Symmetry breaking in quantum chaotic systems

We show, using semiclassical methods, that as a symmetry is broken, the transition between universality classes for the spectral correlations of quantum chaotic systems is governed by the same parametrization as in the theory of random matrices. The theory is quantitatively verified for the kicked rotor quantum map. We also provide an explicit substantiation of the random matrix hypothesis, namely that in the symmetry-adapted basis the symmetry-violating operator is random.     

Symmetry breaking and finite-size effects in quantum many-body systems

We consider a quantum many-body system on a lattice which exhibits a spontaneous symmetry breaking in its infinite-volume ground states, but in which the corresponding order operator does not commute with the Hamiltonian. Typical examples are the Heisenberg antiferromagnet with a Néel order and the Hubbard model with a (superconducting) off-diagonal long-range order. In the corresponding finite system, the symmetry breaking is usually obscured by quantum fluctuation and one gets a symmetric ground state with a long-range order. In such a situation, Horsch and von der Linden proved that the finite system has a low-lying eigenstate whose excitation energy is not more than of orderN ^–1, whereN denotes the number of sites in the lattice. Here we study the situation where the broken symmetry is a continuous one. For a particular set of states (which are orthogonal to the ground state and with each other), we prove bounds for their energy expectation values. The bounds establish that there exist ever-increasing numbers of low-lying eigenstates whose excitation energies are bounded by a constant timesN ^–1. A crucial feature of the particular low-lying states we consider is that they can be regarded as finite-volume counterparts of the infinite-volume ground states. By forming linear combinations of these low-lying states and the (finite-volume) ground state and by taking infinite-volume limits, we construct infinite-volume ground states with explicit symmetry breaking. We conjecture that these infinite-volume ground states are ergodic, i.e., physically natural. Our general theorems not only shed light on the nature of symmetry breaking in quantum many-body systems, but also provide indispensable information for numerical approaches to these systems. We also discuss applications of our general results to a variety of interesting examples. The present paper is intended to be accessible to readers without background in mathematical approaches to quantum many-body systems.     

Mutually unbiased projectors and duality between lines and bases in finite quantum systems

Quantum systems with variables in the ring Z(d)$\mathbb{Z}\left(d\right)$ are considered, and the concepts of weak mutually unbiased bases and mutually unbiased projectors are discussed. The lines through the origin in the Z(d)×Z(d)$\mathbb{Z}\left(d\right)\times \mathbb{Z}\left(d\right)$ phase space, are classified into maximal lines (sets of d$d$ points), and sublines (sets of d_i$d_i$ points where d_i|d$d_i|d$). The sublines are intersections of maximal lines. It is shown that there exists a duality between the properties of lines (resp., sublines), and the properties of weak mutually unbiased bases (resp., mutually unbiased projectors).     

Dynamical symmetry breaking of lambda- and vee-type three-level systems on quantization of the field modes

We develop a scheme to construct the Hamiltonians of the lambda-, vee- and cascade-type three-level configurations using the generators of SU(3) group. It turns out that this approach provides a well-defined selection rule to give different Hamiltonians for each configuration. The lambda- and vee-type configurations are exactly solved with different initial conditions while taking the two-mode classical and quantized fields. For the classical field, it is shown that the Rabi oscillation of the lambda model is similar to that of the vee model and the dynamics of the vee model can be recovered from lambda model and vice versa simply by inversion. We then proceed to solve the quantized version of both models by introducing a novel Euler matrix formalism. It is shown that this dynamical symmetry exhibited in the Rabi oscillation of two configurations for the semiclassical models is completely destroyed on quantization of the field modes. The symmetry can be restored within the quantized models when both field modes are in the coherent states with large average photon number which is depicted through the collapse and revival of the Rabi oscillations.      

Confined states in two-dimensional flat elliptic quantum dots and elliptic quantum wires

The energy spectrum and corresponding wave functions of a flat quantum dot with elliptic symmetry are obtained exactly. A detailed study is made of the effect of ellipticity on the energy levels and the corresponding wave functions. The analytical behavior of the energy levels in certain limiting cases is obtained.     

Absence of symmetry breaking for systems of rotors with random interactions

We prove that Gibbs states for the Hamiltonian , with thes _x varying on theN-dimensional unit sphere, obtained with nonrandom boundary conditions (in a suitable sense), are almost surely rotationally invariant if withJ _xy i.i.d. bounded random variables with zero average, 1 in one dimension, and 2 in two dimensions.     

