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Phase Transitions and Edge Scaling of Number Variance in Gaussian Random Matrices Ricardo Marino, Satya N. Majumdar, Grégory Schehr, and Pierpaolo Vivo Laboratoire de Physique Théorique et Modéles Statistiques (UMR 8626 du CNRS), Université Paris-Sud, Bâtiment 100, 91405 Orsay Cedex, France (Received 2 April 2014; published 26 June 2014) We consider N × N Gaussian pffiffiffi pffiffiffi random matrices, whose average density of eigenvalues has the Wigner semicircle form over ½− 2; 2. For such matrices, using a Coulomb gas technique, we compute the large N behavior of the probability P N;L ðN L Þ that N L eigenvalues lie within the box ½−L; L. This probability scales as P N;L ðN L ¼ κL NÞ ≈ exp ð − βN 2 ψ L ðκL ÞÞ, where β is the Dyson index of the ensemble and rate function ψ L ðκ L Þ is a β-independent pffiffiffi pffiffiffi that we compute exactly. We identify three regimes pffiffiffi as L is varied: (i) N −1 ≪ L < 2 (bulk), (ii) L ∼ 2 on a scale of OðN −2=3 Þ (edge), and (iii) L > 2 (tail). We find a dramatic nonmonotonic behavior of the number variance V N ðLÞ as a function of L: after a logarithmic growth ∝ lnðNLÞ in the bulk (when L ∼ Oð1=NÞ), V N ðLÞ decreases pffiffiffi abruptly as L approaches the edge of the semicircle before it decays as a stretched exponential for L > 2. This “dropoff” of V N ðLÞ at the edge is described by a scaling function V~ β that smoothly interpolates between the bulk (i) and the tail (iii). For β ¼ 2 we compute V~ 2 explicitly in terms of the Airy kernel. These analytical results, verified by numerical simulations, directly provide for β ¼ 2 the full statistics of particle-number fluctuations at zero temperature of 1D spinless fermions in a harmonic trap. DOI: 10.1103/PhysRevLett.112.254101
PACS numbers: 05.45.Mt, 02.10.Yn, 02.50.-r, 24.60.-k
There has been enormous progress in the last decade on the experimental manipulation of cold atoms [1,2], which generated several interesting theoretical questions concerning the interplay between quantum and statistical behaviors in many-body systems. These experiments are usually carried out in the presence of optical laser traps that confine the particles in a limited region of space. In particular, 1D systems, such as spinless fermions in presence of a harmonic trap, have played a crucial role in these recent developments [2–7]. One important observable that has been studied is the number of fermions N L in the ground state (T ¼ 0) within a given box ½−L; þL. The variance of N L , denoted by V N ðLÞ, characterizes the quantum fluctuations in the ground state of this many-body system. This quantity has also been studied recently in a number of other quantum systems, including several lattice models of fermions [8,9]. The variance V N ðLÞ as a function of the box size L turns out to be highly nontrivial even for the simplest possible many-body quantum system, namely 1D spinless fermions in a harmonic trap. In this case, it was numerically found that V N ðLÞ has a rather rich nonmonotonic dependence on L—it first increases with L and then drops rather dramatically when L exceeds some threshold value [4,7]. Analytically deriving this dependence on L is thus a challenging problem. In this Letter, exploiting a connection of this fermionic system at T ¼ 0 to the Gaussian Unitary Ensemble (GUE) of random matrices, we obtain, for large N, this variance V N ðLÞ exactly for arbitrary L, which explains its nonmonotonic behavior. In addition, using a Coulomb gas technique for random matrices, we are able to calculate the full probability distribution of N L in the large N limit. 0031-9007=14=112(25)=254101(5)
