Single-particle Density Matrix Research Articles

Hybrid programming-model strategies for GPU offloading of electronic structure calculation kernels.

To address the challenge of performance portability and facilitate the implementation of electronic structure solvers, we developed the basic matrix library (BML) and Parallel, Rapid O(N), and Graph-based Recursive Electronic Structure Solver (PROGRESS) library. The BML implements linear algebra operations necessary for electronic structure kernels using a unified user interface for various matrix formats (dense and sparse) and architectures (CPUs and GPUs). Focusing on density functional theory and tight-binding models, PROGRESS implements several solvers for computing the single-particle density matrix and relies on BML. In this paper, we describe the general strategies used for these implementations on various computer architectures, using OpenMP target functionalities on GPUs, in conjunction with third-party libraries to handle performance critical numerical kernels. We demonstrate the portability of this approach and its performance in benchmark problems.

Non-Markovian master equation for quantum transport of fermionic carriers.

We propose a simple, yet feasible, model for quantum transport of fermionic carriers across tight-binding chain connecting two reservoirs maintained at arbitrary temperatures and chemical potentials. The model allows for elementary derivation of the master equation for the reduced single particle density matrix in a closed form in both Markov and Born approximations. In the Markov approximation the master equation is solved analytically, whereas in the Born approximation the problem is reduced to an algebraic equation for the single particle density matrix in the Redfield form. The non-Markovian equation is shown to lead to resonant transport similar to Landauer's conductance. It is shown that in the deep non-Markovian regime the transport current can be matched with that obtained by the non-equilibrium Green's function method.

Lindbladian route towards thermalization of a Luttinger liquid

We study the nonequilibrium dynamics of a Luttinger liquid after a simultaneous quantum quench of the interaction and dissipative quench to the environment within the realm of the Lindblad equation. When the couplings to the environment satisfy detailed balance, the system is destined to thermalize, which we follow using bosonization. The thermodynamic entropy of the system, which also encodes information about the entanglement with the environment, either exhibits a maximum and minimum or grows monotonically before reaching its thermal value in a universal fashion. The single-particle density matrix reveals interesting features in the spatiotemporal ``phase diagram,'' including thermal, prethermal, and dissipation enhanced sudden quench Luttinger liquid behavior, as well as dissipation, enhanced Fermi liquid response.

Quantum Manifestation of the Classical Bifurcation in the Driven Dissipative Bose-Hubbard Dimer.

We analyze the classical and quantum dynamics of the driven dissipative Bose-Hubbard dimer. Under variation of the driving frequency, the classical system is shown to exhibit a bifurcation to the limit cycle, where its steady-state solution corresponds to periodic oscillation with the frequency unrelated to the driving frequency. This bifurcation is shown to lead to a peculiarity in the stationary single-particle density matrix of the quantum system. The case of the Bose-Hubbard trimer, where the discussed limit cycle bifurcates into a chaotic attractor, is briefly discussed.

Solitary waves explore the quantum-to-classical transition

How the classical appearance of our environment arises from the underlying quantum many-body theory is an open fundamental question. We propose that phenomena involved in the quantum-to-classical transition can be probed in collisions of bright solitary waves in Bose-Einstein condensates, where thousands of atoms form a large compound object at ultra cold temperatures. For the experimentally most relevant quasi-1D regime, where integrability is broken through effective three-body interactions, we find that ensembles of solitary waves exhibit complex interplay between phase coherence and entanglement generation in beyond mean-field simulations using the truncated Wigner method: An initial state of two solitons with a well-defined relative phase looses that phase coherence in the ensemble, with its single-particle two-mode density matrix exhibiting similar dynamics as a decohering two-mode superposition. This apparent decoherence is a prerequisite for the formation of entangled superpositions of different atom numbers in a subsequent soliton collision. The necessity for the solitons to first decohere is explained based on the underlying phase-space of the quintic mean-field equation. We show elsewhere that superpositions of different atom numbers later further evolve into spatially entangled solitons. Loss of ensemble phase coherence followed by system internal entanglement generation appear in an unusual order in this closed system, compared to a typical open quantum system.

