Ground state energy of the dilute Bose-Hubbard gas on Bravais lattices

GR OUND ST A TE ENER GY OF THE DILUTE BOSE-HUBBARD GAS ON BRA V AIS LA TTICES NORBER T MOKRZA ´ NSKI, MAR CIN NAPI ´ ORKO WSKI, AND JACEK WOJTKIEWICZ Abstract. W e study interacting bosons on a three–dimensional Bra v ais lattice with positive hopping amplitudes and on-site repulsive interactions. W e prove that, in the dilute limit ρ → 0, the ground state energy density satisﬁes e ( ρ ) = 4 π a ρ 2  1 + O ( ρ 1 / 6 )  where a is the lattice scattering length deﬁned through the corresponding tw o–b o dy problem. This establishes the analogue of the Dyson and Lieb–Yngv ason theorems for the Bose-Hubbard gas. Our result sho ws that the leading- order energy is universal: although the lattice geometry aﬀects the microscopic disp ersion relation, it enters the leading order asymptotics only through the scattering length. In particular, it is independent of other features of the underlying Bra v ais lattice. 1. Introduction Understanding the ground state energy of in teracting quan tum many–bo dy systems is a central problem in mathematical ph ysics. Although a complete description is generally out of reach, rigorous results can b e obtained in suitable asymptotic regimes. One particularly tractable regime is the dilute limit, where the particle densit y ρ is suﬃciently small compared to the in teraction scale. In this setting, Bose gases exhibit a remark able univ ersality: to leading order, the ground state energy dep ends only on a single eﬀective parameter, the t wo–bo dy scattering length a . F or three–dimensional contin uum Bose gases with repulsive interactions, this is seen in the so-called Lee-Huang- Y ang-W u [24, 41] formula e 0 ( ρ ) = 4 π a ρ 2  1 + 128 15 √ π ( ρ a 3 ) 1 / 2 + 8( 4 π 3 − √ 3) ρ a 3 ln( ρ a 3 ) + . . .  , whic h captures in the dilute regime ρ a 3 → 0 the correct ground state energy p er unit volume, up to corrections of order a ρ 2 ( ρ a 3 ), whic h are expected to no longer be univ ersal in a . The leading-order term w as rigorously established b y Dyson [10] as an upp er b ound and, ov er 40 y ears later, b y Lieb and Yngv ason [26] as a low er b ound (see also [43]). An upp er b ound matching the second order term was established by Y au and Yin [42] (see also [11, 4, 3]), while a low er b ound ﬁnishing the pro of of the Lee-Huang-Y ang conjecture w as established b y F ournais and Solo vej in [15] (see also [16]). In [21, 22], a new, simpler pro of that also establishes the free energy expansion in the p ositiv e temp erature case w as giv en (see also [38, 44, 2]). Finally , recently , Bro oks, Olden brug, Saint Aubin and Sc hlein [7] established for the ﬁrst time an upper b ound that includes the third order term (so-called W u term). A natural question is whether this universalit y p ersists in discrete settings. Bose gases realized in optical lattices are describ ed by lattice Hamiltonians, most prominently b y the Bose–Hubbard mo del [17, 14], which has b ecome a standard eﬀectiv e mo del for interacting b osons and has been extensively studied in both theory and exp erimen t. Suc h systems arise on a v ariet y of lattice geometries [34] b ey ond the simple cubic case, which motiv ates the con- sideration of general Brav ais lattices. In this setting the single–particle disp ersion and the asso ciated low–energy kinematics dep end strongly on the geometry and hopping amplitudes of the underlying lattice. In contrast to the con tinuum, b oth the in teraction and the lattice structure inﬂuence the tw o–b o dy problem, and it is not a priori clear whether the leading-order energy retains a universal form indep enden t of the microscopic details. In this work we show that such universalit y indeed survives on lattices as far as the leading order term of the energy is concerned. W e consider in teracting b osons on an arbitrary three–dimensional Brav ais lattice with p ositive hopping amplitudes and on-site repulsive interactions and pro ve that, in the dilute limit e 0 ( ρ ) = 4 π a ρ 2 (1 + O ( ρ 1 / 6 )) where a denotes the lattice scattering length (cf. App endix C). Thus all microscopic information — b oth the in teraction strength and the lattice geometry — is absorb ed into this eﬀectiv e parameter, and the leading-order energy is indep endent of other details of the underlying lattice. This provides a discrete analogue of Dyson and Lieb–Yngv ason theorems for the Bose–Hubbard gas. It is w orth emphasizing that this universalit y is sp eciﬁc to the leading-order term. While in the contin uum the second and (exp ectedly) third order terms also exhibit a univ ersal structure dep ending only on the scattering length, in the lattice setting one exp ects higher-order terms to dep end explicitly on the single-particle disp ersion and hence 1 2 N. MOKRZA ´ NSKI, M. NAPI ´ ORKO WSKI, AND J. WOJTKIEWICZ on the geometry of the underlying lattice. In particular, b ey ond order a ρ 2 the energy is not determined solely b y the scattering length. Our result therefore iden tiﬁes the precise regime in which lattice eﬀects are completely absorb ed in to this eﬀective parameter. W e expect that the same leading-order univ ersalit y holds for more general short-range lattice p otentials. Related univ ersalit y results hav e b een obtained for fermionic systems, b oth in the con tin uum [25, 13, 18, 19, 8] and on the cubic lattice (with nearest n eighbor hopping) [20, 39], where the leading-order energy is again determined b y an appropriate scattering parameter. The av ailable lattice pro ofs for fermions employ techniques that do not readily transfer to bosons. In particular, arguments based on Dyson-type lemmas hav e no direct counterparts. Therefore, in order to prov e our main result, w e rely on tec hniques that ha ve b een developed more recen tly in the con text of contin uous b osonic systems. F or the low er b ound we use a localization method coupled with Bogoliub ov theory [5, 32, 9]. T o mak e it work we need to dev elop estimates for the eigen v alues of Neumann Laplacians on general Brav ais lattices. In fact, these b ounds lead to the relative error of order O ( ρ 1 / 6 ) in the low er b ound. The upp er b ound is an adaptation of the argumen t in [11] that allows to include lattice disp ersion relations which are not radial. The remainder of the pap er is organized as follows. In Section 2 we in tro duce the lattice framework and state the main result. The upper b ound is obtained via suitable grand-canonical trial states and the equiv alence of ensembles. This is presented in Section 3. The corresp onding low er bound is pro ved in Section 4. Sev eral auxiliary results (including the discussion on Brav ais lattices and the lattice scattering length) are collected in the Appendices. Data a v ailabilit y . Data sharing is not applicable to this article as no new data were created or analyzed in this study . Ac kno wledgemen ts. The work of NM and MN was supported b y the National Science Centre (NCN) grant Sonata Bis 13 (pro ject num b er 2023/50/E/ST1/00439). 2. The model and the main resul t W e start by presenting basic deﬁnitions and ob jects of interest. W e refer to App endix A for the discussion concerning those asp ects. Our main ob ject of in terest is a three dimensional (monoatomic) crystal lattice. T o deﬁne it w e need to specify the underlying Brav ais lattice and the neigh b orho o d relation b etw een the points on the lattice. T o this end we ﬁrst ﬁx three linearly indep endent vectors a 1 , a 2 , a 3 ∈ R 3 and a matrix A comp osed of those v ectors as columns. W e consider a Bra v ais lattice Λ deﬁned as (2.1) Λ = A Z 3 = { m 1 a 1 + m 2 a 2 + m 3 a 3 : m 1 , m 2 , m 3 ∈ Z } In this context vectors a i are called the primitiv e (translation) vectors of the lattice Λ. F or a given even num b er L ∈ 2 N we consider a ﬁnite version of the Brav ais lattice (2.1) of size L , denoted Λ L and deﬁned as Λ L = A ( Z ∩ [ − L/ 2 , L/ 2]) 3 =  m 1 a 1 + m 2 a 2 + m 3 a 3 : m 1 , m 2 , m 3 = − L 2 , − L 2 + 1 , . . . , L 2 − 1 , L 2  . (2.2) W e equip Λ L with p erio dic b oundary condition (i.e Λ L ≃ ( A Z 3 ) / (( L + 1) A Z ) 3 ) and with the standard counting measure, hence we can deﬁne the (one particle) Hilb ert space H L of a particle on the lattice Λ L as H L = L 2 (Λ L ) with the inner product ⟨ ψ , φ ⟩ H L = X x ∈ Λ L ψ ( x ) φ ( x ) . F or a given natural num b er N w e also deﬁne N -particle b osonic Hilbert space H N L as H N L = N O sym H L , i.e. the functions of N v ariables x 1 , x 2 , . . . , x N ∈ Λ L symmetric under the p ermutations of those v ariables. The inner product on this space is deﬁned for simple tensors as * N O j =1 ψ j , N O j =1 φ j + H N L = N Y j =1 ⟨ ψ j , φ j ⟩ H L , whic h can b e extended to the whole H N L b y linearit y . GROUND ST A TE ENERGY OF THE DILUTE BOSE-HUBBARD GAS ON BRA V AIS LA TTICES 3 No w w e will deﬁne the neigh b or relation on Λ and Λ L . Let D b e a set of all ”p ositive directions” D = { m 1 a 1 + m 2 a 2 + m 3 a 3 ∈ Λ \ { 0 } : the ﬁrst non-zero m j is positive } . Note that D ∪ ( − D ) = Λ \ { 0 } and D ∩ ( − D ) = ∅ . T o eac h