Unitary transformations in weakly nonideal Bose gases

Unitary transformations in weakly nonideal Bose gases

a/._- 9 February __ BB 1998 PHYSICS ELSEVIER Physics Letters A 238 (1998) LETTERS A 258-264 Unitary transformations in weakly nonideal Bose ...

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a/._-

9 February

__ BB

1998

PHYSICS

ELSEVIER

Physics Letters A 238 (1998)

LETTERS

A

258-264

Unitary transformations in weakly nonideal Bose gases Q. Shi, V.L. Safonov ‘, M. Mino, H. Yamazaki 2 Department of‘ Physics. Faculty Received

of Science, Okayama University, Okayama 700. Japan

1 April 1997; revised manuscript received I October 1997; accepted for publication 21 November Communicated

1997

by A.R. Bishop

Abstract We developed a method for constructing nonlinear unitary transformations in many-body systems. It is demonstrated how to eliminate the three-boson interaction terms which describe forbidden processes in one- and two-component Bose gases.

The corresponding effective four-boson interaction amplitudes are calculated. 0 PACS: 05.30.5; 75.30.D Keywords: Unitary transformation; Boson; Interaction;

1998 Published by Elsevier Science

B.V.

Wave

1. Introduction Unitary transformations play an important role in solid state physics [ 11. They make it possible to simplify the problem of interacting quasiparticles by eliminating “inconvenient” interaction terms from a Hamiltonian and constructing the corresponding effective interaction terms. Typical inconvenient terms in the Hamiltonians of the theory of spin waves and magneto-elastic waves in magneto-ordered crystals describe three-boson processes which are forbidden by the laws of conservation of energy and momentum. Consider, for example, the Hamiltonian

where A(k) is the Kronecker delta function (A(k) = 0 if k law of conservation of momentum. Such three-boson terms with exchange and dipole-dipole interactions. For the positive interaction terms in (1) play no role in the magnon dynamics &k, f&k>

+&kj

f

0

# 0 and A(0) = I) which (7fj3’) appear in the theory energy spectrum &k > 0 it in the first approximation as

corresponds to the of a ferromagnet is obvious that the far as

for any kl f k2 -t k3 = 0.

’ On leave from Russian Research Center “Kurchatov Institute”. Institute. 2-12-I Hisakata, Tenpaku, Nagoya 468, Japan. ’ Corresponding author. E-mail: [email protected] 0375-9601/98/$19.00 @ 1998 Published PII SO375-9601(97)00949-3

123182 Moscow, Russia. Present address:

by Elsevier Science B.V. All rights reserved.

ISML, Toyota Technological

Q. Shi et al. /Phyic.c

Letters

A 238 (1998)

25X-264

This means that the initial Hamiltonian tij3’ has an inconvenient form wave dynamics. Applying the corresponding unitary transformation to ( 1 ), three-boson terms and construct the effective four-boson interactions. The develop a regular method of constructing the unitary transformations which the Hamiltonian of weakly interacting Bose gases. 1.1. Tlw prirzcipal idea

qf the

25’)

for the theoretical study of spinwe can eliminate the “ineffective” purpose of the present paper is to eliminate “forbidden” terms from

method

A general form of unitary transformation

can be written as

A( 0) = ,“R’He-“7“,

(2)

where R is an anti-Hermitian of the equation

operator

(‘72’ = -R)

and 0 a formal parameter.

This expression

is the solution

$7&H) =[R,fi(fu] with the initial condition R(O) = 3-1. We can write the most general form of R( 0) as an expansion in terms of operator combinations with unknown H-dependent coefficients, and the most general form of R as anti-Hermitian operator combinations. After substituting these expressions into (3), one obtains a set of linear differential equations by comparing coefficients in analogous operator compositions. Solving these equations with the initial conditions, we get the transformed Hamiltonian R(0). In order to eliminate inconvenient terms one needs to put their coefficients (for example, for 0 = 1) equal to zero. This condition defines the choice of 72. The above procedure has first been proposed for the spin Hamiltonian diagonalization 13.3] and it was successfully used in physics of magnetic excitations [ 41 and in the theory of superconductivity [ 5 1. One can easily check that the quadratic Hamiltonian of Bose system in the k-space. 7-L”’ = x[Akblbk

+ ;(Bkb:bt,

+ Bib&k)

13)

1,

k

is transformed

to a diagonal

I/‘FI’“lJ-’

= xEpb;bk,

form by &k =

k

Sigtl(&)

J

A; --

(5)

lBk12.

