Table Of Content

Landau superfluids as non equilibrium stationary states 4 1 0 2 Walter F. Wreszinski l u Instituto de Fisica USP J 2 Rua do Mataõ, s.n., Travessa R 187 05508-090 São Paulo, Brazil ] h [email protected] p - h t July 3, 2014 a m [ Abstract 3 v 5 We define a superfluid state to be a nonequilibrium stationary 8 state (NESS), which, at zero temperature, satisfies certain metastabil- 2 ity conditions, which physically express that there should be a suffi- 2 . ciently small energy-momentum transfer between the particles of the 1 0 fluid and the surroundings (e.g., pipe). It is shown that two models, 4 theGirardeaumodelandtheHuang-Yang-Luttinger (HYL)modelde- 1 scribe superfluids in this sense and, moreover, that, in the case of the : v HYLmodel,themetastability conditionisdirectlyrelatedtoNozières’ i X conjecture that, due to the repulsive interaction, the condensate does r not suffer fragmentation into two (or more) parts, thereby assuring a its quantum coherence. The models are rigorous examples of NESS in which the system is not finite, but rather a many-body system. 1 Introduction and Summary Superfluidity of a Bose fluid (e.g. Helium IV) remains an outstanding and fascinating theoretical problem (see the complementary reviews [Kad13] and [Leg99], as well as the book [Leg06]). In particular, Kadanoff in [Kad13] (see also [BP12]) recently rather sharply questioned the relevance of Landau’s 1 criterion([LL67],[HDCZ09], [WdSJ05])tothesuperfluid property. Thelatter may be roughly stated in the following way: by the flow of a fluid along a pipe, momentum may be lost to the walls, if the modulus of the velocity |~v| is greater than ǫ(p~) v ≡ min , (1.1) c ~p |p~| where ǫ(p~) are the energies of the “elementary excitations” generated by friction. This indicates that the concept of superfluidity still lacks a clear and precise theoretical foundation. One reason, argued in [Kad13], Kadanoff, is that “given the many mechanisms for broadening the distributions of both energy and momentum, it seems very implausible that a condition like (1.1) can begin to account for the very long-lived nature of the flow of superfluid Helium” — see also [BP12]. This argument is essentially supported by the results of the present paper (see remark 3). In the sequel, however, Kadanoff suggests [Kad13] that superfluidity is brought about by the existence of a coherent, “macroscopic” wave-function of type Ψ(~x) = exp iχ(~x) (1.2) ~ h i withχpossiblycomplex; seealsothemorecompleteanalysisof([Leg99],[Leg06]). This macroscopic wave-function is precisely the complex classical field oc- curring in the definition of ODLRO, and explains the two-fluid model and the London-Landausuperfluid hydrodynamics ([Lon54],[Kha65], [Lan41]; see also [MR04] for a nice textbook treatment, and [SW09], pg. 7, for a related discussion). Thus, ODLRO and the associated coherence properties of the condensate wave-function would suffice as the basis of the phenomenon of translational superfluidity, which would, then, not even require an interaction, at least conceptually, since the free Bose gas also exhibits ODLRO (see, e.g., [MR04], pg. 119). It seems to us, however, that a crucial element is missing in the above discussion, namely, the stability of the condensate, a point brought up em- phatically by Nozières in [‘er03]. Consider the situation in which one asks whether, instead of having at T = 0 a condensate strictly in the lowest energy state, one could fragment it into two states 1 and 2, of arbitrarily close energies in the thermodynamic limit, with populations N and N = N−N . 1 2 1 Naturally, only the potential energy is able to distinguish between these two 2 choices. Adopting a Hartree (mean-field) approximation with interaction Hamiltonian UN2 H = , (1.3) N,V 2V with U > 0, he suggests that the energy costs U[(N1+N2)2−N12−N22] = UN1N2, 2V V i.e., extensive in both N and N in the thermodynamic limit, where U > 0 1 2 is the strength of the interaction, assumed repulsive, and thus that it is the Coulomb interaction which interdicts the fragmentation, thereby assuring the condensate’s quantum coherence, with the corresponding wave-function becoming a macroscopic observable. His argument in the above form is not directly relevant to superfluidity, because it is easy to see that the mean- fieldinteractionHamiltonian(1.3)doesnotexhibitsuperfluidity (inLandau’s sense), seeSection3. Weshallshowinthatsection, however, thataneffective