Proof of Bose condensation for weakly interacting lattice bosons
aa r X i v : . [ c ond - m a t . qu a n t - g a s ] J un Proof of Bose condensation for weakly interacting lattice bosons
D. P. Sankovich ∗ V. A. Steklov Mathematical Institute, Gubkin str. 8, Moscow, Russia. (Dated: February 9, 2020)
Abstract
A weakly interacting Bose gas on a simple cubic lattice is considered. We prove the existence ofthe standard or zero-mode Bose condensation at sufficiently low temperature. This result is validfor sufficiently small interaction potential and small values of chemical potential. Our methodexploits infrared bound for the suitable two-point Bogolyubov’s inner product. We do not use thereflection positivity or some expansion methods.
PACS numbers: 05.30.Jp, 03.75.Fi, 67.40.-w NTRODUCTION
Bose condensation (BC) is one of the most amazing phenomena exhibited by macroscopicsystems. The study of condensation is fundamental because it is a clue to our understand-ing of phase transitions. BC was first described for an ideal gas of free bosons. In threespatial dimensions, BC occurs at low enough temperature when there is a nonzero densityof particles in zero-momentum state: n = lim V →∞ V D b † b E > , (1)where b † k , b k are the creation and annihilation operators for the single-particle state of mo-mentum ~ k , V is the volume of system and h . . . i denotes thermal averaging. It is recognizedthat the BC is a common quantum property of the many-particle systems. For interactingsystems the standard criterion (1) was reexamined. Onsager and Penrose [1] proposed toidentity condensation with an off-diagonal long-range order in the one-particle reduced den-sity matrix. This criterion shows that the thermal average of particle operator for the mode p = 0 can still be used as a characterization of a BC (in this case the thermal average in (1)is taken over interacting Hamiltonian.) Subsequently, a more detailed classification of thedifferent types of BC was proposed [2]. We will rely on the standard definition. Accordingto the classification of the work [2], this type of BC is called the conventional condensationof type I.The experimental creation of BC (for a review of the theory of trapped Bose gases withextensive references in the literature see [3]) has sparked interest in their properties. Oneof the most hard and comprehensive problem is connected with the rigorous proof of theexistence (or absence) of BC for realistic non-ideal Bose systems. However it is believed thatthis the solvable issue. We will not dwell on the many aspects of BC and refer the readerto monographs [4–8]. In the following we consider a gas of interacting bosons in a lattice.Let us consider this case in more detail.Lattice systems in theoretical physics has long been regarded as an idealization of anatural crystal, subsystems which can be in a finite number of states [9]. Mathematicalphysics applies lattice models to approximate the quantum fields and make sense of thevarious formal considerations [10]. After 1995, the development of experimental physics[11–13] allowed to look at the system as a grid of the observed physical objects, allowingfor a convenient practical implementation [3]. The successes in the experimental study of2C are numerous and impressive. Nevertheless, the rigorous theoretical justification for thisphenomenon remains, as before, not completely solved, even for weakly interacting systemswith pair interaction.The basic model of interacting lattice bosons is the Bose–Hubbard (BH) model [14, 15].The possibility of applying this model to gases of alkali atoms in optical traps was firstrealized in [16]. Theoretical investigation of phase transitions in the BH model is mainlybased on the application of some or other approximate or numerical methods. We willnot discuss these methods and consider a small number of rigorous results. Note that allthese results relate mainly to some simplifying modifications of the BH model (see a verycomplete review [17].) In [18] the lattice infinite-range hopping BH model has been studiedfor all temperatures and chemical potentials. A model with a hard-core BH potential wasstudied by rigorous perturbation theory in [19]. A related model with extra chessboardpotential is considered in [20]. The authors of the last article used the equivalence of themodel considered by them and the XY model of spin 1 / MODEL AND GAUSSIAN DOMINATION