The ground state many-body wave function of N 1D spinless fermions in a harmonic potential UðxÞ ¼ ð1=2Þmω2 x2 pffiffiffiffiffi is ffi given by the Slater determinant Ψ0 ð~xÞ ¼ ð1= N!Þ det½φi ðxj Þ, where φn ðxÞ is the single particle harmonic oscillator wave function, φn ðxÞ ∝ 2 Hn ðxÞe−x =2 (we have set ℏ ¼ m ¼ ω ¼ 1), where Hn ðxÞ are Hermite polynomials. By explicitly evaluating this determinant, it is easy to see that 1 − PN x2i Y i¼1 jΨ0 ð~xÞj2 ¼ e ðxj − xk Þ2 ; ð1Þ ZN jj where ZN;β is a normalization constant and E½λ ¼ P P ðN=2Þ Ni¼1 λ2i − ð1=2Þ j≠k ln jλj − λk j. It is well known [10] that, for large N, the average density ρN ðλÞ of eigenvalues (normalized to unity) of a Gaussian random matrix approaches the celebrated semicircle law pffiffiffi Wigner’s pffiffiffi on p the compact support ½− 2; 2, ρN ðλÞ → ρsc ðλÞ ¼ ffiffiffiffiffiffiffiffiffiffiffiffi −1 π 2 − λ2 . The typical distance between eigenvalues is thus of order ∼ Oð1=NÞ near the center of the semicircle. The average hN L i for large N can be computed pffiffiffiffiffiffiffiffiffiffiffiffiffias RL hN L i ¼ N −L ρsc ðxÞdx ¼ Nκ ⋆L , where κ ⋆L ¼ ½L 2 − L2 þ pffiffiffi 2arcsinðL= 2Þ=π. The variance V N ðLÞ was also computed in [12,13], but only in the bulk limit, i.e., when L ∼ Oð1=NÞ (the box size is of the order of the interparticle spacing near the center). On this scale, setting L ¼ Δ=N, V N ðLÞ was shown to grow logarithmically with Δ, V N ðLÞ ∼ ð2=βπ 2 Þ lnðΔÞ, for Δ ≫ 1. In contrast, numerical simulation in the fermionic system shows a nonmonotonic behavior of V N ðLÞ as L increases beyond Oð1=NÞ. A natural question is then: can one calculate V N ðLÞ for all L? In this Letter we indeed compute V N ðLÞ for all L in the large N limit, which exhibits a striking “dropoff” effect near the semicircular edge (see Fig. 1). Our method also allows us to compute, for arbitrary L, the full PDF of N L for large N, p which was ffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi known to be a Gaussian but only on a scale Oð lnðNLÞÞ, in the regime L ∼ Oð1=NÞ [14–17]. Our results can be summarized as follows. We find that the number variance V N ðLÞ for an interval ½−L; L behaves as pffiffiffi 8 2 2 32 N −1 ≪ L < 2 > 2 ln ðNLð2 − L Þ Þ; βπ > < pffiffiffi 2 V N ðLÞ ∼ V~ β ðsÞ; ð3Þ L ¼ 2 þ psffiffi2 N −3 > > pffiffiffi : exp ½−βNϕðLÞ; L > 2; where the scaling function V~ β ðsÞ is computed explicitly in (12) for β ¼ 2—its asymptotic behaviors for generic β are given in (13)—and the function ϕðLÞ is given in (10) (in this third ∼ stands pffiffiregime ffi pffiffifor ffi a logarithmic equivalent). In (3), L≷ 2 means jL − 2j ≫ N −2=3 . We thus identify three qualitatively different pffiffiffiregimes −1 dependingpon the value of L: (i) N ≪ L < 2 (bulk), ffiffiffi (ii) L ∼ p2ffiffiffi, on a scale of OðN −2=3 Þ (edge), and (iii) L > 2 (tail). Moreover, we are able to obtain the full probability distribution P N;L ðN L Þ of N L , for large N.
FIG. 1 (color online). Number variance V N ðLÞ as a function of L. Theoretical result Eq. (3) in solid p blue ffiffiffi line. Inset: edge scaling behavior of the variance around L ∼ 2 (described pffiffiffi pffiffiffi by the scaling function V~ 2 ðsÞ, in (12), with s ¼ ðL − 2Þ 2N 2=3 ) together with numerical simulations for N ¼ 5000 and β ¼ 2 (and averaged over 30 000 matrices).
Calling κ L ¼ N L =N the fraction of eigenvalues in ½−L; L we obtain [18] P N;L ðN L ¼ κ L NÞ ≈ exp ð − βN 2 ψ L ðκ L ÞÞ;
ð4Þ
for 0 ≤ κ L ≤ 1, where the rate function ψ L ðκ L Þ is βindependent and can be explicitly computed in terms of single integrals (see Eq. (56) in [19]). The rate function ψ L ðκ L Þ is convex and has a minimum (zero) at κL ¼ κ ⋆L (see Fig. 3). Thus the distribution of N L is peaked around N L ¼ κ⋆L N which is precisely its mean value hN L i ¼ κ⋆L N for large N. This distribution has non-Gaussian tails and even near its peak in hN L i it exhibits an anomalous quadratic behavior which is modulated here by a logarithmic singularity. The probability distribution of N L —considered here for simplicity as a continuous variable—can be written by integrating the jpd (2) with a delta constraint P δðN L − Ni¼1 1½−L;L ðλi ÞÞ, obtaining [19] P N;L ðN L Þ ∝