Hamiltonian for dilute systems — the Huang-Yang-Luttinger (HYL) model ([KHL57], [Hua87], [MvdBP]) — does reproduce Nozières’ heuristics in a precise sense. The homogeneous Bose gas in a cubic box with periodic boundary conditions (b.c.) at zero temperature has also been recently studied in a paper by Cornean, Derezinski and Zin [HDCZ09]. For the excitation spectrum of the weakly interacting Bose gas, see [Sei11]. Inorder thatthetheorydoesnotdependonthedetailsoffinitesystems in a box, but only onquantities which remain fixed inthe thermodynamic limit, weformulateourframeworktakingthisintoaccountinSection2. Wedefinea superfluid state to be a non-equilibrium stationary state (NESS) at zero temperature, satisfying certain metastability conditions, which physically express that there should be a sufficiently small energy-momentum transfer between the particles of the fluid and the surroundings (e.g., the pipe). In Section 3 it is shown that two models, the Girardeau model [Gir60] and the Huang- Yang-Luttinger (HYL) model [KHL57] describe superfluids in this sense and, moreover, that, in the case of the HYL model, the metastability condition is directly related to the previously mentioned Nozières’ conjecture. The models are rigorous examples of NESS in which the system is not finite, but rather a many-body system. Our results do not rely on any assumptions on states of infinite systems. 3 2 The general framework 2.1 General considerations The Landau condition (1.1) is supposed to lie at the bottom of a stability condition for the system in uniform motion with respect to a fixed, arbitrary inertial system. Although Landau formulated the latter in terms of ”elementary excitations”, we propose, along the lines of [SW09], to regard it in a preliminary step as a spectral condition imposed on a Hamiltonian describing a finite system. In order that this condition behaves smoothly when the thermodynamic limit is taken, it is necessary to consider, in some sense, the ”Hamiltonian of an infinite system”, and not just the thermodynamic limit of the ground state energy. A standard way of doing this is to consider the time dependent Green’s functions [Bay69]. A fundamental requirement for this, in a rigorous approach, is to define an (algebra of) observables ”of the infinite system”. We consider Bosons in translational motion, to begin with in finite re- gions,which we take to be cubes, generically denoted by Λ. The thermodynamic limit will be taken along the sequence of cubes Λ = [−nL,nL]d of side L = 2nL with n = 1,2,3,··· (2.1) n n where d is the space dimension. The number L is arbitrary, and the sequence {n = 1,2,3,···}maybereplacedbyanysubset oftheset ofpositiveintegers. Let H = L2 (Λ ) for j = 1,2,3,··· (2.2.1) Λj per j denote the Hilbert space consisting of functions on Rd, with f ∈ L2(Λ ) and j such that f is periodic with period L = 2jL with j = 1,2,3,··· in each of j the variables ~x = (x ) with i = 1,··· ,d, i.e., f(x + 2jL) = f(x ) with 1 = i i i 1,··· ,d and ~x = (x ),i = 1,··· ,d. Let i F denote the symmetrical Fock space over H with j = 1,2,3,··· Λj Λj (2.2.2) We shall sometimes consider F = ⊗ H with j = 1,2,3,··· (2.2.3) Λj,Nj s,N Λj , the symmetrized tensor product of H corresponding to N particles in Λj j Λ . Our local algebras of observables will be taken as the (von Neumann) j 4 algebras of bounded operators on F (or on F where N /|Λ | = ρ, ρ is Λj Λj,Nj j j the density and |Λ| denotes the volume of Λ): A = B(F ) (2.3) Λj Λj By the choice (2.1) and (2.3) we have the isotony property A ⊂ A for k < l or Λ ⊂ Λ (2.4) Λk Λl k l Indeed, a general element of H may be written f(~x) = c exp(2π~n· Λk ~n∈Zd ~n ~x/L ) (or more precisely as a limit in the L2 topology ofPfinite sums), with k |c |2 < ∞. Since functions with a given period are always periodic ~n∈Zd ~n Pwith a larger period, but not conversely, if k < l or Λ ⊂ Λ , f is also an k l element of H , but, not conversely, so that the inclusion is strict. By the Λl Riesz lemma, the same happens for the algebras A . No changes in this Λk argument arise on passing to F or finally to F . Defining Λj,Nj Λj A = ∪ A (2.5.1) L Λk Λk as our local algebra, the norm-closed algebra A = A¯ (2.5.2) L is a C* algebra called the quasi-local algebra of the infinite system, where the bar denotes the C*-inductive limit [KR86], Proposition 11.4.1): the isotony property (2.4) is crucial here. Since the local agebras are separable, because the Hilbert spaces