Properties of Bose gas in a lattice model (strong interaction)
aa r X i v : . [ c ond - m a t . o t h e r] O c t Properties of Bose gas in a lattice model(strong interaction)
E.G. Batyev ∗ A.V. Rzhanov Institute of Semiconductor Physics , 630090 Novosibirsk, Russia
The doubts concerning validity of gas approximation for strong interaction (for example, hardspheres) are expressed. A contradictory example - a Bose system in a lattice model - is considered.Namely, the X − Y model for spin 1 / PACS numbers: 67.10.-j
INTRODUCTION
As is known, a Bose gas model with a weak interaction(repulsion), considered by Bogoliubov [1], is generalizedfor a case of an arbitrary interaction value provided thata so-called gas approximation is correct (for example, see[2]). The main point of this generalization is a transitionfrom true interaction to scattering length (which is smallin comparison with inter-particle distance within the gasapproximation). However, the gas approximation, be-ing good for a classical case, is not always suitable for aquantum one. The reason is the absence of a conceptionabout trajectories (no quasi-classics) in ultra-quantumlimit (which takes place for Bose system). Therefore onecan hardly speak about binary collisions. In fact, inter-action of every particle with all the particles at once israther probable: when particle number is small, mostparticles are in Bose condensate, i.e. these particles arecharacterized by an infinite wave length. The aim ofthe present work is to show this phenomenon within theframework of a simple model. Namely, a Bose gas inthe lattice model with infinite interaction is considered,so that no more than one particle can be at each cite(description of interaction like for hard spheres). Thismodel is equivalent to X-Y model for spin 1 / /
2) corresponds toa particle. An average energy value is calculated using atrial function at given particle number written for the X-Y model and a ratio of a mean-square energy deviation toan average energy value is shown to tend to zero within amacroscopic limit. The result of calculation of the basicstate energy via the traditional approach (according tothe accepted rules) is shown to be different.Bose gas in the hard spheres model was considered, forexample, in the work [3], where pseudo-potential methodwas used. The pseudo-potential value is selected so thatthe scattering of two particles at each other was the same as in the case of hard spheres. It is the pseudo-potential,for which the corresponding multi-particle Hamiltonian iswritten, then the interaction is presented in a more sim-ple form and, finally, the basic state energy value (calcu-lated from this simple form) is given ([3], section 1). Thismodel and mathematical treatment is given in [4] also.Nevertheless, no accuracy estimate of the traditional ap-proach was made as nobody used the trial function forinitial interaction (hard spheres). This raises the ques-tion of whether this approach is correct in the case ofBose condensate, when no conception about particle tra-jectory and accordingly about binary collisions exists.
THE MODEL
The Hamiltonian of the X − Y model is: H = − t X
TRIAL FUNCTION
Above mentioned taken into account, the trial functionhas the following form:Φ N = (cid:16) S + (cid:17) N | ↓ > (cid:18) S + ≡ N X n =1 S + ( n ) (cid:19) . (6)This approximation can be expected to be good (at leastfor small particle number N << N ), as all the particlesare in Bose condensate.First let us consider normalization, i.e. the value(Φ N , Φ N ). The initial state | ↓ > (vacuum to particles)corresponds to maximum system spin ( N /
2) with max-imum negative projection ( − N / S + operator raises the projection by a unitywithout changing of the full spin value. Matrix elementsof the operator (calculated by the normalized functions)are known from the general courses of quantum mechan-ics: (cid:16) S + (cid:17) M,M − = p ( S + M )( S − M + 1) . Here M is S z spin projection value. For example, theeffect of operation on the particle’s vacuum ( M − − N / , S = N /