Here U = exp

c( 4

R,b:btp

(6)

- Rib&,)

and tanh(41Rkl)

2. One-component

= -/Bkl

Ak ’

arg( Rk ) = arg( Bk >

(7)

Bose gas

In order to eliminate Hamiltonian as follows,

the forbidden

three-boson

terms from

( 1) we shall write the general

form of the

Q. Shi et aLlPhysics

260 “;i(@[email protected]‘k+; k

A 238 (1998) 258-264

[%(k~,k~,k~;8)b~,b~7b:1+h.c.]A(k,+k2+k3)

c kl ,kz,k3

~(k,,k2,k3,k4;8)b~,b~~bklbl,d(k,

c

+ ;

Letten

+ kz - kx - k4).

(8)

ki,ks,k>.k4

Here and below we omit all terms containing five and more Bose operators. Before proceeding let us make some general remarks. (a) As long as we consider a weakly interacting Bose gas, the amplitudes of interactions have to decrease with increasing the number of Bose operators in the Hamiltonian. For example, for (8) &k >> qi/3 > &,/2 In this case the order of the creation and annihilation operators in the interaction terms is not so important. If we change the order of operators, say b!s,bk,b;zbk,

=bl,b;2bk,bk,

+b~,bk,A(k2

- k3),

then the role of additional terms will be negligibly small relative to the “bare” quadratic terms. (b) It is easy to show that the result of commutation of combinations of Bose operators, [ bt -v

bt

. . b , b+ b+ b . b] , -w ml 112 m2

b .

4

leads to a sum of compositions containing ni + ml + n2 + m2 - 2 Bose operators: ni + 122- 1 creation and ml +rnz - 1 annihilation operators. For example, for n2 = m2 = 1 the result of commutation contains compositions with the same numbers of creation (ni) and annihilation (mz) operators. This means that if we want to remove a term

x

et.,b+,-b ’

. b +h.c.

.

ml

nt

from the initial Hamiltonian, R, ,b+.,. b;u

we should use the anti-Hermitian

-h.c. nr1

PiI 2.1. Three-boson

annihilation

Thus, in order to remove the ineffective with the anti-Hermitian operator RI = Calculating

c IR,(q,,q2,q3)b~,b~,bb, 414?4

three-boson

-h.c.lAtq,

term from (8)) we shall use the unitary transformation

+qz +e).

(9)

the commutators

Mq,,q2,q3)b; [RI(q,,q2,q3)bfi from (3)

operator of the form

- h.c., ekb;bk]. / b;_bll q b;?bil -h.c.,

9,(k,,k2,k3;B)b~,b~*b~~

obtain the

$%(k,,kz,kG’)

=--3(&k,

+&k3)Rl(kl,k2>k3),

+h.c.l,

Q. Shi et al. /Physics

~m(k,.k?,k?.kd;H)=-6R,(k,,kl,-k,

Letten

A 238

(1998)

258-264

261

-k?_)9;(k~,kq,-k7-k4;8) c IO)

-6RT(k3,kj,-k~-kj)9,(k,,k*,-k,-k?;8). Solving these equations 91(0= I) =O, we get

Rl(q,,qz,q,) =

with the initial

conditions

[email protected](0 = 0) = Pr.

$( 0 = 0) = (), and requiring

~,(q,,q,7q,) 3(Eq,

+E

II)

)

+s

42

that

Ql

and &k,,kz,ki,k4;1)

= -P,(k,,kz,-k,

- k4)

1

1

X

C12)

+

(

Sk,

+

&k> +

&-k,-k>

1

=; k,

&k3

( 1 ) contains

If the initial Hamiltonian x(4’ ,

- kz)W;(k3,k4,-k3

+

Ekq

+

&-kq-k4

>

a “bare” four-boson

~(k,,k?.kj,kq)b:,b:lbk,bt,A(k,

interaction +kz

-ki

(13)

- kj),

.k:.ki.kJ

then the effective amplitude

of boson-boson

scattering

in the transformed

ei”’ (@ = 1) is t 14)

2.2. Tile con$uence and decay processes Let us now consider ‘FI’“’ = 1

c

the three-boson

interactions

[~~(k,,k2,k3)btlb:lbk;

energy of the form

+h.c.lA(k,

i IS)

+ k2 - kj).

kl,kn.kq

It is obvious

that this Hamiltonian

Sk, + ek> -

Ekl

=

0,

However, the Hamiltonian A&k,,

kz, kj) =

Ek,

contains

(15) contains +

Ek>

-

Ek>

#

0

%(4,~c72~43)

Eq, + -%I, - Eq,

for kl + k2 - k? = 0. processes

- h.c.lA(q,

We introduce a general form of Hamiltonian analogous to the above one gets

=

process

also terms for which

three-boson

c [R?(q,,q2.q3)b~,b~2bq, 41.4241

&(q,,qz.q3)

the three-boson

k, + kz - kj = 0.

