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We use high dimensional bosonization to derive an effective field theory that describes the Pomeranchuck transition in two-dimensional Fermi liquids. The bosonization approach explicitly retains all low-energy degrees of freedom of the system. The resultant theory has dynamical exponent z=2 at tree level and upper critical dimension d_c=2, thus in 2D the system is at the upper critical dimension. These results differ from those of an earlier study based on integrating out fermions.

We introduce a new operator algebra for the description of the low-energy physics of one-dimensional, integrable, multicomponent quantum liquids. Considering the particular case of the Hubbard chain in a constant external magnetic field and with varying chemical potential, we show that at low energy its Bethe-ansatz solution can be interpreted in terms of the new {\it pseudoparticle operator algebra}. Our algebraic approach provides a concise interpretation of and justification for several recent studies of low-energy excitations and transport which have been based on detailed analyses of specific Bethe-ansatz eigenfunctions and eigenenergies. A central point is that the {\it exact ground state} of the interacting many-electron problem is the non-interacting pseudoparticle ground state. Furthermore, in the pseudoparticle basis, the quantum problem becomes perturbative, {\it i.e.}, the two-pseudoparticle forward-scattering vertices and amplitudes do not diverge, and one can define a many-pseudoparticle perturbation theory. We write the general quantum-liquid Hamiltonian in the new basis and show that the pseudoparticle-perturbation theory leads, in a natural way, to the generalized Landau-liquid approach.

We discuss a nexus among quantum topology, quantum physics and quantum computing.

Concentrating on zero temperature Quantum Monte Carlo calculations of electronic systems, we give a general description of the theory of finite size extrapolations of energies to the thermodynamic limit based on one and two-body correlation functions. We introduce new effective procedures, such as using the potential and wavefunction split-up into long and short range functions to simplify the method and we discuss how to treat backflow wavefunctions. Then we explicitly test the accuracy of our method to correct finite size errors on example hydrogen and helium many-body systems and show that the finite size bias can be drastically reduced for even small systems.

Concentrating on zero temperature Quantum Monte Carlo calculations of electronic systems, we give a general description of the theory of finite size extrapolations of energies to the thermodynamic limit based on one and two-body correlation functions. We introduce new effective procedures, such as using the potential and wavefunction split-up into long and short range functions to simplify the method and we discuss how to treat backflow wavefunctions. Then we explicitly test the accuracy of our method to correct finite size errors on example hydrogen and helium many-body systems and show that the finite size bias can be drastically reduced for even small systems.

We use a density-functional theoretical approach to set up a computationally simple self-consistent scheme to calculate the pair distribution functions and the effective interactions in quantum Coulomb liquids. We demonstrate the accuracy of the approach for different statistics and space dimensionalities by reporting results for a two-dimensional electron gas and for a three-dimensional boson plasma over physically relevant ranges of coupling strength, in comparison with Monte Carlo data.

We apply density functional theory to study the freezing of superfluid {$^{4}\rm{He}$}, charged bosons and charged fermions at zero temperature. We employ accurate Quantum Monte Carlo data for the linear response function in the uniform phase of these systems, a quantity that has different behavior for large values of the wavevector than previously assumed. We find that, as a result of this {\it{exact}} behavior, different approximations in the density functional theory of freezing that involve linear response, all fail to correctly describe the crystallization in {\it{three dimensions}}, while yielding satisfactory predictions in {\it{two dimensions}}. This demonstrates the shortcomings of the currently popular density functional approximate theories to describe $3d$-freezing in the quantum regime. We also investigate the consequences of the exact asymptotic behavior of response functions on the form of effective interactions and polarization potentials in the electron gas, at small distances.

We describe spin correlations in the vicinity of a generalized impurity in a wide class of fractionalized spin liquid states. We argue that the primary characterization of the impurity is its electric charge under the gauge field describing singlet excitations in the spin liquid. We focus on two gapless U(1) spin liquids described by 2+1 dimensional conformal field theories (CFT): the staggered flux (sF) spin liquid, and the deconfined critical point between the Neel and valence-bond-solid (VBS) states. In these cases, the electric charge is argued to be an exactly marginal perturbation of the CFT. Consequently, the impurity susceptibility has a 1/T temperature dependence, with an anomalous Curie constant which is a universal number associated with the CFT. One unexpected feature of the CFT of the sF state is that an applied magnetic field does not induce any staggered spin polarization in the vicinity of the impurity (while such a staggered magnetization is present for the Neel-VBS case). These results differ significantly from earlier theories of vacancies in the sF state, and we explicitly demonstrate how our gauge theory corrects these works. We discuss implications of our results for the cuprate superconductors and organic Mott insulators.

A phenomenological theory is presented for two-dimensional quantum liquids in terms of the Fermi surface geometry. It is shown that there is a one-to-one correspondence between the properties of an interacting electron system and its corresponding Fermi surface. By doing this, the concept of Fermi surface is generalized to include different topologies. It is shown that for a Fermi liquid the corresponding Fermi surface is rough. In the presence of a condensate, the Fermi surface is faceted, while for a ferromagnetc instability, the Fermi surface becomes a frozen solid. I also determine the surface tension, the step free energy, low lying excitations and other surface and transport properties of the Fermi surface. The different transitions between these phases are also determined. A non-Fermi liquid phase is shown to be a pre-roughening state of the Fermi surface, the properties of which are briefly analyzed.

A simple and physically transparent magnetoelasticity theory is proposed to describe linear dynamics of incompressible fractional quantum Hall states. The theory manifestly satisfies the Kohn theorem and the $f$-sum rule, and predicts a gaped intra-Landau level collective mode with a roton minimum. In the limit of vanishing bare mass $m$ the correct form of the static structure factor, $s(q)\sim q^4$, is recovered. We establish a connection of the present approach to the fermionic Chern-Simons theory, and discuss further extensions and applications. We also make an interesting analogy of the present theory to the theory of visco-elastic fluids.

We show that the recently developed {\it pseudoparticle operator algebra} which generates the low-energy Hamiltonian eigenstates of multicomponent integrable systems also provides a natural operator representation for the the Virasoro algebras associated with the conformal-invariant character of the low-energy spectrum of the these models. Studying explicitly the Hubbard chain in a non-zero chemical potential and external magnetic field, we establish that the pseudoparticle perturbation theory provides a correct starting point for the construction of a suitable critical-point Hamiltonian. We derive explicit expressions in terms of pseudoparticle operators for the generators of the Virasoro algebras and the energy-momentum tensor, describe the conformal-invariant character of the critical point from the point of view of the response to curvature of the two-dimensional space-time, and discuss the relation to Kac-Moody algebras and dynamical separation.

A comprehensive microscopic dynamical theory is presented for the description of quantum fluids as they transform into glasses. The theory is based on a quantum extension of mode-coupling theory. Novel effects are predicted, such as reentrant behavior of dynamical relaxation times. These predictions are supported by path integral ring polymer molecular dynamics simulations. The simulations provide detailed insight into the factors that govern slow dynamics in glassy quantum fluids. Connection to other recent work on both quantum glasses as well as quantum optimization problems is presented.