Spontaneous symmetry breaking in coupled Bose-Einstein condensates
aa r X i v : . [ c ond - m a t . s t a t - m ec h ] N ov Spontaneous symmetry breaking in coupled Bose-Einsteincondensates
Hal Tasaki ∗ We study a system of two hardcore bosonic Hubbard models weakly coupled witheach other by tunneling. Assuming that the single uncoupled model exhibits off-diagonal long-range order, we prove that the coupled system exhibits spontaneoussymmetry breaking (SSB) in the infinite volume limit, in the sense that the twosubsystems maintain a definite relative U(1) phase when the tunneling is turnedoff. Although SSB of the U(1) phase is never observable in a single system, SSBof the relative U(1) phase is physically meaningful and observable by interferenceexperiments. The present theorem is made possible by the rigorous theory of low-lying states and SSB in quantum antiferromagnets developed over the years.
The essence of Bose-Einstein condensation is off-diagonal long-range order (ODLRO) related tothe global quantum mechanical U(1) phase [1, 2, 3, 4]. It is known however that spontaneoussymmetry breaking (SSB) of the U(1) symmetry, which is predicted by mean filed-theories, isnever observable because of the law of particle number conservation [2, 5]. Although such a lackof SSB is sometimes regarded as paradoxical, there is indeed nothing problematic. A ground statewith long-range order (LRO) but without SSB is perfectly understood by the rigorous theory oflow-lying states and SSB developed mainly for quantum antiferromagnets [6, 7, 8, 9, 10, 11, 12, 13].In the present paper, we demonstrate, through a new rigorous result, that SSB of the relativeU(1) symmetry may take place when two Bose-Einstein condensates are weakly coupled [5]. Moreprecisely we consider two hardcore bosonic Hubbard models coupled with each other throughweak tunneling. By making full use of the rigorous theory of low-lying states, we prove that,in the infinite volume limit, the two subsystems maintain definite relative U(1) phase whenthe tunneling is turned off, provided that the single hardcore bosonic Hubbard model exhibitsODLRO. We stress that the SSB of relative U(1) phase is physically meaningful, and directlyrelated to the observation of interference patterns in cold atom experiments [14, 15]. See [16] forrelated phenomena in a classical setting,We have formulated our theorem for the hardcore bosonic Hubbard model mainly for sim-plicity. It is in principle possible to extend the result to a broader class of boson systems, butthat requires extra (nontrivial) work to extend the theorems in [11, 12].
We start by defining the basic (uncoupled) hardcore bosonic Hubbard model, and state ourassumption about ODLRO. Let Λ be the d -dimensional L × · · · × L hypercubic lattice, and let B be the corresponding set of bonds ( x, y ), i.e., ordered pairs of neighboring sites x, y ∈ Λ. Weimpose periodic boundary conditions. For each site x ∈ Λ we denote by ˆ a x and ˆ a † x the annihilationand the creation operators, respectively, of a bosonic particle at site x . They satisfy the standardcommutation relations [ˆ a x , ˆ a y ] = [ˆ a † x , ˆ a † y ] = 0 and [ˆ a x , ˆ a † y ] = δ x,y for for any x, y ∈ Λ. The number ∗ Department of Physics, Gakushuin University, Mejiro, Toshima-ku, Tokyo 171-8588, Japan x is defined by ˆ n x = ˆ a † x ˆ a x , and the total number operator by ˆ N = P x ∈ Λ ˆ n x . Wedenote by | Φ vac i the unique state such that ˆ a x | Φ vac i = 0 for any x ∈ Λ.Let N = 1 , , . . . , L d be the total boson number. When we vary the system size, we alwaysfix the density ρ = N/L d , and make both L and N large. We define the Hilbert space H N with N hardcore bosons as the space spanned by all states ˆ a † x ˆ a † x · · · ˆ a † x N | Φ vac i such that x i = x j if i = j . Then our Hamiltonian is ˆ H = − ˆ P hc X ( x,y ) ∈B ˆ a † x ˆ a y , (1)where ˆ P hc is the projection operator onto H N . We denote by | Φ GS i ∈ H N the unique groundstate of ˆ H . The uniqueness is proved by the standard Perron-Frobenius argument [17, 18]. See[13].To test for possible ODLRO, we define the order operators byˆ O + = X x ∈ Λ ˆ a † x , ˆ O − = X x ∈ Λ ˆ a x , (2)and also the self-adjoint order operators byˆ O (1) = ˆ O + + ˆ O − , ˆ O (2) = ˆ O + − ˆ O − i . (3)It is readily found that these operators are transformed by the global U(1) phase rotation ˆ U θ = e − iθ ˆ N as ˆ U † θ ˆ O (1) ˆ U θ = cos θ ˆ O (1) − sin θ ˆ O (2) , ˆ U † θ ˆ O (2) ˆ U θ = cos θ ˆ O (2) + sin θ ˆ O (1) . (4)This means that the pair ( ˆ O (1) , ˆ O (2) ) transforms precisely as a vector. We shall assume through-out the present paper that, for some dimension d and the density ρ ∈ (0 , | Φ GS i of the hard-core bosonic Hubbard model (1) exhibits ODLRO in the sense that h Φ GS | (cid:16) ˆ O (1) L d (cid:17) | Φ GS i = h Φ GS | (cid:16) ˆ O (2) L d (cid:17) | Φ GS i ≥ λ ( ρ ) , (5)for any L with the order parameter λ ( ρ ) > This is expected to be valid for a large rangeof ρ for any d ≥
2, but is proved rigorously only for d ≥ ρ = 1 / | Φ GS i ∈ H N implies h Φ GS | ˆ O (1) L d | Φ GS i = h Φ GS | ˆ O (2) L d | Φ GS i = 0 , (6)for any d , ρ , and L .The ground state | Φ GS i therefore exhibits (OD)LRO as in (5) but no SSB as in (6). Such LROwithout SSB has been studied intensively in the context of quantum antiferromagnets, and it isknown that, in such a situation, there inevitably appears a series of low-lying states, i.e., stateswith very low excitation energies, and physical ground states with SSB are linear combinationsof the low-lying states [6]. See, e.g., [7]. For a discussion of similar phenomena in nuclear physics,see Chapter 11 of [24]. By now a fully rigorous theory of low-lying states has been developed[8, 9, 10, 11, 12, 13]. We recommend [12], which can be read as a compact review. We here take the “statistical mechanical point of view”, and regard the Bose-Einstein condensation as aphenomenon in the infinite volume limit. Symmetry breaking ground states in the uncoupled system
We now review the theory of low-lying states and symmetry breaking in the context of hardcorebosonic Hubbard model. To develop a theory parallel to that for quantum antiferromagnets, weintroduce the extended Hilbert space H = L d M K =0 H K , (7)which contains all possible particle numbers. We then take the Hamiltonianˆ H µ = ˆ H − µ ˆ N , (8)on H , and adjust the chemical potential µ so that the ground state of ˆ H µ coincides with theprevious ground state | Φ GS i which has the fixed density ρ . We assume that such tuning of µ ispossible. When the target density is ρ = 1 /
2, it is known rigorously that the right choice is µ = 0[13, 17, 18, 25, 26].Following [11, 12], we define trial states | Γ M i = ( ˆ O + ) M | Φ GS ik ( ˆ O + ) M | Φ GS ik , | Γ − M i = ( ˆ O − ) M | Φ GS ik ( ˆ O − ) M | Φ GS ik , (9)for M = 1 , , . . . . The following theorem, which establishes the existence of the series of low-lyingstates, was proved in [11] for the case ρ = 1 /
