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The current understanding of some important nonlinear collective processes in quantum plasmas with degenerate electrons is presented. After reviewing the basic properties of quantum plasmas, we present model equations (e.g. the quantum hydrodynamic and effective nonlinear Schr\"odinger-Poisson equations) that describe collective nonlinear phenomena at nanoscales. The effects of the electron degeneracy arise due to Heisenberg's uncertainty principle and Pauli's exclusion principle for overlapping electron wavefunctions that result in tunneling of electrons and the electron degeneracy pressure. Since electrons are Fermions (spin-1/2), there also appears an electron spin current and a spin force acting on electrons due to the Bohr magnetization. The quantum effects produce new aspects of electrostatic (ES) and electromagnetic (EM) waves in a quantum plasma that are summarized in here. Furthermore, we discuss nonlinear features of ES ion waves and electron plasma oscillations (ESOs), as well as the trapping of intense EM waves in quantum electron density cavities. Specifically, simulation studies of the coupled nonlinear Schr\"odinger (NLS) and Poisson equations reveal the formation and dynamics of localized ES structures at nanoscales in a quantum plasma. We also discuss the effect of an external magnetic field on the plasma wave spectra and develop quantum magnetohydrodynamic (Q-MHD) equations. The results are useful for understanding numerous collective phenomena in quantum plasmas, such as those in compact astrophysical objects, in plasma-assisted nanotechnology, and in the next-generation of intense laser-solid density plasma interaction experiments.

The entanglement entropy of a pure quantum state of a bipartite system is defined as the von Neumann entropy of the reduced density matrix obtained by tracing over one of the two parts. Critical ground states of local Hamiltonians in one dimension have an entanglement entropy that diverges logarithmically in the subsystem size, with a universal coefficient that is is related to the central charge of the associated conformal field theory. Here I will discuss recent extensions of these ideas to a class of quantum critical points with dynamic critical exponent $z=2$ in two space dimensions and to 2D systems in a topological phase. The application of these ideas to quantum dimer models and fractional quantum Hall states will be discussed.

In these notes I review the basic concepts of the effects of interactions on quantum particles. I focuss here mostly on the case of fermions, but several aspects of interacting bosons are mentioned as well. These notes have been voluntarily kept at an elementary level and should be suitable for students wanting to enter this field. I review the concept of Fermi liquid, and then move to a description of the interaction effects, as well as the main models that are used to tackle these questions. Finally I study the case of one dimensional interacting particles that constitutes a fascinating special case.

We explore the equations of state and other properties of various quantum fluids (3He, 4He, their mixtures, and H_2) confined within individual carbon nanotubes. Above a threshold number of particles, N_a, the fluid density near the axis begins to grow above a negligibly small value. The properties of this axial fluid phase are sensitive to the tube size and hence to the transverse compression in the case of a bundle of nanotubes. We consider He at zero temperature and H_2 at low temperatures.

A numerical investigation for the stability of the incompressible slip flow of normal quantum fluids (above the critical phase transition temperature) inside a microslab where surface acoustic waves propagate along the walls is presented. Governing equations and associated slip velocity and wavy interface boundary conditions for the flow of normal fluids confined between elastic wavy interfaces are obtained. The numerical approach is an extension (with a complex matrix pre-conditioning) of the spectral method. We found that the critical Reynolds number ($Re_{cr}$ or the critical velocity) decreases significantly once the slip velocity and wavy interface effects are present and the latter is dominated ($Re_{cr}$ mainly depends on the wavy interfaces).

We consider the canonical quantization of an ordinary fluid. The resulting long-distance effective field theory is derivatively coupled, and therefore strongly coupled in the UV. The system however exhibits a number of peculiarities, associated with the vortex degrees of freedom. On the one hand, these have formally a vanishing strong-coupling energy scale, thus suggesting that the effective theory's regime of validity is vanishingly narrow. On the other hand, we prove an analog of Coleman's theorem, whereby the semiclassical vacuum has no quantum counterpart, thus suggesting that the vortex premature strong-coupling phenomenon stems from a bad identification of the ground state and of the perturbative degrees of freedom. Finally, vortices break the usual connection between short distances and high energies, thus potentially impairing the unitarity of the effective theory.