Z Y N i¼1
Z dλi
dξ expð−βE½λ; ξ; N L Þ: 2π
ð5Þ
The energy E½λ; ξ; N L of a configuration fλg is then given P by E½λ;ξ;N L ¼E½λþðξ=2ÞðN L − Ni¼1 1½−L;L ðλi ÞÞ, where a Lagrange multiplier ξ is introduced to take care of the delta constraint. Written in this form, (5) is just the grandcanonical partition function of a 2D fluid of charged particles (the eigenvalues) confined to a line. The system is in equilibrium at inverse temperature β under competing interactions: a confining quadratic potential and a logarithmic all-to-all repulsion term. In addition, a fraction κL ¼ N L =N of particles is constrained within the box ½−L; L. Introducing the normalized density of eigenvalues
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PHYSICAL REVIEW LETTERS 27 JUNE 2014 PRL 112, 254101 (2014) P ρðλÞ ¼ N −1 Ni¼1 δðλ − λi Þ, one converts the multiple integral in (5) to a functional integral over ρ, which is then evaluated for large N using a saddle point method. This procedure, originally introduced by Wigner and Dyson [12,20], has been successfully employed in recent works on the top eigenvalue of Gaussian and Wishart matrices [21–25], conductance fluctuations in mesoscopic systems [26] or bipartite entanglement of quantum systems [27–29]. For the present problem, a similar Coulomb gas analysis was performed in Ref. [15] but it was restricted to the narrow regime L ∼ Oð1=NÞ. Here we explore pffiffiffi instead the full range L ≫ N −1 up to the edge, L ∼ 2, and beyond. Skipping details [19], the resulting expression for P N;L ðN L ¼ κL NÞ reads to leading order for large N Z β 2 FIG. 2 (color online). Phase diagram of the Coulomb gas (7) in P N;L ðN L ¼ κL NÞ ∝ D½ρdξdη exp − N S½ρ ; ð6Þ 2 the (κ , L) plane. The equilibrium density ρ⋆ ðxÞ, plotted in insets, L
where the action (depending on L and κL ¼ N L =N) is Z ZZ 2 S½ρ ¼ dxx ρðxÞ − dxdx0 ρðxÞρðx0 Þ ln jx − x0 j Z Z L þη dxρðxÞ − 1 þ ξ dxρðxÞ − κL : ð7Þ −L
Here η is another Lagrange multiplier enforcing the normalization of the density, and the first integrals run over ð−∞; ∞Þ. We now evaluate the integral (6) for large N with a saddle point method. Differentiating (7) functionally with respect to ρ and then with respect to x, we obtain a singular integral equation for ρ⋆ ðxÞ (depending on L and κ L ) Z ρ⋆ ðx0 Þ 0 dx ; x ∈ suppðρ⋆ Þ and x ≠ L; ð8Þ x ¼ Pr x−x0 where Pr denotes the Cauchy principal part, and suppðρ⋆ Þ the region on the real line where ρ⋆ ðxÞ R L > ⋆0. Equation (8) is to be solved with the constraints −L ρ ðxÞdx ¼ κ L and R∞ ⋆ ρ ðxÞdx ¼ 1. This integral equation (8) can be solved −∞ using the resolvent method [19]. The normalized density has generally a three-cut support (see Fig. 2), sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 ðx2 − b2 Þðx2 − a2 Þ ρ⋆ ðxÞ ¼ ; ð9Þ π L2 − x2 which is valid for x belonging to any of the intervals in the support. The value of the edges a and b are determined by 2 2 2 the relation R L ⋆ a þ b ¼ L þ 2 together with the constraint −L ρ ðxÞdx ¼ κL . For a fixed value of L, the equilibrium density will take three different shapes (see Fig. 2) according to the fraction κL of particles stacked in the box. If κL ¼ κ⋆L (solid blue line in Fig. 2), as many particles are stacked in the box as naturally expected from (5) without any constraint. Thus the equilibrium density is just the semicircle. For the cases κL > κ ⋆L (κ L < κ⋆L ), an excess of particles accumulates inside (outside) the box, giving rise to three disconnected
exhibits a transition between two distinct shapes as the critical solid blue line κ ⋆L is crossed, along which ρ⋆ is given by the Wigner’s semicircle density.
