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Bulletin of the American Physical Society

Quantum electrodynamical time-dependent density functional theory for many-electron systems on a lattice

Mehdi Farzanehpour, I. V. Tokatly

Published Mar 6, 2015• 0 citations
OpenAlex

Abstract

We present a rigorous formulation of the time-dependent density-functional theory for interacting lattice electrons strongly coupled to cavity photons. We start with an example of one particle on a Hubbard dimer coupled to a single photonic mode, which is equivalent to the single mode spin-boson model or the quantum Rabi model. For this system we prove that the electron-photon wave function is a unique functional of the electronic density and the expectation value of the photonic coordinate, provided the initial state and the density satisfy a set of well defined conditions. Then we generalize the formalism to many interacting electrons on a lattice coupled to multiple photonic modes and prove the general mapping theorem. We also show that for a system evolving from the ground state of a lattice Hamiltonian any density with a continuous second time derivative is locally v representable.

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Quantum electrodynamical time-dependent density-functional theory for many-electron systems on a lattice

Mehdi Farzanehpour, I. V. Tokatly

We present a rigorous formulation of the time-dependent density-functional theory for interacting lattice electrons strongly coupled to cavity photons. We start with an example of one particle on a Hubbard dimer coupled to a single photonic mode, which is equivalent to the single mode spin-boson model or the quantum Rabi model. For this system we prove that the electron-photon wave function is a unique functional of the electronic density and the expectation value of the photonic coordinate, provided the initial state and the density satisfy a set of well defined conditions. Then we generalize the formalism to many interacting electrons on a lattice coupled to multiple photonic modes and prove the general mapping theorem. We also show that for a system evolving from the ground state of a lattice Hamiltonian any density with a continuous second time derivative is locally $v$ representable.

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Density-Potential Mapping in the Standard and Quantum Electrodynamical Time-Dependent Density Functional Theory

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This thesis is devoted to the formulation and implications of the time-dependent density functional theory (TDDFT). The work is divided into two main parts. In the first part we develop rigorous theorems for the density-potential mapping in quantum many-body systems on a lattice. We prove the uniqueness of the TDDFT map and demonstrate that a given density is v-representable if the initial many-body state and the density satisfy certain well defined conditions. In particular, we show that for a system evolving from its ground state any density with a continuous second time derivative is v-representable. Then the lattice TDDFT formulation is extended to cover system of interacting lattice electrons strongly coupled to cavity photons.We prove that under some mathematical conditions the electron-photon wave function is a unique functional of the electronic density and the expectation value of the photonic coordinate. Then we further generalize the ground state v representability theorem to include the ground state of a general lattice electron-photon Hamiltonian.

•Jan 1, 2015•0 citations

Time-dependent density functional theory on a lattice

Mehdi Farzanehpour, I. V. Tokatly

Time-dependent density functional theory (TDDFT) for quantum many-body systems on a lattice is formulated rigorously. We prove the uniqueness of the density-to-potential mapping and demonstrate that a given density is $v$ representable if the initial many-body state and the density satisfy certain well-defined conditions. In particular, we show that for a system evolving from its ground state any density with a continuous second time derivative is locally in time $v$ representable and therefore the lattice TDDFT is guaranteed to exist. The TDDFT existence and uniqueness theorem is valid for any connected lattice, independently of its size, geometry, and/or spatial dimensionality. General statements of the existence theorem are illustrated on a pedagogical exactly solvable example, which displays all the details and subtleties of the proof in a transparent form. In conclusion we briefly discuss remaining open problems and directions for future research.