We suggest a more general than quantum statistical mechanics ($QSM$) microdescription of objects in a heat bath taken into account a vacuum as an object environment - modification of quantum mechanics at finite temperatures; we call it $(\hbar, k)$-dynamics ($ \hbar kD$). This approach allows us in a new manner to calculate some important macroparameters and to modify standard thermodynamics. We create an effective apparatus for features description of nearly perfect fluids in various mediums. As an essentially new model of an object environment we suppose a quantum heat bath and its properties, including cases of cold and warm vacuums, are studied. We describe the thermal equilibrium state in place of the traditional density operator in term of a wave function the amplitude and phase of which have temperature dependence. We introduce a new generative operator, Schroedingerian, or stochastic action operator, and show its fundamental role in the microdescription. We demonstrate that a new macroparameter, namely the effective action, can be obtained through averaging of the Schroedingerian over the temperature dependent wave function. It is established that such different parameters as internal energy, effective temperature, and effective entropy and their fluctuations can be expressed through a single quantity - the effective action.

Recent advances in cold atom experimentation suggest that studies of quantum two-dimensional melting of dipolar molecules, with dipoles aligned perpendicular to ordering plane, may be on the horizon. An intriguing aspect of this problem is that two-dimensional \emph{classical} aligned dipoles (already studied in great detail in soft matter experiments on magnetic colloids) are known to melt via a two-stage process, with an intermediate hexatic phase separating the usual crystal and isotropic fluid phases. We estimate here the effect of quantum fluctuations on this hexatic phase, for both dipolar systems and charged Wigner crystals. Our approximate phase diagrams rely on a pair of Lindemann criteria, suitably adapted to deal with effects of thermal fluctuations in two dimensions. As part of our analysis, we determine the phonon spectra of quantum particles on a triangular lattice interacting with repulsive $1/r^3$ and $1/r$ potentials. A large softening of the transverse and longitudinal phonon frequencies, due to both lattice effects and quantum fluctuations, plays a significant role in our analysis. The hexatic phase is predicted to survive down to very low temperatures.

We briefly review a theoretical picture of the quantum Hall effect in which gauge symmetry of non-relativistic quantum mechanics, the representation theory of U(1)-current algebra, and the study of odd, positive integral quadratic forms on N-dimensional integral lattices play fundamental roles. We present a classification of so-called minimal incompressible quantum Hall fluids for low values of the lattice dimension N. The classification contains several new universality classes of quantum Hall fluids which are physically interesting.

Strongly correlated quantum fluids are phases of matter that are intrinsically quantum mechanical and that do not have a simple description in terms of weakly interacting quasiparticles. Two systems that have recently attracted a great deal of interest are the quark gluon plasma, a plasma of strongly interacting quarks and gluons produced in relativistic heavy ion collisions, and ultracold atomic Fermi gases, very dilute clouds of atomic gases confined in optical or magnetic traps. These systems differ by 19 orders of magnitude in temperature, but were shown to exhibit very similar hydrodynamic flows. In particular, both fluids exhibit a robustly low shear viscosity to entropy density ratio, which is characteristic of quantum fluids described by holographic duality, a mapping from strongly correlated quantum field theories to weakly curved higher dimensional classical gravity. This review explores the connection between these fields, and also serves as an introduction to the focus issue of New Journal of Physics on Strongly Correlated Quantum Fluids: From Ultracold Quantum Gases to Quantum Chromodynamic Plasmas . The presentation is accessible to the general physics reader and includes discussions of the latest research developments in all three areas.

