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arXiv:cond-mat/0406029 · Physical Review B

Random-matrix and quantum-critical crossovers for interacting electrons in quantum dots

Ganpathy N. Murthy

Published Oct 15, 2004• 10 citations
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Abstract

Quantum dots with a large Thouless number $g$ embody a regime where both disorder and interactions can be treated nonperturbatively using large-$N$ techniques (with $N=g$) and quantum phase transitions can be studied. Here we focus on dots where the noninteracting Hamiltonian is drawn from a crossover ensemble between two symmetry classes, where the crossover parameter introduces a tunable energy scale independent of and much smaller than the Thouless energy. We show that the quantum critical regime, dominated by collective critical fluctuations, can be accessed at this energy scale. The nonperturbative physics of this regime can only be described by the large-$N$ approach, as we illustrate with two experimentally relevant examples.

DOI
10.1103/physrevb.70.153304

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Quantum dots with a large Thouless number ggg embody a regime where both disorder and interactions can be treated nonperturbatively using large-NNN techniques (with N=gN=gN=g) and quantum phase transitions can be studied. Here we focus on dots where the noninteracting Hamiltonian is drawn from a crossover ensemble between two symmetry classes, where the crossover parameter introduces a tunable energy scale independent of and much smaller than the Thouless energy. We show that the quantum critical regime, dominated by collective critical fluctuations, can be accessed at this energy scale. The nonperturbative physics of this regime can only be described by the large-NNN approach, as we illustrate with two experimentally relevant examples.
Interacting electrons in quantum dots with large Thouless number ggg in the three classical random matrix symmetry classes are well understood. When a specific type of spin-orbit coupling known to be dominant in two-dimensional semiconductor quantum dots is introduced, we show that an interacting quantum critical crossover energy scale emerges and low-energy quasiparticles generically have a decay width proportional to their energy. The low-energy physics of this system is an example of a universal interacting crossover regime.
We study a system of two tunnel-coupled quantum dots with the first dot containing interacting electrons (described by the universal Hamiltonian) not subject to spin-orbit coupling whereas the second contains noninteracting electrons subject to spin-orbit coupling. We focus on describing the behavior of the system near the Stoner transition. Close to the critical point quantum fluctuations become important and the system enters a quantum-critical regime. The large-NNN approximation allows us to calculate physical quantities reliably even in this strongly fluctuating regime. In particular, we find a scaling function to describe the crossover of the quasiparticle decay rate between the renormalized Fermi-liquid regime and the quantum-critical regime.
We provide a framework for analyzing the problem of interacting electrons in a ballistic quantum dot with chaotic boundary conditions within an energy ET{E}_{T}ET​ (the Thouless energy) of the Fermi energy. Within this window we show that the interactions can be characterized by Landau Fermi-liquid parameters. When g, the Thouless number (also the dimensionless conductance when it has strong coupling to leads) of the dot, is large, we find that the disordered interacting problem can be solved in a saddle-point approximation which becomes exact as $\stackrel{\ensuremath{\rightarrow}}{g}\ensuremath{\infty}$ (as in a $\mathrm{l}\mathrm{a}\mathrm{r}\mathrm{g}\mathrm{e}\ensuremath{-}N$ theory). The infinite g theory has two phases as a function of the Landau parameter um{u}_{m}um​ in a channel with angular momentum m: A weak-coupling phase where constant charging and exchange interactions dominate the low-energy physics, as in previous ``universal Hamiltonian'' treatments, and a strong-coupling phase characterized by the same order parameter as in the Pomeranchuk transition in clean systems (a spontaneous interaction-induced Fermi-surface distortion), but smeared and pinned by disorder. Thus, both interactions and disorder are crucial to the existence of these phases. At finite g, the two phases and critical point evolve into three regimes in the ${u}_{m}\ensuremath{-}1/g$ plane---weak- and strong-coupling regimes separated by crossover lines from a quantum-critical regime controlled by the quantum-critical point. In this, the first of a two part series, we focus on the consequences of this picture for Coulomb Blockade experiments. We employ analytical and numerical methods to predict the statistics of single-particle levels, Coulomb Blockade peak spacings, conductance peak heights, and quasiparticle widths. We show that in the strong-coupling and quantum-critical regions, the quasiparticle acquires a width of the same order as the level spacing $\ensuremath{\Delta}$ within a few $\ensuremath{\Delta}\mathrm{'}\mathrm{s}$ of the Fermi energy due to coupling to collective excitations. In the strong-coupling regime if m is odd, the dot will (if isolated) crossover from the orthogonal to unitary ensemble for an exponentially small external flux or will (if strongly coupled to leads) break time-reversal symmetry spontaneously. For any m, the peak-spacing distribution becomes broader than expected in previous works and even has support at negative values, which in turn is correlated with small peak heights. Ballistic/chaotic quantum dots afford us unrivalled theoretical and experimental control over the problem of simultaneous disorder and interactions due to the 1/g1/g1/g expansion and our ability to vary disorder and interaction much more readily than in the bulk.
The chaotic mixing by random two-body interactions of many-electron Fock states in a confined geometry is investigated. Two regimes are distinguished in the dependence of the typical number of Fock states that are mixed into an eigenstate on the interaction strength V, the excitation energy varepsilon, and the level spacing Delta. In both regimes the number is large (indicating delocalization in Fock space). However, only the large- V regime is described by the golden rule (indicating chaotic mixing). The crossover region is characterized by a maximum in a scaling function that becomes more pronounced with increasing excitation energy. The scaling parameter that governs the transition is (varepsilonV/Delta(2))ln(Delta/V).
We study the prethermal dynamics of an interacting quantum field theory with an N-component order parameter and O(N) symmetry, suddenly quenched in the vicinity of a dynamical critical point. Depending on the initial conditions, the evolution of the order parameter, and of the response and correlation functions, can exhibit a temporal crossover between universal dynamical scaling regimes governed, respectively, by a quantum and a classical prethermal fixed point, as well as a crossover from a Gaussian to a non-Gaussian prethermal dynamical scaling. Together with a recent experiment, this suggests that quenches may be used in order to explore the rich variety of dynamical critical points occurring in the nonequilibrium dynamics of a quantum many-body system. We illustrate this fact by using a combination of renormalization group techniques and a nonperturbative large-N limit.