An explicit large deviation analysis of the spatial cycle Huang-Yang-Luttinger model

Here we introduce a family of spatial cycle HYL-type models in which the counter-term only affects cycles longer than some cut-off that diverges in the thermodynamic limit. We derive large deviation principles and explicit pressure expressions for these models, and use the zeroes of the rate functions to study Bose-Einstein condensation.


Introduction
Since London's proposal [Lon38] that the super-fluid phase-transition in He 4 is an example of Bose-Einstein condensation (BEC), it has been of interest to know how, in theory, interaction potentials affect the condensation of bosons. London himself conjectured that the momentum-space condensation of bosons is enhanced by spatial repulsion between particles. To address this question, Huang, Yang and Luttinger [HYL57] introduced a model (HYL) of a bosons with a hard-sphere repulsion which displays enhanced condensation. The Hamiltonian for that model is given in terms of the energy occupation counts (number of particles occupying an eigenstate), and this version has been studied by the Dublin group in the early 1990s [BLP88,BDLP90]. In particular, they show that the density of zero-momentum condensation (ground state energy) in momentum space is a function of the interaction parameters, that is, the mean field parameter and the parameter of the negative counter term in the Hamiltonian.
In this paper we study a corresponding model of spatial cycles with HYL-type interaction and we analyse the condensation of parts of the system into 'infinitely long cycles'. The model is given by so-called bosonic cycle counts λ Λ = (λ k ) k∈N where |Λ|λ k denotes the number of cycles of length k in finite volume Λ R d . Each cycle of length k is a closed continuous path with time horizon [0, kβ] and describes k bosons. The bosonic cycle counts are random numbers and their probability weights are obtained from a spatial marked Poisson point process, see [ACK11,AD18]. For the spatial cycle version of the HYL model one needs only the cycle counts and the detailed geometric structures of the cycles do not enter the equation. The Hamiltonian of the spatial cycle HYL model in Λ R d with interaction parameter a > b ≥ 0 and chemical potential µ ∈ R is defined as results below are different for the three possible regimes where either κ = 0, κ ∈ (0, ∞), or κ = ∞.
Our main results is a representation of the thermodynamic limit of the pressure via a large deviation principle for the pair empirical particle density M Λ = ( m Λ −1 k=1 kλ k , k≥m Λ kλ k ), see Theorem 2.2. Both the rate function and the pressure depend on the parameter κ ∈ [0, ∞] characterising the scale on which the counter term cycle lengths grow to infinity. The pressure is given as a variational problem (2.2) over R 2 , equal to the representation given in [BLP88,BDLP90], where now the second entry refers to the density of particles in 'infinitely long cycles'. We obtain all the zeroes for the rate functions for all parameter regimes and identify critical parameter with two distinct zeroes, see Theorem 2.5. This analysis provides the pressure representation along with the phase transition (BEC) as well as the density of 'infinitely long cycles' in the phase transition regime as a function of the chemical potential and the interaction parameter a > b. The relevant parameter is the chemical potential µ in the Hamiltonian (1.1) and the net chemical potential is µ + α, and we have some control on the temperature (β) dependence in Theorem 2.7 leading to phase diagrams in the (µ − β) -plane, see Figure 1. Finally, we compare the density of 'infinetely long cycle' given by the zeroes of the rate functions in Corollary 2.10 with density order parameter defined as the double limit, thermodynamic limit for the density in cycles of length above a lower followed by taking the lower bound to infinity, see Theorem 2.11. We find that the densities equal as long as κ ∈ [0, ∞) whereas for κ = ∞ the counter terms diverge too quickly to infinity to lead to any non-vanishing density of 'infinitely long cycles', see Remark 2.12.
In previous study of the momentum-space model the relevant occupation number is the lowest index opposite to our spatial cycle model, e.g., in [BLP88] the authors need a technical constraint for the counter term up to an index growing to infinity slower than the volume. This technical gap has been closed in [BDLP90] where the all counter term indices are incorporated. However, in those cases for the momentum-space model, the higher indices are not relevant for the zero-momentum condensation, whereas for our model the precise asymptotic of counter terms towards 'infinitely long cycles' play a crucial role.