H are separable, and the dual B(H ) is isomorphic to Λj Λj H bytheRieszlemma, thequasi-localalgebracontainsacountable(norm-) Λj dense set (upon approximation by a suitable A for sufficiently large n). Λn The necessity of this construction arises because for us it is imperative to define a momentum operator for the finite system, which requires some space translation invariance. Making the system into a torus is already a form of rending it ”infinite” : the periodic b.c. are of global nature. For this reason, the isotony property does not hold in general, and a special choice such as (2.1) had to be made. If, however, L is irrational, the sequence x = nL with n = 1,2,3,··· is uniformly distributed modulo 1 by Weyl’s n criterion ([KN74], Example 2.1), so that the above special choice does not seem to imply any essential loss of generality. In Λ (now generally simply called Λ) we consider a generic conservative j system of N identical particles of mass m. In units in which ~ = m = 1, the 5 generator of time translations, the Hamiltonian, will be denoted by H and Λ the generator of space translations, the momentum, will be denoted by P~ . Λ As operators on F they take the standard forms Λ − N ∆ H = r=1 r +V(~x ,...,~x ) (2.6) Λ P2 1 N with V a potential satisfying V ≥ 0 (2.7) i.e., only repulsive interactions will be considered, and N P~ = −i ∇ (2.8) Λ r X r=1 with usual notations for the Laplacean ∆ and the gradient ∇ acting on the r r coordinates of the r-th particle. We assume that H and P~ are self-adjoint Λ Λ ~ operators acting on F , with domains D(H ) and D(P ), and Λ,N Λ Λ D(P~ ) ⊃ D(H ). (2.9) Λ Λ For “sufficiently regular” potentials in (2.6), (see [MR04], and for precise conditions, ([RS78], Chapter XIII.12 ), as well as for the models we shall treat, H has a unique ground state (g.s) Ω , i.e., Λ Λ H Ω = E Ω , (2.10.1) Λ Λ Λ Λ where E ≡ infspec(H ). (2.10.2) Λ Λ Unicity of the ground state and space-translation invariance of the Hamilto- nian (2.6) imply that P~ Ω =~0. (2.10.3) Λ Λ We shall consider the so-called physical hamiltonian, normalizing the g.s. energy to zero, i.e. H ≡ H −E (2.11.1) Λ Λ Λ which is therefore positive: e H ≥ 0 (2.11.2) Λ e 6 By thermodynamic stability, E ≥ −c|Λ|, (2.12) Λ where |Λ| is the volume of Λ, and c is a positive constant. In general, E Λ is of order of O(−d|Λ|) for some d > 0, and in order to obtain a physical Hamiltoniansatisfying (2.11.2)it isnecessary to performthe renormalization (2.11.1) (infinite in the thermodynamic limit). For positive Hamiltonians with repulsive potential (2.7), this is also necessary, otherwise the spectrum would tend to (plus) infinity in the thermodynamic limit. For instance, in the Girardeau model E = (N−N−1)(πρ)2 where E denotes the ground N,L 6 N,L state energy of N particles in a periodic box of length L and ρ = N is the L density. Note that the renormalization (2.11.1) is also physically imperative, because we shall be concerned with the spectrum of H in a neighborhood Λ of the ground state. Of course, the operators in the Heisenberg picture are not affected by the renormalization (2.11.1). Let,now, Sd ≡ 2π~n |~n ∈ Zd (2.13.1) Λ L (cid:8) (cid:9) and, given ~v ∈ Rd, let lim ~ ~v =~v = k (2.13.2) L ~nL,L ~nL,L such that ~ ~ |k −~v| = inf |k −~v | ~nL,L lim ~k∈SΛd (2.13.3) ~ If there is more than one k satisfying (2.13.3), we pick any one of them. ~nL,L We have: lim ~v =~v , (2.14) L lim N,L→∞ where N,L → ∞ will be always taken to mean the thermodynamic limit, whereby N N → ∞, L → ∞, = ρ with 0 < ρ < ∞, (2.15) Ld 7 where ρ is a fixed density. The unitary operator of Galilei transformations appropriate to velocity ~v is given by L U~v ≡ exp(i~v ·(~x +...+~x )) . Λ L 1 N (2.16) We shall assume (2.9) and that U~v maps D(H ) into D(H ). From now on Λ Λ Λ we shall write ~v for ~v : the notation (2.14) assures that no confusion arises. L It follows from (2.16), (2.6) and (2.8) that, on D(H ), Λ (U~v)†H U~v = Λ Λ Λ N(~v)2 H +~v ·P~ +e , Λ Λ 2 e (2.17) which is the expression of Galilean covariance. In addition to the standard model (2.6), we shall also consider effective Hamiltonians, for dilute Bose systems, which we define as follows. Assume that the limit e(ρ) ≡ lim L−dE , (2.18.1) Λ N,L→∞ exists, where E is the ground state energy defined in (2.10.2), and a denotes Λ the scattering length corresponding to the potential V in (2.6), which is also a measure of the interaction range; the mean particle distance is ρ−1/3, now