blobs and a divergence of the density ρ⋆ around the inner (outer) box walls (see Fig. 2). Evaluating (6) at the saddle point, we obtain a large deviation decay of the probability for large N of the form in Eq. (4), where the rate function is given by ψ L ðκ L Þ ¼ ð1=2Þ½S½ρ⋆ − S½ρsc where ρsc is the Wigner’s semicircle density. The second term comes from the large N behavior of the normalization constant ZN;β in (2) and needs to be subtracted. The rate function ψ L ðκ L Þ is therefore determined by the action (7) at the saddle point S½ρ⋆ . Its full expression is rather cumbersome (see [19]) but can be easily plotted as shown in Fig. 3 for two different values of L. In order to extract the variance V N ðLÞ from the rate function, we expand ψ L ðκL Þ around its minimum κL ¼ κ ⋆L . −1 We first Þ ffiffihas pffiffiffi notice that ψ L ðκLp ffi a different shape for N p≪ ffiffiffi L < 2 (bulk) and L > 2 (tail), while the edge L ∼ 2 needs a separate treatment. pffiffiffi The bulk N −1 ≪ L < 2.—In this case, ψ L ðκ L Þ is twosided, and setting κ L ¼ κ ⋆L − δ [19], we find that close to its minimum κ ⋆L , it behaves like ψ L ðκ L ¼ κ ⋆L − δÞ ∼ ðπ 2 =4Þδ2 = ln ðLð2 − L2 Þ3=2 =jδjÞ as δ → 0. Therefore, around the critical value κ ⋆L for “sufficiently large" Lð2 − L2 Þ3=2 ≫ jδj, the rate function is nonanalytic and displays a quadratic behavior modulated by a logarithmic singularity. The physical origin of this nonanalytic behavior is linked to a phase transition in the associated Coulomb gas when κ L crosses the critical value κ⋆L (see Fig. 2). Inserting this behavior (close to κ ⋆L ) into (4), using that δ ¼ κL − κ L ¼ Oð1=NÞ, we find that P N;L ðN L Þ has a Gaussian behavior around κ ⋆L N, with a variance growing as in Eq. (3) (first line), thus recovering Dyson’s bulk behavior away from the edge. This Gaussian limiting distribution is thus valid on a scale pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ∼Oð lnðNLð2 − L2 Þ3=2 ÞÞ around κ⋆L N. However, beyond
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FIG. 3 (color online). Behavior of the rate function ψ L ðκ L Þ as a functionpffiffiof ffi κL ∈ ½0; 1 for two different values pffiffiffi of L: L ¼ 0.6 < 2 (green, two-sided) and L ¼ 1.6 > 2 (red, one-sided). The solid blue line in the plane (L, κ L ) is the critical line κ⋆L , where ψ L ðκ L Þ has a minimum (zero).
this scale, the fluctuations of κ L N are instead described by the full large deviation function in Eq. (4) which has nonGaussian tails [19]. analysis holds in the bulk, pffiffiThis ffi pffiffiffi for a fixed N −1 ≪ L < 2p, ffiffibut breaks down for L ∼ 2 (edge) ffi and in the tail, L p> ffiffiffi 2. The tail, L > 2.—In this regime the widthp offfiffiffithe box is much larger than the semicircle sea, L > 2, and κ ⋆L freezes to the value 1 (see Fig. 2), since on average all the eigenvalues are contained within the box. Therefore the rate function ψ L ðκL Þ is one-sided. Setting κL ¼ 1 − δ and expanding ψ L ðκL Þ to leading order in δ > 0, one obtains a linear behavior [19] ψ L ðκ L ¼ 1 − δÞ ∼ ϕðLÞδ, as δ → 0, where pffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi ϕðLÞ ¼ L L2 − 2=2 þ ln ððL − L2 − 2Þ= 2Þ: ð10Þ This function ϕðLÞ turns out to be identical to the large deviation function describing the right tail of the top eigenvalue λmax , Prob:½λmax > L ≈ exp½−βNϕðLÞ. Here NϕðLÞ is the energy cost to pull out one particle at a distance larger than L away from the Wigner sea, while the density of the rest (N − 1) charges remains of the standard semicircular form [24]. To understand this connection with the right tail of λmax one can extrapolate our Coulomb gas calculation of ψ L ðκ L Þ to the case where there is a discrete number of particles outside the interval ½−L; L. Because our Coulomb gas calculation preserves the symmetry ρ ðxÞ ¼ ρ ð−xÞ, which is always true in the continuum limit, this extrapolation can only be done for an even number of particles, say two of them: one in the interval ð−∞; −L and the other one in the interval ½L; þ∞Þ, hence in this case δ ¼ 2=N. Therefore, the energy cost of such a configuration, given by βN 2 ψ L ðκ L ¼ 1 − 2=NÞ, is
