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arXiv:1201.3541

Pinning effect and QPT-like behavior for two particles confined by a core-shell potential

Pier Paolo Marchisio, J. P. Coe, Irene D’Amico

Published Jan 17, 2012• 0 citations

Abstract

We study the ground state entanglement, energy and fidelities of a two-electron system bounded by a core-shell potential, where the core width is varied continuously until it eventually vanishes. This simple system displays a rich and complex behavior: as the core width is varied, this system is characterized by two peculiar transitions where, for different reasons, it displays characteristics similar to a few-particle quantum phase transition. The first occurrence corresponds to something akin to a second order quantum phase transition, while the second transition is marked by a discontinuity, with respect to the driving parameter, in the first derivatives of quantities like energy and entanglement. The study of this system allows to shed light on the sudden variation of entanglement and energy observed in S. Abdullah et al. [Phys. Rev. B 80, 235302 (2009)]. We also compare the core-shell system with a system where a core well is absent: this shows that, even when extremely narrow, the core well has a relevant `pinning' effect. Interestingly, depending on the potential symmetry, the pinning of the wavefunction may either halve or double the system entanglement (with respect to the no-core-well system) when the ground state is already bounded to the outer (shell) well. In the process we discuss the system fidelity and show the usefulness of considering the particle density fidelity as opposed to the more commonly used -- but much more difficult to access -- wavefunction fidelity. In particular we demonstrate that -- for ground-states with nodeless spatial wavefunctions -- the particle density fidelity is zero if and only if the wavefunction fidelity is zero.

DOI: 10.48550/arxiv.1201.3541

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We study the ground state entanglement, energy and fidelities of a two-electron system bounded by a core-shell potential, where the core width is varied continuously until it eventually vanishes. This simple system displays a rich and complex behavior: as the core width is varied, this system is characterized by two peculiar transitions where, for different reasons, it displays characteristics similar to a few-particle quantum phase transition. The first occurrence corresponds to something akin to a second order quantum phase transition, while the second transition is marked by a discontinuity, with respect to the driving parameter, in the first derivatives of quantities like energy and entanglement. The study of this system allows to shed light on the sudden variation of entanglement and energy observed in S. Abdullah et al. [Phys. Rev. B 80, 235302 (2009)]. We also compare the core-shell system with a system where a core well is absent: this shows that, even when extremely narrow, the core well has a relevant `pinning' effect. Interestingly, depending on the potential symmetry, the pinning of the wavefunction may either halve or double the system entanglement (with respect to the no-core-well system) when the ground state is already bounded to the outer (shell) well. In the process we discuss the system fidelity and show the usefulness of considering the particle density fidelity as opposed to the more commonly used -- but much more difficult to access -- wavefunction fidelity. In particular we demonstrate that -- for ground-states with nodeless spatial wavefunctions -- the particle density fidelity is zero if and only if the wavefunction fidelity is zero.

We model a core-shell quantum dot using a one dimensional system of two interacting electrons with a potential that can vary from one corresponding to a dot with comparable width of core and shell to one where the core well width becomes very small, but is always non-zero, while the shell becomes wide. We calculate the entanglement due to the spatial degrees of freedom and also the site or local entanglement found in a partition of the potential accessible to measurement. We quantify both using the linear entropy of the reduced density matrix. A sharp variation of the entanglement had been seen in earlier work for the transformation from a one dimensional model core-shell dot to a double dot [1]. We investigate whether the spatial entanglement and energy derivatives display any discontinuity as the core well width becomes small and if the two-electron state is always bound to the core well.

We consider a spin ladder model which is known to have matrix product states as exact ground states with spin liquid characteristics. The model has two critical-point transitions at the parameter values

u=0

and $\ensuremath{\infty}$. We study the variation of entanglement and fidelity measures in the ground states as a function of

u

and specially look for signatures of quantum phase transitions at

u=0

and $\ensuremath{\infty}$. The two different entanglement measures used are

S(i)

(the single-site von Neumann entropy) and

S(i,j)

(the two-body entanglement). At the quantum critical point (QCP) $u=\ensuremath{\infty}$, the entanglement measure

E

[=S(i),S(i,j)]

vanishes but remains nonzero at the other QCP

u=0

. The first and second derivatives of

E

with respect to the parameter

u

and the entanglement length associated with

S(i,j)

are further calculated to identify special features, if any, near the QCPs. We further determine the GS fidelity

F

and a quantity

\text{ln}|D|

related to the second derivative of

F

and show that these quantities calculated for finite-sized systems are good indicators of QPTs occurring in the infinite system.