2, and in [12] for general ρ . Theorem 1
Suppose that the ground state | Φ GS i exhibits ODLRO as in (5) . Then there areconstants C and C which depend only on d , ρ , and λ ( ρ ) . For any L and M such that | M | ≤ C L d/ , the state | Γ M i is well-defined, and satisfies h Γ M | ˆ H µ | Γ M i ≤ h Φ GS | ˆ H µ | Φ GS i + C | M | L d . (10)We proceed to construct a new trial state by summing up these low-lying states. For θ ∈ R and an integer valued function M max ( L ) > M max ( L ) ≤ C L d/ , let | Ξ θ i = 1 p M max ( L ) + 1 M max ( L ) X n = − M max ( L ) e − inθ | Γ n i , (11)where we set | Γ i = | Φ GS i . Let x, y ∈ Z d be neighboring sites, and let ˆ h x,y = − ˆ P hc (ˆ a † x ˆ a y + ˆ a † x ˆ a y )be the local Hamiltonian. By using (10) and the translation invariance, one finds thatlim L ↑∞ h Ξ θ | ˆ h x,y | Ξ θ i = ǫ GS ( µ ) , (12)where ǫ GS ( µ ) = lim L ↑∞ h Φ GS | ˆ H µ | Φ GS i / ( dL d ) is the ground state energy per bond. Thus the state | Ξ θ i can be regarded essentially as a ground state when L is large. Then the following theoremwas proved in [12], improving the results in [11]. If ρ = 1 /
2, the second term in the right-hand side can be replaced by C M /L d . We expect that the samebound is possible for general ρ , but cannot prove it for technical reasons. heorem 2 If M max ( L ) diverges to infinity not too rapidly as L ↑ ∞ , one has lim L ↑∞ h Ξ θ | ˆ O ± L d | Ξ θ i = m ∗ e ± iθ , (13)lim L ↑∞ h Ξ θ | ˆ O ( α ) L d | Ξ θ i = ( m ∗ cos θ ( α = 1) m ∗ sin θ ( α = 2) , (14)lim L ↑∞ h Ξ θ | (cid:16) ˆ O ( α ) L d (cid:17) | Ξ θ i = ( ( m ∗ cos θ ) ( α = 1)( m ∗ sin θ ) ( α = 2) , (15) where the symmetry breaking order parameter m ∗ is defined by m ∗ := lim k ↑∞ lim L ↑∞ n h Φ GS | (cid:16) ˆ O ( α ) L d (cid:17) k | Φ GS i o / (2 k ) , (16) with α = 1 , , and satisfies m ∗ ≥ p λ ( ρ ) . The theorem shows that the state | Ξ θ i , which is essentially a ground state of (8), exhibitsODLRO and also fully breaks the U(1) phase symmetry. The symmetry breaking is manifest inthe remarkable relations h Ξ θ | ˆ a † x | Ξ θ i ≃ m ∗ e iθ , h Ξ θ | ˆ a x | Ξ θ i ≃ m ∗ e − iθ , (17)which hold for large L because of (13). In the state | Ξ θ i , the U(1) phase is “pointing” in thespecific direction θ .Comparing (14) and (15), we find that ˆ O ( α ) /L d , which is the density of the order operator,exhibits vanishing fluctuation as L becomes large. Vanishing fluctuation is usually a sign that thestate is a physically realistic macroscopic state. In fact the state corresponding to | Ξ θ i is regardedas a realizable ground state in quantum antiferromagnets. Moreover the (near) ground state | Ξ θ i is similar in many aspects to ground states obtained by mean-field theories for bosons [1, 2, 4, 27].Nevertheless we cannot regard | Ξ θ i as a “realistic” state of an isolated Bose-Einstein condensatesince it is a superposition of states with different particle numbers. (See, e.g., section III.D.1 of[2] and [5].) Suppose that one confines exactly N particles in a container and then cool themdown to the ground state. It is never possible to generate a superposition as in (11) by allowedphysical processes. But a more important point is that the phase θ characterizing | Ξ θ i is aphysically meaningless quantity, which can never be measured experimentally.In this sense the exact ground state | Φ GS i , which exhibits ODLRO but no symmetry breaking,may be a better model of an isolated Bose-Einstein condensate. We should note