precisely twice (as there are two particles) the energy pffiffiffi NϕðLÞ to pull out one charge, at a distance L > 2, outside the Wigner sea (ignoring correlation effects between the two particles). Hence N 2 ψ L ðκ L ¼ 1 − 2=NÞ∼ 2NϕðLÞ, in agreement with ψ L ðκ L ¼ 1 − δÞ ∼ ϕðLÞδ, as δ → 0. By using a similar energetic argument [24], one can further show that P N;L ðN L ¼ N − kÞ ∼ Ae−kNβϕðLÞ , with A ¼ ð1 − e−NβϕðLÞ Þ [19], which is valid for k ≪ N. The number variance V N ðLÞ can be easily computed from this (discrete) exponential distribution, yielding in Eq. (3). V N ðLÞ ∼ e−NβϕðLÞ , as announced pffiffiffi The edge, jL − 2j ∼ OðN −2=3 Þ.—This regime smoothly connects the other two ones. The number variance suddenly drops down to p zero ffiffiffi when L approaches the edge of the semicircle L ∼ 2. In this regime, the probability distribution P N;L ðN L ¼ N − kÞ can be expressed, for β ¼ 1; 2 and 4 [30,31], in terms of Fredholm determinants, or equivalently as integrals involving a special solution of the Painlevé II equation. Computing the number variance from these expressions is however quite difficult. A simpler way to compute V N ðLÞ in this regime is to resort to a finite N calculation and then take the large N limit in the edge scaling limit. We illustrate this approach in the case of GUE (β ¼ 2)—but it could be extended to β ¼ 1 and 4. For β ¼ 2, V N ðLÞ is given by [10,11,32] Z L Z L Z L V N ðLÞ ¼ dxK N ðx;xÞ − dx dy½K N ðx;yÞ2 : ð11Þ −L
−L
−L
Here K N ðx; yÞ is the GUE kernel, expressed in terms of Hermite polynomials [19]. in the vicinity of the edge, setting L ¼ pffiffiWe ffi now pffiffizoom ffi [11] and find that when N → ∞, 2þ 2Np2=3 pffiffis=ð ffi ffiffiffi Þ 2=3 V N ð 2 þ s=ð 2N ÞÞ → V~ 2 ðsÞ where [19] Z ∞ ZZ dxK Ai ðx; xÞ − 2 dxdy½K Ai ðx; yÞ2 : V~ 2 ðsÞ ¼ 2 s
½s;∞2
ð12Þ Here, K Ai ðx; yÞ is the Airy kernel given by K Ai ðx; yÞ ¼ ½AiðxÞAi0 ðyÞ − AiðyÞAi0 ðxÞ=ðx − yÞ where AiðxÞ is the Airy function and, at coinciding points, K Ai ðx; xÞ ¼ ðAi0 ðxÞÞ2 − xAi2 ðxÞ. One can show that V~ 2 ðsÞ behaves asymptotically as V~ 2 ðs → −∞Þ ∼ 3=ð2π 2 Þ ln jsj [19,33] 3=2 and V~ 2 ðs → ∞Þ ∼ ð8πÞ−1 s−ð3=2Þ e−ð4=3Þs [19]. These asymptotic behaviors ensure a perfect matching with p theffiffiffi behaviors of V N ðLÞ on both sides of p theffiffiffi edge for L≷ 2 in pffiffi(3). ffi For instance, when L approaches 2 from above, L → 2þ one can substitute of (3) the behavior pffiffiin ffiþ the last line 7=4 pffiffiffi of ϕðLÞ when L → 2 , ϕðLÞ ∼ ð2 =3ÞðL −pffiffi2ffi Þ3=2 . Hence, for β ¼ 2, V N ðLÞ ∼ exp ½−Nð211=4 =3ÞðL − 2Þ3=2 , which after a rearrangement of the argument coincides ~ with pffiffithe ffi 2=3asymptotic pffiffiffi behavior of V 2 ðs → ∞Þ, with s ¼ 2N ðL − 2Þ. We can similarly show that the
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pffiffiffi matching also holds when L approaches 2 from below. Assuming that this matching holds for all values of β, one expects the asymptotic behaviors (3 ln jsj; s → −∞ βπ 2 ð13Þ V~ β ðsÞ ∼ 3=2 exp ð− 2β s Þ; s → ∞: 3 In Fig. 1 we show a plot of V~ 2 ðsÞ together with a comparison with numerical simulations, showing an excellent agreement. In conclusion, our exact large N results in Eq. (3) for GUE random matrix (with β ¼ 2) provide an analytical description of V N ðLÞ for 1D spinless fermions in the ground state, which was computed before only numerically [4,7]. They may also be relevant to the statistics of entanglement entropy in such systems [4]. Even though most experiments in cold atoms are performed in bosonic systems (rather than fermionic), our results are expected to apply to bosonic systems in presence of very strong repulsive interactions between bosons [34]. We also expect a similar “dropoff” effect (see Fig. 1) to occur in the statistics of the so-called index, i.e., the number of positive eigenvalues in a given interval. The index is relevant for instance to describe the energy landscape of complex and glassy systems [35,36] and was thus recently studied for different ensembles [36–39]. Here, we have shown that the presence of the edge induces the “dropoff” effect. Hence it would be interesting to compute the number variance for random matrix ensembles without an edge, such as the Cauchy ensemble (see for instance [40] and references therein). We acknowledge valuable correspondence with Viktor Eisler and Ettore Vicari. S. N. M. and G. S. acknowledge support by ANR Grant No. 2011-BS04-013-01 WALKMAT and in part by the Indo-French Centre for the Promotion of Advanced Research under Project No. 4604-3. This work is supported by “Investissements d’Avenir” LabEx PALM (ANR-10-LABX-0039-PALM) (G. S. and P. V.).