direction v ∈ D and its rev erse direction ( − v ) w e will assign a weigh t t ( v ) = t ( − v ) ≥ 0. The neighborho o d relation on Λ is deﬁned as (2.3) x ∼ y ⇐ ⇒ t ( y − x ) > 0 . This relation is symmetric and equips b oth the inﬁnite lattice Λ and the ﬁnite lattice Λ L with the weigh ted graph structure, where in the latter y − x is understo o d in the sense of p erio dic b oundary condition, i.e. as the element of the ( A Z 3 ) / (( L + 1) A Z ) 3 ) group. W e will make tw o additional assumptions. The ﬁrst one is that (2.4) # { v ∈ D : t ( v )  = 0 } < ∞ , meaning that w e only consider a ﬁnite distance hopping. This assumption is satisﬁed in most commonly encountered crystal systems in ph ysics. F or the future purp oses we will also deﬁne a parameter R 0 ( t ) called the hopping length as (2.5) R 0 ( t ) = min { L ∈ 2 N : ∀ x ∼ 0 x ∈ Λ L } , that is the smallest L such that all the neigh b ors of p oin t x = 0 in the sense of (2.3) b elong to Λ L . In the up coming pro ofs we will consider only L ≥ R 0 ( t ) as this condition will allow us to capture all the p ossible hoppings within the ﬁnite volume. T o state the second assumption we will ﬁrst denote (2.6) D 1 := { a 1 , a 2 , a 3 } ⊂ D . W e will assume (2.7) t ( v )  = 0 for v ∈ D 1 that is hopping along the primitive vectors of the lattice Λ is allow ed. The Bose-Hubbard Hamiltonian H N ,L of the N particle system, acting on H N L , is given by H N ,L = − N X i =1 ∆ i + U N X i 0 (i.e. the interaction is repulsiv e). The ground state energy E 0 ( N , L ) of the system is deﬁned b y (2.9) E 0 ( N , L ) = inf ψ ∈H N L ∥ ψ ∥ =1 ⟨ ψ , H N ,L ψ ⟩ H N L . The inner pro duct abov e is called the exp ectation v alue of the op erator H N in the state ψ . In general, the exp ectation v alue of some op erator T acting on the Hilb ert space H in the state (i.e. normalized vector) ψ ∈ H is deﬁned as (2.10) ⟨ T ⟩ ψ = ⟨ ψ , T ψ ⟩ H . W e will use this notation when there will b e no ambiguit y on which Hilb ert space this exp ectation is ev aluated. W e are interested in the ground state energy per unit volume, i.e. (2.11) e 0 ( ρ ) = lim N →∞ L →∞ N/ | Λ L |→ ρ E 0 ( N , L ) | Λ L | . Existence of this limit (under some more general assumptions and even in some broader setting) is known, w e refer e.g. to [36] for the details. It is also kno wn that e ( ρ ) is a con tinuous (up to the b oundary ρ = 0) and conv ex function of ρ . 4 N. MOKRZA ´ NSKI, M. NAPI ´ ORKO WSKI, AND J. WOJTKIEWICZ In order to state the main theorems w e introduce the sc attering length of the p otential whic h we will denote a (see Appendix C for more details). F or the on-site interaction p oten tial that we are dealing with it is deﬁned as (2.12) 8 π a = U U γ + 1 , with γ = 1 2 | b Λ | − 1 Z b Λ dp ε ( p ) and ε ( p ) b eing the lattice disp ersion relation, given by (2.13) ε ( p ) = X v ∈ D 2 t ( v )  1 − cos( v · p )  = 4 X v ∈ D t ( v ) sin 2  v · p 2  , p = ( p 1 , p 2 , p 3 ) ∈ b Λ . Here b Λ is the Brillouin zone of the lattice Λ b Λ = B T 3 =  b 1 t 1 + b 2 t 2 + b 3 t 3 : − 1 2 ≤ t i < 1 2  with perio dic boundary conditions , where T 3 = [ − 1 2 , 1 2 ) 3 is a three dimensional unit torus (this identiﬁcation also allows to identify the Haar measure dp in the in tegral as the Leb esgue measure), | b Λ | denotes the measure of this set and B is a matrix comp osed of columns b 1 , b 2 , b 3 satisfying a i · b j = 2 π δ i,j , that is these are the primitiv e vectors of the recipro cal lattice Λ ∗ . W e refer to Appendix A for a more detailed discussion. The expression ε ( p ) can b e seen as the eigenv alue corresp onding to the eigenfunction χ p ( x ) = e ip · x , x ∈ Λ of the discrete Laplacian deﬁned in (2.8). The sum in (2.13) is ﬁnite due to the assumption (2.4). The formula (2.12) is derived explicitly in App endix C. Due to the assumption (2.7) there exist p 0 > 0 suc h that we hav e the estimate (2.14) ε ( p ) ≥ c | p | 2 for | p | < p 0 with | p | b eing the Euclidean norm of the vector p ∈ R 3 and with the constant c indep endent of p (one can take c = 1 2 min { t ( a 1 ) , t ( a 2 ) , t ( a 3 ) } > 0). This in particular implies that γ is well deﬁned as the function 1 / ε ( p ) is in tegrable near zero. Moreo ver, as we will see in the pro ofs of the following prop ositions, this inequality will b e crucial for obtaining the desired results. The assumption (2.7) itself can b e c hanged in suc h a wa y that (2.14) still holds true, for example assuming that some certain other hopping constan ts are non-zero (this how ev er would require some mo diﬁcations in the pro ofs). F or the purp ose of this pap er we will stick to (2.4) as this is the simplest case when (2.14) holds. The main theorem that we prov e is the following Theorem 2.1. In the setting as ab ove, in p articular with assumptions (2.4) and (2.7) , we have e 0 ( ρ ) = 4 π a ρ 2  1 + O  ρ 1 / 6  ρ → 0  , W e will pro ve Theorem 2.1 by proving the upp er and the low er b ound separately . In fact the upp er b ound (see b elo w) pro vides a better error estimate that the one coming from the low er bound. As men tioned in the introduction, the pro of of the upper b ound will b e based on [11], adapted to the lattice setting. The main p oint of this approac h is that instead of constructing a sequence of states on H N L , i.e. states with ﬁxed num b er of particles, w e will work in the grand canonical setting. More precisely , we will construct a sequence of states { Ψ L,N } L,N on the F o ck spaces F L := F ( H L ) = ∞ M n =0 H n L , ( H 0 L = C ) with eac h ha ving ﬁxed av erage num b er of particles ⟨N ⟩ Ψ N,L = N . W e will denote the grand canonical Hamiltonian (i.e. the second quantization of H N ,L ) as H GC L . This op erator acts on the F o ck space F L , for Ψ = (Ψ ( n ) ) n ∈ N ∈ F L its action is giv en b y (2.15) ( H GC L Ψ) ( n ) = H n,L Ψ ( n ) . W e are going to pro ve the following result. GROUND ST A TE ENERGY OF THE DILUTE BOSE-HUBBARD GAS ON BRA V AIS LA TTICES 5 Prop osition 2.2. L et ρ > 0 b e smal l enough. F or any se quenc es N → ∞ , L → ∞ with N | Λ L | → ρ ther e exists a se quenc e of trial states Ψ L ∈ F L with ⟨N ⟩ Ψ L = N such that lim L →∞ N →∞ N/ | Λ L |→ ρ ⟨ H GC L ⟩ Ψ L | Λ L | = 4 π a ρ 2  1 + O ( ρ 1 / 2 )  . The upp er b ound (with the same error term as ab ov e) follows then from the v ariational principle and the equiv alence of ensem bles. The adaptation of this w ell known argument for the discrete setting will b e presen ted at the end of Section 3, after the pro of of Proposition 2.2. As for the low er b ound we will use the metho d of dividing the large (thermo dynamic) lattice Λ L in to smaller sub-lattices Λ ℓ with a prop erly c hosen length scale ℓ . In the standard pro of of the corresp onding b ound for the con tinuous (i.e. non-discrete) setting presented e.g. in [28, Chapter 2.2] the length scale ℓ is chosen is suc h a wa y that ρ 1 / 3 ≪ ℓ ≪ ρ − 1 / 2 , which allows to eﬀectiv ely use the Dyson lemma and obtain the desired result. Since we w ere unable to pro ve the discrete analogue of this lemma that w ould b e of use to us, w e propose a diﬀerent approach and choose ℓ ∼ ρ − 1 / 2 , whic h is commonly known as the Gross-Pitaevskii length scale. Then w e will use the metho d from [32] (also recently used in [9]) to get the op erator inequality b ounding H n,ℓ from b elow for certain v alues of n and ℓ . In the end we will use the obtained b ound and the method from [5] to get the following result. Prop osition 2.3. F or smal l enough ρ > 0 the gr ound state ener gy density in the thermo dynamic limit satisﬁes e 0 ( ρ ) ≥ 4 π a ρ 2  1 − O ( ρ 1 / 6 )  . Let us stress that the w orse error term than in the upper b ound is a consequence of sp ectral estimates that we deriv e for general Brav ais lattices with general neighbor relations. F or example, this error can b e impro ved to b e of order O ( ρ 1 / 2 ln ρ ) if one considers cubic lattices with nearest neighbor hopping. The pro of of Prop osition 2.3 is giv en Section 4. This will ﬁnish the pro of of Theorem 2.1. In the rest of the pap er we use the conv ention that C denotes a generic constant (indep enden t of relev ant parameters) whic h ma y c hange from line to line. 3. The upper bound This section is devoted the pro of of Prop osition 2.2 and the corresp onding upp er b ound. As mentioned before, the idea of the proof will follow the one in [11], but with some adaptation to the discrete setting. In particular, we will use metho ds that do not rely on the spherical symmetry of the system. The proof will be done in a few steps, man y of them b eing by now standard in the ﬁeld. 3.1. Momen tum representation of the Hamiltonian. It will b e con v enient to rewrite the grand canonical Hamiltonian (2.15) in the formalism of creation and annihilation op erators - w e refer to [40] and [31] for more details concerning second quantization and Bogoliub ov transformations. W e will also use notation from Appendix A. W e will ﬁx L ∈ 2 N satisfying L ≥ R 0 ( t ) (recall deﬁnition (2.5)) and w ork within the momen tum representation. F ollowing App endix A (in particular section A.5) w e denote b Λ L =    d X j =1 m j b j L + 1 : m j = − L 2 , − L 2 + 1 , . . . , L 2 − 1 , L 2    , where b j ae primitiv e v ectors of the recipro cal lattice Λ ∗ . F or p ∈ b Λ L w e denote b y a p and a ∗ p annihilation and creation operators of a particle with a momentum p ∈ b Λ L , that is a p = a ( χ p ) , a ∗ p = a ∗ ( χ p ) , where χ p ( x ) = 1 | Λ L | 1 / 2 e ip · x , x ∈ Λ L . Direct computation of the matrix elements of the one- and tw o-b o dy op erators in H N ,L in the ab ov e basis yields (3.1) H GC L = X p ε ( p ) a ∗ p a p + U 2 | Λ L | X p,q ,k a ∗ p + k a ∗ q − k a q a p , where indices p, q , k run o ver b Λ L and ε ( p ) is deﬁned analogously as in (2.13) but only for discrete v alues of p : (3.2) ε ( p ) = X v ∈ D 2 t ( v )  1 − cos( v · p )  , p = ( p 1 , p 2 , p 3 ) ∈ b Λ L . 