We consider a many boson system in equilibrium, at a given temperature T ( β = ( kT ) − , where k is the Boltzmann constant), a given chemical potential µ , and with given interac-tions. The system is described in the grand canonical formalism. Let us be more preciseand introduce the mathematical framework. Let Λ ⊂ Z be a finite cube of volume V = | Λ | .Introduce the bosonic Fock space F = ⊕ N ≥ H Λ ,N , where H Λ ,N is the Hilbert space of sym-metric complex functions on Λ N . Creation and annihilation operators for a boson at site j ∈ Λ are denoted by a † j and a j , respectively. Then n j = a † j a j is the one-site numberoperator, and N Λ = P j ∈ Λ n j is the total number operator.The basic Hamiltonian of the uniform Bose–Hubbard model is H Λ = t X n,i ( a † n − a † n + δ i )( a n − a n + δ i )+ U X n a † n a n ( a † n a n − − µN Λ , (2)where i is summed from 1 to 3, and n is summed over Λ. Here δ i is the unit vector whose i -th component is 1. We shall consider the periodic boundary conditions, soΛ = { n ∈ Z : − L i / ≤ n i < L i / , i = 1 , , } is a domain of Z wrapped onto a torus. Then the setΛ ∗ = { k i = 2 πl/L i : l = 0 , ± , . . . , ± L i / , i = 1 , , } is dual to Λ with respect to Fourier transformation on the domain Λ = L × L × L . Thefirst term in the Hamiltonian (2) corresponds to the hopping interaction of bosons betweenneighboring sites. The hopping parameter t is chosen to be positive. The second term in(2) is the on-site repulsive interaction ( U > H Λ ≥ U | Λ | N − (cid:18) µ + U (cid:19) N Λ . The superstability condition implies the convergence of the grand-partition function for any µ ∈ R , β >
0. Notice that attraction (
U <
0) makes the model unstable (in contrast to thefermion case.) 4et b † p , b p are creation and annihilation Bose operators with the wave vector p ∈ Λ ∗ , b † p = 1 p | Λ | X n ∈ Λ a † n e ipn , b p = 1 p | Λ | X n ∈ Λ a n e − ipn . In terms of b † p , b p , the Hamiltonian H Λ becomes H Λ = X p ω p b † p b p + U | Λ | X p,q,k b † p b † q b p + k b q − k − µN Λ , where N Λ = P p b † p b p and ω p = 4 t X α =1 sin p α ≡ tǫ p ≥ . Since the summation of p, q, k is always restricted to Λ ∗ , we will not explicitly specify it.First, we define the Bogolyubov inner product (cid:0) b † p , b p (cid:1) = Z − Z Tr (cid:0) b † p e − xβH Λ b p e − (1 − x ) βH Λ (cid:1) dx, where Z Λ = Tr exp( − βH Λ ) is the grand-canonical partition function. Note, that the thermalexpectation of the double commutator c p ≡ (cid:10) [ b † p , [ βH Λ , b p ]] (cid:11) H Λ ≥ . This follows from Bogolyubov’s inequality or by an eigenfunction expansion [31]. The non-negativity of c p means µ ≤ U n Λ , (3)where n Λ = h N Λ i H Λ / | Λ | .Consider the family of operators H Λ ( h ) = X p ω p ( b † p − h ∗ p )( b p − h p )+ U | Λ | X p,q,k b † p b † q b p + k b q − k − µN Λ and introduce the function f ( U ) = Tr e − βH Λ ( h ) − Tr e − βH Λ (0) , (4)where h p ∈ C . We will explore the region of sufficiently small U . In virtue of inequality (3) letus focus on the situation with a small and non-negative chemical potential, µ = U λ + o ( U ),where λ ≥
0. It is easy to verify that f (0) = 0. The derivative of f ( U ) with respect to U is5 ′ ( U ) = − β Tr " | Λ | X p,q,k b † p b † q b p + k b q − k − λ X p b † p b p ! (cid:0) e − βH Λ ( h ) − e − βH Λ (0) (cid:1) . (5)By (5), we obtain f ′ (0) = − β Tr | Λ | X p,q,k h ∗ p h ∗ q h p + k h q − k + 2 | Λ | X p h ∗ p h p N Λ − λ X p h ∗ p h p ! e − βH (0)Λ , where H (0)Λ = X p ω p b † p b p . This implies that f ′ (0) ≤ λ ≤ n (0) , where n (0) = h N Λ i H (0)Λ / | Λ | . We conclude that f ( U ) ≤ U and µ small enough. The inequality f ( U ) ≤ BOSE CONDENSATION