We define the probability weights for the bosonic cycle counts in Section 1.1, and show in Section 1.2 and Appendix A how these weights are obtained from a marked Poisson point process description of the ideal Bose gas with free and Dirichlet boundary conditions. Both section can be skipped by the reader at first reading. In Section 2 we present all our results along with Figure 1 of the phase diagram, and in Section 3 we present all our proofs.
1.1. Probability weights. The ideal Bose gas at thermodynamic equilibrium with inverse temperature β > 0 and chemical potential α < 0 defines the probability weights for the bosonic cycle counts when there is no interaction. The cycle counts themselves are actually random functions of marked point configurations ω ∈ Ω as we will describe in more detail below in Section 1.2, that is, λ Λ = λ Λ (ω). For x = (x k ) k∈N ∈ 1 (R + ) with |Λ|x k ∈ N 0 , we denote by Q the probability distribution of the reference process, that is, the probability that the bosonic cycle counts are equal to x is given by As shown in [AD18], the thermodynamic pressure of the ideal Bose gas is given as The spatial cycle HYL model is given by the Gibbs distribution P Λ in Λ defined via its Radon-Nikodym density with respect to the reference process Q Λ in Λ, A vital ingredient in our result is the free energy of the ideal Bose gas given as the Legendre-Fenchel transform of the pressure, and, as the pressure p 0 (β, α) is differentiable for α < 0 and diverges for α > 0, we obtain the version of the free energy derived in [Ada08].
1.2. Reference process. We define the underlying marked Poisson point reference process in the following as background information, since our results depend solely on the measure P Λ defined above.
The reference process is a marked Poisson point process, see [ACK11,AD18]. The space of marks is defined as where, for k ∈ N, we denote by C k,Λ the set of continuous functions f : [0, kβ] → R d satisfying f (0) = f (kβ) ∈ Λ, equipped with the topology of uniform convergence. We call the marks cycles. By : E → N we denote the canonical map defined by (f ) = k if f ∈ C k,Λ . We call (f ) the length of f ∈ E. We consider spatial configurations that consist of a locally finite set ξ ⊂ R d of particles, and to each particle x ∈ ξ we attach a mark f x ∈ E satisfying f x (0) = x. Hence a configuration is described by the counting measure ω = x∈ξ δ (x,fx) on R d × E.
Consider on C k,Λ the canonical Brownian bridge measure Here B = (B t ) t∈[0,kβ] is a Brownian motion in R d with generator ∆, starting from x under P x . Then µ (kβ) x,y is a regular Borel measure on C k,Λ with total mass equal to the Gaussian density, Let ω P = x∈ξ P δ (x,Bx) be a Poisson point process on R d × E with intensity measure equal to ν, whose projection onto R d × C k,Λ is equal to Alternatively, we can conceive ω P as a marked Poisson point process on R d , based on some Poisson point process ξ P on R d , and a family (B x ) x∈ξ P of i.i.d. marks, given ξ P . The intensity of ξ P is q (α) (1.2). Denote Q denote the distribution of ω P and denote by E the corresponding expectation. Note that our reference process is a countable superposition of Poisson point processes and, as long as q (α) < ∞ is finite, this reference process is a Poisson point process as well. The bosonic cycle counts are averages of the random number of points in Λ with mark length equal to k, i.e., and λ k (ω) = N k (ω)/|Λ|.

1.3.
Notations. Throughout the whole text when we write Λ ↑ R d we mean the limit for a sequence of centred finite-volume boxes with unbounded volume. Furthermore, we write f (λ) ∼ g(λ) whenever lim λ→∞ f (λ) g(λ) = 1.

