\) ^{50}. However, as shown in the Methods section, not all the approximations for the exchangecorrelation potential for the Hubbard model respect this property. Using the KS Hamiltonian Eq. (2), we write the full manybody Hamiltonian as \(\hat{H}={\hat{H}}_{KS}+{\rm{\Delta }}\hat{H}\), where the perturbative term is then defined by
For obtaining the mean value 〈W〉, we suggest three related protocols (as illustrated in Fig. 1).
‘Zeroorder’ approximation protocol
We write the interacting Hamiltonian as \(\hat{H}={\hat{H}}_{0}+{\rm{\Delta }}\hat{H}\) where \({\hat{H}}_{0}\) is a (formally) noninteracting Hamiltonian and \({\rm{\Delta }}\hat{H}\equiv \hat{H}{\hat{H}}_{0}\). We will consider the case in which \({\hat{H}}_{0}={\hat{H}}_{KS}\) and compare it with the case in which \({\hat{H}}_{0}\) corresponds to the standard noninteracting approximation to \(\hat{H}\), see Results section. Then we approximate the initial thermal state of the system as the noninteracting one, \({\hat{\rho }}_{0}=\exp \,[\beta {\hat{H}}_{0}(t=\mathrm{0)}]/{Z}_{0}\), and the timedependent evolution is calculated according to the (formally) noninteracting \({\hat{H}}_{0}\mathrm{(}t\mathrm{)}\) up to the final time τ. The quantum work will then be estimated based on this time evolution and on the spectra of the initial and final zeroorder Hamiltonians. We note that in this protocol the time dependency in \({\hat{H}}_{0}\mathrm{(}t\mathrm{)}\) is included only through the (explicitly) timedependent term appearing in \(\hat{H}\). For \({\hat{H}}_{0}={\hat{H}}_{KS}\), we expect this method to reduce the magnitude of the perturbation \({\rm{\Delta }}\hat{H}\) as manybody interactions are already partially accounted in the zeroorder term \({\hat{H}}_{KS}\), see Eqs (2), (3) and (4). We note that indeed the zero order term \({\hat{H}}_{KS}\) of this approximation already reproduces a very important property of the manybody system, namely the groundstate site occupation distribution. We therefore expect this to produce more accurate results with respect to the standard perturbation expansion.
Protocol with firstorder correction to eigenenergies
Here we propose to use the same formally noninteracting dynamics of the previous protocol, but now to include the first order correction in the estimate of the eigenenergies associated to the initial and final Hamiltonians of the system, with the corresponding correction to the approximation of the initial thermal state. As we will show, a better estimate of the eigenenergies may be important to preserve agreement with exact results for certain regimes, and especially so when the zeroorder and exact Hamiltonians present qualitatively different eigenstate degeneracies, as in the case study below. While, in this paper, we will consider firstorder corrections only, it should be possible to further improve accuracy by including higher order corrections to the eigenenergies.
Protocol including timeevolution effects of manybody interactions
This third protocol applies to the case in which \({\hat{H}}_{0}={\hat{H}}_{KS}\). Here we include an implicit timedependency in V _{ H } and V _{ xc }. These quantities are functionals of the site occupation, so time dependence can be included via the time dependence of n _{ i }. This implies a timedependence which is local in time, i.e. that has no memory and could then not describe accurately nonMarkovian processes. Nevertheless, it allows to mimic, at least in part, the variation with time of the interaction effects due to the particles dynamics. As we will show, this significantly improves results in certain regimes. The timedependent site occupation n _{ i }(t) will be obtained by solving the system selfconsistently. This protocol takes inspiration from the adiabatic LDA approximation for timedependent DFT^{63,64,65}. It may be further enhanced by improving the approximation for the eigenenergies of the initial and final time Hamiltonians, as described in the previous subsection.
We will consider two approximations for the KS exchangecorrelation potential (see Results and Methods sections). We will compare the results from these protocols to the exact results and to the results obtained for sameorder standard perturbation theory.
