String theory of the roton

Physics Letters A 161 (1992) 541—544 North-Holland

PHYSICS LETTERS A

String theory of the roton Veit Elser Laboratory ofAtomic and SolidState Physics, Cornell University, Ithaca, NY 14853-2501, USA Received 29 May 1991; accepted for publication 20 November 1991 Communicated by A.A. Maradudin

An effective string theory of vortex ring excitations in a boson superfluid can in principle be derived using the coherent-state functional integral. The details of this derivation are complicated but lead unambiguously to the existence of a purely imaginary term in the string’s effective action. This term is responsible for stabilizing a microscopic vortex ring against collapse if its momentum is sufficiently great.

In a material with a translationally invariant ground state the low energy vibrational excitations are described by a dispersion relation (p). If the material is a solid this function shows considerable structure, particularly when p is near the zone boundary. Nevertheless, this structure in e (p) is a simple consequence of a purely harmonic Hamiltonian. The same cannot be said if the material is superfluid 4He. There, although the smallp excitations phonons are indeed manifestations of small amplitude fluctuations of the Bose-condensate phase angle the corresponding harmonic Hamiltonian [1] does not reproduce the pronounced “dip” in ~(p) when p is large. To understand the excitations in this dip rotons one is therefore led to consider anharmonicity and in particular the topological nature of the phase angle degree of freedom when fluctuations in ~ are comparable to 27t. The idea ofthe roton in 4He as “the ghost ofa vanished vortex ring” has had a long and distinguished history (for a historical account, see ref. [2]). Since macroscopic, classically propagating vortex rings certainly exist, it is of interest to consider the sense in which there might be a “smallest” such ring and whether that object corresponds to the roton. To investigate this question one requires a quantum theory ofvortex dynamics a theory that correctly treats the zero-point fluctuations of the ring. This Letter shows how a quantum theory of vortices can be derived from the boson Hamiltonian. Not surprisingly, —

—

the dynamical variable is the string formed by the core of the vortex. An essential ingredient of the theory is a purely imaginary term in the string’s action which implements quantization in the dynamical phase space. The existence of such a term was recently conjectured by Davis and Shellard [3] as the basis of the Magnus force. A convenient starting point in the derivation is the coherent-state functional integral representation of the many-boson partition function with action [4]

~,

—

S[

w]

=

J

dr d3x

(

~

2m

I vwi2

—

—

Elsevier Science Publishers B.V.

—

I Vi

2+

Al Vi 14)~

(1)

where h = 1, m is the boson mass, /2 is the chemical potential, and the two-body interaction has been approximated by a delta-function pseudopotential of strength A=2Ea/m, where a is the s-wave scattering length [5]. The region of integration in (1) is a space—time box with periodic boundary conditions having extent 1 /kBT in the imaginary time, or ~ dimension. A dot above a field variable denotes differentiation with respect to r. Upon making the familiar substitution v= e the action S[p, ~,] decomposes into purely imaginary and real parts i~[p, ~] + V[p, Vq~J, where ‘~,

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Volume 161, number 6

ø[p, ç~]=

PHYSICS LETTERS A

$ $

did 3x p~,

(2)

(~-I Vp 2 +p I Vç~12)

V[p, V~~] = di d3x [~—

—

(3)

~+2p2I .

The term <1 is clearly responsible for the quantization of the conjugate variablesp(t, x) and ~(r, x) at each point x in space. We would now like to “integrate out” the p field and obtain an effective action involving only the phase angle Since p’s at different i are uncoupled, the necessary functional three-dimensional. Unfortunately this integrals task is farare from being a Series of Gaussian integrals since V is not quadratic in p and moreover the integration measure requires p~ 0. On the other hand, since our primary interest is the problem of how an effective string theory can be derived in principle, it is appropriate at this point to resort to an approximation. We proceed by assuming the r derivatives of ~ are in some sense small, The saddle point in the p integration is then determined by minimizing V (with qi fixed). The behavjour of the saddle point function j3 itself a functional ofV~ is characterized by the value Po= /2/22 it assumes when I V~ I is small and a healing length scale ç~—~1~ 1 For example, if the ~, field forms a simple line vortex then ~ will be zero at the core and differ significantly from Po only in a cylindrical region of radius Expanding V to quadratic order in fluctuations about ~ and performing the resulting Gaussian integral, we arrive at an effective action of the form ~,,

did 3x d 3y

X K—’ [P1(x,y)

~~ ( t,

(4)

x)

b(r, Y)

and K1 is the inverse of the operator K[~](~,~) =ô3(x—y) >< 542

nj[8m’\. —L

—

-Ytfl

,

(V /~\ -~-

—

5,

j

+ 2].

