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arXiv:2211.09820 · SciPost Physics

The quantum perfect fluid in 2D

Aurélien Dersy, Andrei Khmelnitsky, Riccardo Rattazzi A

Published Jul 24, 2024• 7 citations
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Abstract

We consider the field theory that defines a perfect incompressible 2D fluid. One distinctive property of this system is that the quadratic action for fluctuations around the ground state features neither mass nor gradient term. Quantum mechanically this poses a technical puzzle, as it implies the Hilbert space of fluctuations is not a Fock space and perturbation theory is useless. As we show, the proper treatment must instead use that the configuration space is the area preserving Lie group {S\mathrm{Diff}} S D i f f . Quantum mechanics on Lie groups is basically a group theory problem, but a harder one in our case, since {S\mathrm{Diff}} S D i f f is infinite dimensional. Focusing on a fluid on the 2-torus T^2 T 2 , we could however exploit the well known result {S\mathrm{Diff}}(T^2)\sim SU(N) S D i f f ( T 2 ) ∼ S U ( N ) for N\to ∞ N → ∞ , reducing for finite N N to a tractable case. SU(N) S U ( N ) offers a UV-regulation, but physical quantities can be robustly defined in the continuum limit N\to∞ N → ∞ . The main result of our study is the existence of ungapped localized excitations, the vortons, satisfying a dispersion \omega \propto k^2 ω ∝ k 2 and carrying a vorticity dipole. The vortons are also characterized by very distinctive derivative interactions whose structure is fixed by symmetry. Departing from the original incompressible fluid, we constructed a class of field theories where the vortons appear, right from the start, as the quanta of either bosonic or fermionic local fields.

