Pis'ma v ZhETF, vol. 98, iss. 11, pp. 853-863 © 2013 December 10
Tkachenko waves
In memory of V.K. Tkachenko
E. B. Sonin1
Racah Institute of Physics, Hebrew University of Jerusalem, 91904 Jerusalem, Israel Submitted 5 November 2013
This is a short review of theoretical and experimental studies of Tkachenko waves starting from their theoretical prediction by Tkachenko about 50 years ago up to their unambiguous experimental observation in the Bose-Einstein condensate of cold atoms.
DOI: 10.7868/S0370274X13230215
1. Introduction. In the year 2013 Vladimir Kon-stantinovich Tkachenko passed away in the age of 76. This sad event urges to review the legacy of this brilliant scientist in physics.
His active life in physics unfortunately was very short because of health problems. He published not more than about 10 papers but what papers! Tkachenko, being nominally (and really) an experimentalist, published the papers, which Dyson [1] called "a tour de force of powerful mathematics". Tkachenko's seminal works on a vortex lattice in superfluid helium and its oscillation were written about 50 years ago but up to now they remain actual and challenging in various areas of physics, superfluid liquids, cold-atom Bose-Einstein condensates, and astrophysics among them.
The series of Tkachenko's papers on dynamics of vortex lattices started from the paper [2], in which he calculated exactly the energy of an arbitrary periodic vortex lattice and showed that the triangular lattice has the lowest energy as in the mixed state of type II superconductors. In the second paper [3] he found (also exactly) the spectrum of waves in the vortex lattice for all wave vectors in the Brillouin zone. These waves are now called Tkachenko waves. Finally in his third paper [4] he demonstrated that in the long-wavelength limit the Tkachenko wave is nothing else as a transverse sound wave in the vortex lattice and its frequency is determined by the shear elastic modulus.
The following short review addresses the original theory of Tkachenko waves suggested for superfluid 4He and its nowadays extension on Tkachenko waves in the Bose-Einstein condensate of cold atoms, and also overviews a long and controversial story of attempts to detect Tkachenko waves experimentally first in liquid 4He and pulsars and then in Bose-Einstein cold-atom
condensates, which culminated in an unambiguous observation of Tkachenko waves.
2. Tkachenko waves from the elasticity theory of a two-dimensional vortex crystal. We start not from the exact solution but from a more transparent approach deriving the Tkachenko wave from the elasticity theory of the vortex lattice.
The equation of motion in the continuous elasticity theory for atoms in the crystal lattice is the second Newton law:
d2yL r <-\\
pW =f' (1)
where p is the mass density, u is the atom displacement, and the force f is defined as a functional derivative of the elastic energy of the crystal:
f _ ÖE _ dE ( dE Ju du \ dVjU
Vi
dE <9V,u
(2)
We took into account translational invariance, which eliminates the dependence of the energy from the constant displacement u.
Like in the elasticity theory, in vortex dynamics one can also introduce a continuous medium approximately describing an array of discrete vortex lines. This means that one carries out averaging (coarse-graining) of the equations of hydrodynamics over rather long scales of the order of intervortex distance. The approach is accurate enough as far as parameters of the medium do not vary essentially at the intervortex distance. This approach was called in Ref. [5] macroscopic hydrodynamics. In contrast to the elasticity theory of atomic crystals, the equation of vortex motion connects the force on the vortex not with an acceleration but with velocities:
-p ■ 2n x (vL - v) = f,
(3)
e-mail: sonin@cc.huji.ac.il
where ^ is the angular velocity vector, vL = du/dt is the vortex velocity, and v is the average velocity of the
nictMa b X3TO TOM 98 BHn. 11-12 2013
853
liquid. We consider the T = 0 case when the center-of-mass velocity coincides with the superfluid velocity. The angular velocity Q determines the vortex density nv = 2Q/k, where k = h/m is the circulation quantum and m is the mass of a particle. The forces in Eq. (3) are forces on all vortices piercing a unit area. In classical hydrodynamics the left-hand side of Eq. (3) is called Magnus force. But in the theory of superfluidity and superconductivity they usually relate the Magnus force only with the term proportional to the vortex velocity vL, while the term proportional to the fluid current pv is called Lorentz force.
The equations for vortex displacements must be supplemented by the continuity equation,
dp dt
+ V • (pv) = 0,
(4)
and by the Euler equation, which in the rotating coordinate frame is [5]
d v „ _
—- + 2H x vL = -V/x.
dt
(5)
The continuity and the Euler equations allow to determine the liquid velocity v and the chemical potential p.