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Phase Transitions and Edge Scaling of Number Variance in Gaussian Random Matrices Ricardo Marino, Satya N. Majumdar, Grégory Schehr, and Pierpaolo Vivo Laboratoire de Physique Théorique et Modéles Statistiques (UMR 8626 du CNRS), Université Paris-Sud, Bâtiment 100, 91405 Orsay Cedex, France (Received 2 April 2014; published 26 June 2014) We consider N × N Gaussian pffiffiffi pffiffiffi random matrices, whose average density of eigenvalues has the Wigner semicircle form over ½− 2; 2. For such matrices, using a Coulomb gas technique, we compute the large N behavior of the probability P N;L ðN L Þ that N L eigenvalues lie within the box ½−L; L. This probability scales as P N;L ðN L ¼ κL NÞ ≈ exp ð − βN 2 ψ L ðκL ÞÞ, where β is the Dyson index of the ensemble and rate function ψ L ðκ L Þ is a β-independent pffiffiffi pffiffiffi that we compute exactly. We identify three regimes pffiffiffi as L is varied: (i) N −1 ≪ L < 2 (bulk), (ii) L ∼ 2 on a scale of OðN −2=3 Þ (edge), and (iii) L > 2 (tail). We find a dramatic nonmonotonic behavior of the number variance V N ðLÞ as a function of L: after a logarithmic growth ∝ lnðNLÞ in the bulk (when L ∼ Oð1=NÞ), V N ðLÞ decreases pffiffiffi abruptly as L approaches the edge of the semicircle before it decays as a stretched exponential for L > 2. This “dropoff” of V N ðLÞ at the edge is described by a scaling function V~ β that smoothly interpolates between the bulk (i) and the tail (iii). For β ¼ 2 we compute V~ 2 explicitly in terms of the Airy kernel. These analytical results, verified by numerical simulations, directly provide for β ¼ 2 the full statistics of particle-number fluctuations at zero temperature of 1D spinless fermions in a harmonic trap. DOI: 10.1103/PhysRevLett.112.254101
PACS numbers: 05.45.Mt, 02.10.Yn, 02.50.-r, 24.60.-k
There has been enormous progress in the last decade on the experimental manipulation of cold atoms [1,2], which generated several interesting theoretical questions concerning the interplay between quantum and statistical behaviors in many-body systems. These experiments are usually carried out in the presence of optical laser traps that confine the particles in a limited region of space. In particular, 1D systems, such as spinless fermions in presence of a harmonic trap, have played a crucial role in these recent developments [2–7]. One important observable that has been studied is the number of fermions N L in the ground state (T ¼ 0) within a given box ½−L; þL. The variance of N L , denoted by V N ðLÞ, characterizes the quantum fluctuations in the ground state of this many-body system. This quantity has also been studied recently in a number of other quantum systems, including several lattice models of fermions [8,9]. The variance V N ðLÞ as a function of the box size L turns out to be highly nontrivial even for the simplest possible many-body quantum system, namely 1D spinless fermions in a harmonic trap. In this case, it was numerically found that V N ðLÞ has a rather rich nonmonotonic dependence on L—it first increases with L and then drops rather dramatically when L exceeds some threshold value [4,7]. Analytically deriving this dependence on L is thus a challenging problem. In this Letter, exploiting a connection of this fermionic system at T ¼ 0 to the Gaussian Unitary Ensemble (GUE) of random matrices, we obtain, for large N, this variance V N ðLÞ exactly for arbitrary L, which explains its nonmonotonic behavior. In addition, using a Coulomb gas technique for random matrices, we are able to calculate the full probability distribution of N L in the large N limit. 0031-9007=14=112(25)=254101(5)