Criteria for the absence of quantum fluctuations after spontaneous symmetry breaking

The lowest-energy state of a macroscopic system in which symmetry is spontaneously broken, is a very stable wavepacket centered around a spontaneously chosen, classical direction in symmetry space. However, for a Heisenberg ferromagnet the quantum groundstate is exactly the classical groundstate, there are no quantum fluctuations. This coincides with seven exceptional properties of the ferromagnet, including spontaneous time-reversal symmetry breaking, a reduced number of Nambu–Goldstone modes and the absence of a thin spectrum (Anderson tower of states). Recent discoveries of other non-relativistic systems with fewer Nambu–Goldstone modes suggest these specialties apply there as well. I establish precise criteria for the absence of quantum fluctuations and all the other features. In particular, it is not sufficient that the order parameter operator commutes with the Hamiltonian. It leads to a measurably larger coherence time of superpositions in small but macroscopic systems.     

Quantum fluctuations in quantum lattice systems with continuous symmetry

We discuss conditions for the absence of spontaneous breakdown of continuous symmetries in quantum lattice systems atT=0. Our analysis is based on Pitaevskii and Stringari's idea that the uncertainty relation can be employed to show quantum fluctuations. For one-dimensional systems, it is shown that the ground state is invariant under a continuous transformation if a certain uniform susceptibility is finite. For the two- and three-dimensional systems, it is shown that truncated correlation functions cannot decay any more rapidly than|r| ^–d+1 whenever the continuous symmetry is spontaneously broken. Both of these phenomena occur owing to quantum fluctuations. Our theorems cover a wide class of quantum lattice systems having not-too-long-range interactions.     

A finite-dimensional quantum model for the stock market

We present a finite-dimensional version of the quantum model for the stock market proposed in C. Zhang and L. Huang [A quantum model for the stock market, Physica A 389 (2010) 5769]. Our approach is an attempt to make this model consistent with the discrete nature of the stock price and is based on the mathematical formalism used in the case of the quantum systems with finite-dimensional Hilbert space. The rate of return is a discrete variable corresponding to the coordinate in the case of quantum systems, and the operator of the conjugate variable describing the trend of the stock return is defined in terms of the finite Fourier transform. The stock return in equilibrium is described by a finite Gaussian function, and the time evolution of the stock price, directly related to the rate of return, is obtained by numerically solving a Schrödinger type equation.     

Finite size scaling of the spontaneous symmetry breaking model of X-chromosome inactivation

X-Chromosome inactivation is the process whereby one of the two X-chromosomes in female cells is silenced to prevent the cell producing too much of any X-linked proteins and RNA. The proposed blocking-factor mechanism of X-inactivation is not well understood and hence is the subject of much current research. In this paper we investigated the nature of the phase transition predicted to exist in the spontaneous symmetry breaking model of X-inactivation proposed by Nicodemi and Prisco [Mario Nicodemi, Antonella Prisco, Symmetry breaking model for x-chromosome inactivation, Phs. Rev. Lett. 98 (2007) 108104]. Finite size effects were investigated by using an on lattice Monte Carlo simulation. From the scaling it is concluded that the transition is in general abrupt. The critical temperature of the system was determined to be 1.68±0.01E₀/k_B in the thermodynamic limit when the concentration C=0.025 blocking-factors per lattice site.     

Spontaneous symmetry breaking in quantum many-body systems (A solid-state physicist’s view of Goldstone’s theorem and all that)

The concept of spontaneous symmetry breaking arose first in the context of superconductivity, before it became important for elementary particle physics. Starting with its original discovery, a comparison of the workings of the Goldstone mechanism in relativistic quantum fixed theory on the one hand and in quantum statistical mechanics on the other is given. The roles of locality and of long range forces are traced. For condensed matter physics, an approach using functional integral methods and macroscopic order parameter fields, valid near critical points is outlined. A possibly more widely valid approach is also presented, to complete this review of the Goldstone theories in quantum statistical mechanics. Talk delivered at the International Symposium on Theoretical Physics, Bangalore, November 1984.     

Dynamic symmetry breaking in kscn: A 39K and 14N NMR study

Abstract

The ³⁹K ½ → ?½ NMR angular rotation patterns in KSCN show no symmetry change on going through T_c . We have directly determined with ¹⁴N NMR the SCN head-tail flipping rate. The results show that the absence of a symmetry change in ³⁹K angular rotation patterns is due to the fact that the time scale for the SCN fluctuations is much longer than the characteristic time scale of this experiment. The same is true for the diffuse neutron scattering data where both above and below T_c the SCN groups appear ordered and static and disorder is restricted to domain boundaries. KSCN thus seems to be the first known example where a structural phase transition can be observed in the slow motion and not in the fast motion regime and where a dynamic breaking of the symmetry of the high temperature phase takes place.     