Results
We will focus on a Hubbard dimer with two electrons of opposite spin (half filling), which is known to display a rich physical behaviour^{66,67,68}, including an analogue to the Mott metalinsulator transition^{67, 68}. Because of its nontrivial dynamics, this model is ideal as a test bed for assessing the accuracy of approximations in reproducing quantities related to quantum fluctuations and quantum thermodynamics. When the system is reduced to a dimer with halffilling, the Eq. (1) can be written in the subspace basis set \(\{\uparrow \downarrow ,\,0\rangle ,\,\uparrow ,\,\downarrow \rangle ,\,\downarrow ,\,\uparrow \rangle ,\,\mathrm{0,}\,\uparrow \downarrow \rangle \}\) as
We will calculate the average work along the dynamics driven by the linear timedependent onsite potentials \({{\rm{\Delta }}}_{1}={{\rm{\Delta }}}_{2}={{\rm{\Delta }}}_{0}({{\rm{\Delta }}}_{0}{{\rm{\Delta }}}_{\tau })t/\tau \), with the parameterized initial and final values Δ_{0} = 0.5J, Δ_{ τ } = 5J, at the parameterized temperature T = J/(0.4K _{ B }). The combined values of U and τ will then determine the dynamical regime (sudden quench to intermediate to adiabatic, see discussion of Exact results). Due to the small Hilbert space associated to (5), the quantum dynamics generated by it can be exactly solved by directly integrating the timedependent Schrödinger equation. The Hamiltonian (5) can describe various physical systems, including, but not limited to, two gatedefined quantum dots^{69,70,71}, cold atoms^{67}, etc.
A schematic description of how we will apply to the Hubbard dimer the protocols proposed is provided in Fig. 1.
Exact results
Average work
In the panel (a) of Fig. 2 we display the exact work that can be extracted from an interacting Hubbard dimer at halffilling when operated according to the dynamics described by the Hamiltonian (5). By varying the interaction U and the time length of the dynamics τ, we can access very different regimes: from noninteracting to very strongly interacting; from sudden quench, to intermediate and to adiabatic dynamics.
Crossover between nonadiabatic to adiabatic dynamics
This system has three characteristic energy scales, U, 1/τ, and J. For the parameter region for which \(U\mathop{ > }\limits_{ \tilde {}}J\), the crossover between nonadiabatic to adiabatic dynamics depends mostly on the interplay between U and 1/τ, and occurs when the two energy scales become comparable, that is for \(U\propto 1/\tau \), see behaviour of dashedred curve in Fig. 2(a). For each given U, when \(\tau \gg 1/U\) the work does not depend on the time length of the dynamics, showing that, for that particular U the work has converged to its adiabatic value. For \(\tau \mathop{ < }\limits_{ \tilde {}}1/U\) and a given interparticle interaction, the work instead strongly depends on τ, increasing with τ up to the maximum allowed for that particular interparticle interaction. For \(U\mathop{ < }\limits_{ \tilde {}}J\), the J energy scale starts to influence the crossover between nonadiabatic to adiabatic regime, so that the simple relation between U and τ described above breaks down in Fig. 2(a).
The contour curves for equal average work are strikingly different between the adiabatic and nonadiabatic regime: in the U, τ plane, they can be well approximated by U = constant for the adiabatic regime, while they rapidly and almost linearly increase with τ for nonadiabatic dynamics. The behaviour of these contour curves mirrors the fact that in the nonadiabatic regime the final state of the system, and so the work, is strongly influenced by the details of the dynamics, and hence strongly dependent on the timescale on which the timedependent driven protocol occurs; however, by definition, in an adiabatic (energylevel transitionless) process the system remains at all times with the same energylevel occupation of the instantaneous Hamiltonian as in the initial thermal state, which means that the final state of the system –and hence the work – is completely defined, independently from the time taken by the system for going from the initial to the final state. We note that, as the Hamiltonian changes due to the driven protocol, the final state in the adiabatic regime is not an equilibrium state at the inverse temperature β.