The expression for the elastic force can be obtained on the phenomenological basis taking into account hexagonal symmetry of the triangular lattice. We consider a 2D problem in the xy plane normal to the angular velocity vector Q (the axis z) with no dependence on z. The general expression for the elastic energy density in the 2D case is [6]
+
Cm 2
(Vyux + Vxuy) - 4V
xuxVy uy
(6)
Here Cn is the inplane compressibility modulus and C66 is the shear modulus. We used here Voigt's notations for elastic moduli [7] adopted in the theory of superconductivity. Equation (6) is a particular case of a more general expression given in Refs. [5, 8], which took into account the z dependence. From Eqs. (2) and (6) one obtains an expression for the force on vortices:
f = (C11 - Cee)V(V • u) + CeeAu.
(7)
The term proportional to the divergence V • u can be neglected in the low frequency (long wavelength) limit. Then the components of the force f = —Vj a^ are determined by the stress tensor
= -C66(ViUj + Vj ui)
(8)
for purely shear deformation. Here subscripts i and j take only two values x and y corresponding to the two axes in the xy plane. Then the Eq. (3) of vortex motion becomes
du
~dt
. C66 . 1 Vi = V 2f2p X '
(9)
It is convenient to divide the vortex displacement field u(r) and the fluid velocity field v(r) into longitudinal and transverse parts ( u = up + u^, v = v|| + v^), so that V • u^ = V • v^ =0 and V x up = V x vp = 0. In an incompressible liquid vp = 0 and the liquid velocity v = v^ is purely transverse. Then Eq. (5) after integration over time yields
-[2H x uii].
(10)
After exclusion of v Eq. (9) yields the equations for longitudinal and transverse displacements u» and u^:
du» C66 r„ . ,
——1 =-z x Au i ,
dt 2 ilp1 J'
du^
—2Qx un
(11)
(12)
Excluding the small longitudinal displacement up from equations one obtains an equation similar to that for the transverse sound in the conventional elasticity theory:
d2u± dt2
with
ct
4Auj p
(13)
(14)
being the velocity of the Tkachenko wave <x eik r-iWi with the sound-like spectrum w = cTk. Vortices in the Tkachenko wave move on elliptical paths, but the longitudinal component up parallel to the wave vector k is proportional to a small factor w/Q (see Eq. (12)). Thus it is fairly accurate to consider the Tkachenko wave to be a transverse sound wave in the two-dimensional lattice of rectilinear vortices [4]. Comparing Eqs. (10) and (12) one can see that in our approximation the liquid and the vortices move with the same velocity. The phe-nomenological approach cannot provide the value of the shear modulus. But it is clear that the elastic energy is in fact the kinetic energy of the velocity field induced by the vortices, and scaling estimations show that the shear modulus should be on the order of C66 ~ pkQ. Its exact value can be obtained from the exact value of the energy of the vortex lattice obtained by Tkachenko (Sec. 3).
v
All experiments on Tkachenko waves dealt with finite cylindric liquid samples, and it is necessary to know the boundary conditions for Tkachenko cylindric waves. We restrict ourselves with axisymmetric modes.
The field of transverse displacements u^ may be determined by a vector potential Ф = Ф1:
u^ = V x ^ = -z x V^. The potential ^ must satisfy the wave equation
84
- еТДФ
0.
(15)
(16)
Axisymmetric modes with the sound-like spectrum w = cTk correspond to a cylindrical wave
0,
Ф = Ф0 J0(kr)e-iut,
дФ
dr
(17)
where subscripts r and y denote radial and azimuthal components in the cylindric coordinate frame (r, y).
Suppose that no external force acts upon the liquid, which fills a cylinder of the radius R. Then eigenfrequen-cies are defined by the condition that the total angular momentum M does not vary. Since in the Tkachenko wave the fluid and vortices move together
M = 2np / vvr2dr = -2iwnp /
00
= -2iwnp^0R2J2(kR)e-
uvr2dr
The condition M
0 yields eigenfrequencies ct
-32,;
R
(18)
(19)
where j2,i denotes the ith zero of the Bessel function J2(z). For the fundamental frequency j2,i = 5.14. This is a result obtained by Ruderman [9] who discussed Tkachenko modes in pulsars (see Sec.4).
The condition M = 0 is equivalent to the boundary condition that the azimuthal component of the momentum flux through the liquid boundary r = R vanishes. This momentum flux is given by the relevant stress tensor component avr in cylindrical coordinates:
(r) = —pcT
dr
>(r)
The condition avr (R) = 0 requires that
duv(R) uv(R)
dr
R
0.
(20)
(21)
This yields the same spectrum Eq. (19) as the condition M = 0.
3. Exact solution of Tkachenko. Tkachenko has found an exact solution for the vortex lattice and its oscillation using the theory of elliptic functions on the complex plane [2, 3]. It is well known that a two-dimensional vector r(x, y) can be presented as a complex variable z = x + iy. Then the velocity field v(z) = = vx + ivy induced by vortices located in nodes of a vortex lattice with position vectors zki =
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