The theory relevant to the study of matter in equilibrium with the radiation field is thermal quantum electrodynamics (TQED). We present a formulation of the theory, suitable for non relativistic fluids, based on a joint functional integral representation of matter and field variables. In this formalism cluster expansion techniques of classical statistical mechanics become operative. They provide an alternative to the usual Feynman diagrammatics in many-body problems which is not perturbative with respect to the coupling constant. As an application we show that the effective Coulomb interaction between quantum charges is partially screened by thermalized photons at large distances. More precisely one observes an exact cancellation of the dipolar electric part of the interaction, so that the asymptotic particle density correlation is now determined by relativistic effects. It has still the $r^{-6}$ decay typical for quantum charges, but with an amplitude strongly reduced by a relativistic factor.

Motivated by recent experimental results, we reconsider the theory of the edge excitations for the fractional Hall effect at filling factors $\nu=p/(2np+1)$. We propose to modify the standard $u(1)\otimes su(p)$ edge theory for this series by introducing twist fields which change the boundary conditions of the bosonic fields and simulate the effect of fractions of flux quanta $\phi_0/p$. This has the effect of removing the conserved charges associated to the neutral modes while keeping the right statistics of the particles. The Green function of the electron in presence of twists decays at long distance with an exponent varying continuously with $\nu$.

Motivated by recent experimental results, we reconsider the theory of the edge excitations for the fractional Hall effect at filling factors $\nu=p/(2np+1)$. We propose to modify the standard $u(1)\otimes su(p)$ edge theory for this series by introducing twist fields which change the boundary conditions of the bosonic fields and simulate the effect of fractions of flux quanta $\phi_0/p$. This has the effect of removing the conserved charges associated to the neutral modes while keeping the right statistics of the particles. The Green function of the electron in presence of twists decays at long distance with an exponent varying continuously with $\nu$.

We employ recently developed approximation methods in the hybrid quantization of the Gowdy $T^3$ model with linear polarization and a massless scalar field to obtain physically interesting solutions of this inhomogeneous cosmology. More specifically, we propose approximate solutions of the quantum Gowdy model constructed in such a way that, for the Hamiltonian constraint, they effectively behave as those corresponding to a flat homogeneous and isotropic universe filled with a perfect fluid, even though these quantum states are far from being homogeneous and isotropic. We analyze how one can get different perfect fluid effective behaviors, including the cases of dust, radiation, and cosmological constant.

We study the one-point and two-point Green's functions in a complex random matrix model to sub-leading orders in the large N limit. We take this complex matrix models as a model for the two-state scattering problem, as applied to spin dependent scattering of impurities in quantum Hall fluids. The density of state shows a singularity at the band center due to reflection symmetry. We also compute the one-point Green's function for a generalized situation by putting random matrices on a lattice of arbitrary dimensions.

We describe an analog model for quantum gravity effects in condensed matter physics. The situation discussed is that of phonons propagating in a fluid with a random velocity wave equation. We consider that there are random fluctuations in the reciprocal of the bulk modulus of the system and study free phonons in the presence of Gaussian colored noise with zero mean. We show that in this model, after performing the random averages over the noise function a free conventional scalar quantum field theory describing free phonons becomes a self-interacting model.

We formally derive the hydrodynamic limit of a system modelling a bosons gas having a condensed part, made of a quantum kinetic and a Gross-Pitaevskii equation. The limit model, which is a two-fluids Euler system, is approximated by an isentropic system, which is then studied. We find in particular some conditions for the hyperbolicity, and we study the weak solutions. A numerical example is given at the end.

The elementary excitations of fractional quantum Hall (FQH) fluids are vortices with fractional statistics. Yet, this fundamental prediction has remained an open experimental challenge. Here we show that the cross current noise in a three-terminal tunneling experiment of a two dimensional electron gas in the FQH regime can be used to detect directly the statistical angle of the excitations of these topological quantum fluids. We show that the noise also reveals signatures of exclusion statistics and of fractional charge. The vortices of Laughlin states should exhibit a ``bunching'' effect, while for higher states in the Jain sequences they should exhibit an ``anti-bunching'' effect.