however thatthe distinction between | Ξ θ i and | Φ GS i becomes subtle when we restrict the class of observables.More precisely, if ˆ A is a local observable such that [ ˆ N , ˆ A ] = 0, we expect that h Φ GS | ˆ A | Φ GS i ≃ h Ξ θ | ˆ A | Ξ θ i . (18)We also note that the | Φ GS i is obtained from | Ξ θ i by | Φ GS i = p M max ( L ) + 1 ˆ P N | Ξ θ i , (19)where ˆ P N is the projection onto H N , or by | Φ GS i = p M max ( L ) + 12 π Z π dθ | Ξ θ i . (20)It is interesting that the exact ground state | Φ GS i can be regarded as a superposition of thesymmetry breaking (near) ground states | Ξ θ i for all possible θ . See [13] for further discussionabout the relation between | Φ GS i and | Ξ θ i . 4 Coupled Bose-Einstein condensates
We now consider a system of two Bose-Einstein condensates coupled weakly by tunneling. Wethen find that a spontaneous symmetry breaking that fixes the relative U(1) phase takes place.Unlike the phase θ discussed above, the relative phase ϕ is experimentally measurable. Thissituation corresponds, e.g., to the experimental setup where bosons are trapped in a double-wellpotential [14, 15]. The clear interference pattern observed experimentally [14] is a manifestationof a fixed relative U(1) phase. We consider two exact copies of the d -dimensional hyper cubic lattice Λ, and call them Λ a and Λ b . Lattice sites are denoted as ( x, ν ) ∈ Λ ν where ν = a , b and x ∈ Λ. On each lattice wedefine the same system of hardcore bosons as before. We further assume that there is a tunnelingHamiltonian which weakly couples the two subsystems on Λ a and Λ b . The total Hamiltonian isthus ˆ H tot ε = ˆ H a + ˆ H b + ε ˆ H tunnel . (21)Here we set, for ν = a , b, ˆ H ν = − ˆ P hc X ( x,y ) ∈B ˆ a † ( x,ν ) ˆ a ( y,ν ) , (22)which are the exact copies of (1), andˆ H tunnel = − X x ∈ Λ (cid:0) e iϕ ˆ a † ( x, a) ˆ a ( x, b) + e − iϕ ˆ a ( x, a) ˆ a † ( x, b) (cid:1) . (23)Here the phase factor ϕ ∈ R is introduced to make clear the physical picture; it is most naturalto set ϕ = 0. We treat this problem in a physically realistic Hilbert space where the total number of particlesin the coupled system is exactly 2 N . If we denote the copy of the N particle Hilbert space H N as H νN for ν = a , b, the whole Hilbert space is H tot2 N = N M K =0 H a K ⊗ H b2 N − K . (24)In other words, we assume that the two Bose-Einstein condensates can exchange particles in acoherent manner, while completely isolated from the outside world. This may be a reasonableidealization of realistic situations in cold atom experiments.Let | Φ totGS ,ε i ∈ H tot2 N be the unique ground state of the total Hamiltonian (21) with ε > Theorem 3
Assume the existence of ODLRO (in the single uncoupled system) as in (5) . Thenfor any x ∈ Z d , we have lim ε ↓ lim L ↑∞ h Φ totGS ,ε | ˆ a † ( x, a) ˆ a ( x, b) | Φ totGS ,ε i = ˜ m e − iϕ , (25)lim ε ↓ lim L ↑∞ h Φ totGS ,ε | ˆ a ( x, a) ˆ a † ( x, b) | Φ totGS ,ε i = ˜ m e iϕ , (26) with ˜ m ≥ m ∗ ≥ p λ ( ρ ) . The interference pattern is observed after switching off the trapping potential and letting the particles evolvealmost freely. We should note that there is an essentially different class of interference phenomena between twoBose-Einstein condensates. It is known that two condensates which have no fixed relative phase (and hence wellapproximated by | Φ GS i ⊗ | Φ GS i ) also exhibit interference. See, e.g., [28]. The general case reduces to ϕ = 0 by replacement e iϕ ˆ a ( x, b) → ˆ a ( x, b) for all x ∈ Λ.