6 N. MOKRZA ´ NSKI, M. NAPI ´ ORKO WSKI, AND J. WOJTKIEWICZ W e also note that creation and annihilation operators satisfy standard canonical comm utation relations [ a p , a k ] = [ a ∗ p , a ∗ k ] = 0 , [ a p , a ∗ k ] = δ p,k . 3.2. Construction of the states. W e pro ceed to the construction of the trial state Ψ L ∈ F L with ⟨N ⟩ Ψ L = N . As for this moment v alues of N and L are ﬁxed, we will simplify notation and omit index L in some of the ob jects (i.e. Ψ L = Ψ etc.). Consider N 0 satisfying 0 ≤ N 0 ≤ N and for p ∈ b Λ L \ { 0 } consider a (ﬁnite) sequence of real num b ers ( c p ) ⊂ R , suc h that | c p | < 1 and c p = c − p . Deﬁne the state Ψ as (3.3) Ψ =   e − N 0 / 2 Y p  =0 (1 − c 2 p ) 1 / 4   e 1 2 P p  =0 c p a ∗ p a ∗ − p + √ N 0 a ∗ 0 | 0 ⟩ , where p belongs to Λ ∗ . Here the exp onent should b e understo o d as a notation for the prop er series expansion. One can recognize that Ψ is a so-called Bogoliub ov trial state, that is it is of the form Ψ = W U ∗ | 0 ⟩ , where W = W  p N 0 / | Λ |  = e √ N 0 ( a ∗ 0 − a 0 ) is the W eyl op erator build up on constan t function p N 0 / | Λ | and U is the Bogoliub ov transformation giv en by (3.4) U = exp   X p ∈ b Λ \{ 0 } − artanh c p  a ∗ p a ∗ − p − a p a − p    . The state Ψ is normalized and conserv es momentum, meaning that (3.5) p  = q ⇒ ⟨ a ∗ p a q ⟩ Ψ = 0 and p  = − q ⇒ ⟨ a p a q ⟩ Ψ = ⟨ a ∗ p a ∗ q ⟩ Ψ = 0 . 3.3. Computation of the energy. W e will no w ﬁnd the energy ⟨ H GC L ⟩ Ψ of the system in the state Ψ. This is a w ell-known computation, so will only state the main steps. It follo ws from the prop erties of the W eyl and Bogoliubov transformations that (3.6) ⟨ a ∗ 0 a 0 ⟩ Ψ = N 0 , ⟨ a ∗ 0 a ∗ 0 a 0 a 0 ⟩ Ψ = ⟨ a 0 a 0 ⟩ Ψ = N 2 0 and for p  = 0 (3.7) ⟨ a ∗ p a p ⟩ Ψ = c 2 p 1 − c 2 p , ⟨ a ∗ p a ∗ − p ⟩ Ψ = ⟨ a p a − p ⟩ Ψ = c p 1 − c 2 p . The same computation as in [30, Appendix A.] leads, using (3.6) and (3.7), to the following expression ⟨ H GC L ⟩ Ψ = X p  =0 ε ( p ) ⟨ a ∗ p a p ⟩ + U 2 | Λ L | X p,q  =0  ⟨ a ∗ p a ∗ − p ⟩⟨ a q a − q ⟩ + 2 ⟨ a ∗ p a p ⟩⟨ a ∗ q a q ⟩  + U N 0 2 | Λ L | X p  =0  2 ⟨ a p a − p ⟩ + 4 ⟨ a ∗ p a p ⟩  + U N 2 0 2 | Λ L | = X p  =0 ε ( p ) c 2 p 1 − c 2 p + U 2 | Λ L | X p,q  =0 " c p c q (1 − c 2 p )(1 − c 2 q ) + 2 c 2 p c 2 q (1 − c 2 p )(1 − c 2 q ) # + U N 0 | Λ L | X p  =0 " c p 1 − c 2 p + 2 c 2 p 1 − c 2 p # + U N 2 0 2 | Λ L | . (3.8) As men tioned in the statement of Proposition 2.2 we will only consider states Ψ ∈ F with ﬁxed exp ectation v alue of particle num b ers N := ⟨N ⟩ Ψ and later consider v alues of N and L such that ⟨N ⟩ Ψ | Λ L | = N L 3 → ρ when N → ∞ and L → ∞ . By (3.6) and (3.7) w e hav e ⟨N ⟩ Ψ = N 0 + X p  =0 c 2 p 1 − c 2 p GROUND ST A TE ENERGY OF THE DILUTE BOSE-HUBBARD GAS ON BRA V AIS LA TTICES 7 hence for considered v alues of N and L we get (3.9) ρ = N 0 | Λ L | + 1 | Λ L | X p  =0 c 2 p 1 − c 2 p + o (1) L →∞ , where o (1) L →∞ denotes the expression con v erging to zero as L → ∞ . W e also observ e that (as ρ is ﬁxed) (3.10) 1 | Λ L | 2    N 2 0 + 2 N 0 X p  =0 c 2 p 1 − c 2 p +   X p  =0 c 2 p 1 − c 2 p   2    = ρ 2 + o (1) L →∞ . F rom no w on we will assume that parameters N 0 and c p are c hosen in suc h a w ay that (3.9) is satisﬁed. In the follo wing part it will b e conv enient to rewrite the exp ectation v alue ⟨ H GC L ⟩ Ψ in (3.8) to express it in terms of the total densit y ρ . The observ ations (3.9), (3.10) and X p,q  =0 c 2 p c 2 q (1 − c 2 p )(1 − c 2 q ) =   X p  =0 c 2 p 1 − c 2 p   2 sho w that ⟨ H GC L ⟩ Ψ = X p  =0 ε ( p ) c 2 p 1 − c 2 p + U 2 | Λ L | X p,q  =0 c p c q (1 − c 2 p )(1 − c 2 q ) + U   ρ − 1 | Λ L | X q  =0 c 2 q 1 − c 2 q + o (1) L →∞   X p  =0 c p + c 2 p 1 − c 2 p + U 2 | Λ L |  ρ 2 + o (1) L →∞  , whic h after rewriting giv es ⟨ H GC L ⟩ Ψ = X p  =0 ε ( p ) c 2 p 1 − c 2 p + U ρ ( c p + c 2 p ) 1 − c 2 p + U 2 | Λ | X p,q  =0 c p c q − 2 c 2 q ( c p + c 2 p ) (1 − c 2 p )(1 − c 2 q ) + o (1) L →∞ · X p  =0 c p + c 2 p 1 − c 2 p + U 2 | Λ |  ρ 2 + o (1) L →∞  (3.11) W e expect (and pro v e it in further steps) that with prop er selection of co eﬃcients c p the v alue of X p,q  =0 c 2 p c 2 q (1 − c 2 p )(1 − c 2 q ) =   X p  =0 c 2 p 1 − c 2 p   2 will be negligible in the thermo dynamic limit in the dilute regime. T o this end we rewrite (using symmetry of summation with resp ect to indices p and q ) X p,q  =0 c p c q − 2 c 2 q ( c p + c 2 p ) (1 − c 2 p )(1 − c 2 q ) = X p,q  =0 ( c p − c 2 p )( c q − c 2 q ) (1 − c 2 p )(1 − c 2 q ) − 3 X p,q  =0 c 2 p c 2 q (1 − c 2 p )(1 − c 2 q ) =   X p  =0 c p − c 2 p 1 − c 2 p   2 − 3   X p  =0 c 2 p 1 − c 2 p   2 . With this result expression (3.11), after further simpliﬁcations c p − c 2 p 1 − c 2 p = c p 1+ c p and c p + c 2 p 1 − c 2 p = c p 1 − c p b ecomes ⟨ H GC L ⟩ Ψ = X p  =0 ε ( p ) c 2 p 1 − c 2 p + U ρc p 1 + c p + U 2 | Λ L |   X p  =0 c p 1 − c p   2 − 3 U 2 | Λ L |   X p  =0 c 2 p 1 − c 2 p   2 + o (1) L →∞ · X p  =0 c p 1 − c p + U 2 | Λ L |  ρ 2 + o (1) L →∞  (3.12) 8 N. MOKRZA ´ NSKI, M. NAPI ´ ORKO WSKI, AND J. WOJTKIEWICZ As a ﬁnal mo diﬁcation of this expression we will replace the squared term in the ﬁrst line with a linear one at exp ense of some another negligible term in the low density limit. The main idea is to add and subtract a ρw (0) term to every element of the sum (also recall that w (0) is giv en in (C.8)). W e will denote (3.13) s p = c p 1 + c p and for conv enience w e will additionally deﬁne s 0 = 0. Next w e write   X p  =0 s p   2 =   X p ∈ b Λ L s p   2 =   X p ∈ b Λ L  s p + ρw (0)    2 − 2 | Λ | ρw (0) X p ∈ b Λ L s p − | Λ L | 2 ρ 2 w (0) 2 =   X p ∈ b Λ L  s p + ρw (0)    2 − 2 | Λ L | ρw (0) X p  =0 s p − | Λ L | 2 ρ 2 w (0) 2 . Ev entually w e obtain the follo wing expression for the energy in the state Ψ ⟨ H GC L ⟩ Ψ = X p  =0 ε ( p ) c 2 p 1 − c 2 p + U ρ c p 1 − c p − U ρw (0) c p 1 + c p ! − U 2 | Λ L | ρ 2 w (0) 2 + U 2 | Λ L |   X p ∈ b Λ L  s p + ρw (0)    2 − 3 U 2 | Λ L |   X p  =0 c 2 p 1 − c 2 p   2 + o (1) L →∞ · X p  =0 c p 1 − c p + U 2 | Λ L |  ρ 2 + o (1) L →∞  . (3.14) 3.4. Minimalization. W e pro ceed to the minimalization pro cedure. F or a start, we are in terested in term-by-term minimization of the sum in the ﬁrst line of (3.14), that is the w ant to minimize: (3.15) ε ( p ) c 2 p 1 − c 2 p + U ρ c p + c 2 p 1 − c 2 p − U ρw (0) c p 1 + c p . As mentioned b efore, it will turn out that the quadratic terms (second line in (3.14)) will b e negligible with the selection of c p minimizing this expression. In a more concrete manner, we will start with pro ving the following follo wing Lemma. Lemma 3.1. The minimal value of (3.15) is (3.16) 1 2  p ( ε ( p ) + 2 U ρ ) ( ε ( p ) + 2 U ρw (0)) − ε ( p ) − U ρ (1 + w (0))  . The explicit values of c p c an b e r e c over e d fr om r elation (3.19) . Pr o of. In order to minimize (3.15) it will b e conv enient to rewrite it in terms of v ariable s p in tro duced in (3.13). The in verse relation is given by (3.17) c p = s p 1 − s p and since c p ∈ ( − 1 , 1) we hav e s p ∈ ( −∞ , 1 2 ). Direct computation yields c 2 p 1 − c 2 p = s 2 p 1 − 2 s p , c p 1 − c p = s p 1 − 2 s p , so (3.15) expressed in terms of s p b ecomes (3.18) ε ( p ) s 2 p 1 − 2 s p + U ρ s p 1 − 2 s p − U ρw (0) s p . No w the minimalization problem reduces to ﬁnding minimum of the function F ( x ) := A x 2 1 − 2 x + B x 1 − 2 x − C x. for A, B , C > 0 on the domain x < 1 2 . A straightforw ard analysis shows that the minimal v alue of the function F is attained at x 0 = 1 2 − 1 2  1 + 2 B − C A + 2 C  1 / 2 GROUND ST A TE ENERGY OF THE DILUTE BOSE-HUBBARD GAS ON BRA V AIS LA TTICES 9 and is equal to F ( x 0 ) = 1 2  p ( A + 2 B )( A + 2 C ) − ( A + B + C )  . Going bac k to the original minimalization problem, we hav e A = ε ( p ) , B = U ρ, C = U ρw (0) , so the minimal v alue of the expression (3.15) is exactly as in (3.16) and is attained at (3.19) s p = 1 2 − 1 2  ε ( p ) + 2 U ρ ε ( p ) + 2 U ρw (0)  1 / 2 = 1 2 − 1 2  1 + 2 U ρ (1 − w (0)) ε ( p ) + 2 U ρw (0)  1 / 2 . □ Ha ving minimized the lo cal part of the energy (3.14) now we will show that the remaining parts are negligible in the dilute limit ρ → 0. Lemma 3.2. F or s p chosen as in (3.19) (and r esp e ctively chosen c p as in (3.17) ) we have asymptotic b ounds (3.20) 1 | Λ L |    U 2 | Λ L |   X p  =0  s p + ρw (0)    2 − 3 U 2 | Λ L |   X p  =0 c 2 p 1 − c 2 p   2    ≲ ρ 