2) is: (cid:16) S + (cid:17) M,M − = p N → S + Φ = p N Φ ( M = − N / . Consequently, having denoted the corresponding to Φ N normalized function by e Φ N ( e Φ N ≡ D N Φ N ), one can de-rive: 1 D N = N − N / Y M =1 − N / p ( S + M )( S − M + 1) == N Y n =1 p n ( N − n + 1) . And, finally: e Φ N = D N Φ N ; (7) D N = s ( N − N )! N ! N ! . Auxiliary relations
Some relations are necessary in what follows. The mostsimple is the calculation of, for example, (cid:16) Φ N , Φ N ( n ) (cid:17) ,where Φ N ( n ) ≡ S + ( n )Φ N − . This value does not de-pend on the cite number n , as all the cites are equivalent.Therefore one can write:Φ N ( n ) ≡ S + ( n )Φ N − ; (cid:16) Φ N , Φ N ( n ) (cid:17) = 1 N X n (cid:16) Φ N , Φ N ( n ) (cid:17) = 1 N (Φ N , Φ N ) , (8) (cid:16) Φ N , Φ N ( n, n ′ ) (cid:17) = 1 N ( N −
1) (Φ N , Φ N ) . Analogous relation can be written for the case of bothfunctions containing the cite number, for example: (cid:16) Φ N ( n ′ ) , Φ N ( n ) (cid:17) n = n ′ = 1 N − (cid:26)X n ′ (cid:16) Φ N ( n ′ ) , Φ N ( n ) (cid:17) −− (cid:16) Φ N ( n ) , Φ N ( n ) (cid:17)(cid:27) == 1 N − (cid:26)(cid:16) Φ N , Φ N ( n ) (cid:17) − (cid:16) Φ N ( n ) , Φ N ( n ) (cid:17)(cid:27) . As for the value with coinciding numbers ( n ′ = n ), onecan notice that one of these states is occupied a fortiori,so a norm of the state is got using the relation (7) andsubstituting N → ( N − , N → ( N − (cid:16) Φ N ( n ) , Φ N ( n ) (cid:17) = 1 N N (Φ N , Φ N ) ; (9) (cid:16) Φ N ( n ′ ) , Φ N ( n ) (cid:17) n = n ′ = N − N N ( N − (cid:16) Φ N , Φ N (cid:17) . The correctness of the relation is tested by substitutionof N = 1. ENERGY
Now an average energy value can be calculated: E = ( e Φ N , H e Φ N ) = D N (Φ N , H Φ N ) = = − tD N N ν (cid:16) Φ N +1 ( n ′ ) , Φ N +1 ( n ) (cid:17) n ′ = n . Hence taking into account (7), (9) the following relationis derived: E = − tD N N ν ( N + 1)( N − (cid:16) Φ N +1 , Φ N +1 (cid:17) = (10)= ( − tν ) N ( − N − N − ) . Note natural symmetry at the substitution N → ( N − N ). The contribution linear in relation to the particlenumber N is just particle energy at the band bottom,quadratic contribution is a result of particle interactionbeing taken into account (the test of correctness: trueresult after substitution of N by 1).It is easy to see, that similar energy value is obtainedby simplified approach, namely, when using the trialfunction in the form:Φ (0) = N Y n =1 ( u + vS + ( n )) | ↓ > (11)( u + v = 1 , v = N/N ) ; E (0) = (Φ (0) , H Φ (0) ) = ( − tN ν )( uv ) == ( − tν ) N ( − NN ) . The function (11) is a self-consistent field approximation.It is interesting to note, that foregoing is true in a two-dimensional case (square lattice and three-dimensionalspin).
Distribution function
A distribution function can be found for the state (11).The trial function (11) can be rewritten using Bose par-ticles and presented in the form:Φ (0) → N Y n =1 ( u + vA + n ) | > . The number of particles with given quasi-momentum is: < A + ( k ) A ( k ) > = 1 N X n,n ′ < A + n A n ′ > ×× exp (cid:26) i k h R ( n ′ ) − R ( n ) i(cid:27) ; X n,n ′ = X n = n ′ + X n = n ′ . The forbidding of two(many)fold occupation of thecites should be taken into account. The result of thecalculation is: n (0) = < A + (0) A (0) > = N (cid:18) − NN (cid:19) + (cid:18) NN (cid:19) , (12) n ( k ) = < A + ( k ) A ( k ) > (cid:12)(cid:12)(cid:12) k =0 = (cid:18) NN (cid:19) . The summation gives the required result: < A + (0) A (0) > + X k =0 < A + ( k ) A ( k ) > = N (state number N − k = 0). Then the energy value is the same: E (0) → − tν < A + (0) A (0) > + X k =0 ǫ ( k ) < A + ( k ) A ( k ) > . It should be emphasized, that though most particlesare in the condensate (see (12)), the approximate wavefunction of the system cannot be written in the form( A + (0)) N | > (in contrast to weak interaction [1]). Oth-erwise, the forbidden case can take place, i.e. the parti-cles can meet at one cite. The traditional approach
It is interesting to compare obtained energy value withthe one obtained using the traditional approach (see [2])within the gas approximation (in our case it takes placeat