Results
Our main results concern a complete large deviation analysis and pressure representation of the spatial cycle HYL-model. Recall that the partition function Z Λ (β, α) for any Hamiltonian, H, of marked point configurations can be written as an expectation with respect to the reference process. This follows using the the Feynman-Kac formula for traces with symmetrised initial and terminal conditions (see [BR97,AK08]). AD18]). Let β ∈ (0, ∞), α < 0 and Λ R d . Assume that H Λ : Ω → R is measurable and bounded from below. Then, The Hamiltonian of the spatial cycle HYL model is given in (1.1) as a function of the bosonic cycle counts λ(ω) ∈ 1 (R + ).
The thermodynamic pressure for the spatial cycle HYL-model is where the expectation is with respect to the reference Poisson process and the ideal Bose gas pressure, p 0 (β, α), is given by (1.3), see [AD18]. The limit on the right hand side of (2.1) is obtained via a large deviation analysis for the pair empirical particle density M Λ : Ω → R 2 , distributed under the measure µ (2) Λ := P Λ • M −1 Λ with P Λ being the distribution of the Gibbsian point process of the spatial cycle HYL-model in Λ, i.e. dP Λ dQ Λ = e −βH Λ /E e −βH Λ . Theorem 2.2 (Large deviation principles and pressure representation). Let β > 0, α < 0, Then the sequence of measures (µ (2) Λ ) Λ R d satisfies the large deviation principle on R 2 + with rate β|Λ| and good rate function . This is due to the sequence of counter terms forcing the density in the second component of the pair empirical density to be either zero or greater or equal to κ. Λ are 0 and m Λ |Λ| . This case then forces all non-zero permitted values of the cycle counts to diverge to infinity, and thus they are naturally unlikely, and the 'greater than m Λ ' states aren't allowed to support any particle mass. Therefore the spatial cycle HYL-model in this case is essentially equal to the particle mean field model studied for example in [BCMP05,AD18].
(c) The empirical cycle counts (λ Λ ) Λ R d under the reference process satisfy, for any d ≥ 1 and α < 0, a large deviation principle with rate β|Λ| and rate function see [AD18]. This large deviation analysis require to deal with the lack of continuity of the particle density D(x) = k∈N kx k for x ∈ 1 (R), see similar problems in [ACK11]. The splitting into the pair empirical density allows to obtain the upper and lower bounds in our large deviation principles separately with continuous Hamiltonian functions. The large deviation principles in Theorem 2.2 provide more explicit representations of the pressure allowing a more precise phase transition analysis below in Theorem 2.5.
The pressure representations on the right hand side of (2.2) and their analysis is the key step in revealing the phase transition phenomenon also known as Bose-Einstein condensation.
Corollary 2.4. The right hand side of (2.2) can, for κ = 0, be written as where I 0 is the rate function for the ideal Bose gas and D(λ) = k∈N kλ k is the density, see [AD18].
Proof. The first equality follows from optimising over y ≥ 0, whereas the second can be proved by conditioning on the value of D(λ).
The analysis of the pressure is related to finding the zeroes of the rate functions. In the following theorem we identify thus a critical parameter regime for the pressure in terms of the so-called chemical potential µ ∈ R. This analysis is followed by showing that the critical parameter is the phase transition point when positive particle density is carried by so-called 'infinitely' long cycles, see Theorem 2.11 below. Recall the critical density of the ideal Bose gas, see [Ada08,AD18], For the following results, we define the function s β : (0, ∞) → R by Here and thereafter we let p 0 (β, s) and f 0 (β, x) denote the s and x partial derivatives respectively.
The zeroes in the previous theorem lead immediately to the following pressure representations We define the sub-critical and the super-critical pressure respectively as Corollary 2.6 (Pressure representation). For κ = 0 we have and for κ ∈ (0, ∞), for µ ≥ max{µ r , µ * } , We have some control over the transition potentials µ * κ , κ ∈ [0, ∞), in the following theorem. Denote Theorem 2.7. For d = 2 and all β > 0, Otherwise we describe the high and low temperature behaviours.
Low temperature: High temperature: Furthermore, for κ ∈ (0, +∞), we have A second conclusion from our results above is the asymptotic limit for the pair empirical particle density.