The ground state many-body wave function of N 1D spinless fermions in a harmonic potential UðxÞ ¼ ð1=2Þmω2 x2 pffiffiffiffiffi is ffi given by the Slater determinant Ψ0 ð~xÞ ¼ ð1= N!Þ det½φi ðxj Þ, where φn ðxÞ is the single particle harmonic oscillator wave function, φn ðxÞ ∝ 2 Hn ðxÞe−x =2 (we have set ℏ ¼ m ¼ ω ¼ 1), where Hn ðxÞ are Hermite polynomials. By explicitly evaluating this determinant, it is easy to see that 1 − PN x2i Y i¼1 jΨ0 ð~xÞj2 ¼ e ðxj − xk Þ2 ; ð1Þ ZN jj where ZN;β is a normalization constant and E½λ ¼ P P ðN=2Þ Ni¼1 λ2i − ð1=2Þ j≠k ln jλj − λk j. It is well known [10] that, for large N, the average density ρN ðλÞ of eigenvalues (normalized to unity) of a Gaussian random matrix approaches the celebrated semicircle law pffiffiffi Wigner’s pffiffiffi on p the compact support ½− 2; 2, ρN ðλÞ → ρsc ðλÞ ¼ ffiffiffiffiffiffiffiffiffiffiffiffi −1 π 2 − λ2 . The typical distance between eigenvalues is thus of order ∼ Oð1=NÞ near the center of the semicircle. The average hN L i for large N can be computed pffiffiffiffiffiffiffiffiffiffiffiffiffias RL hN L i ¼ N −L ρsc ðxÞdx ¼ Nκ ⋆L , where κ ⋆L ¼ ½L 2 − L2 þ pffiffiffi 2arcsinðL= 2Þ=π. The variance V N ðLÞ was also computed in [12,13], but only in the bulk limit, i.e., when L ∼ Oð1=NÞ (the box size is of the order of the interparticle spacing near the center). On this scale, setting L ¼ Δ=N, V N ðLÞ was shown to grow logarithmically with Δ, V N ðLÞ ∼ ð2=βπ 2 Þ lnðΔÞ, for Δ ≫ 1. In contrast, numerical simulation in the fermionic system shows a nonmonotonic behavior of V N ðLÞ as L increases beyond Oð1=NÞ. A natural question is then: can one calculate V N ðLÞ for all L? In this Letter we indeed compute V N ðLÞ for all L in the large N limit, which exhibits a striking “dropoff” effect near the semicircular edge (see Fig. 1). Our method also allows us to compute, for arbitrary L, the full PDF of N L for large N, p which was ffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi known to be a Gaussian but only on a scale Oð lnðNLÞÞ, in the regime L ∼ Oð1=NÞ [14–17]. Our results can be summarized as follows. We find that the number variance V N ðLÞ for an interval ½−L; L behaves as pffiffiffi 8 2 2 32 N −1 ≪ L < 2 > 2 ln ðNLð2 − L Þ Þ; βπ > < pffiffiffi 2 V N ðLÞ ∼ V~ β ðsÞ; ð3Þ L ¼ 2 þ psffiffi2 N −3 > > pffiffiffi : exp ½−βNϕðLÞ; L > 2; where the scaling function V~ β ðsÞ is computed explicitly in (12) for β ¼ 2—its asymptotic behaviors for generic β are given in (13)—and the function ϕðLÞ is given in (10) (in this third ∼ stands pffiffiregime ffi pffiffifor ffi a logarithmic equivalent). In (3), L≷ 2 means jL − 2j ≫ N −2=3 . We thus identify three qualitatively different pffiffiffiregimes −1 dependingpon the value of L: (i) N ≪ L < 2 (bulk), ffiffiffi (ii) L ∼ p2ffiffiffi, on a scale of OðN −2=3 Þ (edge), and (iii) L > 2 (tail). Moreover, we are able to obtain the full probability distribution P N;L ðN L Þ of N L , for large N.
FIG. 1 (color online). Number variance V N ðLÞ as a function of L. Theoretical result Eq. (3) in solid p blue ffiffiffi line. Inset: edge scaling behavior of the variance around L ∼ 2 (described pffiffiffi pffiffiffi by the scaling function V~ 2 ðsÞ, in (12), with s ¼ ðL − 2Þ 2N 2=3 ) together with numerical simulations for N ¼ 5000 and β ¼ 2 (and averaged over 30 000 matrices).