N << N ). The system energy can be estimatedwithin the gas approximation using the scattering am-plitude. This means to find a vertex function in stairapproximation and then to write interaction energy inmain approximation, provided that all the particles arein the condensate.For this purpose Bose particles and their interactionaccording to Hubbard is used (see (3)). The relation forinteraction energy is: H int = U X n A + n A + n A n A n = (13)= UN X p + p = p + p A + ( p ) A + ( p ) A ( p ) A ( p ) (cid:12)(cid:12)(cid:12) U →∞ . According to [2] a full vertex function Γ, describingmutual scattering of two particles, should be found in gasapproximation. It is Γ, that should be used for estimationof the interaction role within the gas limit instead of theinitial interaction U . For this purpose diagram techniqueis used and calculations in stair approximation are made(sum frequency is equal to double particle energy in theband bottom, total momentum is zero):Γ = U + 2 i U N < GG > + ... = U − i ( U/N ) < GG > ; Γ (cid:12)(cid:12) U →∞ → iN < GG > ,< GG > = X p Z dω π G (Ω + ω, p ) G ( − ω, − p ) ,G ( ω, p ) = 1 ω − ǫ ( p ) + iδ . Here sum frequency is Ω = 2 ǫ (0). The result of thecalculation is:Γ − = − iN < GG > = 1 N X p ǫ ( p ) − ǫ (0) ; (14)Γ − → . t . The last value is given for simple cubic lattice. The ex-pression for energy (all the particles are in the condensate A + (0) = A (0) → √ N ) is: E → ǫ (0) < A + (0) A (0) > + Γ N < A + (0) A + (0) A (0) A (0) > = − tν N + Γ N N = − tν N (cid:18) − . ν NN (cid:19) . One can see, that contribution of interaction for cubiclattice ( ν = 6) is one and a half times less than for usedtrial function. It should be emphasized, that it is theconsequence of binary collision approximation. Accuracy evaluation
Corrections to energy can be estimated using the func-tions resulting from Hamiltonian action on the functionΦ N . Thus: S − ( n )Φ N ≡ S − ( n ) S + Φ N − = h S + S − ( n ) − S z ( n ) i Φ N − . First, the value S z ( n )Φ N is found. It is easy to see, that: S z ( n )Φ N = Φ N ( n ) + S + n S z ( n )Φ N − o . From this recurrent relation follows: S z ( n )Φ N = N Φ N ( n ) −
12 Φ N . (15)It is verified directly or by summation by n . Thus: S − ( n )Φ N = Φ N − − N − N − ( n )+ S + n S − ( n )Φ N − o . From this recurrent relation follows: S − ( n )Φ N = N Φ N − − N ( N − N − ( n ) . (16)It is verified by a direct substitution as well as at N =1 , N = 2.The result is: H Φ N = − t (cid:26) N ν Φ N − N ( N − X
Within the framework of the model used, the conven-tional Bose gas theory for the particles with strong inter-action was shown to be inconsistent. If most particles arein Bose condensate, description of their interaction basedon binary collisions (using binary scattering amplitude)does not suit, as there is no quasi-classics for the parti-cles with an infinite wave length. It turns out, as if everyparticle interacts with all the particles at once. This factleads to the increase of the interaction energy, as shownin the model used here.It should be emphasized, that in present work the con-clusion about the accuracy of the approach is made af-ter writing of a trial function. In [3] first the problem is simplified by a transition to a pseudo-potential, thensolved by the perturbation theory. However, there are nosuccessful attempts to write a multi-particle function forinitial interaction (hard spheres).Note, that the results obtained suit for two-dimensional case (three-dimensional spin).So far it is not clear, how the spectrum of elementaryexcitations can be got. Perhaps, a diagram Belyaev - typetechnique (see, for example, [5]) or its modified form (?)should be used.Acknowledgement. The author gratefully acknowl-edges the discussion of A.V. Chaplik, M.V. Entin andV.M. Kovalev and also the financial support of RFBR(Grant 11-02-00060) and Russian Academy of Sciences(Programs). ∗ Electronic address: [email protected][1] N.N. Bogoliubov, J. Phys. U.S.S.R. , 23 (1947).[2] E.M. Lifshitz, L.P. Pitaevskii, Statistical physics, part 2 (Pergamon, 1980).[3] T.D. Lee, K. Huang and C.N. Yang, Phys. Rev. , 1135(1957).[4] Kerson Huang,
Statistical mechanics (John Wiley & Sons,Inc., New York - London, 1963).[5] A.A. Abrikosov, L.P. Gorkov, and I.E. Dzyaloshinski,