Transition to ‘Mott insulator’
The very strongly interacting parameter region \(U\mathop{ > }\limits_{ \tilde {}}5J\) corresponds to the twoparticle equivalent of the Mottinsulator^{67}, where double site occupation becomes energetically very costly. Hence for the Hubbard dimer, in this regime both double and zero site occupation are highly suppressed. As the Hilbert space available to the system dynamics reduces across the transition, we observe a corresponding decreasing of the average work that can be extracted from the system.
Entropy production and irreversibility
The entropy production \(\langle {\rm{\Sigma }}\rangle \) is defined in terms of the dissipated work in the outofequilibrium dynamics, \(\langle {\rm{\Sigma }}\rangle =(\beta \langle W\rangle {\rm{\Delta }}F)\), where \({\rm{\Delta }}F=(\mathrm{1/}\beta )\,\mathrm{ln}\,({Z}_{\tau }/{Z}_{0})\) is the free energy variation in the protocol. It is related with an uncompensated heat, that is the energy that will be dissipated to the environment in order for the outofequilibrium system to get back to the thermal equilibrium. In this sense \(\langle {\rm{\Sigma }}\rangle \) quantifies the degree of irreversibility of the dynamical process at hand^{22, 72}. It is then instructive to look at it as the system undergoes different dynamic regimes. In general the degree of irreversibility will be related to the size of the Hilbert space available to the dynamics. For the system at hand both entering the adiabatic regime and the Mott metalinsulator transition contribute to the reduction of the available Hilbert space, and hence to the decrease of entropy production. This phenomenon shows well in the data plotted in the panel (b) of Fig. 2, where we observe a combined decrease in the entropy production as τ increases and the adiabatic regime is entered and as U increases and the system tends to ‘freeze’ towards the n _{1} = n _{2} = 1 Mottinsulator configuration.
Results from ‘zeroorder’ approximations (standard noninteracting and KohnSham based)
In this section we compare results from the protocol which uses a zeroorder approximation, where the dynamics is propagated according to the Hamiltonian \({\hat{H}}_{0}(t)\) and time dependency is included only through the actual driving term Δ_{ i }(t). As the zeroorder Hamiltonian, \({\hat{H}}_{0}(t)\), is always formally the sum of singleparticle Hamiltonians, the dynamics it generates will then correspond to the dynamics of noninteracting systems. Being formally noninteracting, this dynamics is easier to calculate numerically and would be easier to simulate and measure experimentally (in a quantum simulator) than the one originated by the manybody Hamiltonian \(\hat{H}(t)\). We underline once more that the formally noninteracting systems in \({\hat{H}}_{0}(t)\) represent physical systems (noninteracting particles) in the case of standard zeroorder perturbation \({\hat{H}}_{0}(t)={\hat{H}}_{NI}(t)\equiv J{\sum }_{i=\mathrm{1,}L;\sigma =\uparrow ,\downarrow }\) \(({\hat{c}}_{i,\sigma }^{\dagger }{\hat{c}}_{i+\mathrm{1,}\sigma }+{h}{.c}\mathrm{.)}+{\sum }_{i=\mathrm{1,}L}\,{{\rm{\Delta }}}_{i}(t){\hat{n}}_{i}\), while represent fictitious systems (‘KohnSham particles’) when \({\hat{H}}_{0}={\hat{H}}_{KS}\).