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The last two decades have witnessed increasing experimental interest in the study and control of many-electron systems strongly interacting with quantum electromagnetic fields. This includes notable experiments in the areas of cavity and circuit quantum electrodynamics, quantum computing via photon mediated atom entanglement, electromagnetically induced transparency, quantum plasmonics, quantum simulators and chemistry. The description of realistic coupled matter-photon systems requires combining electronic structure methods from material science with quantum-optical methods. In this work, we propose a formally exact and numerically feasible approach, termed quantum electrodynamical time-dependent density functional theory (QED-TDDFT), that generalizes successful TDDFT for electronic structure calculations, to the electron-photon coupling. We develop a general framework for the description of coupled matter-photon systems in all possible realizations of interest, ranging from the fully relativistic case, to well-known quantum-optical model (e.g., the Rabi model). Of particular interest for condensed-matter applications is the density functionalisation of non-relativistic QED. The underlying theory in fact corresponds to standard non-relativistic quantum mechanics for the electrons, but with additional terms, which correct for relativity and electron-photon interactions. However, any application of density-functional theory requires approximations to the exchange-correlation (xc) functional. In this work we construct such an approximation for the description of electron-photon interaction effects in an optical cavity (the theory of reference is QED-TDDFT for many-electron systems coupled to cavity photons). The derived electron-photon optimized effective potential (OEP) reduces in the static limit to the Lamb shift of the ground state. The OEP is then tested on the Rabi model from weak to strong coupling regimes. The results are promising and open a path for describing complex strongly coupled matter-photon systems. Further, we consider the inclusion of relativistic spin-spin effects into the xc functional of standard spin density functional theory. In this way we provide quantum corrections that are essential for the description of magnetic inhomogeneities at the nanoscale. Such an approach improves over the current (classical) micromagnetic treatment of domain walls and skyrmions for spintronic applications.

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Geometric theory of constrained Schrödinger dynamics with application to time-dependent density-functional theory on a finite lattice

Eric Cancès B, Théo Duez, Jari van Gog +3 more

Time-dependent density-functional theory (TDDFT) is a central tool for studying the dynamical electronic structure of molecules and solids, yet aspects of its mathematical foundations remain insufficiently understood. In this work, we revisit the foundations of TDDFT within a finite-dimensional setting by developing a general geometric framework for Schrödinger dynamics subject to prescribed expectation values of selected observables. We show that multiple natural definitions of such constrained dynamics arise from the underlying geometry of the state manifold. The conventional TDDFT formulation emerges from demanding stationarity of the action functional, while an alternative, purely geometric construction leads to a distinct form of constrained Schrödinger evolution that has not been previously explored. This alternative dynamics may provide a more mathematically robust route to TDDFT and may suggest new strategies for constructing nonadiabatic approximations. Applying the theory to interacting fermions on finite lattices, we derive novel Kohn--Sham schemes in which the density constraint is enforced via an imaginary potential or, equivalently, a nonlocal Hermitian operator. Numerical illustrations for the Hubbard dimer demonstrate the behavior of these new approaches.