with d = 3. It has been made plausible (see, e.g., [Hua87], pg. 231) that at extremely low temperatures it is possible to describe a weakly interacting Bose gas only in terms of three parameters, λ, ρ−1/3 and a, where λ is the thermal wavelength. In the zero temperature limit we are reduced to ρ−1/3 and a, and, if a ≪ ρ−1/3, or ρa3 ≪ 1 (2.18.2) we speak of a dilute Bose gas. It has been conjectured that e(ρ) has an asymptotic expansion (see [Yin12] and references given there): k e(ρ) = e(j)(ρ)+O((ρ)1/2)e(k)(ρ) (2.18.3) X j=1 as ρ → 0, in (ρ)1/2, with e1(ρ) = 4πaρ2 (2.18.4) 8 and 128(ρa3)1/2 e2(ρ) = e1(ρ) . (2.18.5) 15 (π) p We shall refer to the k-th approximant in (2.18.3), as approximant of order (k) and to k as order of the approximation. Several rigorous results sup- port (2.18.4), it was rigorously proved by Lieb and Yngvason in a seminal paper [LY98] that e(ρ) = 4πa(ρ)2+o(a(ρ)2), and various rigorous additional results are given and reviewed in [Yin12], to which we also refer for the references. The famous second order correction, (2.18.5), was first conjectured by Lee and Yang in 1957 [LY57], on the basis of the pseudopotential approximation of Lee, Huang and Yang [TDLY57], and the binary-collision expansion method [Hua87]. Definition 1WecallaHamiltonianH aneffective Hamiltonian toorder Λ k for the dilute Boson system, if it satisfies both (2.18.6), (2.17) (Galilean covariance) and space-translation invariance. In particular, if the ground state is unique, as in the HYL model, (2.10.3) also holds. The article ”an” in definition 2.1 expresses the fact that the definition does not specify a unique Hamiltonian but rather a family of Hamiltonians, which, unlike (2.6), depend explicitlt on N,L and are therefore less fundamental (hence the name ”effective”), but do retain the basic physical symme- tries (this latter fact is explicitly used in proposition 2 on the HYL model). We may therefore hope that effective Hamiltonians provide an approximate description of the physical properties of a dilute system. Asweshallsee, theBogoliubovapproximationdoesnotdefine aneffective Hamiltonian according to the above definition, because it lacks Galilean covariance. The pseudopotential approximation does define an effective Hamil- tonian: for j = 1, see Section 3.2, for j = 2, see the conclusion. We shall henceforth assume that the ground state Ω is unique. Λ From (2.17) we may be led, as in [SW09], to ask whether the Hamiltonian H = H +~v ·P~ (2.19.1) ~v,Λ Λ Λ f is the one appropriate to describe the Bose fluid in uniform motion with velocity ~v. By (2.17), N(~v)2 H ≥ −∆E (Λ) = − . (2.19.2) ~v,Λ ~v 2 9 On the other hand, by (2.17), H = (U~v)†H U~v ~v,Λ Λ Λ Λ −∆E (Λ), e~v (2.19.3) from which spec(H ) = spec(H )−∆E (Λ) ~v,Λ Λ ~v e (2.19.4) follows. By (2.19.4), −∆E (Λ) ∈ spec(H ), which, together with (2.19.2), ~v Λ implies that e −∆E (Λ) = infspec(H ). (2.20) ~v ~v,Λ Incidently, this proves that the Bogoliubov approximation (BA) is not Galilean covariant: this is a consequence of the fact that, in the BA, the operator H is non-negative for |~v| sufficiently small (see [ZB01]), contra- ~v,Λ dicting (2.20), which was derived on the basis of Galilean covariance. By (2.20), one might be led to consider the vector Ψ corresponding ~v,Λ to the lowest eigenvalue (2.20), and the corresponding state hΨ ,·Ψ i, ~v,Λ ~v,Λ with the renormalization H → H + ∆E (Λ) ≥ 0 as describing the ~v,Λ ~v,Λ ~v ground state of the Bose fluid in motion with velocity ~v, which would yield e an equilibrium state in the thermodynamic limit N,L → ∞. We have, however, the following result. Lemma 1 U−~v|Ω i is the unique eigenvector of H corresponding to Λ Λ ~v,Λ the eigenvalue −∆E (Λ). ~v Proof Apply (2.19.3) to (U~v)†|Ω ) = U−~v|Ω ). Λ Λ Λ Λ Lemma 1 shows that Ψ = U−~v|Ω i represents a state of the Bose fluid ~v,Λ Λ Λ in uniform motion with velocity −~v. Our aim is, however, to look at the system, initiallyinanequilibrium(ground)state, |Ω ), whensetintouniform Λ motion with velocity ~v, not −~v, or, alternatively, to look at it from the point of view of a moving observer with the opposite velocity −~v : by (2.17), (2.11.1) and (2.10.3), its initial energy will be the kinetic energy ∆E (Λ) ~v associated to this velocity field. According to Landau, in order to detect frictionin this system, one should compare this quantity with the eigenvalues (or ”elementary excitations”) of the operator U~v)†H U~v, which, again by by Λ Λ Λ e 10