In Fig. 3 we present the contour plots of the relative error, with respect to the exact average extractable work for all ‘zeroorder’ approximations. In the panel (a) we show the results for standard perturbation theory, where the dynamics is propagated according to the noninteracting part of the manybody Hamiltonian, H _{ NI }(t). In the panels (b) and (c) of Fig. 3 we present our results for the case where the Hamiltonian is approximated to zeroorder by \({\hat{H}}_{KS}(t)\). We approximate the exchangecorrelation potential in two ways: using the pseudo local density approximation (PLDA) as proposed by Gunnarsson and Schönhammer^{61} (Fig. 3, panel (b)) and using the recent parametrization to the exact correlation energy E _{ c } proposed by Carrascal et al.^{68} (PAR) (Fig. 3, panel (c)). The two approximations PLDA and PAR are described in the ‘Methods’ section. It is known that the PLDA is not a particularly good approximation for the Hubbard model^{50}, but we chose it as we wish to show that even this provides already a good improvement over standard zeroorder perturbation theory (compare panel (a) and (b) of Fig. 3). This is true especially for small and intermediate values of τ: for example, for \(\tau \approx 0\) (sudden quench), the use of PLDA instead of standard zeroorder perturbation doubles the Uinterval for which the relative error in the average work is below 10%, and makes it almost four times larger for τ~2/J (nonadiabatic/adiabatic crossover region).
The parametrization to E _{ c } by Carrascal et al.^{68} reproduces the exact correlation potential V _{ c } for the Hubbard dimer quite accurately; as such, Fig. 3, panel (c), is close to the best results we can expect for this system from the approximation with \({\hat{H}}_{0}={\hat{H}}_{KS}\) we are proposing. We see that now the average work is reproduced to high accuracy in most of the (U, τ) parameter region, even in parameter regions with strong many body interactions and/or corresponding to a dynamics very far from adiabaticity.
For very small τ (the sudden quench dynamics) we can reproduce the extractable work within 10% accuracy for interaction strengths larger than U = 4J. This is an enormous improvement over results obtained from the noninteracting zeroorder dynamics (Fig. 3, panel (a)), where, for the same values of τ, the exact extractable work could be reproduced within 10% error only for \(U\mathop{ < }\limits_{ \tilde {}}0.5J\). In the nonadiabatic/adiabatic crossover region, \(\tau \approx \mathrm{2/}J\), we reproduce very well the exact average extractable work up to interactions \(U\approx 6J\), while standard perturbation theory does poorly, accounting for interactions only up to \(U\approx 1J\) for \(\tau \approx 2\) and up to \(U\approx 3J\) for \(\tau \approx 3\). In this respect we wish to remark that even the ‘lighter patch’ occurring in the panel (c) in Fig. 3 within the region \(3\mathop{ < }\limits_{ \tilde {}}U\mathop{ < }\limits_{ \tilde {}}5\), \(1.5\mathop{ < }\limits_{ \tilde {}}\tau \mathop{ < }\limits_{ \tilde {}}3.5\) still corresponds to very good accuracy, with a maximum relative error of 12% for the τ = 2 cut and of 14% for the U = 4 cut. Finally, in the adiabatic regime, results from Carrascal’s parametrization still outperforms substantially standard perturbation, almost doubling the 10% relative error accuracy region, which now extends to interaction strengths of \(U\approx 4.5J\), against the limit of \(U\approx 2.5J\) for standard perturbation.
We note that, at least for the Hubbard dimer, a ‘zeroorder’type of approximation will always start to deteriorate as the system enters the Mott metalinsulatortype transition, and that this is independent of how well manybody interactions are accounted for in \({\hat{H}}_{0}\). In fact \({\hat{H}}_{0}\), and hence \({\hat{H}}_{KS}\), by definition, does not include a manybody interaction term formally written as \(U={\sum }_{i=\mathrm{1,}L}\,{\hat{n}}_{i,\uparrow }{\hat{n}}_{i,\downarrow }\): this leads to a spectrum where the singlet and triplet eigenstates, \(\frac{1}{2}[\uparrow ,\,\downarrow \rangle \downarrow ,\,\uparrow \rangle ]\) and \(\frac{1}{2}[\uparrow ,\downarrow \rangle +\downarrow ,\uparrow \rangle ]\), are always nondegenerate. However, in the Mottinsulatortype regime described by the actual manybody system of Hamiltonian \(\hat{H}\), only the two aforementioned states remain energetically accessible and, most importantly, they become degenerate. Why the first feature may be mimicked (e.g. this is done by the exact \({\hat{H}}_{KS}\) to reproduce the exact ground state siteoccupation profile) the intrinsic qualitative difference in degeneracy between the interacting and the formally noninteracting spectra determines the failure of any ‘zeroorder’type of approximation in the Mottinsulatortype region, which is what we observe in Fig. 3.