Physical Review A•May 6, 2026•0 citations
We present a rigorous formulation of the time-dependent density-functional theory for interacting lattice electrons strongly coupled to cavity photons. We start with an example of one particle on a Hubbard dimer coupled to a single photonic mode, which is equivalent to the single mode spin-boson model or the quantum Rabi model. For this system we prove that the electron-photon wave function is a unique functional of the electronic density and the expectation value of the photonic coordinate, provided the initial state and the density satisfy a set of well defined conditions. Then we generalize the formalism to many interacting electrons on a lattice coupled to multiple photonic modes and prove the general mapping theorem. We also show that for a system evolving from the ground state of a lattice Hamiltonian any density with a continuous second time derivative is locally v representable.
We present a rigorous formulation of the time-dependent density-functional theory for interacting lattice electrons strongly coupled to cavity photons. We start with an example of one particle on a Hubbard dimer coupled to a single photonic mode, which is equivalent to the single mode spin-boson model or the quantum Rabi model. For this system we prove that the electron-photon wave function is a unique functional of the electronic density and the expectation value of the photonic coordinate, provided the initial state and the density satisfy a set of well defined conditions. Then we generalize the formalism to many interacting electrons on a lattice coupled to multiple photonic modes and prove the general mapping theorem. We also show that for a system evolving from the ground state of a lattice Hamiltonian any density with a continuous second time derivative is locally vvv representable.
This thesis is devoted to the formulation and implications of the time-dependent density functional theory (TDDFT). The work is divided into two main parts. In the first part we develop rigorous theorems for the density-potential mapping in quantum many-body systems on a lattice. We prove the uniqueness of the TDDFT map and demonstrate that a given density is v-representable if the initial many-body state and the density satisfy certain well defined conditions. In particular, we show that for a system evolving from its ground state any density with a continuous second time derivative is v-representable. Then the lattice TDDFT formulation is extended to cover system of interacting lattice electrons strongly coupled to cavity photons.We prove that under some mathematical conditions the electron-photon wave function is a unique functional of the electronic density and the expectation value of the photonic coordinate. Then we further generalize the ground state v representability theorem to include the ground state of a general lattice electron-photon Hamiltonian.
Time-dependent density functional theory (TDDFT) for quantum many-body systems on a lattice is formulated rigorously. We prove the uniqueness of the density-to-potential mapping and demonstrate that a given density is vvv representable if the initial many-body state and the density satisfy certain well-defined conditions. In particular, we show that for a system evolving from its ground state any density with a continuous second time derivative is locally in time vvv representable and therefore the lattice TDDFT is guaranteed to exist. The TDDFT existence and uniqueness theorem is valid for any connected lattice, independently of its size, geometry, and/or spatial dimensionality. General statements of the existence theorem are illustrated on a pedagogical exactly solvable example, which displays all the details and subtleties of the proof in a transparent form. In conclusion we briefly discuss remaining open problems and directions for future research.
The last two decades have witnessed increasing experimental interest in the study and control of many-electron systems strongly interacting with quantum electromagnetic fields. This includes notable experiments in the areas of cavity and circuit quantum electrodynamics, quantum computing via photon mediated atom entanglement, electromagnetically induced transparency, quantum plasmonics, quantum simulators and chemistry. The description of realistic coupled matter-photon systems requires combining electronic structure methods from material science with quantum-optical methods. In this work, we propose a formally exact and numerically feasible approach, termed quantum electrodynamical time-dependent density functional theory (QED-TDDFT), that generalizes successful TDDFT for electronic structure calculations, to the electron-photon coupling. We develop a general framework for the description of coupled matter-photon systems in all possible realizations of interest, ranging from the fully relativistic case, to well-known quantum-optical model (e.g., the Rabi model). Of particular interest for condensed-matter applications is the density functionalisation of non-relativistic QED. The underlying theory in fact corresponds to standard non-relativistic quantum mechanics for the electrons, but with additional terms, which correct for relativity and electron-photon interactions. However, any application of density-functional theory requires approximations to the exchange-correlation (xc) functional. In this work we construct such an approximation for the description of electron-photon interaction effects in an optical cavity (the theory of reference is QED-TDDFT for many-electron systems coupled to cavity photons). The derived electron-photon optimized effective potential (OEP) reduces in the static limit to the Lamb shift of the ground state. The OEP is then tested on the Rabi model from weak to strong coupling regimes. The results are promising and open a path for describing complex strongly coupled matter-photon systems. Further, we consider the inclusion of relativistic spin-spin effects into the xc functional of standard spin density functional theory. In this way we provide quantum corrections that are essential for the description of magnetic inhomogeneities at the nanoscale. Such an approach improves over the current (classical) micromagnetic treatment of domain walls and skyrmions for spintronic applications.
In the present thesis, a rigorous mathematical formulation of time-dependent density-functional theory for lattice systems is derived, serving a formally exact approach towards solving time-dependent many-particle Schrödinger problems. After introducing the necessary mathematical foundations, a one-to-one mapping from external potentials onto electronic densities obtained by solving the Schrödinger equation is introduced. Its properties and the related Banach spaces of scalar potentials and electronic densities are discussed in detail. It allows to precisely formulate the Kohn-Sham Iteration scheme, mapping a given many-particle to an effective single-particle Schrödinger problem, both generating the exact same density. Employing the Banach fixed point theorem, the Kohn-Sham Iteration scheme is proven to be convergent depending on the chosen initial density.
Time-dependent density-functional theory (TDDFT) is a central tool for studying the dynamical electronic structure of molecules and solids, yet aspects of its mathematical foundations remain insufficiently understood. In this work, we revisit the foundations of TDDFT within a finite-dimensional setting by developing a general geometric framework for Schrödinger dynamics subject to prescribed expectation values of selected observables. We show that multiple natural definitions of such constrained dynamics arise from the underlying geometry of the state manifold. The conventional TDDFT formulation emerges from demanding stationarity of the action functional, while an alternative, purely geometric construction leads to a distinct form of constrained Schrödinger evolution that has not been previously explored. This alternative dynamics may provide a more mathematically robust route to TDDFT and may suggest new strategies for constructing nonadiabatic approximations. Applying the theory to interacting fermions on finite lattices, we derive novel Kohn--Sham schemes in which the density constraint is enforced via an imaginary potential or, equivalently, a nonlocal Hermitian operator. Numerical illustrations for the Hubbard dimer demonstrate the behavior of these new approaches.