Corollary 2.8 (Thermodynamic limit of the pair particle density). For every δ > 0, for each unique minimiser (x, y) in Theorem 2.5, that is, when either µ = µ * in case κ = 0, µ = µ * κ for κ ∈ (0, ∞), and for all µ ∈ R when κ = ∞. (b) Our analysis above shows that at the critical values of the chemical potential, µ = µ * for κ = 0 and µ = µ * κ when κ ∈ (0, ∞), we do not have a unique zero. Our large deviation analysis shows that for every δ-neighbourhood Denoting (x (i) c , y (i) c ), i = 1, 2, the zeroes at these critical points, the concentration of measure problem ask whether there are λ i ∈ [0, 1], i = 1, 2, such that λ 1 + λ 2 = 1 and This requires finer asymptotic analysis going beyond the large deviation analysis studied here. We devote future work to analyse the fluctuation behaviour of our model, e.g. similar to the analysis in [CD14] for the ideal Bose gas, as well as the concentration of measure problem at criticality.
A third observation from our results above is that the expected density of particles in unbounded cycles is related to the expected density of Corollary 2.10 (Density in 'infinitely long cycles'). (i) κ = 0, for µ > µ * , and (κ) (µ) is continuous at µ = µ r and discontinuous at µ = µ * κ .
The reader maybe be tempted to identify the expect density (κ) (µ) as the density of particle in unbounded ('infinitely long cycles') cycles which may corresponds to the density of the Bose-Einstein condensate. To answer this conjecture we first define the condensate density, a definition which goes back to [Gir60], and which is frequently used, e.g. in [Lew86,BLP88,BDLP90].
For any K ∈ N define D K as the random particle density of cycles with length greater than K, Then the particle density in infinitely long cycles is defined as Theorem 2.11 (Condensate Density). (a) Let β > 0, α < 0, µ ∈ R, and a > b > 0. For κ = 0 and κ = ∞, Remark 2.12. Since m Λ > K eventually, it follows that ∆ (κ) (µ) ≥ (κ) (µ). However, if κ > 0 we have parameter ranges in which this inequality is strict. This indicates that a positive condensate density is held on cycles whose lengths diverge slower than m Λ , and there therefore not directly affected by the counter-term. In the κ = ∞ case, m Λ diverges too quickly to affect any positive density, see Remark 2.3, and we approach the mean field model. The case κ = ∞ is special in the sense that the density of 'infinitely long cycles' as the zero of the rate function vanishes but the above order parameter does not vanish when µ ≥ a c .

Proofs
The proof of the main large deviation theorem is in Section 3.1, and all remaining proofs about the zeroes of the rate function and the condensate density are in Section 3.2.
3.1. Proof of the large deviation principles, Theorem 2.2. To prove the large deviation principle in Theorem 2.2, we adapt and extend the ideas in [BLP88]. Our model is on spatial cycle structures and thus our method is different as the so-called condensate resides in cycles of unbounded length. For the negative counter term in the Hamiltonian the long cycles are the relevant ones, whereas in [BLP88] the energy indices with low values are relevant. Though technically slightly more challenging, our method allows us to investigate the different ways the counter term scales with the volume -in this way providing insight how the cycle condensate is scaled with the volume. The standard Varadhan Lemma approach for our model does not work directly as, due to the counter term, lower semicontinuity is missing. The general idea is to find lower and upper bounds for the Hamiltonian, and then prove large deviation principles for the two bounds individually. The final step is then to identify the two bounds. The proof for the upper bound is in Section 3.1.1 and the one for the lower bound in Section 3.1.2.
3.1.1. Large Deviation Upper Bound. We split the proof in two parts. In the first one we derive the large deviation principle for pair empirical particle density under the reference measure, i.e., with no interaction. In the second step we will apply Varadhan's lemma to a lower bound of the Hamiltonian. Proof.
Given the properties of the weights q k , we immediately see convergence of the first term on the right hand side for any value of κ This gives divergence for s < −α or t > −α. The Legendre -Fenchel transform is then
(3.1) and can easily obtain the large deviation upper bound.