In order to remove these forbidden anti-Hermitian operator Rz =

terms which describe

we shall use the unitary

+ 42 - 4).

containing

transformation

with the

c 16)

‘Ffk”’ with @z( kl , kz, k3; 0). and after calculations

( 17)

Q. Shi et al. /Physics

262

The corresponding

additional

Letters A 238 (1998) 258-264

part to the effective four-boson

amplitude

( 14) has the form

1 p2(kl>

kZ, kl

+

k2)p;(k3,

k4.

k3 +

k4)

1 +

( &k,

Ek> -

Eki + &ka -

Ek,+kj

I

-ly2(kl,k3_k,,k3)~y;(kj,k?--kq,k2)

(

+ &k,-k,

Ek,

-~~(kl,k4-kl,k4)~~(k3,k2-k3,k2)

-

-

-1V?_(kz,k4-k2,k4)~‘2+(k3,kl -k3,kl) This expression

-

Ekz >

-

&k? >

-

&k,

-

Ek,

1

+

&k, + &kl-kq

1

+ Ek?

+Ekyk _Ek 42

&k4 + Ekz-ka

&kq

+ Sk,-kd 1

+ 4

&k, + Q-k3

.

(18)

Bose gas

NOW we turn our attention [email protected]

to a two-component

+ ek&k>

k +

?

&kq

I

holds &k > lfnf*/Aa.

3. Two-component

‘Ft =

Ek

Ekl

+

‘=kl+kd >

+

Bose gas with the Hamiltonian

[Wj(kl, k2, kdbl,bL2cii

c

+ h.c.lA(k,

+ k2 + k3)

kl ,kz.k?

c

[lyq(kl,kZ.k~)b:,b~*ck,

+

h.c.lA(kl

+ k2 - k3)

kl.kz.ky +2

c

+h.c.lA(k,

[~~(k,,k?,ki)b:,bk,c:~

- kz+k?),

(19)

ki.kz,k3

where ek is the energy and ~1, Ck are the corresponding Bose operators which commute with bl, b,. We are interested in eliminating the forbidden three-boson processes from (19) and in obtaining the additional parts to the four-boson amplitudes in the Hamiltonian (13) and

c

7-l?’=

?Yk,, k2, k3, kdb$$,ck,A(k,

+ kz - kg - k4).

(20)

kl.k?.kx.k4

Such a problem is very important, for example, for constructing the effective Hamiltonian in the theory of parametric resonance of magnons [ 61. It is convenient to remove the ineffective three-boson terms from (19) by means of several unitary transformations. The anti-Hermitian operators can be taken in the forms R3=

c

[R3(q1,q2,q3)b~,b~*c~~ -h.c.lA(q,

+e+e)t

QI4243 R4

=

c

[R4(q,,q2,q3)b~,b~,cq7

-

h.c.lA(q, + 42 - 4x)+

qI424 Rs = c QI4243

[Rg(q,,q2,q3)b~,b42c~1- h.c.lA(q,

- q2 + 43).

(21)

Q. Shi et al. /Physics

After calculations

The corresponding

21y,(q,,q,,q3)

%, - %, + eP7

additional

kz,

- ps(kl,

parts to @( ki , k2, k3, k4) in ( 13 ) can be written as

- kdP;(k3,

k2, -kl

+ pJ(ki.

kl

+ kz)P;(h,

k.l, k3 - kl)P;(kq.

p5(kz,k3,k3

-

pstkz,h,k4 -

-kl

k2, k2 - k4)

k2)P;(k3,kl,kl

- k3)

- kl,k4)ly,(k3,kz

Ek,

+ Ek2 -

Ek

Concluding

e-k,-k:

&ks +

&kA +

&kl + &kd -

1

I-

Ek4 -

eki-k,

?-

Ek,

-

Ek? +

+ Sk4

-

Ek,

Ek,

-

.?kl

ekz-k,

>

+

ekl_k4

1

+

&kq +

>

1

ek,-k>

1 -

EkZ + ek:_k4

+

&k? +

kd >

1

&k4 + a-k,

1

ekii

1

+

&k3 f

e-k,-k4

1

+

ek,tk:

1

(23)

ekJ-k2

-

k2,

k3,

kdW;(kl

1 Ek,

+

ek,_ki

- k4r k,, kq)

&kl

(

+

Sk,-k,

+ ekz

I-

Eklfkz

-

ekA

2 -

&k,

Ek3 + ek2

+

ekJ !

