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arXiv:cond-mat/0401492

Quantum Hydrodynamics of Fermi Fluids

Girish S. Setlur

Published Jan 26, 2004• 1 citation
arXivHTMLPDF

Abstract

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.

DOI
10.48550/arxiv.cond-mat/0401492

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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.
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The work in this thesis is written with the following question in mind: does the physics investigated in colliders like the one at CERN have its counterpart in phenomena like superfluidity and superconductivity? \nMore precisely, this thesis focuses on the condensed matter parallel of non-Abelian gauge theories exploited in high-energy physics. These theories capture the notion of parallel transport and offer the possibility of deconfinement, i.e., the existence of quarks and gluons. In the first part of the thesis, the notion of parallel transport in the context of spin-orbit coupled systems is studied, which have attracted much attention in recent years. The question arises whether hydrodynamic conservation laws for non-Abelian spin currents exist. The answer is yes, provided the spin system is in a superfluid state.\nIn the second part, the doped Mott insulator is studied, which is relevant for the problem of high-temperature superconductivity. A gauge theory considering the electron as being split up into spinons and holons (a condensed matter analogue of quarks and gluons) is studied. It is shown how this theory, when some subtle issues are treated correctly, leads to physical predictions which can be (and, in some cases, have been) confirmed experimentally.
We study thermal conductivity in one-dimensional electronic fluids combining kinetic [R. Samanta, I. V. Protopopov, A. D. Mirlin, and D. B. Gutman, Thermal transport in one-dimensional electronic fluid, Phys. Rev. Lett. 122, 206801 (2019)] and hydrodynamic [I. V. Protopopov, R. Samanta, A. D. Mirlin, and D. B. Gutman, Anomalous hydrodynamics in one-dimensional electronic fluid, Phys. Rev. Lett. 126, 256801 (2021)] theories. The kinetic approach is developed by partitioning the Hilbert space into bosonic and fermionic sectors. We focus on the regime where the long-living thermal excitations are fermions and compute thermal conductivity. From the kinetic theory standpoint, the fermionic part of thermal conductivity is normal, while the bosonic one is anomalous, that scales as ω–1/3 and thus dominates in the infrared limit. The multi-mode hydrodynamic theory is obtained by projecting the fermionic kinetic equation on the zero modes of its collision integral. On a bare level, both theories agree and the thermal conductivity computed in hydrodynamic theory matches the result of the kinetic equation. The interaction between hydrodynamic modes leads to renormalization and consequently to anomalous scaling of the transport coefficients. In a four-mode regime, all modes are ballistic and the anomaly manifests itself in Kardar-Parisi-Zhang-like broadening with asymmetric power-law tails. “Heads” and “tails” of the pulses contribute equally to thermal conductivity, leading to ω–1/3 scaling of heat conductivity. In the three-mode regime, the system is in the universality class of a classical viscous fluid [Herbert Spohn, Nonlinear fluctuating hydrodynamics for anharmonic chains, J. Stat. Phys. 154, 1191 (2014); O. Narayan and S. Ramaswamy, Anomalous heat conduction in one-dimensional momentum-conserving systems, Phys. Rev. Lett. 89, 200601 (2002)].
This thesis considers out-of-equilibrium dynamics of strongly interacting non-relativistic Fermi gases in several two and three dimensional geometries. The tools of second-order hydrodynamics and gauge-gravity duality will be utilized to address this system. Many of the themes of this work are motivated by the observed similarities in transport properties between strongly interacting Fermi gases and other very different strongly interacting quantum fluids such as the quark-gluon plasma, high temperature superconductors, and quantum field theories described by gauge-gravity duality. In particular, these systems all nearly saturate the conjectured lower bound on the ratio of shear viscosity to entropy density η/s≥ℏ/(4πkB)η/s \geq \hbar/(4 πk_B)η/s≥ℏ/(4πkB​) coming from the AdS/CFT correspondence. Among other things, this observation, in conjunction with current experiment and data analysis in atomic, condensed matter, and nuclear physics lends itself to the following questions: How perfect of a fluid is the strongly interacting Fermi gas, and can one find a more stringent constraint on η/sη/sη/s in Fermi gases? Do the similarities in transport properties among strongly interacting quantum systems extend beyond dynamics controlled by the hydrodynamical shear viscosity? In regards to the first question, by utilizing second-order hydrodynamics, it will be demonstrated that higher-order collective modes of a harmonically trapped Fermi gas may serve as a more sensitive probe of the shear viscosity. For the second question, both second-order hydrodynamics and a gravity dual theory are used to make predictions about dynamics occurring on short timescales where hydrodynamics is expected to break down. In particular the appearance of a class of "non-hydrodynamic" collective modes not contained within a Navier-Stokes description of the strongly interacting Fermi gas will be discussed.
In Chapter 1 we start with general Landau scheme of the conservation laws for the hydrodynamics of classical liquids and superfluids. On the basis of Landau scheme we consider hydrodynamics of rotating superfluids with large number of vortices. We study different regimes of slow and fast rotations. In Chapter 4 we consider complicated and yet unresolved problem of chiral anomaly in superfluid 3He-A and present two competing approaches. The first one is based on supersymmetric hydrodynamics, another one on the formal analogy with Dirac equation in QED. In Chapters 6 and 7 we study composed fermions and bosons which correspond to the different bound states consisting of two fermions, two bosons and fermion and boson in ultracold quantum gasses and mixtures. We also elucidate the exact solutions of the Skorniakov-Ter-Martirosian integral equations. We present the basic ideas of Nozieres-Schmitt Rink and Legget theory on BCS-BEC crossover in quantum gasses and construct the phase diagrams for the s-wave and p-wave pairings. In Chapters 9 and 10 we discuss non-phonon mechanisms of superconductivity of the Kohn-Luttinger type for different models in 3D and 2D electron systems at low densities including Fermi-gas, Hubbard and Shubin-Vonsovsky models. We complete the superconductive phase-diagrams of p-wave and d-wave pairing and discuss the possibility to increase Tc already at low density. In Chapters 15 and 16 the mechanisms of electron nanoscale phase-separation in manganite-type oxide materials are analyzed. The instability of magnetic (or charge) homogeneous ordering towards the formation of droplet structures (magnetic polarons) is predicted for a wide parameter range on the phase-diagram. The transport properties of these materials in the phase-separated state are also discussed.