We note that the large improvement provided by the ‘zeroorder’, KSbased approximations comes at no additional computational cost with respect to standard perturbation, as in both cases we are propagating formally noninteracting Hamiltonians.
Adding first order perturbation corrections to the initial and final energy spectra
At the end of the previous section we have discussed how, in regimes where the extent of extractable work is dominated by the spectrum and details of the system dynamics become less relevant, the KSbased ‘zeroorder’ approximation protocol may be seriously limited, and especially so if there exist different degeneracy patterns between the exact and the ‘zeroorder’ approximation spectra. For the Hubbard dimer this happens in the Mottinsulatortype parameter region. In this section we explore if a potential solution to this issue could be to lift this degeneracy by applying higher order perturbation to the initial (t = 0) and final (t = τ) spectra, while performing the system dynamics according to the ‘zeroorder’ Hamiltonian. In this paper, we have consider first order perturbation (FOP) corrections, and results are provided in Fig. 4. FOP corrections to the energy spectra can be considered accurate only for relatively low interactions \(U\mathop{ < }\limits_{ \tilde {}}1J\). For these values of U we see indeed either an improvement in accuracy or, where results were already within the 10% of the exact ones, this accuracy is maintained.
For larger values of the interaction, we can give a qualitative explanation of the influence of modifying the energy spectra. Let us first consider the adiabatic regime (\(\tau \gg \mathrm{1/}U\)): here results for the average work are dominated by the accuracy of the spectrum as the system – in a perfectly adiabatic case – would remain at any t in a thermal state characterized by the same occupation probabilities determined at time t = 0. So for \(2J\mathop{ < }\limits_{ \tilde {}}U\mathop{ < }\limits_{ \tilde {}}5J\), as the spectra provided by the FOP are increasingly quantitatively worsening, we see that this correction reduces the accuracy of the results. However, for \(U\mathop{ > }\limits_{ \tilde {}}5J\) the system undergoes the analogue to the Mott metalinsulator transition, which, as discussed in the previous section, cannot be properly accounted for by ‘zeroorder’ protocols because there is a qualitatively different degeneracy between the zeroorder and the exact spectra. In this parameter region then the protocol using the FOP spectrum, quantitatively inaccurate but qualitatively correct, provides a substantial improvement over the ‘zeroorder’ protocol, as can be seen in Fig. 4. In particular, FOP corrections are very important at very large particleparticle interaction strength U, \(U\approx 10J\): here as long as U is accounted for in the eigenenergy splitting, the system freezes in the ground state and the FOP approximation is then enough to reproduce the exact result W = 0 shown in Fig. 2. In the same parameter region, without the FOP correction, the standard noninteracting zeroorder approximation, completely independent from U, would predict maximum average work (W ≥ 3.3J for \(\tau \mathop{ > }\limits_{ \tilde {}}\mathrm{2.5/}J\)), while the KSbased zeroorder protocols provide some improvement over this result but still predict a waytoohigh average work (W ≥ 2.1J).
For \(U\mathop{ > }\limits_{ \tilde {}}1J\), and for nonadiabatic and transition regime (\(\tau \mathop{ < }\limits_{ \tilde {}}\mathrm{1/}U\)), both spectrum and dynamics contribute to the average work. Here results from the FOP corrections seem to depend on how well the ‘zeroorder’ protocol was already reproducing the exact dynamics. In particular, for the KSbased zero order protocol which uses the accurate parametrization of the exact E _{ c } (panel (c) of Fig. 4), the contribution of the quantitatively incorrect spectra from the FOP worsen the results.