(7)

1xJ’I

it

where c= ~J8mApo,we find that a plane wave mode with vector has inertia proportional 2m/ 2 +wave ic2). With thep stiffness of the same modeto(given (p by V) proportional to p2/2m, we arrive at the Bogoliubov spectrum [1]: ~

=

~—~

(8)

(6)

~.

$

2 exp(—KIx—yI) —mp0

diameter in the boson—boson potential. However, since a vortex core has structure on the scale ~ l/~J~,we see that Seff is valid even for vortex excitations in the dilute (a3p 0—*0) limit. Having thetoobvious offorthe theory we understood are now ready take thelimitations final step in

~

T[~, ~] =

K~[po](X,Y)=

(5)

—

Vc~]=i~[~5, ç~]+ T[~, ~] + V[ã, Vci1~ where the kinetic energy term T is given by

To see how a spectrum of harmonic excitations emerges from (4) one considers field configurations where I V~I 0 everywhere. One may then replace ~ by Po since the deviations are in any case small compared to the Gaussian fluctuations that were integrated over. The term 1[po, g~]is now independent of the dynamics of the q.’ field and can be neglected. What remains is a quadratic term in each of T[p 0, ço] and V[p0, Vp]. Since

If we loosely interpret #c~—~ as a kind of “zone boundary” scale then the upward curvature of e,~(p) in the superfluid is opposite the corresponding downward curvature in solids. The difference can be traced to the reduced inertia of zone boundary modes in the former as opposed to a reduced stiffness in the latter. We note that e~ (p) rises sharply for momenta p> K, where the linear phonon dispersion has given way to a single-particle type of spectrum. Turning now to the possibility of vortex ring configurations in the ci field it is worth remarking that neither at the core of a vortex nor at the space—time event where a ring is created or destroyed is there a singular contribution from Seff. In eithercase the singular behavior in ~i is always associated with a vanishing of ~ so that Seff remains well behaved. As a related concern, one might question the legitimacy of our expression for the action in the vicinity of a vortex when our use of the pseudopotential at the outset limits our theory to length scales certainly larger than the scale a the corresponding hard-core

—

5eff[~,

20 January 1992

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Volume 161, number 6

PHYSICS LETI’ERS A

20 January 1992

mulating an effectivestring theory. The goal is to derive an effectiveaction for a theory in which the only surviving degree of freedom is the mathematical curve in space where ci is singular (and ~ vanishes). Clearly this entails “integrating-out” the ci field subject to the constraint that ci has the required singularity structureon a given world sheet in space—time. Recognizing that such a calculation presents even greater difficulties than the earlier problem of dimmatingp, we will be content in making some general observations about the effective string action Sst,.,ng. Considering for the moment only the terms T and V

Neglecting the contribution to (I) from the space—time region traversed by the tube, we have

111 Se~ it

of vorticity, the ±1 change in w(x) at the surface v corresponds to the orientation of the surface normal ~‘~< o0v. The integral of w(x) over space is then (up to an additive constant) just the signed volume enclosed by v or

is clear that Sstnng will be nonlocal in time as well as space. There will be long-range interactions between pieces of vortex line of two kinds: instantaneous “magnetostatic” interactions and retarded interactions corresponding to the exchange of a phonon. Neither of these will be too important in the propagation of a single microscopic vortex ring. The contribution of Tto ~ will merely be to penalize world sheets with large i derivatives while V will impose an energetic cost on large rings. It is important to realize that the contribution of V is probably a monotonic function of ring size. In the saddle point approximation this statement has been demonstrated rigorously by Jones and Roberts [6] ~‘ who showed that V has no nontrivial local minima. Thus minimizing Vwith the boundary condition of a circular vortex ring of radius r will produce a potential Vring(r) that is monotonic all the way to r= 0. Moreover, Vnng( 0+) is still substantial ~ ~ will vanish at a point and differ significantly from Po in a region of size This means that rotons if indeed they correspond to vortex rings haveisenergy 2a. What missat least order A ~Ap~ ~Jpo/m ing fromofthis analysis, ofcourse, is a mechanism that keeps small vortex rings from shrinking to a point and vanishing altogether. The stability of a vortex ring is entirely attributable to 0, the purely imaginary term in S~ 0-.The role ofthis term is best appreciated in the limit of a large ring where it acquires a simple geometrical interpretation. For such a ring ~ is nearly Pa except inside a relatively thin tube of radius ~ surrounding the ring. ~.