Lemma 3.2. For any closed set C ⊂ R 2 , Proof. The upper bound for closed sets C ⊂ R 2 , follows for example with Baldi's Lemma (see [DZ09]). The exponential tightness follows from the fact that the finite domain of Λ * contains a neighbourhood of the origin. We improve this bound for (x, y) ∈ K (κ) by noting that the image of Λ ∈ E = 0 for all Λ sufficiently large.
Step 2: The crucial step is to bound Hamiltonian in (1.1) from below. We have Proof.
The function H in (3.2) is continuous and bounded below, thus the statement is simply a matter of applying Varadhan's Lemma. Taking C = R 2 gives the bound on the thermodynamic pressure by using the Legendre-Fenchel transform for the pressure.
In Section 3.1.2 we obtain the corresponding lower bound on the thermodynamic pressure, i.e., confirming (2.2). Then Lemma 3.3, in conjunction with the pressure representation in (2.2), gives the large deviation upper bound 3.1.2. Large Deviation Lower Bound. The large deviation lower bound is more delicate as we shall find an upper bound on the energy, that is, the counter term cannot be replaced by the square of the sum of the single terms. In the following steps we derive a more detailed splitting of the empirical particle density which is the novel step in this type of large deviation proofs. Our splitting is based on properties derived for the pair empirical particle density splitting in Section 3.1.1. The rate function J κ , defined in (3.1), is a good rate function which follows from the fact that the origin is in the interior of the domain where the limiting logarithmic moment generating function is finite as well as the fact that J κ is convex and lower semi-continuous. We combine this with the local nature of the large deviation lower bound, that is, we will show that for any (x, y) ∈ R 2 , lim δ↓0 lim inf where B δ ((x, y)) is the open ball of radius δ > 0 around (x, y). In the first step we define the detailed splitting which is local as it depends on (x, y). In the second step we prove a large deviation lower bound under the reference measure for the new splitting of the empirical particle into a quadruple, and, using the derived upper bound on the energy, the lower bound for the quadruple splitting follows with Varadhan's Lemma. In a final step we employ the contraction principle to derive the lower bound in (3.3).
Marginal ν (4) Λ,1 : The function Λ * 1 is strictly convex on [0, c ). For x ∈ [0, c ), we can proceed by standard Gärtner-Ellis type arguments by tilting the measure. Let η < 0 be the unique solution to ηx − p 0 (β, η) = sup s<0 {sx − p 0 (β, s)}, and define the tilted measure by where p Λ (s) := 1 β k<m Λ q k e βsk . Taking the limits, we obtain (3.4) Now, using the strict convexity and the Gärtner-Ellis upper bound (see [DZ09,Theorem 2.3.6]), the last term on the right hand side of (3.4) vanishes as ν (4) Λ,η,1 (B δ (x)) → 1 as Λ ↑ R d , and For x > c , the function Λ * 1 is no longer strictly convex, and thus the standard argument fails to show that the last term on the right hand side of (3.4) vanishes. Our new method here is to directly use the Poisson nature of the distribution of the bosonic cycle counts to obtain individual estimates. Pick x > c , and proceed as above to derive the lower bound in (3.4). As we cannot exploit the strict convexity of Λ * 1 , we need to estimate the last term on the right hand side of (3.4) directly: pick r ∈ N, then eventually m Λ > r. Denote η c := k∈N kq k e βηk , η c,r := k<r kq k e βηk , and let Q η be the distribution of the reference Poisson process with parameters q k e ηβk k∈N . Then for x > η c,r , Now the mean and variance of |Λ| r−1 k=1 kλ k are equal to |Λ| η c,r , so Chebyshev's inequality implies that Independence of the Poisson variables implies that For the remaining factor we use Stirling's formula to get . Combing the individual estimates and taking r → ∞ gives lim inf Since η < 0 can be chosen arbitrarily close to zero, we obtain (3.5) for all x ≥ 0.