To model isotropic homogeneous quantum turbulence in superfluid helium, we have performed Direct Numerical Simulations (DNS) of two fluids (the normal fluid and the superfluid) coupled by mutual friction. We have found evidence of strong locking of superfluid and normal fluid along the turbulent cascade, from the large scale structures where only one fluid is forced down to the vorticity structures at small scales. We have determined the residual slip velocity between the two fluids, and, for each fluid, the relative balance of inertial, viscous and friction forces along the scales. Our calculations show that the classical relation between energy injection and dissipation scale is not valid in quantum turbulence, but we have been able to derive a temperature--dependent superfluid analogous relation. Finally, we discuss our DNS results in terms of the current understanding of quantum turbulence, including the value of the effective kinematic viscosity.

The problem of time is an unsolved issue of canonical General Relativity. A possible solution is the Brown-Kuchar mechanism which couples matter to the gravitational field and recovers a physical, i.e. non vanishing, observable Hamiltonian functional by manipulating the set of constraints. Two cases are analyzed. A generalized scalar fluid model provides an evolutionary picture, but only in a singular case. The Schutz' model provides an interesting singularity free result: the entropy per baryon enters the definition of the physical Hamiltonian. Moreover in the co-moving frame one is able to identify the time variable tau with the logarithm of entropy.

From a simple analysis of particle orbits and fluid flows in presence or not of dissipation, some connections between apparently uncorrelated research areas are made. The main results point out for a deep relation between quantization of classical conservative systems and the dissipative version of these systems.

We consider thermodynamics of the van der Waals fluid of quantum systems. We derive general relations of thermodynamic functions and parameters of any ideal gas and the corresponding van der Waals fluid. This provides unambiguous generalization of the classical van der Waals theory to quantum statistical systems. As an example, we apply the van der Waals fluid with fermi statistics to characterize the liquid-gas critical point in nuclear matter. We also introduce the Bose-Einstein condensation in the relativistic van der Waals boson gas, and argue, that it exhibits two-phase structure separated in space.

For infinite (bulk) quantum fluids of particles interacting via pairwise sufficiently smooth interactions, the Wigner-Kirkwood formalism provides a semiclassical expansion of the Boltzmann density in configuration space in even powers of the thermal de Broglie wavelength $\lambda$. This result permits one to generate an analogous $\lambda$-expansion for the bulk free energy and many-body densities. The present paper brings a technically nontrivial generalization of the Wigner-Kirkwood technique to semi-infinite quantum fluids, constrained by a plane hard wall impenetrable to particles. In contrast to the bulk case, the resulting Boltzmann density involves also position-dependent terms of type $\exp(-2x^2/\lambda^2)$ ($x$ denotes the distance from the wall boundary) which are non-analytic in $\lambda$. Under some condition, the analyticity in $\lambda$ is restored by integrating the Boltzmann density over configuration space; however, in contrast to the bulk free energy, the semiclassical expansion of the surface part of the free energy (surface tension) contains odd powers of $\lambda$, too. Explicit expressions for the leading quantum corrections in the presence of the boundary are given for the one-body and two-body densities. As model systems for explicit calculations, we use Coulomb fluids, in particular the one-component plasma defined in the $\nu$-dimensional (integer $\nu\ge 2$) space.

We review the effective field theory treatment of topological quantum fluids, focussing on the Hall fluids.

We use the field theory description of the fractional quantum Hall states to derive the universal response of these topological fluids to shear deformations and curvature of their background geometry, i.e. the Hall viscosity, the Wen-Zee term, and the gravitational Chern-Simons term. To account for the coupling to the background geometry, we show that the concept of flux attachment needs to be modified and use it to derive the geometric responses from Chern-Simons theories. We show that the resulting composite particles minimally couple to the spin connection of the geometry. We derive a consistent theory of geometric responses from the Chern-Simons effective field theories and from parton constructions, and apply it to both abelian and non-abelian states.