3 . Mor e over for the    P p  =0 c p 1 − c p    term we have (3.21) X p  =0 c p 1 − c p ≲ | Λ L | . The notation x ≲ y me ans x ≤ cy for some c onstant c > 0 indep endent of L and ρ . Pr o of. W e will start with analyzing the second term in (3.20). First we chec k that c 2 p 1 − c 2 p = s 2 p 1 − 2 s p = 1 4  1 + 2 U ρ (1 − w (0)) ε ( p ) + 2 U ρw (0)  1 / 2 + 1 4  1 + 2 U ρ (1 − w (0)) ε ( p ) + 2 U ρw (0)  − 1 / 2 − 1 2 . Using the inequality (coming from the T aylor expansion) (1 + x ) 1 / 2 + (1 + x ) − 1 / 2 ≤ 2 + x 2 4 w e can estimate c 2 p 1 − c 2 p ≤ U 2 ρ 2 (1 − w (0)) 2 4( ε ( p ) + 2 U ρw (0)) 2 . No w w e will deduce that (3.22) 1 | Λ | X p  =0 c 2 p 1 − c 2 p ≲ ρ 3 / 2 This result will follow from approximating the sum by the in tegral (as this is a Riemann sum of a con tin uous function on b Λ) and dividing the integration in to regions with small momen ta and momenta separated form zero (w e use the fact that locally near p = 0 the manifold b Λ looks like a subset of the Euclidean space R 3 ). F or suﬃciently large L we hav e 1 | Λ L | X p  =0 c 2 p 1 − c 2 p ≲ | b Λ | − 1 Z b Λ U 2 ρ 2 (1 − w (0)) 2 ( ε ( p ) + 2 U ρw (0)) 2 dp = | b Λ | − 1 Z | p |≤ p 0 U 2 ρ 2 (1 − w (0)) 2 ( ε ( p ) + 2 U ρw (0)) 2 dp + | b Λ | − 1 Z | p | >p 0 U 2 ρ 2 (1 − w (0)) 2 ( ε ( p ) + 2 U ρw (0)) 2 dp. (3.23) Here w e ha ve c hosen the same p 0 as used in (2.14), in particular we hav e the b ound ε ( p ) ≥ c | p | 2 . W e can simplify the upcoming b ounds even more by noting (3.24) U  1 − w (0)  ≤ 1 γ . 10 N. MOKRZA ´ NSKI, M. NAPI ´ ORKO WSKI, AND J. WOJTKIEWICZ Then, b y using spherical co ordinates, w e get | b Λ | − 1 Z | p |≤ p 0 U 2 ρ 2 (1 − w (0)) 2 ( ε ( p ) + 2 U ρw (0)) 2 dp ≤ C ρ 2 Z | p |≤ p 0 1 ( p 2 + 2 U ρw (0)) 2 dp ≤ C ρ 2 Z | p |≤ p 0 1 p 2 ( p 2 + 2 U ρw (0)) dp = C ρ 2 Z p 0 0 1 ( r 2 + 2 U ρw (0)) dr = C ρ 2 1 p 2 U ρw (0) arctan p 0 p 2 U ρw (0) ! ≤ C p U w (0) ρ 3 / 2 . (3.25) The constan t C is dependent only on c from (2.14), U and | b Λ | . The second integral in (3.23) can b e estimated trivially as for | p | > p 0 w e ha ve ε ( p ) > c for some constant c > 0 (dep enden t only on the ﬁxed p 0 ), so | b Λ L | − 1 Z | p | >p 0 U 2 ρ 2 (1 − w (0)) 2 2( ε ( p ) + 2 U ρw (0)) 2 ≤ ρ 2 2 γ ( c + 2 U ρw (0)) 2 ≤ C ρ 2 Com bining the ab ov e results inequalit y (3.22) follo ws. W e conclude that 1 | Λ L |   X p  =0 c 2 p 1 − c 2 p   2 = | Λ L |   1 | Λ L | X p  =0 c 2 p 1 − c 2 p   2 ≲ | Λ L | ρ 3 hence in the thermodynamic limit ( L → ∞ , N → ∞ , N/ | Λ L | → ρ ) we hav e the asymptotics 1 | Λ L | 2   X p  =0 c 2 p 1 − c 2 p   2 ≲ ρ 3 , whic h pro ves this term is indeed negligible. No w we pro ceed to estimate the other term in (3.20). Once again we are dealing with the contin uous function on b Λ, hence for suﬃcien tly large L w e can approximate the sum by the in tegral: (3.26)       1 | Λ L | X p  =0 ( s p + ρw (0))       ≲ | b Λ | − 1      Z T 3 1 2 − 1 2  1 + 2 U ρ (1 − w (0)) ε ( p ) + 2 U ρw (0)  1 / 2 + ρw (0) ! dp      Using equation (C.7) w e ha ve w (0) = Z T 3 b w ( p ) dp = Z T 3 U (1 − w (0)) 2 ε ( p ) dp w e can write | b Λ | − 1      Z T 3 1 2 − 1 2  1 + 2 U ρ (1 − w (0)) ε ( p ) + 2 U ρw (0)  1 / 2 + ρw (0) ! dp      = | b Λ | − 1      Z T 3 1 2 − 1 2  1 + 2 U ρ (1 − w (0)) ε ( p ) + 2 U ρw (0)  1 / 2 + U ρ (1 − w (0)) 2 ε ( p ) ! dp      (3.27) Using the inequality √ 1 + x ≥ 1 + 1 2 x − 1 4 x 2 w e also hav e 1 2 − 1 2  1 + 2 U ρ (1 − w (0)) ε ( p ) + 2 U ρw (0)  1 / 2 ≤ − U ρ (1 − w (0)) 2( ε ( p ) + 2 U ρw (0)) + 1 2  U ρ (1 − w (0)) ε ( p ) + 2 U ρw (0)  2 . Moreo ver − U ρ (1 − w (0)) 2( ε ( p ) + 2 U ρw (0)) + U ρ (1 − w (0)) 2 ε ( p ) = U 2 ρ 2 w (0)(1 − w (0)) ε ( p )( ε ( p ) + 2 U ρw (0)) , hence, after some more straightforw ard estimates (3.27) ≤ | b Λ | − 1 Z b Λ U 2 ρ 2 (1 − w (0)) ε ( p )( ε ( p ) + 2 U ρw (0)) . GROUND ST A TE ENERGY OF THE DILUTE BOSE-HUBBARD GAS ON BRA V AIS LA TTICES 11 Similarly as b efore we will split the in tegration in to the regions | p | ≤ p 0 and | p | > p 0 , where p 0 is still the same as in (2.14). F or | p | > p 0 w e ha ve ε ( p ) > c , hence U 2 ρ 2 w (0)(1 − w (0)) ε ( p )( ε ( p ) + 2 U ρw (0)) ≤ U 2 ρ 2 (1 − w (0)) c ( c + 2 U ρw (0)) ≤ C ρ 2 for constan t C indep endent of ρ . Using this b ound we get Z | p | >p 0 U 2 ρ 2 (1 − w (0)) ε ( p )( ε ( p ) + 2 U ρw (0)) ≤ Z | p | >p 0 U 2 ρ 2 (1 − w (0)) c ( c + 2 U ρw (0)) ≤ Z T 3 U 2 ρ 2 (1 − w (0)) c ( c + 2 U ρw (0)) ≤ C ρ 2 . F or the integral with | p | ≤ p 0 w e use analogous argumen t as in (3.25) to obtain Z | p |≤ p 0 U 2 ρ 2 (1 − w (0)) ε ( p )( ε ( p ) + 2 U ρw (0)) ≤ C U ρ 2 1 p 2 U ρw (0) arctan p 0 p 2 U ρw (0) ! ≤ C ρ 3 / 2 . Using this results in (3.26) we conclude       1 | Λ L | X p  =0  c p 1 + c p + ρw (0)        ≤ C ρ 3 / 2 and so 1 | Λ L |   X p  =0  c p 1 + c p + ρw (0)    2 = | Λ L |   1 | Λ L | X p  =0  c p 1 + c p + ρw (0)    2 ≤ C | Λ L | ρ 3 . In the thermo dynamic limit this gives the asymptotics 1 | Λ L | 2   X p  =0  s p + ρw (0)    2 ≲ ρ 3 . T o pro v e (3.21) we p erform a very similar argument as abov e: ﬁrst we observe       X p  =0 c p 1 − c p       =       X p  =0 1 2  1 + 2 U ρ (1 − w (0)) ε ( p ) + 2 U ρw (0)  − 1 / 2 − 1 2       ≤ X p  =0 " 1 2 − 1 2  1 + 2 U ρ (1 − w (0)) ε ( p ) + 2 U ρw (0)  − 1 / 2 # . Using the inequality (1 + x ) − 1 / 2 ≥ 1 − 1 2 x and approximating sum with the integral w e get       1 | Λ L | X p  =0 c p 1 − c p       ≲ Z b Λ U ρ (1 − w (0)) ε ( p ) + 2 U ρw (0) dp. The in tegral is con vergen t b y similar argumen ts as b efore. Its v alue is indep endent of L (it is dep enden t on ρ , but for this particular bound this fact is irrelev ant), hence the pro of of the lemma is ﬁnished. □ F rom Lemma 3.1 and Lemma 3.2 w e deduce the follo wing corollary concerning the energy . Corollary 3.3. F or the values of c p for which those minima of (3.15) ar e attaine d we have ⟨ H GC L ⟩ Ψ = X p  =0 1 2  p ( ε ( p ) + 2 U ρ ) ( ε ( p ) + 2 U ρw (0)) − ε ( p ) − U ρ (1 + w (0))  + U 2 | Λ L | ρ 2 − U 2 | Λ L | ρ 2 w (0) 2 + | Λ L | · O ( ρ 3 ) ρ → 0 + | Λ L | · o (1) L →∞ . (3.28) 3.5. Thermo dynamic limit. Now w e pass with the expression (3.28) divided by the volume | Λ L | to the thermo- dynamic limit. W e will denote this limit as lim L →∞ 1 | Λ L | ⟨ H GC L ⟩ Ψ = e Ψ The sums in (3.28) are Riemann sums of a con tinuous function on b Λ, hence they con verge to the integrals of the prop er expression. More precisely , we obtain e Ψ = | b Λ | − 1 Z b Λ 1 2  p ( ε ( p ) + 2 U ρ ) ( ε ( p ) + 2 U ρw (0)) − ε ( p ) − U ρ (1 + w (0))  dp + U 2 ρ 2 − U ρ 2 2 w (0) 2 + O ( ρ 3 ) (3.29) 12 N. MOKRZA ´ NSKI, M. NAPI ´ ORKO WSKI, AND J. WOJTKIEWICZ T o see the dep endence on the scattering length a w e recall the deﬁnition (2.12) and write U = 8 π a (1 + U γ ) , hence U 2 ρ 2 = 4 π a ρ 2 + 4 π a U γ ρ 2 , Next note that (e.g. from (C.8)) 4 π a = w (0) 2 γ , so w e can rewrite the last line of (3.29) (b esides the error term) as U 2 ρ 2 − U ρ 2 2 w (0) 2 = 4 π a ρ 2 + 4 π a U γ ρ 2 − U ρ 2 w (0) 2 = 4 π a ρ 2 + U ρ 2 2 w (0)(1 − w (0)) = 4 π a ρ 2 + 1 2 U 2 γ (1 − w (0)) 2 ρ 2 . Recalling also that γ is giv en by the in tegral of the function 1 2 ε ( p ) , this additionally allo ws to rewrite (3.29) as e Ψ = | b Λ | − 1 Z b Λ 1 2  p ( ε ( p ) + 2 U ρ ) ( ε ( p ) + 2 U ρw (0)) − ε ( p ) − U ρ (1 + w (0)) + U 2 (1 − w (0)) 2 ρ 2 2 ε ( p )  dp + 4 π a ρ 2 + O ( ρ 3 ) , (3.30) where w e ha ve joined the previous in tegral with the in tegral deﬁning γ . Let us no w fo cus on ev aluating the ab ov e integral. W e note that the integrand is a p ositive function, which follo ws from the computation p ( ε ( p ) + 2 U ρ ) ( ε ( p ) + 2 U ρw (0)) − ε ( p ) − U ρ (1 + w (0)) + U 2 (1 − w (0)) 2 ρ 2 2 ε ( p ) = 4 U 2 ρ 2 w (0) − U 2 ρ 2 (1 + w (0)) 2 p ( ε ( p ) + 2 U ρ ) ( ε ( p ) + 2 U ρw (0)) + ε ( p ) + U ρ (1 + w (0)) + U 2 (1 − w (0)) 2 ρ 2 2 ε ( p ) = − U 2 ρ 2 (1 − w (0)) 2 p ( ε ( p ) + 2 U ρ ) ( ε ( p ) + 2 U ρw (0)) + ε ( p ) + U ρ (1 + w (0)) + U 2 (1 − w (0)) 2 ρ 2 2 ε ( p ) = U 2 ρ 2 (1 − w (0)) 2  p ( ε ( p ) + 2 U ρ ) ( ε ( p ) + 2 U ρw (0)) − ε ( p ) + U ρ (1 + w (0))  2 ε ( p )  p ( ε ( p ) + 2 U ρ ) ( ε ( p ) + 2 U ρw (0)) + ε ( p ) + U ρ (1 + w (0))  > 0 . (3.31) Next, similarly as b efore, w e will split the integration into tw o regions, but this time in to regions ε ( p ) ≥ δ and ε ( p ) < δ where δ = δ ( ρ ) will be c hosen as a certain function of ρ . F or the ﬁrst region if ρ is suﬃcien tly small (with resp ect to U and w (0)) we can T a ylor expand the square root up to the terms of order ρ 3 . Using the inequality (1 + x ) 1 / 2 ≤ 1 + 1 2 x − 1 8 x 2 + 1 16 x 3 w e get p ( ε ( p ) + 2 U ρ ) ( ε ( p ) + 2 U ρw (0)) = ε ( p ) s  1 + 2 U ρ ε ( p )   1 + 2 U ρw (0) ε ( p )  = ε ( p ) s 1 + 2 U ρ (1 + w (0)) ε ( p ) + 4 U 2 ρ 2 w (0) ε ( p ) 2 ≤ ε ( p ) + U ρ (1 + w (0)) + 2 U 2 ρ 2 w (0) − 1 2 U 2 ρ 2 (1 + w (0)) 2 ε ( p ) + C ε ( p ) 2 ρ 3 ≤ ε ( p ) + U ρ (1 + w (0)) − 1 2 U 2 ρ 2 (1 − w (0)) 2 ε ( p ) + C ( δ ) ρ 3 , where C > 0 is a constan t coming from the T aylor expansion, indep enden t of ρ and (3.32) C ( δ ) = C min ε ( p ) ≥ δ ε ( p ) 2 , GROUND ST A TE ENERGY OF THE DILUTE BOSE-HUBBARD GAS ON BRA V AIS