5s we noted before the assumption of the theorem is rigorously established for d ≥ N = L d / ν = a , b and x ∈ Λ, thelocal order operators ˆ o (1)( x,ν ) and ˆ o (2)( x,ν ) byˆ o (1)( x,ν ) := ˆ a † ( x,ν ) + ˆ a ( x,ν ) , ˆ o (2)( x,ν ) := ˆ a † ( x,ν ) − ˆ a ( x,ν ) i . (27)Exactly as in (3) and (4), the pair (ˆ o (1)( x,ν ) , ˆ o (2)( x,ν ) ) transforms as a vector under the operation ofthe unitary operator ˆ U θ = e − iθ ˆ N . We of course have h Φ totGS ,ε | (ˆ o (1)( x,ν ) , ˆ o (2)( x,ν ) ) | Φ totGS ,ε i = (0 ,
0) againby particle number conservation, but (26) implies thatlim ε ↓ lim L ↑∞ h Φ totGS ,ε | (cid:8) ˆ o (1)( x, a) ˆ o (1)( x, b) + ˆ o (2)( x, a) ˆ o (2)( x, b) (cid:9) | Φ totGS ,ε i = ˜ m cos ϕ. (28)This suggests that the two order operators behaves like two vectors with magnitude ˜ m whichhave a fixed relative angle ϕ . This is more directly seen by noting thatlim ε ↓ lim L ↑∞ h Φ totGS ,ε | (ˆ o (1)( x, a) , ˆ o (2)( x, a) ) (cid:18) cos ϕ sin ϕ − sin ϕ cos ϕ (cid:19) ˆ o (1)( x, b) ˆ o (2)( x, b) ! | Φ totGS ,ε i = ˜ m , (29)which also follows from (26).We conclude that, in the ground state obtained in the double limit lim ε ↓ lim L ↑∞ , the relativeU(1) phase between the two condensates has a definite value ϕ . Since this phase ordering wasachieved by “infinitesimal symmetry breaking field” ε in the tunneling Hamiltonian, we cansay that this is a kind of spontaneous symmetry breaking. As we noted in the beginning,such ordering of relative phase between two weakly coupled Bose-Einstein condensates can beexperimentally observed by means of interference experiments. Note that the Hamiltonian (21) is invariant under the transformationˆ a ( x, a) → e iϕ ˆ a ( x, b) , ˆ a ( x, b) → e − iϕ ˆ a ( x, a) , (30)for all x ∈ Λ, and so is the unique ground state | Φ totGS ,ε i . The invariance implies that h Φ totGS ,ε | e iϕ ˆ a † ( x, a) ˆ a ( x, b) | Φ totGS ,ε i = h Φ totGS ,ε | e − iϕ ˆ a ( x, a) ˆ a † ( x, b) | Φ totGS ,ε i , and hence this quantity is real.The key of the proof is that we can take a state | Θ L,Mϕ i ∈ H tot2 N for ϕ ∈ R with the followingproperties. Like the ground state, it satisfies h Θ L,Mϕ | e iϕ ˆ a † ( x, a) ˆ a ( x, b) | Θ L,Mϕ i = h Θ L,Mϕ | e − iϕ ˆ a ( x, a) ˆ a † ( x, b) | Θ L,Mϕ i ∈ R and this quantity is independentof x . It further satisfies lim M ↑∞ lim L ↑∞ h Θ L,Mϕ | e iϕ ˆ a † ( x, a) ˆ a ( x, b) | Θ L,Mϕ i = ( m ∗ ) (31) Recall that in the ferromagnetic Ising model, for example, one considers a similar double limitlim h ↓ lim L ↑∞ h σ x i to detect possible spontaneous symmetry breaking, where h is the external magnetic field. Although our theorem is about the L ↑ ∞ limit, it suggests that one should start seeing the phase orderingfor ε & L − d when L is large but finite. x . Finally | Θ L,Mϕ i is essentially a ground state of ˆ H tot0 = ˆ H a + ˆ H b in the sense thatlim L ↑∞ L d (cid:8) h Θ L,Mϕ | ˆ H tot0 | Θ L,Mϕ i − E totGS , (cid:9) = 0 , (32)for any fixed M , where E totGS , is the ground state energy of ˆ H tot0 in the space H tot2 N .Let us first assume the existence of | Θ L,Mϕ i , and prove the theorem by following the variationalargument due to Kaplan, Horsch, and von der Linden [9]. Since | Φ totGS ,ε i is the ground state, oneobviously has h Θ L,Mϕ | ˆ H tot ε | Θ L,Mϕ i ≥ h Φ totGS ,ε | ˆ H tot ε | Φ totGS ,ε i . (33)Since ˆ H tot ε = ˆ H tot0 + ε ˆ H tunnel , we have − L d h Φ totGS ,ε | ˆ H tunnel | Φ totGS ,ε i≥ − L d h Θ L,Mϕ | ˆ H tunnel | Θ L,Mϕ i + 1 εL d (cid:8) h Φ totGS ,ε | ˆ H tot0 | Φ totGS ,ε i − h Θ L,Mϕ | ˆ H tot0 | Θ L,Mϕ i (cid:9) ≥ − L d h Θ L,Mϕ | ˆ H tunnel | Θ L,Mϕ i + 1 εL d (cid:8) E totGS , − h Θ L,Mϕ | ˆ H tot0 | Θ L,Mϕ i (cid:9) . (34)By recalling (23) and noting the symmetry, this becomes h Φ totGS ,ε | e iϕ ˆ a † ( x, a) ˆ a ( x, b) | Φ totGS ,ε i ≥ h Θ L,Mϕ | e iϕ ˆ a † ( x, a) ˆ a ( x, b) | Θ L,Mϕ i + 12 εL d (cid:8) E totGS , − h Θ L,Mϕ | ˆ H tot0 | Θ L,Mϕ i (cid:9) . (35)By fixing arbitrary ε >