In this section we want to put the results of the previous section to prove that the BCoccurs in the model (2) for U and µ sufficiently small.From the inequality f ( U ) ≤ − βH Λ ( h )) takes the maximumvalue at { h p = 0 } . A necessary condition for Tr exp( − βH Λ ( h )) to be maximum at { h p = 0 } is represented by inequality (cid:0) b † p , b p (cid:1) H Λ ≤ ( βω p ) − , p = 0 . (6)Infrared bound (6) is essential for our proof of the BC. The proof comes from two points.The first is the Falk–Bruch inequality [32]. We use this inequality to relate Bogolyubov’sinner product to the conventional two-point thermal average. The second is the sum rulefor this average. Using this rule, we obtain some conditions, in which the contribution from p = 0 remains non-vanishing in the thermodynamic limit. As a result, we prove the existenceof one-mode Bose condensate. This is the method of infrared bounds that was originallyintroduced for the classical Heisenberg model in [21]. Later on this method was extended tothe quantum spin systems [31] and to the Bose systems [26, 33].6s noted above, the upper bound for the two-point temperature average (cid:10) b † p b p (cid:11) H Λ can beobtained from the upper bound of ( b † p , b p ) H Λ by the Fulk–Bruch inequality. For any A andself-adjoint H we have the bound b ( A ) ≥ g ( A ) h (cid:18) c ( A )4 g ( A ) (cid:19) , where b ( A ) ≡ ( A † , A ) , g ( A ) ≡ h A † A + AA † i ,c ( A ) ≡ h [ A † , [ βH, A ]] i , and h ( x tanh x ) = x − tanh x. The function h is a well defined strictly monotonically de-creasing convex function from (0 , ∞ ) to (0 ,
1) withlim x → h ( x ) = 1 , lim x →∞ h ( x ) = 0 . Suppose that b ≤ b and c ≤ c . Then g ≤ g , where g = 12 p c b coth r c b . We refer to [31] for the proof of these relations, and for further statements about correlationfunctions. The thermal average of the double commutator is c ( b p ) ≡ c p = β ( ω p + 2 n Λ U − µ ) , p ∈ Λ ∗ , (7)where n Λ is the filling (the thermal average number of particles per site.) The non-negativityof c p means ∆ Λ ≡ n Λ U − µ ≥ . From the Falk–Bruch inequality and bounds (6), (7) we infer that for p = 0, (cid:10) b † p b p (cid:11) H Λ ≤ s ω p + 2 n Λ U − µω p coth β q ω p ( ω p + 2 n Λ U − µ ) − ≡ F Λ ( p, µ ) . From the sum rule 1 | λ | X p (cid:10) b † p b p (cid:11) H Λ = h N i H Λ | Λ | = n Λ , (8)we conclude that the Bose condensate density n = lim | Λ |→∞ | Λ | h b † b i H Λ n = lim | Λ |→∞ n Λ > π Z π − π dk Z π − π dk Z π − π dk F ( k, µ ) , (9)where F ( p, µ ) = lim | Λ |→∞ F Λ ( p, µ ).We will explore the region of non-negative chemical potentials. In this case we have∆ = lim | Λ |→∞ ∆ Λ ≤ nU. Inequality (9) is then more accomplished if will do the followinginequality n > π Z π − π dk Z π − π dk Z π − π dk F ( k, . (10)This follows from the fact that the function F ( p, µ ) is a monotonically increasing functionof ∆.Derive the conditions under which executes the inequality (10). Consider first the caseof zero temperature. Then n > π Z π − π dk Z π − π dk Z π − π dk r nUω k − ! ≡ J ( n ) . If J ′ (0) <
1, then J ( n ) < n for all n ≥
0. Inequality J ′ (0) < Ut ≤ W , (11)where W ≈ .
51 is the Watson’s integral.Consider the case of positive temperatures. Use the estimate coth x ≤ x − in themain inequality (10). Then we have that (10) is accomplished if n − J ( n ) > W βt . (12)We conclude that there is Bose condensation at some finite β whenever (11) and (12) hold.The temperature of the phase transition is estimated from below as θ c ≥ tW [ n − J ( n )] . CONCLUSION
In this paper we have studied a lattice superstable model of imperfect Bose gas. Thepresence of a BC is established for small enough interaction potential U , and small chemicalpotential µ ≥
0. A lower estimate for the critical temperature is received. Our proofs exploit8nfrared bounds, and does not exploit reflection positivity or some expansion methods. Theessential ingredient in the proof of main results is the basic bound (6). Our method relieson the fact that the bound (6) and the sum rule (8) force a macroscopic occupation in the p = 0 mode. This is the conventional BC.We proved (6) for small enough U . Note that (6) is true for small enough t . Indeed,consider (4) as function of t . Then f ′ ( t = 0) = − β Tr "X p ǫ p | h p | e − β ( H Λ | t =0 − µN Λ ) ≤ , for any µ . We see that (6) is true for t small enough ( U is large enough.) Will (6) be truefor any U and t ? This fundamental issue remains open. ∗ [email protected][1] O. Penrose and L. Onsager, Phys. Rev. , 576 (1956).[2] M. van den Berg, J. T. Lewis, and J. V. Pul`e, Helv. Phys. Acta , 1271 (1986).[3] A. S. Parkins and D. F. Walls, Phys. Rep. , 1 (1998).[4] C. Pethick and H. Smith, Bose-Einstein condensation in dilute gases (Cambridge UniversityPress, Cambridge, 2008) p. 569.[5] L. Pitaevskii and S. Stringari,
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