We use the field theory description of the fractional quantum Hall states to derive the universal response of these topological fluids to shear deformations and curvature of their background geometry, i.e. the Hall viscosity, the Wen-Zee term, and the gravitational Chern-Simons term. To account for the coupling to the background geometry, we show that the concept of flux attachment needs to be modified and use it to derive the geometric responses from Chern-Simons theories. We show that the resulting composite particles minimally couple to the spin connection of the geometry. We derive a consistent theory of geometric responses from the Chern-Simons effective field theories and from parton constructions, and apply it to both abelian and non-abelian states.

A central long standing prediction of the theory of fractional quantum Hall (FQH) states that it is a topological fluid whose elementary excitations are vortices with fractional charge and fractional statistics. Yet, the unambiguous experimental detection of this fundamental property, that the vortices have fractional statistics, has remained an open challenge. Here we propose a three-terminal ``T-junction'' as an experimental setup for the direct and independent measurement of the fractional charge and statistics of fractional quantum Hall quasiparticles via cross current noise measurements. We present a non-equilibrium calculation of the quantum noise in the T-junction setup for FQH Jain states. We show that the cross current correlation (noise) can be written in a simple form, a sum of two terms, which reflects the braiding properties of the quasiparticles: the statistics dependence captured in a factor of $\cos\theta$ in one of two contributions. Through analyzing these two contributions for different parameter ranges that are experimentally relevant, we demonstrate that the noise at finite temperature reveals signatures of generalized exclusion principles, fractional exchange statistics and fractional charge. We also predict that the vortices of Laughlin states exhibit a ``bunching'' effect, while higher states in the Jain sequences exhibit an ``anti-bunching'' effect.

In this paper, we pursue our analysis of the W-infinity symmetry of the low-energy edge excitations of incompressible quantum Hall fluids. These excitations are described by (1+1)-dimensional effective field theories, which are built by representations of the W-infinity algebra. Generic W-infinity theories predict many more fluids than the few, stable ones found in experiments. Here we identify a particular class of W-infinity theories, the minimal models, which are made of degenerate representations only - a familiar construction in conformal field theory. The W-infinity minimal models exist for specific values of the fractional conductivity, which nicely fit the experimental data and match the results of the Jain hierarchy of quantum Hall fluids. We thus obtain a new hierarchical construction, which is based uniquely on the concept of quantum incompressible fluid and is independent of Jain's approach and hypotheses. Furthermore, a surprising non-Abelian structure is found in the W-infinity minimal models: they possess neutral quark-like excitations with SU(m) quantum numbers and non-Abelian fractional statistics. The physical electron is made of anyon and quark excitations. We discuss some properties of these neutral excitations which could be seen in experiments and numerical simulations.

We review our recent work on the algebraic characterization of quantum Hall fluids. Specifically, we explain how the incompressible quantum fluid ground states can be classified by effective edge field theories with the W-infinity dynamical symmetry of ``quantum area-preserving diffeomorphisms''. Using the representation theory of W-infinity, we show how all fluids with filling factors $\nu = m/(pm +1)$ and $\nu = m/(pm-1)$ with $m$ and $p$ positive integers, $p$ even, correspond exactly to the W-infinity {\it minimal models}.

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It is shown that gauge theories with fermions are most naturally studied via a polar decomposition of the field variable. This is the fermionic analog of the preprint cond-mat/0210673. The hope is that these two put together will enable the treatment of neutral nonrelativisitc matter composed of electrons and nuclei in a nonperturbative manner with nuclei and electrons treated on an equal footing. We recast the electron-phonon (superconductivity) problem in the hydrodynamic language and indicate how it is solved. In particular we focus on the a.c. conductivity.

We study polar molecules in a stack of strongly confined pancake traps. When dipolar moments point perpendicular to the planes of the traps and are sufficiently strong, the system is stable against collapse but attractive interaction between molecules in different layers leads to the formation of extended chains of molecules, analogously to the chaining phenomenon in classical rheological electro- and magnetofluids. We analyze properties of the resulting quantum liquid of dipolar chains and show that only the longest chains undergo Bose-Einstein condensation with a strongly reduced condensation temperature. We discuss several experimental methods for studying chains of dipolar molecules.