DOI
10.21468/scipostphys.17.1.019

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We consider the field theory that defines a perfect incompressible 2D fluid. One distinctive property of this system is that the quadratic action for fluctuations around the ground state features neither mass nor gradient term. Quantum mechanically this poses a technical puzzle, as it implies the Hilbert space of fluctuations is not a Fock space and perturbation theory is useless. As we show, the proper treatment must instead use that the configuration space is the area preserving Lie group {S\mathrm{Diff}} S D i f f . Quantum mechanics on Lie groups is basically a group theory problem, but a harder one in our case, since {S\mathrm{Diff}} S D i f f is infinite dimensional. Focusing on a fluid on the 2-torus T^2 T 2 , we could however exploit the well known result {S\mathrm{Diff}}(T^2)\sim SU(N) S D i f f ( T 2 ) ∼ S U ( N ) for N\to ∞ N → ∞ , reducing for finite N N to a tractable case. SU(N) S U ( N ) offers a UV-regulation, but physical quantities can be robustly defined in the continuum limit N\to∞ N → ∞ . The main result of our study is the existence of ungapped localized excitations, the vortons, satisfying a dispersion \omega \propto k^2 ω ∝ k 2 and carrying a vorticity dipole. The vortons are also characterized by very distinctive derivative interactions whose structure is fixed by symmetry. Departing from the original incompressible fluid, we constructed a class of field theories where the vortons appear, right from the start, as the quanta of either bosonic or fermionic local 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.
The canonical quantization of the WZNW model provides a complete set of exchange relations in the enlarged chiral state spaces that include the Gauss components $M_\\pm$, ${\\bar M}_\\pm$ of the monodromy matrices MMM, barM\\bar MbarM. Regarded as new dynamical variables, the elements of MMM and barM\\bar MbarM cannot be identified -- they satisfy different exchange relations. Accordingly, the two dimensional theory expressed in terms of the left and right movers' fields does not automatically respect monodromy invariance. Continuing our recent work on the subject, we show that the definition of the appropriate physical subspace of the SU(n)SU(n)SU(n) WZNW model involves an extended quantum group symmetry in a finite dimensional Fock space. In the simplest (n=2n=2n=2) case we explicitly construct the subspace of Uq(sl(2))otimesUq(sl(2))U_q(sl(2))\\otimes U_q(sl(2))Uq​(sl(2))otimesUq​(sl(2)) invariant vectors. Monodromy invariance is also restored in a weak sense.
The fractional quantum Hall fluid is a two-dimensional quantum fluid of electrons subject to a strong magnetic field at low temperatures.Neutral excitations in a fractional quantum Hall droplet define the incompressibility gap of the topological phase.Among these states, there are some specific modes in the long-wavelength limit that can be understood as "spin-2 gravitons".In this thesis, we will introduce a set of analytical results for the energy gap of the graviton modes for model Hamiltonians in the thermodynamic limit, which is governed by a well-defined and universal characteristic tensor.These results can help to construct model Hamiltonians for the graviton modes of different FQH phases and elucidate a hierarchical structure of conformal Hilbert spaces (null spaces of model Hamiltonians) containing the graviton modes and their corresponding ground states.An isomorphism can be defined for these conformal Hilbert spaces, and the mapping between them can be regarded as a more rigorous and general reinterpretation of the composite fermionization of FQH states, with naturally emergent composite fermions (each consisting of one electron and an even number of fluxes).The results from exact diagonalization will be shown, which confirm that for gapped phases, low-lying neutral excitations can undergo a phase transition even when the ground state remains in the same phase.Furthermore, the gaplessness of the Gaffnian state could be testified based on this formalism with further numerical experiments.Recently there has been numerical evidence implying the signature of multiple graviton modes.We will introduce the microscopic theory for the emergence of multiple gravitons in fractional quantum Hall droplets based on composite fermionization and the well-defined particle-hole conjugate within a specific conformal Hilbert space.This reveals the dynamical nature of this phenomenon and provides theoretical insights into the chirality and the "merging and splitting" behaviors of the graviton modes.The experimental relevance of multiple graviton modes will be discussed.The microscopic theory of gravitons can also provide valuable insights into the field-theoretical approaches.
The main purpose of this paper is to further our theoretical understanding of the fractional quantum Hall effect, in particular of spin effects, in two-dimensional incompressible electron fluids subject to a strong, transverse magnetic field. As a prerequisite for an analysis of the quantum Hall effect, the authors develop a general formulation of the many-body theory of spinning particles coupled to external electromagnetic fields and moving through a general, geometrically nontrivial background. Their formulation is based on a Lagrangian path-integral quantization and is valid in arbitrary coordinates, including coordinates moving according to a volume-preserving flow. It is found that nonrelativistic quantum theory exhibits a fundamental, local U(1)\ifmmode\times\else\texttimes\fi{}SU(2) gauge invariance, and the corresponding gauge fields are identified with physical, external fields. To illustrate the usefulness of their formalism, the authors prove a general form of the quantum-mechanical Larmor theorem and discuss some well-known effects, including the Barnett-Einstein-de Haas effect and superconductivity, emphasizing the implications of U(1)\ifmmode\times\else\texttimes\fi{}SU(2) gauge invariance. They then consider two-dimensional, incompressible quantum fluids in more detail. Exploiting U(1)\ifmmode\times\else\texttimes\fi{}SU(2) gauge invariance, they calculate the leading terms in the effective actions of such systems as functionals of the U(1) and SU(2) gauge fields, on large-distance and low-frequency scales. Among the applications of these results are a simple proof of the Goldstone theorem for spin waves and the linearresponse theory of two-dimensional, incompressible Hall fluids, including a Hall effect for spin currents and sum rules for the response coefficients. For two-dimensional, incompressible systems with broken parity and time-reversal symmetry, a particularly significant implication of U(1)\ifmmode\times\else\texttimes\fi{}SU(2) gauge invariance is a duality between the physics inside the bulk of the system and the physics of gapless, chiral modes propagating along the boundary of the system. These modes form chiral $\stackrel{^}{\mathrm{u}}(1)$ and $\mathrm{s}\stackrel{^}{\mathrm{u}}(2)$ current algebras. The representation theory of these current algebras, combined with natural physical constraints, permits one to derive the quantization of the response coefficients, such as the Hall conductivity. A classification of incompressible Hall fluids is outlined, and many examples, including one concerning a superfluid 3He^{3}\mathrm{He}3He-AB\frac{A}{B}BA​ interface, are discussed.
The possibility of approaching quantum simulation of "topological turbulence" is discussed. In the latter, the topological features of turbulent flow are ascribed to a purely group-theoretic and topological view through the group SDiff of volume-preserving diffeomorphisms and the Mapping Class Group for the 2D surfaces that foliate the 3D flow domain. A class of representations of the group of mapping classes for compact boundaryless surfaces is explicitly constructed in terms of the Witt–Virasoro Hopf-coalgebra and implemented in terms of su(1,1). This leads to a complex, nonlinear Hamiltonian naturally associated with the fluid diffeomorphisms that could lend itself to the quantum simulation of turbulence in Fock space.
The Hamiltonian action of a Lie group on a symplectic manifold induces a momentum map generalizing Noether’s conserved quantity occurring in the case of a symmetry group. Then, when a Hamiltonian function can be written in terms of this momentum map, the Hamiltonian is called ‘collective’. Here, we derive collective Hamiltonians for a series of models in quantum molecular dynamics for which the Lie group is the composition of smooth invertible maps and unitary transformations. In this process, different fluid descriptions emerge from different factorization schemes for either the wavefunction or the density operator. After deriving this series of quantum fluid models, we regularize their Hamiltonians for finite ħ $\hbar $ by introducing local spatial smoothing. In the case of standard quantum hydrodynamics, the ħ ≠ 0 ℏ≠0\hbar \ne 0ℏ=0 dynamics of the Lagrangian path can be derived as a finite-dimensional canonical Hamiltonian system for the evolution of singular solutions called ‘Bohmions’, which follow Bohmian trajectories in configuration space. For molecular dynamics models, application of the smoothing process to a new factorization of the density operator leads to a finite-dimensional Hamiltonian system for the interaction of multiple (nuclear) Bohmions and a sequence of electronic quantum states.