Simulating realistic screening clouds around quantum impurities: Role of spatial anisotropy and disorder

Dynamical quantum impurities in metals induce electronic correlations in real space that are difficult to simulate due to their multi-scale nature, so that only s-wave scattering in clean metallic hosts has been investigated so far. However, screening clouds should show anisotropy due to lack of full rotational invariance in two- and three-dimensional lattices, while inherent disorder will also induce spatial inhomogeneities. To tackle these challenges, we present an efficient and robust algorithm based on the recursive generation of natural orbitals defined as eigenvectors of the truncated single-particle density matrix. This method provides well-converged many-body wave functions on lattices with up to tens of thousands of sites, bypassing some limitations of other approaches. The algorithm is put to the test by investigating the charge screening cloud around an interacting resonant level, both on clean and disordered lattices, achieving accurate spatial resolution from short to long distances. We thus demonstrate strong anisotropy of spatial correlations around an adatom in the half-filled square lattice. Taking advantage of the efficiency of the algorithm, we further compute the disorder-induced distribution of Kondo temperatures over several thousands of random realizations, at the same time gaining access to the full spatial profile of the screening cloud in each sample. While the charge screening cloud is typically shortened due to the polarization of the impurity by the disorder potential, we surprisingly find that rare disorder configurations preserve the long range nature of Kondo correlations in the electronic bath.

RTGW2020: An efficient implementation of the multi-orbital Gutzwiller method with general local interactions

In the present paper, we propose an efficient numerical scheme for Gutzwiller method for multi-band Hubbard models with general onsite Coulomb interaction. Following the basic idea of Deng et al. (2009) [26] and extensions by Lanata et al. (2012) [28], the ground state is variationally determined through optimizing the total energy with respect to the variational single particle density matrix (n0), which is called “outer loop”. In the corresponding “inner loop” where n0 is fixed, the non-interacting wave function and the parameters contained in the Gutzwiller projector are determined by a two-step iterative approach. All derivatives of the implementation process have been analytically derived, which allows us to apply some advanced minimization or root-searching algorithms for both the inner and outer loops leading to the highly efficient convergence. In addition, an atomic diagonalization method taking the point group symmetry into account has been developed for the customized design of the Gutzwiller projector, making it convenient to explore many interesting orders at a lower cost of computation. As benchmarks, several different types of correlated models have been studied utilizing the proposed method, which are in perfect agreement with the previous results by DMFT and multi-orbital slave-boson mean field method. Compared with the linear mixing method, Newton's method with analytical derivatives shows much faster convergence for the inner loop. As for the outer loop, the minimization using analytical derivatives also shows much better stability and efficiency compared with that using numerical derivatives.

High Harmonic Generation in Triangular Graphene Quantum Dots

Higher harmonic generation in plane graphene quantum dots initiated by intense coherent radiation is investigated using dynamical Hartree–Fock mean-field theory. A microscopic theory describing the extreme nonlinear optical response of plane graphene quantum dots is developed. The closed set of differential equations for the single-particle density matrix at the multiphoton interaction of graphene quantum dots and strong laser field is solved numerically. The obtained solutions indicate the significance of the type of edge and lateral size and the band gap and laser field strength in the process of high harmonic generation in the triangular graphene quantum dot.