We calculate the spatial entanglement between two electrons trapped in a nanostructure for a broad class of confinement potentials, including single and double quantum dots, and core-shell quantum dot structures. By using a parametrized confinement potential, we are able to switch from one structure to the others with continuity and to analyze how the entanglement is influenced by the changes in the confinement geometry. We calculate the many-body wave function by ``exact'' diagonalization of the time-independent Schr\"odinger equation. We discuss the relationship between the entanglement and specific cuts of the wave function, and show that the wave function at a single highly symmetric point could be a good indicator for the entanglement content of the system. We analyze the counterintuitive relationship between spatial entanglement and Coulomb interaction, which connects maxima (minima) of the first to minima (maxima) of the latter. We introduce a potential quantum phase transition which relates quantum states characterized by different spatial topology. Finally we show that by varying shape, range, and strength of the confinement potential, it is possible to induce strong and rapid variations in the entanglement between the two electrons. This property may be used to tailor nanostructures according to the level of entanglement required by a specific application.

We develop a general theory of the relation between quantum phase transitions (QPTs) characterized by nonanalyticities in the energy and bipartite entanglement. We derive a functional relation between the matrix elements of two-particle reduced density matrices and the eigenvalues of general two-body Hamiltonians of d-level systems. The ground state energy eigenvalue and its derivatives, whose nonanalyticity characterizes a QPT, are directly tied to bipartite entanglement measures. We show that first-order QPTs are signaled by density matrix elements themselves and second-order QPTs by the first derivative of density matrix elements. Our general conclusions are illustrated via several quantum spin models.

We take as a starting point the ground-state electron density in two-electron model atoms in which Coulomb confinement in the He atom is first replaced by harmonic restoring forces. Switching off electron–electron interactions, one readily constructs a third-order differential equation for the ground-state electron density, as in the recent work of March and Ludeña (2004 Phys. Lett. A 330 16). We then switch on two different model interactions, first in the so-called Hookean atom going back to Kestner and Sinanoglu (1962 Phys. Rev. 128 2687), in which e 2 / r 12 is retained as in He, and secondly in the Moshinsky (1968 Am. J. Phys. 36 52) atom in which Kr 2 12 /2 is switched on. Some analyticity properties of the low-order linear homogeneous differential equations which result are next studied. He-like atomic ions are then treated in the limit of large atomic number Z . In this latter case, one identifies both the electron–nuclear cusp, or equivalently Kato's theorem, and the corresponding electron–electron cusp in the ground-state spatial wavefunction Ψ( r 1 , r 2 ). A final comment concerns quantum information and entanglement in relation to the recent work of Amovilli and March (2004 Phys. Rev. A 69 054302).

A quantum phase transition (QPT) in Hermitian systems is independent of the boundary condition in the thermodynamic limit. However, it may happen for a non-Hermitian system, that the QPT strongly depends on the boundary condition. We investigate the many-body ground-state property of a one-dimensional tight-binding ring with an embedded single asymmetrical dimer based on exact solutions. We employ a semilocalization state to describe a quantum phase that is a crossover from extended to localized state. The peculiar feature is that the decay length is of the order of the system size rather than fixed as a usual localized state. In addition, the spectral statistics is nonanalytic as asymmetrical hopping strengths vary, resulting a sudden charge of the ground state. The distinguishing feature of such a QPT is that the density of the ground-state energy varies smoothly due to unbroken symmetry. However, there are other observables, such as the ground-state center of mass and average current, which exhibit the behavior of second-order QPT. This behavior stems from time-reversal symmetry breaking of a macroscopic number of single-particle eigenstates.