LA TTICES 13 whic h is another constan t, dependent only on δ . Using (2.14) w e also note that if δ is suﬃciently small then (3.33) C ( δ ) ≲ 1 δ 4 . No w w e can estimate the in tegrand as follows p ( ε ( p ) + 2 U ρ ) ( ε ( p ) + 2 U ρw (0)) − ε ( p ) − U ρ (1 + w (0)) + U 2 (1 − w (0)) 2 ρ 2 2 ε ( p ) ≤ ε ( p ) + U ρ (1 + w (0)) − 1 2 U 2 ρ 2 (1 − w (0)) 2 ε ( p ) + C ( δ ) ρ 3 − ε ( p ) − U ρ (1 + w (0)) + U 2 (1 − w (0)) 2 ρ 2 2 ε ( p ) = C ( δ ) ρ 3 , hence | b Λ | − 1 Z ε ( p ) ≥ δ 1 2  p ( ε ( p ) + 2 U ρ ) ( ε ( p ) + 2 U ρw (0)) − ε ( p ) − U ρ (1 + w (0)) + U 2 (1 − w (0)) 2 ρ 2 2 ε ( p )  dp ≤ | b Λ | − 1 Z ε ( p ) ≥ δ 1 2 C δ ρ 3 dp ≤ | b Λ | − 1 Z T 3 1 2 C δ ρ 3 dp = 1 2 C ( δ ) ρ 3 . (3.34) W e will explicitly choose δ = δ ( ρ ) after the next step. No w we pro ceed to the integral on the domain ε ( p ) < δ . Using the coarea formula (see e.g. [12, Section 3.4.3]) for some general and suﬃciently regular function f in tegrable near zero we hav e Z ε ( p ) <δ f ( ε ( p )) dp = Z δ 0 f ( r ) Z ε ( p )= r d H 2 ( ξ ) |∇ ε ( ξ ) | ! dr , where H 2 is the tw o dimensional Hausdorﬀ (surface) measure. Using formula (2.13) we see that ε ( p ) ≤ C | p | 2 (1 + | p | 2 ) for some constant C indep endent of p , hence H 2 ( { ε ( p ) = r } ) ≤ C r. Moreo ver |∇ ε ( p ) | > c | p | for some constant c , hence for ξ ∈ { ε ( p ) = r } w e ha ve 1 |∇ ε ( ξ ) | ≤ C √ r and therefore Z ε ( p )= r d H 2 ( ξ ) |∇ ε ( ξ ) | ≤ C r 1 / 2 . As a result, if the function f is non-negative, we get Z ε ( p ) <δ f ( ε ( p )) dp ≤ C Z δ 0 r 1 / 2 f ( r ) dr . W e are going to use this observ ation for the in tegrand as in (3.30), that is f ( ε ( p )) = 1 2  p ( ε ( p ) + 2 U ρ ) ( ε ( p ) + 2 U ρw (0)) − ε ( p ) − U ρ (1 + w (0)) + U 2 (1 − w (0)) 2 ρ 2 2 ε ( p )  . In (3.31) we hav e already noted that this function is positive. W e ha ve | b Λ | − 1 Z ε ( p ) ≤ δ f ( ε ( p )) dp ≤ C Z δ 0 r 1 / 2 f ( r ) dr = C ρ Z δ 0 r 1 / 2 s  r ρ + 2 U   r ρ + 2 U w (0)  − r ρ − U (1 + w (0)) + U 2 (1 − w (0)) 2 ρ 2 r ! dr = C ρ 5 / 2 Z δ /ρ 0 s 1 / 2  p ( s + 2 U ) ( s + 2 U w (0)) − s − U (1 + w (0)) + U 2 (1 − w (0)) 2 2 s  ds ≤ C ρ 5 / 2 Z + ∞ 0 s 1 / 2  p ( s + 2 U ) ( s + 2 U w (0)) − s − U (1 + w (0)) + U 2 (1 − w (0)) 2 2 s  ds, 14 N. MOKRZA ´ NSKI, M. NAPI ´ ORKO WSKI, AND J. WOJTKIEWICZ where in the second to last equalit y we hav e changed the v ariable r := ρs and in the last inequality we used the fact the integrand is well deﬁned and p ositive on R + . Performing similar computation as in (3.31) one can chec k that this integral is con vergen t, in particular w e can conclude | b Λ | − 1 Z ε ( p ) ≤ δ f ( ε ( p )) dp ≤ C ρ 5 / 2 . Com bining this result with (3.34) and (3.32) for any δ ≤ ρ 1 / 8 w e ev entually get | b Λ | − 1 Z b Λ f ( ε ( p )) dp ≤ C ρ 5 / 2 . This allo ws us to estimate e Ψ in (3.30) as e Ψ ≤ 4 π a ρ 2 (1 + C ρ 1 / 2 ) . This ﬁnishes the proof of Proposition 2.2. 3.6. Equiv alence of ensembles. W e will now recall the w ell kno wn argument which shows ho w to use Prop osition 2.2 in order to obtain the low er bound in Theorem 2.1. W e follow the proof from [1, Lemma 3.3.2] with some inspiration from [4, Lemma A.4]. Here, ho wev er, w e will not assume that N | Λ L | is constan t. W e start with the follo wing lemma. Lemma 3.4. F or any N and L lar ger than the hopping length R 0 ( t ) of the lattic e (r e c al l the deﬁnition (2.5) ) we have an ine quality E 0 ( N , L ) | Λ L | ≥ e 0  N | Λ L |  . Pr o of. F or ﬁxed L and for an y k ∈ N denote L ( k ) := k ( L + 1) − 1 and consider a (p erio dic) lattice Λ L ( k ) . This lattice can b e divided into k 3 sub-lattices with each sub-lattice b eing the translation of the original lattice Λ L . 1 Next take any N -particle state ψ N ∈ H N L . Using this state for any k ∈ N we will construct a state on H k 3 N L ( k ) , i.e. the k 3 N -particle Hilb ert space based on a larger lattice Λ L ( k ) . T o this end on each of sub-lattices that Λ L ( k ) can b e divided in to we put a translated, indepe nden t copy of ψ N and deﬁne a state ψ k 3 N ∈ L 2  H k 3 N L ( k )  as the symmetrized result of this pro cedure. No w, since the in teraction p otential of the Bose-Hubbard mo del has zero range, diﬀeren t sub-lattices do not in teract with each other, hence the total interaction potential energy is the sum of p oten tial energies of each sub- lattice. F urthermore, due to the construction of the state, hopping b etw een diﬀerent sub-lattices gives the same con tribution to the kinetic energy as hopping within a single sub-lattice with imp osed perio dic b oundary condition (here we also use the condition L ≥ R 0 ( t )), hence the kinetic energy of the state ψ k 3 N is equal to the sum of kinetic energies of copies of ψ N from eac h sub-lattice. This allo ws us to conclude E 0 ( k 3 N , L ( k )) ≤ ⟨ H k 3 N ,L ( k ) ⟩ ψ k 3 N = k 3 ⟨ H L ⟩ ψ N . Minimizing o ver ψ N giv es E 0 ( k 3 N , L ( k )) ≤ k 3 E 0 ( N , L ) , and therefore E 0 ( N , L ) | Λ L | ≥ E 0 ( k 3 N , L ( k )) | Λ L ( k ) | = E 0 ( k 3 N , L ( k )) k 3 | Λ L | . Since this inequality is v alid for any k , w e can pass to the limit k → ∞ and obtain E 0 ( N , L ) | Λ L | ≥ e 0  N | Λ L |  as desired. □ No w w e can proceed to the main problem. W e ﬁrst observe that trivially E GC 0 ( N , L ) ≤ E 0 ( N , L ) 1 Note that the deﬁnition of L ( k ) is correct as the num b er of points in Λ L is | Λ L | = ( L + 1) 3 , hence | Λ L ( k ) | = k 3 ( L + 1) 3 . GROUND ST A TE ENERGY OF THE DILUTE BOSE-HUBBARD GAS ON BRA V AIS LA TTICES 15 as any canonical trial state with N particles can b e lifted to the grand canonical one, o ccupying only the N -particle sector of the F o ck space (in particular ha ving N as the exp ected num b er of particles). It follo ws that lim sup N →∞ L →∞ N/ | Λ L |→ ρ E GC 0 ( N , L ) | Λ L | ≤ lim sup N →∞ L →∞ N/ | Λ L |→ ρ E 0 ( N , L ) | Λ L | = e 0 ( ρ ) . It remains to pro v e lim inf N →∞ L →∞ N/ | Λ L |→ ρ E GC 0 ( N , L ) | Λ L | ≥ e 0 ( ρ ) . T o this end we introduce a v ariable µ ∈ R (that can b e in terpreted as the chemical p otential) and for an y normalized Ψ ∈ F L with ⟨N ⟩ Ψ = N w e write ⟨ H ⟩ Ψ | Λ L | = 1 | Λ L | [ µ ⟨N ⟩ Ψ + ⟨ H − µ N ⟩ Ψ ] = 1 | Λ L | " µN + ∞ X n =1 ∥ Ψ ( n ) ∥ 2 ( ⟨ H n ⟩ Ψ ( n ) − µn ) # = µ N | Λ L | + ∞ X n =0 ∥ Ψ ( n ) ∥ 2  ⟨ H n ⟩ Ψ | Λ L | − µ n | Λ L |  ≥ µ N | Λ L | + ∞ X n =0 ∥ Ψ ( n ) ∥ 2  E 0 ( n, L ) | Λ L | − µ n | Λ L |  ≥ µ N | Λ L | + ∞ X n =0 ∥ Ψ ( n ) ∥ 2  e 0  n | Λ L |  − µ n | Λ L |  ≥ µ N | Λ L | + ∞ X n =0 ∥ Ψ ( n ) ∥ 2 inf ˜ ρ ≥ 0 ( e 0 ( ˜ ρ ) − µ ˜ ρ ) = µ N | Λ L | + inf ˜ ρ ≥ 0 ( e 0 ( ˜ ρ ) − µ ˜ ρ ) , where in one of the steps w e hav e used the ab ov e Lemma. W e also recognize inf ˜ ρ ≥ 0 ( e 0 ( ˜ ρ ) − µ ˜ ρ ) = − e ∗ 0 ( µ ) , where e ∗ 0 ( µ ) is the Legendre transform of e 0 ( ˜ ρ ) (w e also use notation ˜ ρ in order not to confuse it with ρ ﬁxed in the statemen t of the Theorem 2.1). Since Ψ ∈ F L ab o ve w as arbitrary , w e conclude E GC 0 ( N , L ) | Λ L | ≥ µ N | Λ L | − e ∗ 0 ( µ ) . T aking the limes inferior of b oth sides giv es lim inf N →∞ L →∞ N/ | Λ L |→ ρ E GC 0 ( N , L ) | Λ L | ≥ µρ − e ∗ 0 ( µ ) . F urthermore, as the left hand side is independent of µ , we additionally get lim inf N →∞ L →∞ N/ | Λ L |→ ρ E GC 0 ( N , L ) | Λ L | ≥ sup µ ∈ R ( µρ − e ∗ 0 ( µ )) = e ∗∗ 0 ( ρ ) = e 0 ( ρ ) , where the last equality follows from the fact e 0 is a conv ex and con tinuous (up to a b oundary) function and for suc h functions Legendre transform is an inv olution (i.e. f ∗∗ = f , see e.g. [4, Lemma A.3] for a simple pro of ). The ends the pro of of the upp er bound in Theorem 2.1. 4. The lower bound In this sec tion we will prov e Prop osition 2.3. W e will follo w the strategy describ ed at the b eginning of the pap er. 16 N. MOKRZA ´ NSKI, M. NAPI ´ ORKO WSKI, AND J. WOJTKIEWICZ 4.1. Division into sub-lattices. Similarly as in the pro of of the low er b ound in the contin uous setting (see e.g. [28, Chapter 2]) we will divide the large (thermo dynamic) lattice Λ L in to smaller ones. The up coming lemma is a well-kno wn result, here will give a pro of based on [37, Lemma 5.21] adapted to the lattice setting. Beforehand, in analogy to the deﬁnition (2.9), w e will denote the ground state energy of the N particle system in the b ox of side-length L with Neumann Laplacian as E Neu 0 ( N , L ) = inf ψ ∈H N L ∥ ψ ∥ =1 ⟨ ψ , H Neu N ,L ψ ⟩ with H Neu N ,L = − N X i =1 ∆ Neu Λ L ,i + U N X i s } ) ds. Using this fact for the eigenv alue counting measure of some positive-deﬁnite matrix T and function f ( x ) = 1 /x w e get T r T − 1 = Z ∞ 0 #  1 λ j ( T ) > s  ds = Z ∞ 0 N T ( s ) s 2 ds, where w e denoted as λ j ( T ), j = 1 , 2 . . . , the eigenv alues of T arranged in a non-decreasing order and N T ( t ) = # { j : λ j ( T ) < t } = max { j : λ j ( T ) < t } , that is the sp ectral function counting the eigenv alues (with multiplicit y and the conv ention that max ∅ = 0). Denoting by N Neu ( t ) and N Per ( t ) the sp ectral functions of op erators ( − P + ∆ Neu P + ) and ( − P + ∆ Per P + ) resp ectively w e obtain | T r  ( − P + ∆ Neu P + ) − 1  − T r  ( − P + ∆ Per P + ) − 1  | ≤ Z ∞ 0 | N