Proof. Using (3.6), Then, noting that K is continuous, the statement is simply a matter of applying Varadhan's Lemma.
Step 4: Contraction. Proof. Defineπ : R 4 → R 2 , (x, y 1 , y 2 , z) → (x, y 1 + y 2 + z), and note thatπ is continuous and M Λ =π • π. Then, inf x=X,y 1 +y 2 +z=Y we derive the lower bound for the pressure in Lemma 3.6, and, with the corresponding upper bound in Lemma 3.3, we obtain (2.2) in Theorem 2.2. To obtain the large deviation lower bound, denoted the rate function per Contraction principle J κ (x, y) = infπ−1 (x,y) {J (4) κ + K}. Then, for every open set O ⊂ R 2 , using the lower in Lemma 3.5, and, finally we derive (3.3), where we use the particular quadruple splitting in the last step.
3.2. Variational analysis. Proof of Theorem 2.5. In [BLP88] a related variational principle has been studied. We built on that and extend it to the whole range of the parameter κ ∈ [0, ∞] to obtain an improvement on the results in [BLP88]. To obtain the zeroes of the rate function functions I (κ) it suffices to minimise F µ (x, y) = f 0 (β, x) − µ(x + y) + a 2 (x + y) 2 − b 2 y 2 for (x, y) ∈ K(κ), +∞ otherwise.
For κ = +∞, we have restricted the y-component to y = 0. Therefore we want to optimise F (x, 0). We search for stationary points, and find that they satisfy the equation s β (x) = µ − ax. Since s β is non-decreasing and s β (x) → −∞ as x ↓ 0, and µ − ax is decreasing, there exists a unique stationary point -a minima -given byx 1 .
For κ = 0, we first fix x ≥ 0 and find an optimal choice of y ∈ R + , denotedỹ (x). This gives y (x) = 1 a−b (µ − ax) + , and the derivative of F µ (x,ỹ (x)) is zero precisely when is always a solution and gives a local minima for µ ≤ µ c , whilst x =x 2 (µ) is a solution for µ ≥ µ t and gives a local minima for µ > µ t . These are the only local minima.
Proof of Corollary 2.6. These representations of the pressure follow from substituting the minimisers from Theorem 2.5 into the pressure expression (2.2) in Theorem 2.2.
Proof of Theorem 2.7. We shall use the notation g (u, s) = ∞ k=1 k −s e uk , see (B.1). First note that µ t = µ c if and only if the infimum (2.3) is attained at s = 0, which happens precisely when ap 0 (β, 0) ≤ a−b b . This condition happens if and only if d ≥ 5 and β ≥ β t . In all other cases, there existsᾱ < 0 such that Sinceᾱ is a stationary point, it is given as the unique solution to To deal with d = 2, note that from the infinite sum expression for g (u, s) that for u < 0 we have g (u, 0) = e u 1−e u , and g (u, 1) = − log (1 − e u ). We use these to solve forᾱ and then µ t . For the remaining dimensions, we use (3.7) to determine whether βᾱ → 0 or βᾱ → −∞. Using g (u, s) ∼ e u as u → −∞ for all real s, and the u ↑ 0 asymptotic behaviour of g (u, s) from Appendix B, we get the asymptotic behaviour of µ t (β).
For µ r (β, κ), note that when restricted to (0, c ), the function s β : (0, c ) → (−∞, 0) is a strictly increasing continuous bijection. Furthermore we can re-write From the definition of s β (x) for x ∈ (0, c ], we have Proof of Corollary 2.8. If (x, y) is the unique minimiser of the large deviation rate function I (κ) , then for δ > 0 we have inf I (κ) (u, v) : (u, v) ∈ B δ (x, y) > 0. Since the complement of the open ball B δ (x, y) is closed, the large deviation principle tells us that Q M Λ ∈ B δ (x, y) decays exponentially. Hence the required limit holds.
Proof of Corollary 2.10. This follows from applying Corollary 2.8 with the unique minimisers found in Theorem 2.5.