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Volume 108A, number 8 PHYSICS LETTERS 22 April 1985
QUANTUM THEORY OF THE TWO-DIMENSIONAL BOSE LIQUID
Harry L. MORRISON, Uwe K. ALBERTIN and James V. LINDESAY
Department of Physics, University of California, Berkeley, CA 94720, USA
Received 4 July 1984; accepted for publication 19 February 1985
The local matter density and current density have proven natural coordinates for the study of the elementary excitation spectrum in the Bose liquid. A fundamental decomposition of the current density into algebraic and topological parts is shown to yield both quasiparticle and pseudoparticle excitations.
The algebra of local matter and current densities was introduced long ago by Landau as a basis for his phenom- enological theory of the Bose quantum liquid [1]. Progress toward a microscopic theory of the quasiparticle spec- trum developed from the fact that Dashen and Sharp exhibited the hamiltonian of the interacting Bose system as a functional of the local density and current density [2,3]. The Bogoliubov spectrum was shown to follow natural- ly in terms of a small oscillation theory of these variables [4-6] . Missing from this work, however, was a demon- stration of the existence of the vortex structure of this system.
This becomes particularly significant in the case of two dimensions where symmetry breaking is forbidden for finite temperature systems [7,8]. Moreover, recent progress in the theory of phase transitions indicates a category of two-dimensional systems in which phase t~ansitions proceed through the statistical mechanics of the vortex gas part of the excitation spectrum [9,10]. Through a phenomenological theory of the two-dimensional superfluid, Nelson and Kosterlitz provide strong evidence that the two-dimensional 4He film is in this category [11]. We would like to provide a quantum theory of the Bose liquid which yields both the previous quasiparticle spectrum and, in addition as pseudoparticles, the vortex gas which is the basis of the Nelson-Kosterlitz development.
An equivalent but somewhat simpler approach to the Dashen-Sharp theory is to consider a complex de Broglie field tk(x),x = x , t [12].
1 fd2xlV~(x)12 +1 fd2x fd2x, **(x)~V*(x')v(Ix-x'l)*(x')~(x) ( h = m = 1), (1) H=~ 0
p(x) = Iff(x)l 2 , d(x)=Im ~b*(x)V~k(x). (2)
Now,
ao/at + v . J = o . (3)
We use the identities of Dashen and Sharp
Vp + 2 i J = 2~* V~k, Vp - 2 i J= 2 V~k* ~ . (4)
As recotMed [2], the hamiltonian takes the form:
H = -81 f d 2 x (Vp - 2id) P-~I (Vp + 2iJ) +~-1 fd2xf d2x' p ( x ) o ( I x - x ' l ) p ( x ' ) + c o n s t a n t . (5)
The condition that the system be a liquid implies that p(x) has an equilibrium value <p(x)) = P0. We shall ex-
0.375-9601/85/$ 03.30 © Elsevier Science Publishers B.V. 397 (North-Holland Physics Publishing Division)
Volume 108A, number 8 PHYSICS LETTERS
pand about the equilibrium value: p(x) = PO + ~(x). To lowest order
1 fd2x [IVtS(x)I 2 + 41J(x)l 2] H ~ / t + constant terms =
+21 f d2 x f d2 x, b ( x ) v ( I x - x ' l ) ~ ( x ' ) + constant terms
We Fourier expand:
f i(x) = ~ p ( k ) e i k ' x , J (x ) = ~ J ( k ) e i k ' x , v ( I x - x ' l ) = ~ v (k ) e i k ' ( x - x ' ) . k~O k~O k
Parsevals' identity yields:
/~ - (21r) 2 ~ [Ikl 2 It~(k)l 2 + 41J(k)l 2 + 4PO(2rr)2v(k)lp(k)l 2 ] . 8po k
We make use of the continuity equation: 3p/Dt + V -J = O. In terms of Fourier components, one has
~(k) + i k ' J ( k ) = O, ~ * ( k ) - i k ' J * ( k ) = O.
Hermiticity requires
J * ( k ) = J ( - - k ) , ~*(k) = ~ ( -k ) .
We have
~ ( k ) ~ * ( k ) = [k .J (k ) ] [ k ' J * ( k ) l .
We have the identity:
[k X J(k)]- [k X J*(k)] = [ k l 2 J ( k ) . J * ( k ) - [ k . J * ( k ) ] [k . J (k ) ] .
Therefore,
[J(k)l 2 = ( l / k 2 ) { [ ~ ( k ) [ 2 + [k X J(k)]- [k X J+(k)] }.
The effective hamiltonian becomes,
4 [k X J(k)]- [k X J+(k)] /4 - (2702800 k~0 ~ k21p(k)12 + 4PO(2rr)Ev(k)lp(k)12 + . Ip(k)[2 + ~ -
We observe a decomposition into a quasiparticle part and a pseudoparticle part:
/~ =HQ +/4p.
HQ is the hamiltonian for quasiparticles.
22 April 1985
(6)
(7)
(8)
(9)
(10)
(11)
(12)
(13)
1 I~(k)12 ] . (14) k2po
The diagonalization proceeds through the algebraic method of approximate second quantization [4-6]. The quasiparticle spectrum corresponds to that of the Bogoliubov theory:
w(k) = [T2(k) + 2PO(2n)2v(k)T(k)] 1/2 ,
r(k) = k2/2. (15)
Beyond quasiparticles, we find an additional part of the energy passed over in previous studies of this many
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Volume 108A, number 8 PHYSICS LETTERS 22 April 1985
body system:
Hp = (21r)2 ~ ~ [k X d(k)]" [k X J*(k)] . (16) 2P0 k~0 k 2
Noticing that 1/k 2 is the momentum space inverse to the laplacian, we introduce the two-dimensional Green func- tion and transform back to configuration space. Thus,
A'G(x , x ' ) = ~ (x ' - x )
and
H p - 1 2-o0 fd2x f d2x'tVX S(x)] -C(x,x'). iV'× J(x')]. (17)
' 1 Now G(x, x ) = (2 '0- On Ix - x'l + C) [ 13 ] and V X J(x) = ZiKi 6 (x - xi)~. The (xi} are zeros of the current. The {Ki) are circulations of the current about each zero. Thus we have:
1 C H p - (2~r)(2P0)~ K i K ] l n l x i - x i l (21r)(2P0) ~ KiK / . (18)
This is the harniltonian for a gas of pseudoparticles. The pseudoparticles appear in this theory as vortices. The fun- damental unit of circulation is h/m. The quantization condition is:
f J 'd l = niPoh/m = K i ,
(we restore the constants), n i are integers. In terms of an integer valued circulation density n(x) = ~ini6(x - x i ) , we have
Hp = -rr /90 f d2x f d2x ' n(x)n(x ' ) lnlx - x'l + C f d2x n(x) m2 Ix-x'l>0
Finiteness of the energy over an infinite domain requires ~iKi = 0, or f d2x n(x) = 0 (topological charge neutrali- ty). One has
~2 f d2x ' n(x)n(x') lnlx - x'l Hp = - n m--- T P0 f d2x (19)
Ix-x'l>0
for point vortices. This theory yields the vortex energy expression of Kraiclman [14], whose statistical hydrody- narnies is the basis for the phenomenological theory of Nelson and Kosterlitz. By introducing a spatial extension for the vortices and demanding scale invariance of the statistical mechanics one recovers the Nelson-Kosterlitz en- ergy expression and the associated renormalization group equations. Their phase transition theory is concomitant.
The authors wish to acknowledge useful conversations with Richard Montgomery. One of the authors (U.K.A) wishes to thank the National Science Foundation for fellowship support during this work.
References
[1] L.D. Landau, J. Phys. (Moscow) 5 (1941) 71. [2] R.F. Dashen and D.H. Sharp, Phys. Rev. 165 (1968) 1857. [3] G.A. Goldin, J. Math. Phys. 12 (1971) 462. [4] W. Biertcr and H.L. Morrison, Nuovo Cimento Lett. 1 (1969) 701. [5] R. Vasudavan, R. Sridhar and R. Ranganathan, Phys. Lctt. 29A (1969) 3.
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[6] M. Schwartz, Phys. Rev. A2 (1970) 230. [7] J.C. Garrison, J. Wong and H.L. Morrison, J. Math. Phys. 13 (1972) 1735. [8] J. Fr6hlich and C. Pfister, Commun. Math. Phys. 81 (1981) 277. [9] J.M. Kosterlitz and D.J. Thouless, J. Phys. C6 (1973) 1181.
[10] J.M. Kosterlitz, J. Phys. C7 (1971) 1046. [11] D.R. Nelson and J.R. Kosterlitz, Phys. Rev. Lett. 39 (1977) 1201. [12] S.-I. Tomonaga, Quantum mechanics, Vol. 2 (North-Holland, Amsterdam, 1966) ch. 10, p. 288. [ 13] W.H.J. Fuchs, Theory of functions of one complex variable (Van Nostrand, Princeton, 1967) p. 30. [14] R.H. Kraichnan, J. Fluid Mech. 67 (1975) 155.
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