A generalized diffusion Monte Carlo method for solving the many-body Schr\"odinger equation on curved manifolds is introduced and used to perform a `fixed-phase' simulation of the fractional quantum Hall effect on the Haldane sphere. This new method is used to study the effect of Landau level mixing on the $\nu=1/3$ energy gap and the relative stability of spin-polarized and spin-reversed quasielectron excitations.

We analyze a hydrodynamical model of a polar fluid in (3+1)-dimensional spacetime. We explore a spacetime symmetry -- volume preserving diffeomorphisms -- to construct an effective description of this fluid in terms of a topological BF theory. The two degrees of freedom of the BF theory are associated to the mass (charge) flows of the fluid and its polarization vorticities. We discuss the quantization of this hydrodynamic theory, which generically allows for fractionalized excitations. We propose an extension of the Girvin-MacDonald-Platzman algebra to (3+1)-dimensional spacetime by the inclusion of the vortex-density operator in addition to the usual charge density operator and show that the same algebra is obeyed by massive Dirac fermions that represent the bulk of $\mathbb{Z}^{\,}_{2}$ topological insulators in three-dimensional space.

We analyze a hydrodynamical model of a polar fluid in (3+1)-dimensional spacetime. We explore a spacetime symmetry -- volume preserving diffeomorphisms -- to construct an effective description of this fluid in terms of a topological BF theory. The two degrees of freedom of the BF theory are associated to the mass (charge) flows of the fluid and its polarization vorticities. We discuss the quantization of this hydrodynamic theory, which generically allows for fractionalized excitations. We propose an extension of the Girvin-MacDonald-Platzman algebra to (3+1)-dimensional spacetime by the inclusion of the vortex-density operator in addition to the usual charge density operator and show that the same algebra is obeyed by massive Dirac fermions that represent the bulk of $\mathbb{Z}^{\,}_{2}$ topological insulators in three-dimensional space.

The behavior of quantum fluids (4He and H2) within nanopores is explored in various regimes, using several different methods. A focus is the evolution of each fluid's behavior as pore radius R is increased. Results are derived with the path integral Monte Carlo method for the finite temperature (T) behavior of quasi-one-dimensional (1D) liquid 4He and liquid H2, within pores of varying R. Results are also obtained, using a density functional method, for the T=0 behavior of 4He in pores of variable R.

The behavior of quantum fluids (4He and H2) within nanopores is explored in various regimes, using several different methods. A focus is the evolution of each fluid's behavior as pore radius R is increased. Results are derived with the path integral Monte Carlo method for the finite temperature (T) behavior of quasi-one-dimensional (1D) liquid 4He and liquid H2, within pores of varying R. Results are also obtained, using a density functional method, for the T=0 behavior of 4He in pores of variable R.

We review quantum Monte Carlo results on energetic and structure properties of quantum fluids adsorbed in a bundle of carbon nanotubes. Using realistic interatomic interactions the different adsorption sites that a bundle offer are accurately studied and compared in some cases with strictly one-dimensional geometries. The study is performed quite extensively for $^4$He and restricted to the inner part of a single nanotube for H$_2$ and D$_2$. From a theoretical point of view, nanotubes open the real possibility of a quasi-one-dimensional confinement where to study quantum fluids in extremely reduced dimensionality. The results obtained show that in the narrowest configurations the system is nearly one-dimensional reinforcing the interest on the physics of one-dimensional quantum fluids. Experimental confirmation of the theoretical results obtained is still not in a satisfactory situation due to the difficulties on extracting from the data the dominant adsorption sites.

The kinetic part of the Rasetti-Regge action I_{RR} for vortex lines is studied and links to string theory are made. It is shown that both I_{RR} and the Polyakov string action I_{Pol} can be constructed with the same field X^mu. Unlike I_{NG}, however, I_{RR} describes a Schwarz-type topological quantum field theory. Using generators of classical Lie algebras, I_{RR} is generalized to higher dimensions. In all dimensions, the momentum 1-form P constructed from the canonical momentum for the vortex belongs to the first cohomology class H^1(M,R^m) of the worldsheet M swept-out by the vortex line. The dynamics of the vortex line thus depend directly on the topology of M. For a vortex ring, the equations of motion reduce to the Serret-Frenet equations in R^3, and in higher dimensions they reduce to the Maurer-Cartan equations for so(m).