Dissipative dynamics in the free massive boson limit of the sine-Gordon model

We study the dissipative dynamics of one-dimensional fermions, described in terms of the sine-Gordon model in its free massive boson or semi-classical limit, while keeping track of forward scattering processes. The system is prepared in the gapped ground state, and then coupled to environment through local currents within the Lindblad formalism. The heating dynamics of the system is followed using bosonization. The single particle density matrix exhibits correlations between the left and right moving particles. While the density matrix of right movers and left movers is translationally invariant, the left-right sector is not, corresponding to a translational symmetry breaking charge density wave state. Asymptotically, the single particle density matrix decays exponentially with exponent proportional to -\gamma t|x|\Delta^2−γt|x|Δ2 where \gammaγ and \DeltaΔ are the dissipative coupling and the gap, respectively. The charge density wave order parameter decays exponentially in time with an interaction independent decay rate. The second R'enyi entropy grows linearly with time and is essentially insensitive to the presence of the gap.

Theory of spin–charge-coupled transport in proximitized graphene: an SO(5) algebraic approach

Establishing the conditions under which orbital, spin and lattice-pseudospin degrees of freedom are mutually coupled in realistic nonequilibrium conditions is a major goal in the emergent field of graphene spintronics. Here, we use linear-response theory to obtain a unified microscopic description of spin dynamics and coupled spin–charge transport in graphene with an interface-induced Bychkov–Rashba effect. Our method makes use of an SO(5) extension of the familiar inverse-diffuson approach to obtain a quantum kinetic equation for the single-particle density matrix that treats spin and pseudospin on equal footing and is valid for arbitrary external perturbations. As an application of the formalism, we derive a complete set of drift–diffusion equations for proximitized graphene with scalar impurities in the presence of electric and spin-injection fields which vary slowly in space and time. Our approach is amenable to a wide variety of generalizations, including the study of coupled spin–charge dynamics in layered materials with strong spin–valley coupling and spin–orbit torques in van der Waals heterostructures.

High-order harmonic generation from twisted bilayer graphene driven by a midinfrared laser field

Theoretical calculations of high-harmonic generation (HHG) from commensurate twisted bilayer graphene (tBLG) under intense laser fields are performed. The nonlinear electron dynamics in tBLG is considered by solving the Liouville--von Neumann equation for a single-particle density matrix, which combines the full energy bands and momentum matrix elements within the framework of tight-binding approximation. We show that the pump intensity determines the relative magnitude of two components of the harmonic spectrum parallel and perpendicular to driving polarization. The important dependence of HHG on twisted angles and crystal orientations is also presented. Especially in the absence of the relaxation process, the harmonic emission for twisted angles around 10 \ifmmode^\circ\else\textdegree\fi{} exhibits an evident decrease in efficiency per layer compared to monolayer graphene (MLG), which can be interpreted according to Fermi velocity modification. Our calculation also shows that the relative emission efficiency of different harmonic orders between tBLG and MLG contains redundant information on both the dephasing time and an empirical parameter characterizing the decay of the interlayer electron hopping, thus suggesting an all-optical method for the reconstruction of the two parameters. The reconstruction feasibility is successfully demonstrated by a simple optimization algorithm even if considering the possible experimental uncertainty of both driving pulse parameters and high-harmonic signals. Our results show that HHG spectroscopic characteristics in tBLG might serve as a fingerprint to identify the geometric stacking angle and the electronic interaction between adjacent layers, as well as the strong-field laser induced ultrafast dephasing process.

Few-body nature of Kondo correlated ground states

The quenching of degenerate impurity states in metals generally induces a long-range correlated quantum state known as the Kondo screening cloud. While a macroscopic number of particles clearly take part in forming this extended structure, assessing the number of truly entangled degrees of freedom requires a careful analysis of the relevant many-body wavefunction. For this purpose, we examine the natural single-particle orbitals that are eigenstates of the single-particle density (correlation) matrix for the ground state of two quantum impurity problems: the interacting resonant level model (IRLM) and the single impurity Anderson model (SIAM). As a simple and general probe for few-body versus many-body character we consider the rate of exponential decay of the correlation matrix eigenvalues towards inactive (fully empty or filled) orbitals. We find that this rate remains large in the physically most relevant region of parameter space, implying a few-body character. Genuine many-body correlations emerge only when the Kondo temperature becomes exponentially small, for instance near a quantum critical point. In addition, we demonstrate that a simple numerical diagonalization of the few-body problem restricted to the Fock space of the most correlated orbitals converges exponentially fast with respect to the number of orbitals, to the true ground state of the IRLM. We also show that finite size effects drastically affect the correlation spectrum, shedding light on an apparent paradox arising from previous studies on short chains.