Neu ( s ) − N Per ( s ) | s 2 ds = Z δ 0 | N Neu ( s ) − N Per ( s ) | s 2 ds + Z ∞ δ | N Neu ( s ) − N Per ( s ) | s 2 ds, (4.11) where δ = δ ( L ) will b e c hosen later. W e will b ound eac h of those in tegrals separately . T o b ound the second integral we will use the following fact [23, Corollary 4.3.5]: if A and B are Hermitian n × n matrices with rank( A − B ) ≤ r then λ j ( B ) ≤ λ j + r ( A ) for j = 1 , . . . , n − r and λ j ( B ) ≥ λ j − r ( A ) for j = r + 1 , . . . , n. F rom this fact we can deduce the following b ound on the diﬀerence of sp ectral functions of A and B : for every s > 0 | N A ( s ) − N B ( s ) | ≤ r. T o see it we denote N A ( s ) = k . If k ≤ r then trivially N A ( s ) − N B ( s ) ≤ r, whereas if k > r then s ≥ λ k ( A ) ≥ λ k − r ( B ) ⇒ N B ( s ) ≥ k − r = N A ( s ) − r . T o obtain the second inequalit y we reverse the roles of A and B . Applying this fact to N Neu ( s ) and N Per ( s ) w e get | N Neu ( s ) − N Per ( s ) | ≤ rank(∆ Neu − ∆ Per ) ≤ C L 2 20 N. MOKRZA ´ NSKI, M. NAPI ´ ORKO WSKI, AND J. WOJTKIEWICZ for some constan t C indep endent of L as the diﬀerence ∆ Neu − ∆ Per acts non-trivially only on the boundary ∂ Λ L of Λ L and its nearest neigh b ors, which is the set of cardinalit y of order L 2 . It follo ws that Z ∞ δ | N Neu ( s ) − N Per ( s ) | s 2 ds ≤ C L 2 δ . T o bound the ﬁrst integral in (4.11) w e note that by p oint ( a ) w e hav e | N Neu ( s ) − N Per ( s ) | ≤ N Neu ( s ) − N Per ( s ) ≤ N special ( s ) , where N special ( s ) is the spectral function of − P + ∆ Neu special P + . Moreov er b y p oint ( b ) w e hav e N special ( s ) = 0 for s < c gap L 2 , hence w e can only consider s satisfying (4.12) s ≥ c gap L 2 Recall the explicit form ula for the eigen v alues ε Neu special ( k ) of − ∆ Neu special giv en in (4.8): ε Neu special ( k ) = 4 3 X i =1 t ( a i ) sin 2 p i 2 !      p = π k/ ( L +1) , k ∈ { 0 , 1 . . . , L } 3 . Let | · | ∞ b e the supremum norm of a vector in R 3 . W e observ e that there exist a p 0 > 0 such that for | p | ∞ < p 0 w e ha ve 4 3 X i =1 t ( a i ) sin 2 p i 2 ≥ c | p | 2 ∞ , where c is some constant indep endent of p . Denoting (4.13) s 0 = inf | p | ∞ = p 0 4 3 X i =1 t ( a i ) sin 2 p i 2 ! w e further observe that if s < s 0 then N special ( s ) = # ( k ∈ { 0 , 1 . . . , L } 3 \ { (0 , 0 , 0) } : 4 3 X i =1 t ( a i ) sin 2 k i π 2( L + 1) < s ) ≤ #  k ∈ { 0 , 1 . . . , L } 3 : cπ 2 ( L + 1) 2 | k | 2 ∞ < s  = #  k ∈ { 0 , 1 . . . , L } 3 : | k | ∞ < ( L + 1) π r s c  ≤ C ( Ls 1 / 2 + 1) 3 = C s 3 / 2  L + 1 s 1 / 2  3 ≤ C L 3 s 3 / 2 , where the last inequalit y follows from (4.12). As a ﬁnal result, assuming that δ < s 0 w e get Z δ 0 | N Neu ( s ) − N Per ( s ) | s 2 ds ≤ Z δ c gap /L 2 N special ( s ) s 2 ≤ C L 3 Z δ c gap /L 2 s − 1 / 2 ds ≤ C L 3 δ 1 / 2 . Com bining this with the previous estimate w e even tually obtain 1 | Λ L | | T r  ( − P + ∆ Neu P + ) − 1  − T r  ( − P + ∆ Per P + ) − 1  | ≤ C  δ 1 / 2 + 1 Lδ  , whic h after optimizing in δ yields δ ∼ L − 2 / 3 and 1 | Λ L | | T r  ( − P + ∆ Neu P + ) − 1  − T r  ( − P + ∆ Per P + ) − 1  | ≤ C L − 1 / 3 . This bound is v alid for suﬃciently large L , precisely for suc h L that δ ( L ) ≤ s 0 . □ The method used in the proof of point ( c ) ab o ve can be used to obtain the following useful corollary . Corollary 4.5. Within the setting like in L emma 4.4 for any p ower ν > 3 2 and for suﬃciently lar ge L we have the b ound 1 | Λ L | T r( − P + ∆ Neu P + ) ν ≤ C L 2 − 3 /ν for some c onstant C indep endent of L . GROUND ST A TE ENERGY OF THE DILUTE BOSE-HUBBARD GAS ON BRA V AIS LA TTICES 21 Pr o of. Mimicking the previous proof w e get T r( − P + ∆ Neu P + ) ν = Z δ c gap /L 2 N special ( s 1 /ν ) s 2 ds + Z ∞ δ N special ( s 1 /ν ) s 2 ds ≤ C L 3 Z δ c gap /L 2 s 3 / 2 ν s 2 ds + Z ∞ δ ( L + 1) 3 s 2 ds ≤ C L 3 · ( L − 2 ) − 1+3 / 2 ν + C L 3 ≤ C L 3 · L 2 − 3 /ν . Ab o ve, δ is a ﬁxed constant satisfying δ ν < s 0 , where s 0 w as deﬁned in (4.13). □ 4.3. Analysis on sub-lattices. In this section we will work with the eigen basis of the Neumann Laplacian − ∆ Neu Λ ℓ for the b ox of size ℓ , denoted as { ψ k } k ∈{ 0 , 1 ...,ℓ } 3 . Such basis exists as this is a self-adjoint op erator on a ﬁnite dimensional space. Moreov er it is easy to chec k that the constant function, here denoted as ψ 0 , is an eigenv ector with eigenv alue zero. F urthermore, as the hopping constants deﬁning ∆ Neu Λ L are real, one has ∆ Neu Λ ℓ u ( x ) = ∆ Neu Λ ℓ u ( x ) (in other words this op erator is a complexiﬁcation of a symmetric op erator o ver the real v ector space L 2 R (Λ ℓ ))) and therefore the eigenfunctions ψ k can be c hosen as real-v alued. Our goal is to pro ve the follo wing prop osition. Prop osition 4.6. Assume that (4.14) n ℓ + 1 < c gap 48 π a , wher e c gap is deﬁne d b elow (4.10) . Then the fol lowing op er ator ine quality holds: H n,ℓ ≥ 4 π a n 2 | Λ ℓ | − C  n 2 ℓ 4 log ℓ  − C  n ℓ 3  , for some c onstant C indep endent of n and ℓ . In p articular E Neu 0 ( n, ℓ ) ≥ 4 π a n 2 | Λ ℓ | − C  n 2 ℓ 10 / 3  − C  n ℓ 3  . Pr o of. The idea of the pro of b elow is based on [32, Section 1] and [9, Lemma 5]. Step 1. W e will express the Hamiltonian H n,ℓ in terms of creation and annihilation op erators in the Neumann basis in tro duced ab ov e: for k = ( k 1 , k 2 , k 3 ), k j = 0 , . . . , L denote a k = a ( ψ k ) and a ∗ k = a ∗ ( ψ k ). Next denote by P the pro jection onto the constant function ψ 0 = 1 | Λ ℓ | 1 / 2 ∈ L 2 (Λ ℓ ) and b y m φ the multiplication op erator by a function φ ( x − y ) (i.e. the scattering equation solution) acting on the tw o b o dy Hilb ert space H ⊗ 2 ℓ . W e ﬁrst note that we ha ve an operator inequalit y ( 1 − P ⊗ P m φ ) U δ x,y ( 1 − m φ P ⊗ P ) ≥ 0 . This is equiv alent to U δ x,y ≥ U m φ δ x,y ( P ⊗ P ) + U ( P ⊗ P ) m φ δ x,y − ( P ⊗ P ) m 2 φ δ x,y ( P ⊗ P ) . No w we will take the second quantization of b oth sides of this inequalit y and expressing it in the Neumann basis represen tation. As the system of ψ k ( x ) is the orthonormal basis of the one-b o dy space w e ha ve ⟨ ψ p ⊗ ψ q , ( U m φ δ x,y P ⊗ P ) ψ k ⊗ ψ r ⟩ = ⟨ ψ p ⊗ ψ q , ( U m φ δ x,y ) ψ 0 ⊗ ψ 0 ⟩ δ k, 0 δ r, 0 = δ k, 0 δ r, 0 · U | Λ ℓ | X x,y ∈ Λ L ψ p ( x ) ψ q ( y ) φ ( x − y ) δ x,y = δ k, 0 δ r, 0 δ q ,p · U φ (0) | Λ ℓ | . Here w e ha ve used the fact ψ k can b e c hosen to be real-v alued, so that the complex conjugation in the inner product can be omitted. A similar computation for other matrix elemen ts leads to ⟨ ψ p ⊗ ψ q , ( P ⊗ P U m φ δ x,y ) ψ k ⊗ ψ r ⟩ = δ p, 0 δ q , 0 δ r,k · U φ (0) | Λ ℓ | and ⟨ ψ p ⊗ ψ q , ( P ⊗ P U m 2 φ δ x,y P ⊗ P ) ψ k ⊗ ψ r ⟩ = δ p, 0 δ q , 0 δ k, 0 δ r, 0 · U φ (0) 2 | Λ ℓ | . As the end result we obtain H n,ℓ = X k  =0 ε Neu ( p ) a ∗ k a k + U φ (0) 2 | Λ ℓ | X k  =0 ( a ∗ k a ∗ k a 0 a 0 + a ∗ 0 a ∗ 0 a k a k ) + U (2 φ (0) − φ (0) 2 ) 2 | Λ ℓ | a ∗ 0 a ∗ 0 a 0 a 0 . 22 N. MOKRZA ´ NSKI, M. NAPI ´ ORKO WSKI, AND J. WOJTKIEWICZ Step 2. W e will use the following op erator inequalit y (see [29, Theorem 6.3]): A ( b ∗ k b k + b ∗ − k b − k ) + B ( b ∗ k b ∗ − k b p b − p ) ≥ ( A − p A 2 − B 2 ) [ b k , b ∗ k ] + [ b − k , b ∗ − k ] 2 , v alid for any op erators b k , b − k , b ∗ k and b ∗ − k on F o ck space satisfying [ b k , b − k ] = [ b ∗ k , b ∗ − k ] = 0. Here we will use it for b k = n − 1 / 2 a ∗ 0 a k , b ∗ k = n − 1 / 2 a ∗ k a 0 , k  = 0 . One can chec k that b ∗ k b k ≤ a ∗ k a k , [ b k , b ∗ k ] ≤ 1 , a ∗ k a ∗ k a 0 a 0 = nb ∗ k b ∗ k , a ∗ 0 a ∗ 0 a k a k = nb k b k . No w w e note that the assumption on n ℓ +1 and the low er b ound in (4.10) imply that 24 π a n | Λ ℓ | = 24 π a n ( ℓ + 1) 3 < 1 2 c gap ( ℓ + 1) 2 < 1 2 ε Neu gap ( ℓ ) , hence the (op en) in terv al  16 π a n | Λ ℓ | , 1 2 ε Neu gap ( ℓ ) − 8 π a n | Λ ℓ |  is nonempt y . This allows to choose parameter µ satisfying (4.15) 16 π a n | Λ ℓ | < µ < 1 2 ε Neu gap ( ℓ ) − 8 π a n | Λ ℓ | . In the end w e obtain ( n 0 = a ∗ 0 a 0 ): H n,ℓ = X k  =0 ( ε Neu ( k ) − µ ) a ∗ k a k + U φ (0) 2 | Λ ℓ | X k  =0 ( a ∗ k a ∗ k a 0 a 0 + a ∗ 0 a ∗ 0 a k a k ) + U (2 φ (0) − φ (0) 2 ) 2 | Λ ℓ | a ∗ 0 a ∗ 0 a 0 a 0 + µ N + = 1 2 X k  =0  ( ε Neu ( k ) − µ )( b ∗ k b k + b ∗ k b k ) + nU φ (0) | Λ ℓ | ( b ∗ k b ∗ k + b k b k )  + U (2 φ (0) − φ (0) 2 ) 2 | Λ ℓ | n 0 ( n 0 − 1) + µ N + ≥ − 1 2 X k  =0 " ε Neu ( k ) − µ − s ( ε Neu ( k ) − µ ) 2 − n 2 U 2 φ (0) 2 | Λ ℓ | 2 # + U (2 φ (0) − φ (0) 2 ) 2 | Λ ℓ | n 0 ( n 0 − 1) + µ N + . Recalling that 8 π a = U φ (0) = U 1 + U γ w e see that the square ro ot is well deﬁned as ε Neu ( k ) − µ − nU φ (0) | Λ ℓ | = ε Neu ( k ) − µ − 8 π a n | Λ ℓ | > 1 2 ε Neu gap ( ℓ ) > 0 , in whic h w e ha ve used the upper b ound on µ from (4.15). Step 3. Using the inequality 1 − √ 1 − x ≤ 1 2 x + 1 8 x 2 w e get ε Neu ( k ) − µ − s ( ε Neu ( k ) − µ ) 2 − n 2 U 2 φ (0) 2 | Λ ℓ | 2 ≤ n 2 U 2 φ (0) 2 2 | Λ ℓ | 2 ( ε Neu ( k ) − µ ) + 1 8  n 4 U 4 φ (0) 4 | Λ ℓ | 4 ( ε Neu ( k ) − µ ) 3  . Using the condition µ < 1 2 ε Neu gap ( ℓ ) and (4.10) w e get 1 ε Neu ( k ) − µ = 1 ε Neu ( k ) + µ ε Neu ( k ) 2  1 − µ ε Neu ( k )  ≤ 1 ε Neu ( k ) + π 2 8( ℓ + 1) 2 ε Neu ( k ) 2 ≤ 1 ε Neu ( k ) + C ℓ 2 1 ε Neu ( k ) 2 