The thermodynamic behaviour of a relativistic perfect simple fluid obeying the equation of state $p=(\gamma-1)\rho $, where $0 \le \gamma \le 2$ is a constant, has been investigated. Particular cases include: vacuum($p=-\rho $, $\gamma=0$), a randomly oriented distribution of cosmic strings ($p=-{1 \over 3} \rho $, $\gamma =2/3$), blackbody radiation ($p={1\over 3} \rho$, $\gamma =4/3$) and stiff matter ($p=\rho$, $\gamma=2$). Fluids with $\gamma

We compute directly the entanglement entropy of spatial regions in Chern-Simons gauge theories in 2+1 dimensions using surgery. We use these results to determine the universal topological piece of the entanglement entropy for Abelian and non-Abelian quantum Hall fluids.

We analyze the experimental observation of metastable anisotropy resistance orientation at half filled quantum Hall fluids by means of a model of a quantum nematic liquid in an explicit symmetry breaking potential. We interpret the observed ``rotation'' of the anisotropy axis as a process of nucleation of nematic domains and compute the nucleation rate within this model. By comparing with experiment, we are able to predict the critical radius of nematic bubbles, $R_c\sim 2.6 \mu m $. Each domain contains about $10^4$ electrons.

We analyze the experimental observation of metastable anisotropy resistance orientation at half filled quantum Hall fluids by means of a model of a quantum nematic liquid in an explicit symmetry breaking potential. We interpret the observed ``rotation'' of the anisotropy axis as a process of nucleation of nematic domains and compute the nucleation rate within this model. By comparing with experiment, we are able to predict the critical radius of nematic bubbles, $R_c\sim 2.6 \mu m $. Each domain contains about $10^4$ electrons.

We present a simple approach to calculate the degeneracy and the structure of the ground states of non-abelian quantum Hall (QH) liquids on the torus. Our approach can be applied to any QH liquids (abelian or non-abelian) obtained from the parton construction. We explain our approach by studying a series of examples of increasing complexity. When the effective theory of a non-abelian QH liquid is a non-abelian Chern-Simons (CS) theory, our approach reproduces the well known results for the ground state degeneracy of the CS theory. However, our approach also apply to non-abelian QH liquids whose effective theories are not known and which cannot be written as a non-abelian CS theory. We find that the ground states on a torus of all non-abelian QH liquids obtained from the parton construction can be described by points on a lattice inside a "folded unit cell." The folding is generated by reflection, rotations, etc. Thus the ground state structures on the torus described by the ``folded unit cells'' provide a way to (at least partially) classify non-abelian QH liquids obtained from the parton construction.

We measure the shear viscosity in a two-component Fermi gas of atoms, tuned to a broad s-wave collisional (Feshbach) resonance. At resonance, the atoms strongly interact and exhibit universal behavior, where the equilibrium thermodynamic properties and the transport coefficients are universal functions of the density $n$ and temperature $T$. We present a new calibration of the temperature as a function of global energy, which is directly measured from the cloud profiles. Using the calibration, the trap-averaged shear viscosity in units of $\hbar\,n$ is determined as a function of the reduced temperature at the trap center, from nearly the ground state to the unitary two-body regime. Low temperature data is obtained from the damping rate of the radial breathing mode, while high temperature data is obtained from hydrodynamic expansion measurements. We also show that the best fit to the high temperature expansion data is obtained for a vanishing bulk viscosity. The measured trap-averaged entropy per particle and shear viscosity are used to estimate the ratio of the shear viscosity to the entropy density, which is compared that conjectured for a perfect fluid.