Including implicit timedependency in manybody interaction terms (without and with FOP)
So far we have considered zeroorder Hamiltonians \({\hat{H}}_{0}\) where particleparticle interactions were included at most through timeindependent functionals of the initial siteoccupation. However a more accurate representation of the driven system evolution should be expected by including timedependent functionals. In this subsection we take inspiration from the adiabatic LDA and propose to include a timedependence in these functionals by considering the same functional forms as for the static DFT but calculated at every time using the instantaneous siteoccupation. The timedependence considered is then local in time. To implement this protocol numerically, a selfconsistent cycle to obtain the timedependent siteoccupation n _{ i }(t), and from there the \({V}_{H,i}[{n}_{i}(t)]\) and \({V}_{xc,i}[{n}_{i}(t)]\) functionals, is necessary.
We illustrate this by applying the protocol to the PLDA exchangecorrelation functional and focusing on the nonadiabatic and crossover regimes, \(0\le \tau \le \mathrm{4/}J\). We use as starting point the exact density at the initial time, i.e., \({n}_{i}^{\mathrm{(0)}}(t)={n}_{i}^{(exact)}\mathrm{(0)}\). From this density we obtain the exchangecorrelation energy \({E}_{xc,i}^{\mathrm{(1)}}(t)={E}_{xc,i}^{\mathrm{(1)}}[{n}_{i}^{\mathrm{(0)}}(t)]\) and therefore the KohnSham Hamiltonian \({\hat{H}}_{KS}^{\mathrm{(1)}}(t)={\hat{H}}_{KS}^{\mathrm{(1)}}[{n}_{i}^{\mathrm{(0)}}(t),t]\). We evolve the system using this Hamiltonian and we obtain the state of the system \({\hat{\rho }}^{\mathrm{(1)}}(t)\). From this state we can calculate the next iteration for the siteoccupation \({n}_{i}^{\mathrm{(1)}}(t)={\rm{Tr}}[{\hat{\rho }}^{\mathrm{(1)}}(t){\hat{n}}_{i}]\). Using this, we restart the same cycle calculating the \({E}_{xc,i}^{\mathrm{(2)}}(t)\) and consequently a new KohnSham Hamiltonian \({\hat{H}}_{KS}^{\mathrm{(2)}}(t)\). This cycle is repeated until the convergence criteria \({\sum }_{0 < t < \tau }{n}_{i}^{(k\mathrm{1)}}(t){n}_{i}^{(k)}(t)/N={10}^{6}\) is satisfied, where the time [0, τ] is discretized in N different values of t.
Results are shown in Fig. 5, panel (a), to be compared with the panel (b) of Fig. 3 for \(0\le \tau \le \mathrm{4/}J\). As the system exits the suddenquench regime (\(\tau \mathop{ > }\limits_{ \tilde {}}\mathrm{1/}J\)), and the siteoccupation starts to respond to the dynamics, we notice a marked improvement over using timeindependent functionals. For τ > 1.5/J we now achieve an accuracy of at least 10% up to interaction strength \(5J\le U\le 6J\), while in Fig. 3 it was only up to \(U\le 4J\). The wavypattern in the contour lines for \(\tau > \mathrm{1.5/}J\) reflects the system charge transfer dynamics between the two sites: the PLDA functional is unable to reproduce correctly the Mottinsulator transition so some charge transfer dynamics persists at large values of U.
When including first order corrections to the initial and final energy spectra (Fig. 5, panel (b)), we recover, and for analogous reasons, a behaviour similar to what observed in Fig. 4.
