Srction 2 Quantum-Effect Devices

Chapter 1 Statistical Mechanics of Quantum Dots

Chapter 2 Artificial Atoms

Chapter 3 Coulomb Blockade in Single and Double Quantum Dots

Chapter 4 Superconducting and Quantum-Effect Devices

Chapter 5 Nanostructures Technology, Research, andApplications

Chapter 6 Single-Electron Spectroscopy

45

46 RLE Progress Report Number 138

Chapter 1. Statistical Mechanics of Quantum Dots

Chapter 1. Statistical Mechanics of Quantum Dots

Academic and Research Staff

Professor Boris L. Altshuler, Dr. Nobuhiko Taniguchi

Graduate Students

Anton V. Andreev

1.1 Project Description

SponsorJoint Services Electronics Program

Grant DAAL04-95-1-0038

In 1995, the main objective of our research was thenonuniversal properties of quantum dots. Typicalbehavior of these closed systems of large but finitenumber of electrons is usually called quantumchaos.' This means something opposite to behaviorof quantum integrable systems where spectra canbe characterized by a set of quantum numbers.Quantum chaos can be caused either by disorderor by geometry of a quantum dot or by interactionsbetween electrons. It was intensively studied forseveral years.2 Probably the most popular charac-teristics of chaotic quantum systems are the statis-tics of energy spectra. The statistics that describedistribution of eigenstates in energy and parameterspace were found for systems with well developedchaos to be universal with a pretty high accuracy.This means, e.g., that the correlation functions atsmall energies, after proper rescaling, become inde-pendent from the particular features of the quantumsystem and particular perturbation. Earlier, we eval-uated these functions explicitly.3

The global properties of the spectra (i.e., correlationfunctions at higher energies), are different for dif-ferent systems. We have solved the long timeproblem in the field of quantum chaos byexpressing these features through the properties ofthe underlying classical system.

First, we considered disordered quantum dots.4

Using a nonperturbative approach, we have evalu-ated the large frequency w asymptotics of the two-point correlation function

R(w) = < ij 5(E - Ei) 5(E + CO - Ej) >

where Ei and Ej denote the eigenstates of oursystem (quantum dot) measured in units of themean level spacing and <...> stands for the aver-aging. We found that this function for systems withbroken T-invariance (quantum dot in a magneticfield) can be presented as

R(w) = 6(w) - Rsmooth(w) + Rosc(w) cos(2Trw). (2)

The functions Rsmooth(W) and Rosc(ow) are not uni-versal: they depend on the shape of the quantumdot, its conductance, etc. In the universal limit,

Rsmooth(W) = Rosc(w)1/(4 T 2 W2)

is very important for the understanding of the con-nection between classical and quantum problems;these functions are completely determined by theproperties of classical diffusion in the system.Namely, Rosc(w) is the spectral determinant of thediffusion operator and Rsmooth(W) is the Green'sfunction of the same operator. As a result they areconnected as

Rsmooth(W) = - (1/4Tr 2)a {In [Rosc())] /a Cw2 . (4)

1 M.C. Gutzwiller, Chaos in Classical and Quantum Mechanics (New York: Springer-Verlag, 1990).

2 M.V. Berry, in Chaos in Quantum Physics, eds. M.-J. Gianonni, A. Voros, and J. Zinn-Justin, Les Houches, Session LIl, 1989(Amsterdam: North Holland, 1991), p. 251.

3 For complete discussion see B.L. Altshuler and B.D. Simons, "Universalities: from Anderson Localization to Quantum Chaos," inMesoscopic Quantum Physics, eds. E. Akkermans, G. Montambaux, J-L. Pichard, and J. Zinn-Justin, Les Houches, Session LXI,1994 (Amsterdam: North Holland, 1996), p. 1.

4 A.V. Andreev and B.L. Altshuler, "Spectral Statistics beyond Random Matrix Theory," Phys. Rev. Lett. 75: 902-905 (1995).

Chapter 1. Statistical Mechanics of Quantum Dots

This relation is universal even for quantum dotswithout any disorder.5 We have been able to showthat, in a very general case, there is a classicaloperator whose spectrum completely determinesthe spectral statistics of the quantum dot. This oper-ator describes the time evolution of the classicalcounterpart of our quantum dot. It is crucial for thewhole theory of quantum chaos that this operatorcorresponds to the time irreversible classical dyna-mics. This understanding allowed us to prove aconjecture proposed long ago by Bohigas, Giannoniand Schmit 6 for a certain class of quantum systems.By making use of numerical studies, these authorsfound that the spectral statistics even for simplechaotic systems are very close to universal Wigner-Dyson statistics. From our calculations,5 it followsthat this is true for systems with exponential relaxa-tion. By making use of the nonperturbativeapproach, we can also evaluate nonuniversal cor-rections to this universal behavior, given the set ofthe relaxation times. This connection betweenquantal behavior of the quantum dot and irrevers-ible classical dynamics of its classical counterpart isof a particular fundamental importance for thetheory of both classical and quantum complexsystems.

To complete our study of the universal properties ofquantum dots, we evaluated the statistics of theoscillator strengths, starting from the Wigner-Dysonhypothesis.7 These statistics provide a characteriza-tion of quantum chaos which complements theusual energy level statistics. This theory can beapplied to quantum dots and also to systems likethe hydrogen atom in a strong magnetic field.Without additional assumptions, we have evaluatedexactly the correlation function of oscillatorstrengths at different frequencies and different mag-

5 0. Aham, B. Altshuler, and A.V. Andreev,(1995).

netic fields. We have also discoveredpected differential relation between thestates statistics and those of thestrengths.

an unex-density of

oscillator

Using the mathematical similarity between quantumparticles in one dimension and energy levels, wehave obtained important results that can be appliedfor better understanding of the quantum dots withinteractions between electrons. In particular, wehave discovered a novel relationship between equaltime current and density correlations in the modelknown as Calogero-Sutherland model, where theinteraction between one dimensional fermions isinverse proportional to the distance between them. 8

This relationship exists in addition to the usualWard identities.

1.2 Publications

Agam, O., B.L. Altshuler, and A.V. Andreev. "Spec-tral Statistics: From Disordered to ChaoticSystems." Phys. Rev. Lett. 75: 4389 (1995).

Andreev, A.V., and B.L. Altshuler. "Spectral Statis-tics beyond Random Matrix Theory." Phys. Rev.Lett. 75: 902-905 (1995).

Taniguchi, N., A.V. Andreev, and B.L. Altshuler."Statistics of Oscillator Strength in ChaoticSystems." Europhysics Lett. 29: 515 (1995).

Taniguchi, N., B.S. Shastry, and B.L. Altshuler."Random Matrix Model and Calogero-SutherlandModel: A Novel Current-Density Mapping." Phys.Rev. Lett. 75: 37 (1995).

"Spectral Statistics: From Disordered to Chaotic Systems," Phys. Rev. Lett. 75: 4389

6 0. Bohigas, M.J. Giannoni, and C. Schmidt, Phys. Rev. Lett. 52: 1 (1984).

7 N. Taniguchi, A.V. Andreev, and B.L. Altshuler, "Statistics of Oscillator Strength(1995).

in Chaotic Systems," Europhysics Left. 29: 515

8 N. Taniguchi, B.S. Shastry, and B.L. Altshuler, "Random Matrix Model and Calogero-Sutherland Model: A Novel Current-DensityMapping," Phys. Rev. Lett. 75: 37 (1995).

48 RLE Progress Report Number 138