—

—

~‘

Ref. [6] considers saddle point solutions having general yelocity U; the statement made in the text concerns the special case U=0.

~poJd3x($dt~)

=pojd3x2irw(x),

(9)

where w(x) is a winding number. Discontinuous changes in the integer-valued function w(x) occur at the two-dimensional surface in space v( r, 0) formed by projecting out the time component of the string’s world sheet. With the parameter 0 oriented along the string so that the tangent vector 80v gives the sense

~2itp

~

J

~‘

ô~ VX 8~vdi dO.

(10)

This expression shows that ~ is the direct generalization to strings of the phase acquired by a charged particle moving in a uniform magnetic field. Pursuing the electrodynamic analogy further, one would say that ~ is the result of locally coupling each element of the string’s surface with a fictitious tensor potential B~cc~x”~ corresponding to a three-index “vortomagnetic field” [3] proportional to ç,k. The classical dynamics ofa circular string in this kind of field was studied by Auriuia and Christodoulou [7]; the application to vortices in superfluids was first considered by Davis and Shellard [3]. A simple calculation shows how the term 0 can stabilizeitsa radius vortex ring. Consider a but string that may change and orientation always (for simplicity) remains circular: v(T, 0) =q(i)+r(r) [1(r) cos 0+m(r) sin 0] (11) In (11) q is the position of the ring, r is its radius, 1 and m are orthonormal vectors that define its orientation. Substituting (11) into (10) we find (12) 2n. Combining where n = lx and d(we r, it) = 2icp0 (12) with themterms expect toxbeitrgenerated by T and V io Seff we arrive at a real-time Lagrangian of the form 543

Volume 161, number 6

Lring(E, r,

PHYSICS LETTERS A

1, it, ñ)=d(r, n)E

(13) +~m,(r)~+~m,(r)(n~E)2..., where denotes terms of higher order in q or not involving q at all. Using i3Lrjng/OE=p define the momentum ofthe ring, it follows that the corresponding Hamiltonian has a dependence on p of the form -

...

1 [p d( r, it)] 2m~(r) .[l.....a(r)nn].[p-..d(r,n)]...,

Hring =

—

(14)

20 January 1992

e (p) given by the Bogoliubov spectrum is a harmonic effect and occurs at the scale p—J~7~. In contrast, the branch of vortex ring excitations — an extreme manifestation of anharmonicity begins at momenta exceeding the scale p—.~1/a. Comparing the energy of a vortex ring at this momentum scale, A .,,/po/m2a, with the energy given by the Bogoliuboy spectrum at the same scale, CB 1 / ma 2, we observe that vortex rings have lower energy in the dilute limit (a tPo << 1). Moreover, since vortex rings become strongly admixed with vorticity-free states for momentap
“-j

where a = m 2/ (m, + m2). Viewing the term in (14) as a kind of effective potential for the ring’s dipole moment d(r, it), we see that for sufficiently large p the ring is prevented from collapsing to a point. To make a rough estimate of the required magnitude Pa we note that the radius r must be at least of order ~ if the term 0 is to play a significant role. Vortex rings are thus expected to be stable when momentum 2’-~ 1/a. When the their momentum apexceeds Po—’Po’~ proaches Po from above we can expect a growth of fluctuations where the ring disappears completely and becomes strongly admixed with vorticity-free (phonon/single-particle) states. It is in this sense that one can expect a smooth crossover between the roton and phonon branches ofthe excitation spectrum. In conclusion, we have seen how phonons and vortex ring excitations can be discussed in a common framework. Harmonic as well as anharmonic effects both lead to structure in the dispersion relation but at independent momentum scales. The upturn in

544

I wish to thank the members of the physical sciences theory group at the IBM Almaden Research Center where this work was initiated. Additional support from the David and Lucile Packard Foundation is also appreciated. References [1] N.N. Bogoliubov, Izv. Akad. Nauk SSSR Ser. Fiz. 11(1947) [2] R.J. Donnelly, in: Quantum statistical mechanics in the natural sciences eds. S.L. Mintz and S.M. Widmayer (Plenum, New York, 1974) p. 359. [3] R.L. Davis and E.P.S. Shellard, Phys. Rev. Lett. 63 (1989) 2021. [4] J.W. Negele and H. Orland, Quantum many-particle systems (Addison-Wesley, Reading, 1987). [5] K. Huang, Statistical mechanics (Wiley, New York, 1963). [6] CA. Jones and P.H. Roberts, J. Phys. A 15 (1982) 2599. [7] A. Aurilia and D. Christodoulou, Phys. Lett. B 71 (1977) 90.