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arXiv:1406.4422 · Physical Review Letters

Quantum Field Theory of Fluids

Ben M. Gripaios, Dave Sutherland

Published Feb 18, 2015• 32 citations
arXivHTMLPDF

Abstract

The quantum theory of fields is largely based on studying perturbations around noninteracting, or free, field theories, which correspond to a collection of quantum-mechanical harmonic oscillators. The quantum theory of an ordinary fluid is "freer", in the sense that the noninteracting theory also contains an infinite collection of quantum-mechanical free particles, corresponding to vortex modes. By computing a variety of correlation functions at tree and loop level, we give evidence that a quantum perfect fluid can be consistently formulated as a low-energy, effective field theory. We speculate that the quantum behavior is radically different from both classical fluids and quantum fields.

DOI
10.1103/physrevlett.114.071601

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The quantum theory of fields is largely based on studying perturbations around noninteracting, or free, field theories, which correspond to a collection of quantum-mechanical harmonic oscillators. The quantum theory of an ordinary fluid is "freer", in the sense that the noninteracting theory also contains an infinite collection of quantum-mechanical free particles, corresponding to vortex modes. By computing a variety of correlation functions at tree and loop level, we give evidence that a quantum perfect fluid can be consistently formulated as a low-energy, effective field theory. We speculate that the quantum behavior is radically different from both classical fluids and quantum fields.
We provide a comprehensive picture for the formulation of the perfect fluid in the modern effective field theory formalism at both the classical and quantum level. Due to the necessity of decomposing the hydrodynamical variables (\rho, p, u^\mu) ( ρ , p , u μ ) into other internal degrees of freedom, the procedure is inherently not unique. We discuss and compare the different inequivalent formulations. These theories possess a peculiarity: the presence of an infinite dimensional symmetry implying a vanishing dispersion relation \omega = 0 ω = 0 for the transverse modes. This sets the stage for UV-IR mixing in the quantum theory, which we study in the different formulations focussing on the incompressible limit. We observe that the dispersion relation gets modified by quantum effects to become \omega \propto k^2 ω ∝ k 2 , where the fundamental excitations can be viewed as vortex-anti-vortex pairs. The spectrum exhibits infinitely many types of degenerate quanta. The unusual sensitivity to UV quantum fluctuations renders the implementation of the defining infinite symmetry somewhat subtle. However we present a lattice regularization that preserves a deformed version of such symmetry.
Quantum fluids are fluids, indeed. We have seen in the previous chapter what it means from a classical physics point of view. But quantum fluids are by definition ruled by quantum physics, as well. The goal of this chapter is to introduce some key concepts of quantum physics necessary to understand what makes a fluid a quantum one or rather what quantum property makes an ensemble of particles a quantum fluid. We are here precisely at the edge of our basic intuition of quantum physics. Indeed, one usually associates the notion of quantum object to a microscopic nature and it may thus a priori seem paradoxical to link the word quantum to a macroscopic object, such as the notion of fluid. Thinking it over a bit deeper, quantum physics is certainly present at all scales and we are confronted with quantum mechanics through the many electronic devices which have invaded our everyday life. And it actually turns out that, as we will see later on, most of these quantum devices are directly connected to the notion of quantum fluids, the case of electron fluids being certainly the most common one in such devices. Quantum physics has for sure deeply modified our everyday life. But it has also changed our perception of the world, and as we shall see in this chapter, it is precisely this new vision of the world which allows us to understand the concept of quantum fluids underlying so many physical systems.
The function exp(iwt) describes an oscillating motion. Energy of the oscillator is proportional to the square of w. exp(iwt) is the solution of a differential equation. We have replaced this differential equation by the corresponding finite-time difference equation with d as the smallest span of time. We have, then, symmetrized the equation so that it remains invariant under the change d going to -d. This symmetric equation has solutions in pairs. The angular speed w is modified to w' or w". w' contains a part with an integer. w" contains a part with a half-integer. This corresponds to quantum mechanical oscillator energy levels. F= a.exp(iwt) describes oscillation between -a and +a. If we make w=0, F describes free oscillation between -a and +a. Reciprocal symmetric oscillator, thus, unifies quantum and classical harmonic oscillators on one hand, and a harmonic oscillator and a free particle in a potential well on the other hand. No quantum mechanical postulates are involved.
A quantum fluid is one in which energy quantization has a marked effect on behaviour. Sometimes the effect can be dramatic, as with superfluid helium and with the electron gas in superconducting metals. In this article a simple criterion for distinguishing quantum fluids is obtained, and this is followed by a brief outline of the properties of perfect quantum gases. These ideas are then used as a unifying framework, within which some properties of real quantum fluids sre described and analysed. The approach to the subject is descriptive and physical, with very little dependence on mathematical argument.
For classical field theories with probabilistic initial conditions the classical field observables are an idealization. Their arbitrarily precise values poorly reflect the characteristic uncertainty in the presence of substantial fluctuations. We propose to employ observables based on fluctuating fields. In terms of these "statistical observables" the probabilistic classical field theory becomes a quantum field theory. Non-commuting operators are associated to observables. The quantum rules follow from the laws for classical probabilities. The ``quantum'' is part of the ``classical''. A regularized functional integral guarantees the unitarity of the quantum field theory. We discuss in detail the classical relativistic Klein-Gordon equation with interactions.
In this paper, we examine in a unified fashion dissipative transport in strongly correlated systems. We thereby demonstrate the connection between "bad metals" (such as the high temperature superconductors) and "perfect fluids" (such as the ultracold Fermi gases, near unitarity). One motivation of this work is to communicate to the high energy physics community some of the central unsolved problems in high TcT_cTc​ superconductors. Because of interest in the nearly perfect fluidity of the cold gases and because of new tools such as the AdS/CFT correspondence, this better communication may lead to important progress in a variety of different fields. A second motivation is to draw attention to the great power of transport measurements which more directly reflect the excitation spectrum than, say, thermodynamics and thus strongly constrain microscopic theories of correlated fermionic superfluids. Our calculations show that bad metal and perfect fluid behavior is associated with the presence of a normal state excitation gap which suppresses the effective number of carriers leading to anomalously low conductivity and viscosity above the transition temperature TcT_cTc​. Below TcT_cTc​ we demonstrate that the condensate collective modes ("phonons") do not couple to transverse probes such as the shear viscosity. As a result, our calculated shear viscosity at low TTT becomes arbitrarily small as observed in experiments. In both homogeneous and trap calculations we do not find the upturn in ηηη or η/sη/sη/s (where sss is the entropy density) found in most theories. In the process of these studies we demonstrate compatibility with the transverse sum rule and find reasonable agreement with both viscosity and cuprate conductivity experiments.