Conclusion
We have proposed a new method which uses some tools and concepts from density functional theory to study the nonequilibrium thermodynamics of driven quantum manybody systems, and illustrated it by the calculation of the average extractable work in a driven protocol. The method has the advantage of considering appropriate formally noninteracting systems (KohnSham systems) to approximate the system dynamics, circumventing the theoretical and experimental problems of dealing with actual manybody interactions. It is easily scalable to large systems and can be used at different levels of sophistication, with increasing accuracy.
We have tested it on the Hubbard dimer, a twospin system with a rich dynamics which includes the precursor to a quantum phase transition (Mott metalinsulator transition), and which can be embodied by various physical systems, including coupled quantum dots and cold atom lattices. Our results show that the proposed method reproduces the average extractable work to high accuracy for a very large region of parameter space: for all dynamical regimes (from sudden quench, to the nonadiabatic to adiabatic crossover region, to the adiabatic regime) and up to quite strong particleparticle interactions (\(U\mathop{ < }\limits_{ \tilde {}}6J\)) our results are within 10% of the exact results.
These very encouraging results, together with the simplicity of the method make for a breakthrough in the calculation of nonequilibrium thermodynamic quantities, as the quantum work, in a complex manybody system. Future developments include the possibility to combine the method with quantum simulation techniques and an experimental implementation of quantum simulators based on this method.
Methods
Approximations for the exchangecorrelation energy
The exchangecorrelation energy is a functional of the site occupation density, but its functional form is unknown and needs approximations. In this work we will consider and compare results from two different types of approximations to the exchangecorrelation energy for the Hubbard Model.
The first is the pseudoLDA expression^{61}
Here the homogeneous reference system for the LDA is the threedimensional electron gas, and so exchange is nonzero (for a related discussion see ref. 50).
In the second E _{ xc } = E _{ c }: this is the accurate parametrization to the exact correlation energy recently proposed in refs 68 and 73, and given by
where u = U/2J, \(\delta ={n}_{1}{n}_{2}\mathrm{/2}\), \({f}_{k}(\delta ,u)={g}_{k}(\delta ,u)+u{h}_{k}(\delta ,u)\), \({t}_{s}(\delta )=\sqrt{1{\delta }^{2}}\), and \({e}_{HX}(\delta ,u)=\frac{u(1+{\delta }^{2})}{2}\).
The function g _{ k }(δ, u) can be obtained iteratively from the equation
and using as starting point
Here the coefficients a _{1}(δ, u) and a _{2}(δ, u) are given by
where \({a}_{21}(\delta )=\frac{1}{2}\sqrt{\frac{(1\delta )\delta }{2}}\), \({a}_{11}(\delta )={a}_{21}(\delta )(1+\frac{1}{\delta })\), \({a}_{12}(\delta )=\frac{1\delta }{2}\), and \({a}_{22}(\delta )=\frac{{a}_{12}(\delta )}{2}\).
The functions h _{ k }(δ, u), \(d{g}_{k}(\delta ,u)\), and \(d{h}_{k}(\delta ,u)\) are defined as
In our calculations we used g _{1}(δ, u) to obtain the exchange correlation energy: this already provides good accuracy as shown in ref. 68.
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Acknowledgements
We thank D. J. Carrascal for valuable discussions and K. Capelle for carefully reading our manuscript and useful suggestions. We acknowledge financial support from the University of York, UFABC, CNPq, FAPESP (project no.2014/027781) and from the Royal Society through the Newton Advanced Fellowship scheme (Grant no. NA140436). I.D’. acknowledges support from CNPq (Grant: PVE—Processo: 401414/20140). This research was performed as part of the Brazilian National Institute of Science and Technology for Quantum Information (INCTIQ).
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I.D’. and R.M.S. conceived the idea, M.H. performed the numerical simulations and contributed to refine the model. All authors discussed the results and contributed to the writing of the manuscript.
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Herrera, M., Serra, R.M. & D’Amico, I. DFTinspired methods for quantum thermodynamics. Sci Rep 7, 4655 (2017). https://doi.org/10.1038/s4159801704478y
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