and so, using U 2 φ (0) 2 ≤ 1 γ < C , we get: X k  =0 n 2 U 2 φ (0) 2 4 | Λ ℓ | 2 ( ε Neu ( k ) − µ ) ≤ X k  =0 n 2 U 2 φ (0) 2 4 | Λ ℓ | 2 ε Neu ( k ) + C ℓ 2 X k  =0 n 2 | Λ ℓ | 2 ε Neu ( k ) 2 ≤ X k  =0 n 2 U 2 φ (0) 2 4 | Λ ℓ | 2 ε Neu ( k ) + C n 2 ℓ 9 / 2 , GROUND ST A TE ENERGY OF THE DILUTE BOSE-HUBBARD GAS ON BRA V AIS LA TTICES 23 since b y Corollary 4.5 1 ( ℓ + 1) 3 X k  =0 1 ε Neu ( k ) 2 ≤ C ℓ 1 / 2 . Moreo ver, using n ℓ +1 ≤ C and 1 ε Neu ( k ) − µ ≤ C ε Neu ( k ) w e obtain X k  =0 1 8  n 4 U 4 φ (0) 4 | Λ ℓ | 4 ( ε Neu ( k ) − µ ) 3  ≤ C n 2 ( ℓ + 1) 10 X k  =0 1 ε Neu ( k ) 3 ≤ C n 2 ℓ 6 as (once again from Corollary 4.5) 1 ( ℓ + 1) 3 X k  =0 1 ε Neu ( k ) 3 ≤ C ℓ. F rom the ab ov e it follo ws that 1 2 X k  =0 " ε Neu ( k ) − µ − s ( ε Neu ( k ) − µ ) 2 − n 2 U 2 φ (0) 2 | Λ ℓ | 2 # ≤ X k  =0 n 2 U 2 φ (0) 2 4 | Λ ℓ | 2 ε Neu ( k ) + C n 2 ℓ 9 / 2 . No w, b y Lemma 4.4, we hav e X k  =0 n 2 U 2 φ (0) 2 4 | Λ ℓ | 2 ε Neu ( k ) = X k  =0 n 2 U 2 φ (0) 2 4 | Λ ℓ | 2 ε ( k ) + O  n 2 ℓ 10 / 3  . By observ ation (4.9), the ab o v e sum is a Riemann sum for the integral of the function k 7→ ε ( B k ) on the domain k ∈ [0 , 1 / 2] 3 hence (b y the standard Riemann sum approximation argument) (4.16) X k  =0 n 2 U 2 φ (0) 2 2 | Λ ℓ | 2 ε Neu ( k ) = n 2 U 2 φ (0) 2 2  1 2  3 | Λ ℓ | Z [0 , 1 2 ] 3 dk ε ( B k ) + O  n 2 ℓ 4 log ℓ  . Using the symmetry k j ↔ ( − k j ) of the function under the in tegral and then changing v ariables p = B k we get Z [0 , 1 2 ] 3 dk ε ( B k ) = 1 8 Z [ − 1 2 , 1 2 ] 3 dk ε ( B k ) = 1 8 | det B | Z B [ − 1 2 , 1 2 ] 3 dp ε ( p ) = 1 8 | b Λ | Z b Λ dp ε ( p ) . As a result (4.16) = n 2 U 2 φ (0) 2 2 | Λ ℓ || b Λ | Z b Λ dp ε ( p ) + O  n 2 ℓ 4 log ℓ  = n 2 U 2 φ (0) 2 γ | Λ ℓ | + O  n 2 ℓ 4 log ℓ  . W e also note that all of the obtained previously error terms decay faster than n 2 ℓ 10 / 3 , therefore in the next step we will include all of them in the O  n 2 ℓ 10 / 3  term. Step 4. Gathering all of the estimates we conclude H n,ℓ ≥ µ N + − 1 2 n 2 U 2 φ (0) 2 γ | Λ ℓ | + U (2 φ (0) − φ (0) 2 ) 2 | Λ ℓ | ( n − N + )( n − N + − 1) − C  n 2 ℓ 10 / 3  = µ N + + n 2 2 | Λ ℓ |  − U 2 γ (1 + U γ ) 2 + U + 2 U 2 γ (1 + U γ ) 2  + U + 2 U 2 γ 2 | Λ ℓ | (1 + U γ ) 2  − 2 n N + + N 2 + − n + N +  − C  n 2 ℓ 10 / 3  ≥ µ N + + n 2 | Λ ℓ | U 1 + U γ − 2 U n N + | Λ ℓ | (1 + U γ ) − C  n 2 ℓ 10 / 3  − C  n ℓ 3  = ( µ − 16 n | Λ ℓ | π a ) N + + 4 π a n 2 | Λ ℓ | − C  n 2 ℓ 10 / 3  − C  n ℓ 3  ≥ 4 π a n 2 | Λ ℓ | − C  n 2 ℓ 10 / 3  − C  n ℓ 3  , where in the last inequalit y w e used the lo wer bound from (4.15) and non-negativit y of N + . This ends the proof. □ R emark 4.7 . In the pro of ab o ve w e did not hav e to use the Neumann symmetrization technique used in [9] as in the discrete setting the Neumann Laplacian (B.5) is deﬁned for all functions on Λ ℓ , in particular for the restriction φ | Λ ℓ of the scattering equation solution. W e refer to the Remark B.1 in the App endix for more discussion concerning this fact. 24 N. MOKRZA ´ NSKI, M. NAPI ´ ORKO WSKI, AND J. WOJTKIEWICZ 4.4. Conclusion. Now we can conclude the pro of of Proposition 2.3, hence ﬁnishing the proof of the main Theorem 2.1. The following pro of is based on [5, Corollary 1.3] (which itself is similar to the pro of given in [28, Chapter 2.2]). Pr o of of Pr op osition 2.3. F or ﬁxed ρ > 0 w e deﬁne (4.17) ℓ = ℓ ( ρ ) = &  192 π a c gap ρ  − 1 / 2 ' − 1 , where c gap is once again the constan t from (4.10). As the thermo dynamic limit do es not dep end on the choice of sequences N → ∞ , L → ∞ with N / | Λ L | → ρ w e will consider only the v alues of L suc h that L +1 ℓ +1 is an integer. This will allow us to use the lo calization metho d. W e will also assume that the sequence N | Λ L | tends to ρ from b elow, i.e N | Λ L | ≤ ρ for ev ery considered N and L . This is a purely technical assumption, related to the fact that we are dealing with only discrete v alues of N and L , hence we cannot assume that N | Λ L | = ρ for ev ery N and L , as this w ould signiﬁcan tly restrict the p ossible v alues of ρ . As in the proof of Lemma 4.1 we split the thermo dynamic lattice (of side length L ) into sub-lattices of side length ℓ and in tro duce parameter p deﬁned as (4.18) p := c gap ( ℓ + 1) 48 π a . By Proposition 4.6, for n satisfying n < p we hav e E Neu 0 ( n, ℓ ) ≥ 4 π a  n 2 | Λ ℓ | − C n 2 ℓ 10 / 3 − C n ℓ 3  F or n ≥ p we use the fact the in teraction p otential is non-negative (in particular H n,ℓ is a non-negativ e op erator), so that the ground state energy is sup er-additive: E Neu 0 ( n 1 + n 2 , ℓ ) ≥ E Neu 0 ( n 1 , ℓ ) + E Neu 0 ( n 2 , ℓ ) . With this fact for n ≥ p w e ha ve E Neu 0 ( n, ℓ ) ≥  n p  E Neu 0 ( p, ℓ ) ≥ n 2 p E Neu 0 ( p, ℓ ) . Using Corollary (4.2) we obtain (4.19) E 0 ( N , L ) ≥ 4 π a | Λ L | | Λ ℓ | 2 inf    X n

0 and s ∈ R . W e deﬁne the distributional F ourier transform of ψ , also denoted b y b ψ , as a distribution on b Λ giv en b y (A.6) ⟨ b ψ , f ⟩ = X x ∈ Λ ψ ( x ) ˇ f ( x ) , where f ∈ D ( b Λ) = C ∞ ( b Λ) and ˇ f denotes the inv erse F ourier transform (A.4). Note that if ψ ∈ L 1 (Λ) then this deﬁnition coincides with the standard one (A.3). A.5. Finite Bra v ais lattice. F or a Bra v ais lattice Λ as b efore and for L ∈ 2 N w e deﬁne a ﬁnite Brav ais lattice Λ L as Λ L = ( A Z d ) / ( LA Z d ) =    d X j =1 m j a j : m j = − L 2 , − L 2 + 1 , . . . , L 2 , j = 1 , . . . , d    with perio dic boundary condition. Once again this is an additive group with discrete top ology . Using similar arguments as for the inﬁnite lattice we can sho w that every χ ∈ Hom(Λ L , S 1 ) is of the form (A.7) χ ( x ) = χ p ( x ) = 1 | Λ L | 1 / 2 e ip · x where | Λ L | = ( L + 1) 3 is the num b er of p oints in Λ L and p is the element of (A.8) b Λ L :=    d X j =1 m j b j L + 1 : m j = − L 2 , − L 2 + 1 , . . . , L 2 − 1 , L 2    , where b j are primitive vectors of the recipro cal lattice Λ ∗ . W e w ill use this iden tiﬁcation of b Λ L for the entire pap er. Not that in (A.7) we ha v e in tro duced an additional normalization factor. The reason for it is that when we consider a standard coun ting measure on Λ L as its Haar measure then the system { χ p } p ∈ b Λ L forms an orthonormal basis of L 2 (Λ L ). W e will refer to this system as the momentum basis of L 2 (Λ L ). The correct choice for the Haar measure on b Λ L is again the standard counting measure on b Λ L . Once again we can write the form ulae for the F ourier transform on Λ L and its inv erse on b Λ L explicitly as (A.9) b f ( p ) = 1 | Λ L | 1 / 2 X x ∈ Λ L f ( x ) e − ip · x = ⟨ χ p , f ⟩ L 2 (Λ L ) , p ∈ b Λ L 28 N. MOKRZA ´ NSKI, M. NAPI ´ ORKO WSKI, AND J. WOJTKIEWICZ and (A.10) ˇ g ( x ) = 1 | Λ L | 1 / 2 X p ∈ b Λ L g ( p ) e ip · x = ⟨ χ p , g ⟩ L 2 ( b Λ L ) x ∈ Λ L . As the F ourier transform is unitary w e also note the P arsev al identit y X x ∈ Λ L X x ∈ Λ L f ( x ) g ( x ) = X p ∈ b Λ L b f ( p ) b g ( p ) . Appendix B. Graph calculus B.1. Basic deﬁnitions. It might be useful do work within the graph calculus formalism. A graph G is a couple G = ( V , E ), where V is a ﬁnite 3 set of v ertices and E ⊂ V × V is the set of edges. Note that the edges are directed, that is ( x, y )  = ( y , x ) for x  = y . This approach will alow to deﬁne directional deriv ative. W e will consider only non-orien ted graphs, which in this setting means ( x, y ) ∈ E ⇒ ( y, x ) ∈ E . W e will also assume that there are no self-lo ops (that is there are no edges of the form ( x, x )). W e will sa y that x and y are nearest neighbors if ( x, y ) ∈ E . This deﬁnes a symmetric relation on V that will b e denoted as x ∼ y . Moreo ver to each edge ( x, y ) ∈ E we will assign a positive real num b er t ( x, y ), which gives rise to the weigh ted graph structure. Here we will also assume that t ( x, y ) = t ( y , x ) for every edge ( x, y ). Consider a subset Ω ⊂ V . W e deﬁne the b oundary of Ω, denoted ∂ Ω, as ∂ Ω = { x ∈ Ω : there exists y ∈ Ω , y ∼ x } W e also deﬁne the set of in terior edges E Ω of the set Ω as E Ω = { ( x, y ) ∈ E : x, y ∈ Ω } . It will b e useful to also deﬁne the ”nearest neigh b ors b oundary” of the set Ω deﬁned as ∂ nn Ω := { y ∈ Ω : y ∼ x for some x ∈ ∂ Ω } and the ”nearest neigh b ors closure” of Ω Ω nn := Ω ∪ ∂ nn Ω . With a graph we can asso ciate tw o Hilb ert spaces: the space of functions on the vertices L 2 ( V ) and functions on the edges L 2 ( E ) (b oth with counting measure). F or a function f : V → C we deﬁne its (discrete) gradien t ∇ f : E → C as ∇ f ( x, y ) = p t ( x, y ) ( f ( y ) − f ( x )) . F or a given edge ( x, y ) ∈ E the v alue ∇ f ( x, y )) may b e considered as the directional deriv ativ e in direction x → y . As ∇ : L 2 ( V ) → L 2 ( E ) we can consider its dual ∇ ∗ : L 2 ( E ) → L 2 ( V ) which satisﬁes the prop erty that for any f ∈ L 2 ( V ) and F ∈ L 2 ( E ) we hav e ⟨ F , ∇ f ⟩ L 2 ( E ) = ⟨∇ ∗ F , f ⟩ L 2 ( V ) . W e can also deﬁne ∇ ∗ explicitly b y the form ula (B.1) ∇ ∗ F ( x ) = X y ∼ x p t ( x, y ) ( F ( y, x ) − F ( x, y )) . Next w e can deﬁne the discrete divergence div : L 2 ( E ) → L 2 ( V ) as div = − 1 2 ∇ ∗ and the discrete Laplacian ∆ : L 2 ( V ) → L 2 ( V ) as ∆ = div ◦ ∇ . W e can chec k that the action of ∆ can be written explicitly (B.2) ∆ f ( x ) = X y ∼ x t ( x, y )( f ( y ) − f ( x )) = X y ∼ x ∇ f ( x, y ) . F rom the deﬁnition it is easy to see that the Laplace op erator is self-adjoint on L 2 ( V ). W e will show it for completeness: for f , g ∈ L 2 ( V ) we hav e ⟨ f , ∆ g ⟩ L 2 (Ω) = ⟨ g , − 1 2 ∇ ∗ ∇ f ⟩ L 2 ( E ) = − 1 2 ⟨∇ g , ∇ f ⟩ L 2 ( E ) = ⟨− 1 2 ∇ ∗ ∇ g , f ⟩ L 2 ( V ) = ⟨ ∆ g , f ⟩ L 2 ( V ) 3 W e can also consider inﬁnite, but countable sets of v ertices. This ho wev er requires adding some technical assumptions on summa- bility of functions on v ertices and edges. GROUND ST A TE ENERGY OF THE DILUTE BOSE-HUBBARD GAS ON BRA V AIS LA TTICES 29 W e are in terested in deriving some prop erties of the discrete Laplace op erator resembling the Green identities that hold for the standard (contin uous) Laplacian. Let Ω ⊂ V b e a ﬁxed subset. Then we hav e X x ∈ Ω g ( x )∆ f ( x ) = ⟨ 1 Ω g , ∆ f ⟩ L 2 ( V ) = − 1 2 ⟨ 1 Ω g , ∇ ∗ ∇ f ⟩ L 2 ( V ) = − 1 2 ⟨∇ 1 Ω g , ∇ f ⟩ L 2 ( E ) = − 1 2 X ( x,y ) ∈ E ∇ ( 1 Ω g )( x, y ) · ∇ f ( x, y ) = − 1 2 X ( x,y ) ∈ E Ω ∇ g ( x, y ) ∇ f ( x, y ) + X x ∈ ∂ Ω X y ∈ Ω y ∼ x g ( x ) ∇ f ( x, y ) . (B.3) There is no factor 1 2 in the second term as cases x ∈ Ω, y ∈ Ω and x ∈ Ω, y ∈ Ω are symmetric and give the same con tribution. The factor 1 2 in the ﬁrst term is the side eﬀect of considering the ordered pairs in the deﬁnition of the edge ( x, y ), which essen tially means every b ond b etw een x and y is counted t wice. This result is the discrete analogue to the standard integration by parts form ula Z Ω g ( x )∆ f ( x ) dx = − Z Ω ∇ g ( x ) ∇ f ( x ) dx + Z ∂ Ω g ( x ) ∂ f ∂ n ( x ) dσ ( x ) . B.2. Neumann Laplacian. W e will deﬁne the Neumann Laplacian on some set Ω ⊂ V . T o this end we ﬁrst deﬁne a quadratic form (B.4) Q Neu ( f ) = 1 2 X ( x,y ) ∈ E Ω |∇ f ( x, y ) | 2 . W e note that the v alue of Q Neu ( f ) dep ends only on the restriction of f to the set Ω. The Neumann Laplacian − ∆ Neu Ω is deﬁned as the op erator on L 2 (Ω) associated with this quadratic form, meaning that for every f ∈ L 2 (Ω) there holds ⟨ f , − ∆ Neu Ω f ⟩ L 2 (Ω) = Q ( f ) . W e can write the action of − ∆ Neu Ω explicitly: for f ∈ L 2 (Ω) w e ha ve (B.5) − ∆ Neu Ω f ( x ) = X y ∈ Ω y ∼ x t ( x, y )( f ( x ) − f ( y )) = X y ∈ Ω y ∼ x ∇ f ( y , x ) . R emark B.1 . Note that if the p oin t x is in the interior (i.e. not on the b oundary) of Ω then the action of − ∆ Neu Ω coincides with the action of the standard discrete Laplacian. If x ∈ ∂ Ω then the action of − ∆ Neu Ω lo oks as if the function f satisﬁed an additional condition (B.6) ∀ x ∈ ∂ Ω ∀ y ∈ Ω y ∼ x f ( y ) = f ( x ) . This can b e interpreted as a discrete v ersion of the standard Neumann condition ∂ f ∂ n = 0 on ∂ Ω. W e emphasize ho wev er that here the function f needs to be deﬁned only on the set Ω and not on the set of its nearest neigh b ors. Moreo ver, in some cases, imp osing condition (B.6) migh t be imp ossible – a simple example of suc h situation is Ω = V \ { v 0 } for some v 0 ∈ V , i.e. the set of all but one v ertices. Then for a function f ∈ L 2 (Ω) it is p ossible to imp ose (B.6) if and only if the v alue of f on all neigh b ors of v 0 is the same. This e xample illustrates the fact that the Neumann Laplacian is not the same as the s tandard Laplacian restricted to the functions satisfying Neumann b oundary condition (B.6). How ever, if some function f is supp orted on Ω nn and satisﬁes (B.6) then it is true (by computation similar to the one in (B.3)) that − ∆ Neu Ω f ( x ) = − ∆ f ( x ) for x ∈ Ω . As the ab ov e example sho ws, using the phrase ”Neumann b oundary conditions” is misleading, hence w e will restrain from using that phrase and use the phrase ”Neumann Laplacian” instead. Finally w e will v erify that the op erator − ∆ Neu Ω is self-adjoin t, meaning that for ev ery f , g ∈ L 2 (Ω) w e ha ve ⟨ f , − ∆ Neu Ω g ⟩ = ⟨− ∆ Neu Ω f , g ⟩ . This follows from the fact that − ∆ Neu Ω is the Laplace op erator deﬁned as in (B.2) in the previous subsection for the graph (Ω , E Ω ), so self-adjointness follows from the general consideration of graph Laplace op erators. 30 N. MOKRZA ´ NSKI, M. NAPI ´ ORKO WSKI, AND J. WOJTKIEWICZ Appendix C. The sca ttering equa tion on a la ttice W e will start with deriving the formula for the scattering length (2.12). T o this end w e are in terested in a solution to the equation (deﬁned on Λ = A Z 3 ) (C.1) − ∆ φ ( x ) + U 2 δ x, 0 φ ( x ) = 0 , with the condition (C.2) lim | x |→ + ∞ φ ( x ) = 1 . This equation is called the (zero-energy) scattering equation. W e will see that this equation has a unique solution, hence it is possible to deﬁne the scattering length in a following wa y . Deﬁnition C.1. The sc attering length a is deﬁne d as 4 π a = X x ∈ Λ ∆ φ ( x ) = U 2 φ (0) , wher e φ is the solution to the sc attering e quation (C.1) with c ondition (C.2) . In order to solve the scattering equation for the moment w e will ignore the condition (C.2) and tak e the (distri- butional) F ourier transform (see Appendix A) of its b oth sides. A simple computation leads to (C.3) ε ( p ) b φ + U 2 φ (0) = 0 , where ε ( p ) is the dispersion relation, deﬁned in (2.13). This equation is satisﬁed in the sense of distributions, that is after testing against some smo oth function on b Λ. F or no w we will restrict ourselves to the set b Λ \ { 0 } and test the ab o ve equation with the test function ϕ with supp ϕ not including zero. On this set (2 ε ( p )) − 1 is a well-deﬁned smo oth function and therefore we can multiply b oth sides of the equation (C.3) b y it. It follo ws that b φ = − U φ (0) 2 ε ( p ) . Th us, on this set, w e can identify b φ as a L 1 ( b Λ) function (note that this function would not b e integrable in the dimensions d = 1 and d = 2). By restricting our considerations to the set not containing zero, w e migh t ha ve neglected distributions whose supp ort is the one-point set { 0 } . Since distributions supp orted on one point are the sums of Dirac deltas and their deriv atives, w e conclude that b φ = − U φ (0) 2 ε ( p ) + X α : | α |≤ M c α ∂ α δ 0 , for some M ≥ 0 and c α ∈ C . By equation (C.3) we need to hav e ε ( p ) ·   X α : | α |≤ M c α ∂ α δ 0   = 0 . It follows that M = 1 as the v alue of function ε ( p ) and all of its ﬁrst order deriv atives are zero at p = 0, whereas v alues of second order deriv ativ es at p = 0 are non-zero. A consequence of this observ ation is that b φ = − U φ (0) 2 ε ( p ) + C 0 δ 0 + 3 X j =1 C j ∂ p j δ 0 . Using the inv erse F ourier transform (see equation (A.4) in the Appendix) w e get φ ( x ) = − U φ (0) 2 | b Λ | − 1 Z b Λ e ip · x ε ( p ) dp + C 0 + 3 X j =1 C j x j . The v alue φ (0) is not y et sp eciﬁed, w e need to make sure that this function is self consistent with its v alue at x = 0. Before that we will simplify this expression by using the b oundary condition (C.2) that so far we hav e omitted. By the Riemann-Lebesgue lemma we hav e lim | x |→∞ | b Λ | − 1 Z b Λ e ip · x ε ( p ) dp = 0 , GROUND ST A TE ENERGY OF THE DILUTE BOSE-HUBBARD GAS ON BRA V AIS LA TTICES 31 so this part of the scattering equation solution v anishes. An easy observ ation also leads to conclusion that in order to satisfy (C.2) we need to hav e C 0 = 1 and C j = 0 for j = 1 , 2 , 3. W e ha v e th us simpliﬁed the formula for φ to φ ( x ) = 1 − U φ (0) 2 | b Λ | − 1 Z b Λ e ip · x ε ( p ) dp. Computing the v alue at x = 0 we hav e φ (0) = 1 − U φ (0) 2 | b Λ | − 1 Z b Λ 1 ε ( p ) dp = 1 − U φ (0) γ , where γ = 1 2 | b Λ | − 1 Z b Λ 1 ε ( p ) dp. This leads to (C.4) φ (0) = 1 1 + U γ and (C.5) φ ( x ) = 1 − 1 2 · U 1 + U γ | b Λ | − 1 Z b Λ e ip · x ε ( p ) dp. Using (C.4) in the Deﬁnition C.1 we can explicitly write (C.6) 8 π a = U U γ + 1 , whic h is the deﬁnition used in (2.12) It will also be useful to introduce function w ( x ) := 1 − φ ( x ) or explicitly w ( x ) = 1 2 · U 1 + U γ | b Λ | − 1 Z b Λ e ip · x ε ( p ) dp. This function satisﬁes the equation ∆ w ( x ) + 1 2 U (1 − w ( x )) δ x, 0 = 0 with a condition lim | x |→∞ w ( x ) = 0 . The main adv antage of considering this function instead of φ ( x ) is that its (once again distributional 4 ) F ourier transform b w ( p ) can b e treated as a L 1 ( b Λ) function (and not only as a distribution): (C.7) b w ( p ) = U (1 − w (0)) 2 ε ( p ) = U 1 + U γ · 1 2 ε ( p ) , p  = 0 . 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Ph ys. 141, 683-726 (2010) Dep ar tment of Ma thema tical Methods in Physics, F a cul ty of Physics, University of W arsa w, P asteura 5, 02-093 W arsza w a, Poland Email addr ess : norbert.mokrzanski@fuw.edu.pl Dep ar tment of Ma thema tical Methods in Physics, F a cul ty of Physics, University of W arsa w, P asteura 5, 02-093 W arsza w a, Poland Email addr ess : marcin.napiorkowski@fuw.edu.pl Dep ar tment of Ma thema tical Methods in Physics, F a cul ty of Physics, University of W arsa w, P asteura 5, 02-093 W arsza w a, Poland Email addr ess : jacek.wojtkiewicz@fuw.edu.pl

Ground state energy of the dilute Bose-Hubbard gas on Bravais lattices

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