Symmetries in the multi-configurational time-dependent Hartree wavefunction representation and propagation.

In multi-configurational time-dependent Hartree (MCTDH) approaches, different multi-layered wavefunction representations can be used to represent the same physical wavefunction. Transformations between different equivalent representations of a physical wavefunction that alter the tree structure used in the multi-layer MCTDH wavefunction representation interchange the role of single-particle functions (SPFs) and single-hole functions (SHFs) in the MCTDH formalism. While the physical wavefunction is invariant under these transformations, this invariance does not hold for the standard multi-layer MCTDH equations of motion. Introducing transformed SPFs, which obey normalization conditions typically associated with SHFs, revised equations of motion are derived. These equations do not show the singularities resulting from the inverse single-particle density matrix and are invariant under tree transformations. Based on the revised equations of motion, a new integration scheme is introduced. The scheme combines the advantages of the constant mean-field approach of Beck and Meyer [Z. Phys. D 42, 113 (1997)] and the singularity-free integrator suggested by Lubich [Appl. Math. Res. Express 2015, 311]. Numerical calculations studying the spin boson model in high dimensionality confirm the favorable properties of the new integration scheme.

Localization-delocalization effects of a delocalizing dissipation on disordered XXZ spin chains.

The interplay between interaction, disorder, and dissipation has shown a rich phenomenology. Here, we investigate a disordered XXZ spin chain in contact with a bath which, alone, would drive the system toward a highly delocalized and coherent Dicke state. We show that there exist regimes for which the natural orbitals of the single-particle density matrix of the steady state are all localized in the presence of strong disorders, for either weak interaction or strong interaction. We show that the averaged steady-state occupation in the eigenbasis of the open system Hamiltonian could follow an exponential decay for intermediate disorder strength in the presence of weak interactions, while it is more evenly spread for strong disorder or for stronger interactions. Last, we show that strong dissipation increases the coherence of the steady states, thus reducing the signatures of localization. We capture such signatures of localization also with a concatenated inverse participation ratio that simultaneously takes into account how localized are the eigenstates of the Hamiltonian and how close is the steady state to an incoherent mixture of different energy eigenstates.

Dynamics of entanglement after exceptional quantum quench

We investigate a quantum quench from a critical to an exceptional point. The initial state, prepared in the ground state of a critical hermitian system, is time evolved with a non-hermitian SSH model, tuned to its exceptional point. The single particle density matrix exhibits supersonic modes and multiple light cones, characteristic to non-hermitian time evolution. These propagate with integer multiples of the original Fermi velocity. In the long time limit, the fermionic Green's function decays spatially as $1/x^2$, in sharp contrast to the usual $1/x$ decay of non-interacting fermions. The entanglement entropy is understood as if all these supersonic modes arise from independent quasiparticles (though they do not), traveling with the corresponding supersonic light cone velocity. The entropy production rate decreases with time and develops plateaus during the time evolution, signaling the distinct velocities in the propagation of non-local quantum correlations. At late times, the entanglement entropy saturates to a finite value, satisfying a volume law.

Characterizing many-body localization via state sensitivity to boundary conditions

We introduce characterizations for many-body phase transitions between delocalized and localized phases based on the system's sensitivity to boundary conditions. In particular, we change boundary conditions from periodic to antiperiodic and calculate a shift in the system's energy and shifts in the single-particle density-matrix eigenvalues in the corresponding energy window. We employ the typical model for studying MBL, a one-dimensional disordered system of fermions with nearest-neighbor repulsive interaction where disorder is introduced as randomness on on-site energies. By calculating numerically the shifts in the system's energy and eigenvalues of the single-particle density matrix, we observe that in the localized regime, both shifts are vanishing; while in the extended regime, both shifts are on the order of the corresponding level spacing. We also applied these characterizations of the phase transition to the case of having next-nearest-neighbor interactions in addition to the nearest-neighbor interactions and studied its effect on the transition.