Far-infrared spectroscopy of quantum wires and dots, breaking Kohn’s theorem

We review far-infrared experiments on quantum wires and dots. In particular, we show that with tailored deviations from a parabolic external lateral confinement potential one can break Kohn’s theorem. This allows a detailed investigation of the internal relative motion in quantum dots and wires and the study of electron–electron interaction effects, for example, the formation of compressible and incompressible states in quantum dots and antidots.     

Recovering pure states in two-state quantum systems

In this work, we study the loss and recovery of pure states (i.e., coherence) in two-state molecules and quantum dots. The molecules of two electronic states and a one-dimensional nuclear vibration are modeled by a quantum–classical dynamical model. According to the simulations, pure states of a two-state molecule can be restored by the excitation of the nuclear vibration by a well-defined electromagnetic field. In the case of a quantum dot, pure states can be regained through the modulation of the energy levels through the application of a proper bias voltage on the dot.     

Comment on: “Confined states in two-dimensional flat elliptic quantum dots and elliptic quantum wires” [Physica E 11 (2001) 345]

We argue that the two-dimensional elliptic quantum dot problem with finite barrier cannot be exactly solved, contrary to a recent assertion (van den Broek and Peeters, Physica E 11 (2001) 345. We also prove it explicitly by numerically calculating the correct energy spectrum.     

Impact of symmetry breaking on the Flame Transfer Function of a laminar premixed flame

This work presents a numerical study of the acoustic response of a laminar flame with tunable asymmetry. A V-shaped premixed flame is stabilised in the wake of a cylindrical flame holder that can be rotated. The configuration is symmetric when the flame holder is fixed but increasing its rotation rate breaks the symmetry of the flow. This configuration is submitted to acoustic forcing to measure the effect of rotation of the flame holder on the Flame Transfer Functions. It appears that the asymmetry of the two flame branches changes their respective time delays, resulting in interference in the global unsteady heat release rate fluctuations. Consequently, the Flame Transfer Function exhibits dips and bumps, which are studied via laminar Direct Numerical Simulation. Potential applications for the control of combustion instabilities are discussed.     

Magnetoplasmon excitations in quantum dot arrays

Motivated by the far-infrared transmission experiments of Demel et al., we have investigated the magnetoplasmon excitations in an array of quantum dots within the Thomas–Fermi–Dirac–von Weizsäcker (TFDW) approximation. Detailed calculations of the magnetic dispersion and power absorption from a uniform radiation field unambiguously demonstrates that the noncircular symmetry of the individual dots is responsible for the anticrossing behaviour observed in the experiments. The interdot Coulomb interaction is unimportant at the interdot separation of the samples studied.     

Magnetotransport spectroscopy of a quantum dot: effects of lead opening and phase coherence

We compare open quantum dot magnetoconductance spectra from experiment and theory in the presence of environmental coupling and attributed broadening. Estimates of the phase-breaking time in experiment, and effective broadening in simulation, are determined independently. In a larger, more open dot, with a significantly shorter phase-breaking time, the observed spectrum is broadened, most noticeably about B=0. The required broadening in simulation is characterized by effective temperatures higher than estimates from experiment; however, without accounting for disorder, which will further broaden the spectrum, the agreement is reasonable.     

Coherent dynamics in InGaAs quantum dots and quantum dot molecules

We measure the dephasing time of the exciton ground state transition in InGaAs quantum dots (QD) and quantum dot molecules (QDM) using a sensitive four-wave mixing technique. In the QDs we find experimental evidence that the dephasing time is given only by the radiative lifetime at low temperatures. We demonstrate the tunability of the radiatively limited dephasing time from 400 ps up to 2 ns in a series of annealed QDs with increasing energy separation of 69–330 meV from the wetting layer continuum. Furthermore, the distribution of the fine-structure splitting δ₁ and of the biexciton binding energy δ_B is measured. δ₁ decreases from 96 to with increasing annealing temperature, indicating an improving circular symmetry of the in-plane confinement potential. The biexciton binding energy shows only a weak dependence on the confinement energy, which we attribute to a compensation between decreasing confinement and decreasing separation of electron and hole. In the QDM we measured the exciton dephasing as function of interdot barrier thickness in the temperature range from 5 to 60 K. At 5 K dephasing times of several hundred picoseconds are found. Moreover, a systematic dependence of the dephasing dynamics on the barrier thickness is observed, showing how the quantum mechanical coupling in the molecules affects the exciton lifetime and acoustic-phonon interaction.     

Investigations on finite ideal quantum gases

Recursion formulae of the N-particle partition function, the occupation numbers and its fluctuations are given using the single-particle partition function. Exact results are presented for fermions and bosons in a common one-dimensional harmonic oscillator potential, for the three-dimensional harmonic oscillator approximations are tested. Applications to excited nuclei and Bose–Einstein condensation are discussed.