In recent years the problem of correct description of quantum fluids in the confined geometry at nanoscale length has emerged. It has been recognized that the quantum fluids at these circumstances can be considered as a new state of quantum matter due to close values between characteristic lengths for these quantum liquids and the size of geometrical confinement and significant contribution from the surface atoms. So one has to apply new physics to describe such systems with taking into account their complex nature. For example, last two years the attempts to develop the fractionalized two-fluid hydrodynamics for nanoporous media with fractal dimensions have been made. The actuality of such new hydrodynamics becomes very clear for the last development in chemical synthesis of different kind of aerogels with nanopore structure as well as numerous studies of nanoporous substances. One of the interesting obtained results that density waves (the first sound) and temperature waves (the second sound) become strong coupled even in the absence of viscosity, so it is purely geometric effect of fractal space of nanopores. In the present report we will review the procedure, results and discuss the issues for this approach.

In this paper we address the problem of the quantization of the perfect relativistic fluids formulated in terms of the K\"{a}hler parametrization. This fluid model describes a large set of interesting systems such as the power law energy density fluids, Chaplygin gas, etc. In order to maintain the generality of the model, we apply the BRST method in the reduced phase space in which the fluid degrees of freedom are just the fluid potentials and the fluid current is classically resolved in terms of them. We determine the physical states in this setting, the time evolution and the path integral formulation.

The viscosity of strongly interacting systems is a topic of great interest in diverse fields. We focus here on the bulk and shear viscosities of \emph{non-relativistic} quantum fluids, with particular emphasis on strongly interacting ultracold Fermi gases. We use Kubo formulas for the bulk and shear viscosity spectral functions, $\zeta(\omega)$ and $\eta(\omega)$ respectively, to derive exact, non-perturbative results. Our results include: a microscopic connection between the shear viscosity $\eta$ and the normal fluid density $\rho_n$; sum rules for $\zeta(\omega)$ and $\eta(\omega)$ and their evolution through the BCS-BEC crossover; universal high-frequency tails for $\eta(\omega)$ and the dynamic structure factor $S({\bf q}, \omega)$. We use our sum rules to show that, at unitarity, $\zeta(\omega)$ is identically zero and thus relate $\eta(\omega)$ to density-density correlations. We predict that frequency-dependent shear viscosity $\eta(\omega)$ of the unitary Fermi gas can be experimentally measured using Bragg spectroscopy.

A "quasiclassical" approximation to the quantum spectrum of the Schroedinger equation is obtained from the trace of a quasiclassical evolution operator for the "hydrodynamical" version of the theory, in which the dynamical evolution takes place in the extended phase space $[q(t),p(t),M(t)] = [q_i, \partial_i S, \partial_i \partial_j S ]$. The quasiclassical evolution operator is multiplicative along the classical flow, the corresponding quasiclassical zeta function is entire for nice hyperbolic flows, and its eigenvalue spectrum contains the spectrum of the semiclassical zeta function. The advantage of the quasiclassical zeta function is that it has a larger analyticity domain than the original semiclassical zeta function; the disadvantage is that it contains eigenvalues extraneous to the quantum problem. Numerical investigations indicate that the presence of these extraneous eigenvalues renders the original Gutzwiller-Voros semiclassical zeta function preferable in practice to the quasiclassical zeta function presented here. The cumulant expansion of the exact quantum mechanical scattering kernel and the cycle expansion of the corresponding semiclassical zeta function part ways at a threshold given by the topological entropy; beyond this threshold quantum mechanics cannot resolve fine details of the classical chaotic dynamics.

In this paper, the key ideas of characterizing universality classes of dissipation-free (incompressible) quantum Hall fluids by mathematical objects called quantum Hall lattices are reviewed. Many general theorems about the classification of quantum Hall lattices are stated and their physical implications are discussed. Physically relevant subclasses of quantum Hall lattices are defined and completely classified. The results are carefully compared with experimental data and also with other theoretical schemes (the hierarchy schemes). Several proposals for new experiments are made which could help to settle interesting issues in the theory of the (fractional) quantum Hall effect and thus would lead to a deeper understanding of this remarkable effect.