Vaporization Dynamics of a Dissipative Quantum Liquid.

We investigate the stability of a Luttinger liquid, upon suddenly coupling it to a dissipative environment. Within the Lindblad equation, the environment couples to local currents and heats the quantum liquid up to infinite temperatures. The single particle density matrix reveals the fractionalization of fermionic excitations in the spatial correlations by retaining the initial noninteger power law exponents, accompanied by an exponential decay in time with an interaction dependent rate. The spectrum of the time evolved density matrix is gapped, which collapses gradually as -ln(t). The von Neumann entropy crosses over from the early time -tln(t) behavior to ln(t) growth for late times. The early time dynamics is captured numerically by performing simulations on spinless interacting fermions, using several numerically exact methods. Our results could be tested experimentally in bosonic Luttinger liquids.

Triplet 23S state of a quantum dot in a magnetic field: A ‘Quantal Newtonian’ first law study

Abstract The triplet 2 3 S state of a 2-electron 2-dimensional quantum dot in a magnetic field is studied via a complementary perspective of Schrodinger–Pauli theory. The perspective is that of the individual electron via its equation of motion or ‘Quantal Newtonian’ first law. According to the law, each electron experiences an external and an internal field, the sum of which vanishes. The external field is the sum of the binding and Lorentz fields. The internal field is a sum of the electron-interaction, kinetic, differential density, and internal magnetic fields. The energy is expressed in integral virial form in terms of these fields. The quantal sources of the density; paramagnetic, diamagnetic, and magnetization current densities; pair-correlation density; the Fermi–Coulomb hole charge; and the single-particle density matrix are obtained, and from them the corresponding fields determined. The fields are shown to satisfy the first law.

Flat-band ferromagnetism in twisted bilayer graphene

We discuss twisted bilayer graphene (TBG) based on a theorem of flat band ferromagnetism put forward by Mielke and Tasaki. According to this theorem, ferromagnetism occurs if the single particle density matrix of the flat band states is irreducible and we argue that this result can be applied to the quasi-flat bands of TBG that emerge around the charge-neutrality point for twist angles around the magic angle $\theta\sim1.05^\circ$. We show that the density matrix is irreducible in this case, thus predicting a ferromagnetic ground state for neutral TBG ($n=0$). We then show that the theorem can also be applied only to the flat conduction or valence bands, if the substrate induces a single-particle gap at charge neutrality. Also in this case, the corresponding density matrix turns out to be irreducible, leading to ferromagnetism at half filling ($n=\pm2$).

One-body entanglement as a quantum resource in fermionic systems

We show that one-body entanglement, which is a measure of the deviation of a pure fermionic state from a Slater determinant (SD) and is determined by the mixedness of the single-particle density matrix (SPDM), can be considered as a quantum resource. The associated theory has SDs and their convex hull as free states, and number conserving fermion linear optics operations (FLO), which include one-body unitary transformations and measurements of the occupancy of single-particle modes, as the basic free operations. We first provide a bipartitelike formulation of one-body entanglement, based on a Schmidt-like decomposition of a pure $N$-fermion state, from which the SPDM [together with the $(N-1)$-body density matrix] can be derived. It is then proved that under FLO operations, the initial and postmeasurement SPDMs always satisfy a majorization relation, which ensures that these operations cannot increase, on average, the one-body entanglement. It is finally shown that this resource is consistent with a model of fermionic quantum computation which requires correlations beyond antisymmetrization. More general free measurements and the relation with mode entanglement are also discussed.