Reduced purities as measures of decoherence in many-electron systems

Franco, Ignacio; Appel, Heiko

doi:10.1063/1.4819819

Cited by 8 publications

(23 citation statements)

References 46 publications

(80 reference statements)

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“…The correlation entropy is zero for pure states, and has its maximum for maximally mixed states [38,39,71]. In figure 12 we see that the correlation entropy increases with increasing correlation while the gradient of the correlation entropy functional obeys the intra-system steepening and transitions into the inter-system derivative discontinuity due to the degeneracy of the ground-state for j = p 2 .…”

Section: Exact Correlation Entropy Functionalmentioning

confidence: 93%

Exact maps in density functional theory for lattice models

et al. 2016

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In the present work, we employ exact diagonalization for model systems on a real-space lattice to explicitly construct the exact density-to-potential and graphically illustrate the complete exact density-to-wavefunction map that underly the Hohenberg-Kohn theorem in density functional theory. Having the explicit wavefunction-to-density map at hand, we are able to construct arbitrary observables as functionals of the ground-state density. We analyze the density-to-potential map as the distance between the fragments of a system increases and the correlation in the system grows. We observe a feature that gradually develops in the density-to-potential map as well as in the density-towavefunction map. This feature is inherited by arbitrary expectation values as functional of the ground-state density. We explicitly show the excited-state energies, the excited-state densities, and the correlation entropy as functionals of the ground-state density. All of them show this exact feature that sharpens as the coupling of the fragments decreases and the correlation grows. We denominate this feature as intra-system steepening and discuss how it relates to the well-known inter-system derivative discontinuity. The inter-system derivative discontinuity is an exact concept for coupled subsystems with degenerate ground state. However, the coupling between subsystems as in charge transfer processes can lift the degeneracy. An important conclusion is that for such systems with a neardegenerate ground state, the corresponding cut along the particle number N of the exact density functionals is differentiable with a well-defined gradient near integer particle number. IntroductionOver the last decades ground-state density-functional theory (DFT) has become a mature tool in material science and quantum chemistry [1][2][3][4][5]. Provided that the exact exchange-correlation (xc) functional is known, DFT is a formally exact framework of the quantum many-body problem. In practice, the accuracy of observables in DFT highly depends on the choice of the approximate xc-functional. From the local density approximation (LDA) [6], to the gradient expansions such as the generalized gradient approximations (GGAs), e.g. Perdew-Burke-Enzerhof [7] and the hybrid functionals such as B3LYP [8], to the orbital-functionals such as optimized effective potentials [9] and to the range-separated hybrids such as HSE06 [10], the last decades have seen great efforts and achievements in the development of functionals with more accurate and reliable prediction capability.Nonetheless, available approximate functionals such as the LDA, the GGA's and the hybrid functionals have known shortcomings to model gaps of semiconductors [11], molecular dissociation curves [12], barriers of chemical reactions [13], polarizabilities of molecular chains [14, 15], and charge-transfer excitation energies, particularly between open-shell molecules [16].Practical applications of density functional theory encounter two major problems: (i) while the Hohenberg-Kohn theorem tells us that a...

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Section: Exact Correlation Entropy Functionalmentioning

confidence: 93%

Exact maps in density functional theory for lattice models

et al. 2016

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“…The maximum reduced purity is achieved for a pure electronic state that can be described as a single Slater determinant in some basis. In this case, P 1 = N and P 2 = N (N +1)/2 [23] (when contrasting with the result in [23] note that a superposition of single Slater determinant in which all the determinants differ by one particle transitions, s nm = 1, must also be a single Slater determinant [30,31]). Thus, the maximum value for the distilled purity is given bỹ…”

Section: B Limiting Valuesmentioning

confidence: 73%

“…where the partial trace is over the nuclear degrees of freedom [23]. In the {|E j } eigenbasis, the off-diagonal elements of the electron density matrix are determined by the overlap | χ k (t)|χ j (t) | between the nuclear wavepackets associated with different electronic states.…”

Section: Definitions Of Coherencementioning

confidence: 99%

“…(ii) A chain in the ground vibronic state subject to a laser pulse that is resonant with a specific electronic transition. Case (i) is simple to interpret using reduced purities because the population of the Slater determinants remains approximately constant, and thus the dynamics of the reduced purities reflect the dynamics of the S-coherences [23].…”

Section: Numerical Examplesmentioning

confidence: 99%

“…In analogy with the purity [Eq. (1)], one can define a reduced purity [23] P r (t) = Tr[ (r)Γ2 ] = i (r) λ 2 i ,…”

Section: Introductionmentioning

confidence: 99%

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Quantifying fermionic decoherence in many-body systems

Kar

Franco

2017

The Journal of Chemical Physics

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Practical measures of electronic decoherence, called distilled purities, that are applicable to many-body systems are introduced. While usual measures of electronic decoherence such as the purity employ the full N-particle density matrix which is generally unavailable, the distilled purities are based on the r-body reduced density matrices (r-RDMs) which are more accessible quantities. The r-body distilled purities are derivative quantities of the previously introduced r-body reduced purities [I. Franco and H. Appel, J. Chem. Phys. 139, 094109 (2013)] that measure the non-idempotency of the r-RDMs. Specifically, the distilled purities exploit the structure of the reduced purities to extract coherences between Slater determinants with integer occupations defined by a given single-particle basis that compose an electronic state. In this way, the distilled purities offer a practical platform to quantify coherences in a given basis that can be used to analyze the quantum dynamics of many-electron systems. Exact expressions for the one-body and two-body distilled purities are presented and the utility of the approach is exemplified via an analysis of the dynamics of oligo-acetylene as described by the Su-Schrieffer-Heeger Hamiltonian. Last, the advantages and limitations of the purity, reduced purity, and distilled purity as measures of electronic coherence are discussed.

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Entanglement in the Born–Oppenheimer Approximation

Izmaylov

Franco

2016

J. Chem. Theory Comput.

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The role of electron-nuclear entanglement on the validity of the Born-Oppenheimer (BO) approximation is investigated. Although nonadiabatic couplings generally lead to entanglement and to a failure of the BO approximation, surprisingly the degree of electron-nuclear entanglement is found to be uncorrelated with the degree of validity of the BO approximation. This is because while the degree of entanglement of BO states is determined by their deviation from the corresponding states in the crude BO approximation, the accuracy of the BO approximation is dictated, instead, by the deviation of the BO states from the exact electron-nuclear states. In fact, in the context of a minimal avoided crossing model, extreme cases are identified where an adequate BO state is seen to be maximally entangled, and where the BO approximation fails but the associated BO state remains approximately unentangled. Further, the BO states are found to not preserve the entanglement properties of the exact electron-nuclear eigenstates, and to be completely unentangled only in the limit in which the BO approximation becomes exact.

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Reduced purities as measures of decoherence in many-electron systems

Cited by 8 publications

References 46 publications

Exact maps in density functional theory for lattice models

Exact maps in density functional theory for lattice models

Quantifying fermionic decoherence in many-body systems

Entanglement in the Born–Oppenheimer Approximation

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