0, letting L ↑ ∞ , and then letting M ↑ ∞ , we getlim L ↑∞ h Φ totGS ,ε | e iϕ ˆ a † ( x, a) ˆ a ( x, b) | Φ totGS ,ε i ≥ ( m ∗ ) , (36)where we used (32) and then (31). This becomes the desired (26) if we let ε ↓ | Θ L,Mϕ i . Let | Ξ a θ i and | Ξ b θ i be the exactcopies of | Ξ θ i defined in (11), which is essentially a ground state and also breaks U(1) symmetry.A natural candidate for a state with fixed relative phase is the tensor product | Ξ a θ i ⊗ | Ξ b θ + ϕ i , butthis state is again a superposition of states with different total particle numbers. We can follow(19) or (20) to construct a physical state as | Ξ tot ϕ i ∝ ˆ P tot2 N (cid:0) | Ξ a θ i ⊗ | Ξ b θ + ϕ i (cid:1) , (37)where ˆ P tot2 N is the projection onto H tot2 N , or by phase averaging as | Ξ tot ϕ i ∝ π Z π dθ | Ξ a θ i ⊗ | Ξ b θ + ϕ i . (38)The two constructions lead to exactly the same result, and we get | Ξ tot ϕ i = 1 p M max ( L ) + 1 M max ( L ) X n = − M max ( L ) e inϕ | Γ a n i ⊗ | Γ b − n i , (39)7here | Γ a M i and | Γ b M i are exact copies of low-lying states (9). This is indeed a near groundstate of ˆ H tot0 in which the two condensates have definite relative phase ϕ . For our purpose it isconvenient to replace M max ( L ) by M to define | Θ L,Mϕ i = 1 √ M + 1 M X n = − M e inϕ | Γ a n i ⊗ | Γ b − n i . (40)The desired properties (31) and (32) are proved by using Theorems 1 and 2. See Appendices A.1and A.2 We considered two systems of hardcore bosons weakly coupled with each other by the tunnelingHamiltonian (23). Under the assumption that the uncoupled system exhibits ODLRO, we provedthat, in the infinite volume limit, the two Bose-Einstein condensates maintain definite relativephase ϕ when the tunneling is turned off. This is naturally interpreted as spontaneous breakdownof the relative U(1) phase between the two condensates. Although SSB of the U(1) phase in asingle isolated Bose-Einstein condensates is “observed” only theoretically, SSB in the relativeU(1) phase is realistic and is directly related to experimental observations of interference.Note that the definite relative phase ϕ between the two condensates is realized, as in (39) or(40), by a coherent superposition of states with different divisions of particle numbers betweenthe two subsystems. In other words the two subsystems inevitably entangle if we demand thatthere is a definite relative phase.We conjecture that the ground state obtained through the double limit in (26) resembles (orcoincides with) the large L limit of | Ξ tot ϕ i , although the proof seems very difficult. We also expectthat states realized experimentally in weakly coupled Bose-Einstein condensates resemble | Ξ tot ϕ i .As an alternative approach to give a meaning of symmetry breaking in coupled Bose-Einsteincondensates, a state with a fixed number of particles in a larger system (i.e., the two condensatesand the environment) which is “identical” to | Ξ a θ i ⊗ | Ξ b θ + ϕ i is constructed in [29]. Here “identical”means that the measurement of any observable which conserves the total number of particles inthe two condensates give the same results as | Ξ a θ i ⊗ | Ξ b θ + ϕ i . See also [30] for background. It islikely that our | Ξ tot ϕ i and their states are indistinguishable if we only measure local quantitieswhich preserve the particle number. We wish to thank Akira Shimizu and Masahito Ueda for indispensable discussions and comments whichmade the present work possible, and Tohru Koma and Haruki Watanabe for useful discussions on relatedtopics. The present work was supported by JSPS Grants-in-Aid for Scientific Research no. 16H02211.