+Ekz-k>

-

ek2

1 Ek,

ek:

&-k<-k4

1

+ +

1

Ek _k

Ekq f

+

1

Ek

I

+

+ E-k,-k?

1

- ks.k2)

1

4.

&k> +

Ek

- k4,kzt)

+k?,k’-)~~tk3,k3+k4,k4)

+2Ws(kl,kl

2p5Ck3

- kq)

+

Fk ( I(

k’-)F;(kq,kl,kl

&k,

1

kc+,k3 + k4)

- kz,k2)P;(kvk3

_ z’J’d(k,,h

_

(

1

+

parts to Y( kl , k2, k3, k4) in (20) we get

For the additional -2%(kl,

1

kq, -k3 - kq)

-~s(kl.k4,k4-kl)ly;(k3,k2,k2-k.?)

-

A 238 (1998) 258-264

similar to the above we get

&(4,*42343) =

-Pj(ki.

Lettm

-

Fk3tk4 + ek4 )

1

+ Ek,-k.,

-Ek,

(24) +

Pkl

remarks

WC have demonstrated how to eliminate the three-boson forbidden processes from a Hamiltonian. The procedure was based on the solution of a system of linear differential equations. This made it possible to take into account all terms in the expansion of the unitary operator. Recently Krasitskii [7,8] has developed a method of canonical transformation for the classical weakly nonlinear wave system. In order to remove the forbidden three-wave interaction terms from the Hamiltonian he considered the expansion for complex wave amplitudes ak = bk + I’*‘(k)

+ I”‘(k),

(25)

where /(I’) (k) contains terms with different compositions of n new complex amplitudes. The transformation canonical if certain conditions for the coefficients of these compositions in the expansion (25) are satisfied.

is

264

Q. Shi et al. /Physics Lerters A 238 (1998) 258-264

In our opinion, there is a simpler way to construct a nonlinear canonical transformation for classical According to Bohm’s general theory of collective coordinates [ 91, we can express the old coordinates system in terms of the new variables qold = e

system. of the

-R4neweR=(4+[9r~l+~[[4r~l,~l+...)new.

Here we can use Poisson brackets multiplied by -ifi instead of the commutator. Our results ( 12), ( 18) for the additional part to @(kl, kg, k3, k4) coincide with those in Ref. [7] if we use the additional conditions pl = q’;, ‘[email protected] = [email protected]; and Ek, + &kz= &kz+ E/q for k, + kz = kj + kq. In conclusion, we have developed a method of constructing the unitary transformations to eliminate forbidden three-boson processes from a Hamiltonian and calculated the corresponding effective four-boson interactions. The results are written in a general form and can be applied for any weakly interacting Bose gas (such as a gas of magnons, phonons and so on).

Acknowledgements This work was supported by a Grant-in-Aid for Scientific Research (B) Science, Sport and Culture of Japan and, in part, by RFFI (97-02-17586).

from the Ministry

of Education,

References I I I M. Wagner,

Unitary Transformations in Solid State Physics, in: Modem Problems m Condensed Matter Sciences, Vol. 15, V.M. Agranovich and A.A. Maradudin, eds. (North-Holland, Amsterdam, 1986). 12] V.L. Safonov, preprint KIAE-3691/ I. Moscow, 1982 (in Russian), 13I V.L. Safonov. Phys. Lett. A 97 ( 1983) 164. 141 V.L. Safonov, R.M. Farzetdinova. J. Magn. Magn. Mater. 98 (1991) L235. 15 I V.L. Safonov, Phys. Stat. Solidi (b) 174 ( 1992) 223. 161 V.L. Safonov, H. Yamazaki, J. Magn. Magn. Mat. 161 (1996) 275. 171 V.P. Krasitskii. Sov. Phys. JETP 71 ( 1990) 921. [S I V.P. Krasitskii, J. Fluid Mech. 272 (1994) 1. 19 1 D. Bohm, General theory of collective coordinates, in: The Many Body Problem, Les Houches - session 1958 (Wiley, New York. 1959) p. 401.