A Properties of the state | Θ L,Mϕ i A.1 Expectation value of e iϕ ˆ a † ( x, a) ˆ a ( x, b) Let us prove (31). Here we make use of techniques and results from [12]. For ν = a , b, letˆ o + ν := L − d P x ∈ Λ ˆ a † ( x,ν ) and ˆ o − ν := L − d P x ∈ Λ ˆ a ( x,ν ) . We here abbreviate | Θ L,Mϕ i as | Θ i . From (40)8nd (9), we have | Θ i = 1 √ M + 1 (cid:26) | Φ aGS i| Φ bGS i + M X n =1 e inϕ (ˆ o +a ) n | Φ aGS ik (ˆ o +a ) n | Φ aGS ik (ˆ o − b ) n | Φ bGS ik (ˆ o − b ) n | Φ bGS ik + M X n =1 e − inϕ (ˆ o − a ) n | Φ aGS ik (ˆ o − a ) n | Φ aGS ik (ˆ o +b ) n | Φ bGS ik (ˆ o +b ) n | Φ bGS ik (cid:27) , (41)where | Φ aGS i and | Φ bGS i are exact copies of the ground state | Φ GS i of the Hamiltonian (1) for asingle system with N particles.To evaluate the expectation value h Θ | e iϕ ˆ a † ( x, a) ˆ a ( x, b) | Θ i , we first see that e iϕ ˆ a † ( x, a) ˆ a ( x, b) | Θ i = 1 √ M + 1 (cid:26) M +1 X n =1 e inϕ ˆ a † ( x, a) (ˆ o +a ) n − | Φ aGS ik (ˆ o +a ) n − | Φ aGS ik ˆ a ( x, b) (ˆ o − b ) n − | Φ bGS ik (ˆ o − b ) n − | Φ bGS ik + M − X n =0 e − inϕ ˆ a † ( x, a) (ˆ o − a ) n +1 | Φ aGS ik (ˆ o − a ) n +1 | Φ aGS ik ˆ a ( x, b) (ˆ o +b ) n +1 | Φ bGS ik (ˆ o +b ) n +1 | Φ bGS ik (cid:27) . (42)By using (41) and (42), we get h Θ | e iϕ ˆ a † ( x, a) ˆ a ( x, b) | Θ i = 12 M + 1 (cid:26) M +1 X n =1 (cid:10) (ˆ o − a ) n ˆ a † ( x, a) (ˆ o +a ) n − (cid:11) a (cid:10) (ˆ o +b ) n ˆ a ( x, b) (ˆ o − b ) n − (cid:11) b k (ˆ o +a ) n | Φ aGS ikk (ˆ o +a ) n − | Φ aGS ikk (ˆ o − b ) n | Φ bGS ikk (ˆ o − b ) n − | Φ bGS ik + M − X n =0 (cid:10) (ˆ o +a ) n ˆ a † ( x, a) (ˆ o − a ) n +1 (cid:11) a (cid:10) (ˆ o − b ) n ˆ a ( x, b) (ˆ o +b ) n +1 (cid:11) b k (ˆ o − a ) n | Φ aGS ikk (ˆ o − a ) n +1 | Φ aGS ikk (ˆ o +b ) n | Φ bGS ikk (ˆ o +b ) n +1 | Φ bGS ik (cid:27) , (43)where we wrote h· · · i ν = h Φ ν GS | · · · | Φ ν GS i . Note that translation invariance implies, e.g., h (ˆ o +a ) n ˆ a † ( x, a) (ˆ o − a ) n +1 i a = h (ˆ o + ) n +1 (ˆ o − ) n +1 i , where h· · · i = h Φ GS | · · · | Φ GS i and ˆ o ± := ˆ O ± /L d aredefined for the single system. We can thus rewrite (43) as h Θ | e iϕ ˆ a † ( x, a) ˆ a ( x, b) | Θ i = 12 M + 1 ( M +1 X n =1 (cid:10) (ˆ o − ) n (ˆ o + ) n (cid:11) (cid:10) (ˆ o + ) n (ˆ o − ) n (cid:11)q(cid:10) (ˆ o − ) n (ˆ o + ) n (cid:11) (cid:10) (ˆ o − ) n − (ˆ o + ) n − (cid:11) (cid:10) (ˆ o + ) n (ˆ o − ) n (cid:11) (cid:10) (ˆ o + ) n − (ˆ o − ) n − (cid:11) + M − X n =0 (cid:10) (ˆ o + ) n +1 (ˆ o − ) n +1 (cid:11) (cid:10) (ˆ o − ) n +1 (ˆ o + ) n +1 (cid:11)q(cid:10) (ˆ o + ) n (ˆ o − ) n (cid:11) (cid:10) (ˆ o + ) n +1 (ˆ o − ) n +1 (cid:11) (cid:10) (ˆ o − ) n (ˆ o + ) n (cid:11) (cid:10) (ˆ o − ) n +1 (ˆ o + ) n +1 (cid:11) ) , = 22 M + 1 M X n =1 s (cid:10) (ˆ o − ) n (ˆ o + ) n (cid:11) (cid:10) (ˆ o + ) n (ˆ o − ) n (cid:11)(cid:10) (ˆ o − ) n − (ˆ o + ) n − (cid:11) (cid:10) (ˆ o + ) n − (ˆ o − ) n − (cid:11) . (44)Following [12], let ˆ p = (ˆ o + ˆ o − + ˆ o − ˆ o + ) /
2. Since [ˆ o − , ˆ o + ] = L − d , one finds that (ˆ o − ) n (ˆ o + ) n =ˆ p n + O ( L − d ) and (ˆ o + ) n (ˆ o − ) n = ˆ p n + O ( L − d ). See Lemma 4.1 of [12]. We therefore get h Θ | e iϕ ˆ a † ( x, a) ˆ a ( x, b) | Θ i = 22 M + 1 M X n =1 h ˆ p n ih ˆ p n − i + O ( L − d ) . (45)9t was proved in Lemma 4.2 of [12] thatlim n ↑∞ lim L ↑∞ h ˆ p n ih ˆ p n − i = ( m ∗ ) . (46)This, with (45), implies lim M ↑∞ lim L ↑∞ h Θ | e iϕ ˆ a † ( x, a) ˆ a ( x, b) | Θ i = ( m ∗ ) , (47)which is the desired (31). A.2 Energy expectation value
We shall prove (32). Let E GS be the ground state energy of the Hamiltonian (1) for a singlesystem with N particles. From (10), which shows that | Γ M i has very low excitation energy, onereadily finds that lim L ↑∞ L d (cid:8) h Θ L,Mϕ | ˆ H tot0 | Θ L,Mϕ i − E GS (cid:9) = 0 (48)for any M . This is almost the desired (32) since it is very likely that E totGS , = 2 E GS . But thisequality cannot be proved in general and we need some work.Define the ground state energy density (for the single system) by˜ ǫ ( ρ ) := lim L ↑∞ E GS L d , (49)where L and N always satisfy ρ = N/L d . It is standard that the limit exisits, and ˜ ǫ ( ρ ) extendsto a convex function of ρ ∈ [0 , L ↑∞ E totGS , L d = min δ { ˜ ǫ ( ρ + δ ) + ˜ ǫ ( ρ − δ ) } , (50)where E totGS , is the ground state energy of ˆ H tot0 = ˆ H a + ˆ H b with total particle number 2 N , andwe again fix ρ = N/L d . But the right-hand side is equal to 2˜ ǫ ( ρ ) because of the convexity. Wehave thus proved that lim L ↑∞ L d (cid:8) E GS − E totGS , (cid:9) = 0 . (51)Then (48) and (51) imply the desired (32). References [1] F. Dalfovo, S. Giorgini, L.P. Pitaevskii, and S. Stringari,
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