Calling κ L ¼ N L =N the fraction of eigenvalues in ½−L; L we obtain [18] P N;L ðN L ¼ κ L NÞ ≈ exp ð − βN 2 ψ L ðκ L ÞÞ;
ð4Þ
for 0 ≤ κ L ≤ 1, where the rate function ψ L ðκ L Þ is βindependent and can be explicitly computed in terms of single integrals (see Eq. (56) in [19]). The rate function ψ L ðκ L Þ is convex and has a minimum (zero) at κL ¼ κ ⋆L (see Fig. 3). Thus the distribution of N L is peaked around N L ¼ κ⋆L N which is precisely its mean value hN L i ¼ κ⋆L N for large N. This distribution has non-Gaussian tails and even near its peak in hN L i it exhibits an anomalous quadratic behavior which is modulated here by a logarithmic singularity. The probability distribution of N L —considered here for simplicity as a continuous variable—can be written by integrating the jpd (2) with a delta constraint P δðN L − Ni¼1 1½−L;L ðλi ÞÞ, obtaining [19] P N;L ðN L Þ ∝
Z Y N i¼1
Z dλi
dξ expð−βE½λ; ξ; N L Þ: 2π
ð5Þ
The energy E½λ; ξ; N L of a configuration fλg is then given P by E½λ;ξ;N L ¼E½λþðξ=2ÞðN L − Ni¼1 1½−L;L ðλi ÞÞ, where a Lagrange multiplier ξ is introduced to take care of the delta constraint. Written in this form, (5) is just the grandcanonical partition function of a 2D fluid of charged particles (the eigenvalues) confined to a line. The system is in equilibrium at inverse temperature β under competing interactions: a confining quadratic potential and a logarithmic all-to-all repulsion term. In addition, a fraction κL ¼ N L =N of particles is constrained within the box ½−L; L. Introducing the normalized density of eigenvalues
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PHYSICAL REVIEW LETTERS 27 JUNE 2014 PRL 112, 254101 (2014) P ρðλÞ ¼ N −1 Ni¼1 δðλ − λi Þ, one converts the multiple integral in (5) to a functional integral over ρ, which is then evaluated for large N using a saddle point method. This procedure, originally introduced by Wigner and Dyson [12,20], has been successfully employed in recent works on the top eigenvalue of Gaussian and Wishart matrices [21–25], conductance fluctuations in mesoscopic systems [26] or bipartite entanglement of quantum systems [27–29]. For the present problem, a similar Coulomb gas analysis was performed in Ref. [15] but it was restricted to the narrow regime L ∼ Oð1=NÞ. Here we explore pffiffiffi instead the full range L ≫ N −1 up to the edge, L ∼ 2, and beyond. Skipping details [19], the resulting expression for P N;L ðN L ¼ κL NÞ reads to leading order for large N Z β 2 FIG. 2 (color online). Phase diagram of the Coulomb gas (7) in P N;L ðN L ¼ κL NÞ ∝ D½ρdξdη exp − N S½ρ ; ð6Þ 2 the (κ , L) plane. The equilibrium density ρ⋆ ðxÞ, plotted in insets, L
where the action (depending on L and κL ¼ N L =N) is Z ZZ 2 S½ρ ¼ dxx ρðxÞ − dxdx0 ρðxÞρðx0 Þ ln jx − x0 j Z Z L þη dxρðxÞ − 1 þ ξ dxρðxÞ − κL : ð7Þ −L
Here η is another Lagrange multiplier enforcing the normalization of the density, and the first integrals run over ð−∞; ∞Þ. We now evaluate the integral (6) for large N with a saddle point method. Differentiating (7) functionally with respect to ρ and then with respect to x, we obtain a singular integral equation for ρ⋆ ðxÞ (depending on L and κ L ) Z ρ⋆ ðx0 Þ 0 dx ; x ∈ suppðρ⋆ Þ and x ≠ L; ð8Þ x ¼ Pr x−x0 where Pr denotes the Cauchy principal part, and suppðρ⋆ Þ the region on the real line where ρ⋆ ðxÞ R L > ⋆0. Equation (8) is to be solved with the constraints −L ρ ðxÞdx ¼ κ L and R∞ ⋆ ρ ðxÞdx ¼ 1. This integral equation (8) can be solved −∞ using the resolvent method [19]. The normalized density has generally a three-cut support (see Fig. 2), sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 ðx2 − b2 Þðx2 − a2 Þ ρ⋆ ðxÞ ¼ ; ð9Þ π L2 − x2 which is valid for x belonging to any of the intervals in the support. The value of the edges a and b are determined by 2 2 2 the relation R L ⋆ a þ b ¼ L þ 2 together with the constraint −L ρ ðxÞdx ¼ κL . For a fixed value of L, the equilibrium density will take three different shapes (see Fig. 2) according to the fraction κL of particles stacked in the box. If κL ¼ κ⋆L (solid blue line in Fig. 2), as many particles are stacked in the box as naturally expected from (5) without any constraint. Thus the equilibrium density is just the semicircle. For the cases κL > κ ⋆L (κ L < κ⋆L ), an excess of particles accumulates inside (outside) the box, giving rise to three disconnected
exhibits a transition between two distinct shapes as the critical solid blue line κ ⋆L is crossed, along which ρ⋆ is given by the Wigner’s semicircle density.
blobs and a divergence of the density ρ⋆ around the inner (outer) box walls (see Fig. 2). Evaluating (6) at the saddle point, we obtain a large deviation decay of the probability for large N of the form in Eq. (4), where the rate function is given by ψ L ðκ L Þ ¼ ð1=2Þ½S½ρ⋆ − S½ρsc where ρsc is the Wigner’s semicircle density. The second term comes from the large N behavior of the normalization constant ZN;β in (2) and needs to be subtracted. The rate function ψ L ðκ L Þ is therefore determined by the action (7) at the saddle point S½ρ⋆ . Its full expression is rather cumbersome (see [19]) but can be easily plotted as shown in Fig. 3 for two different values of L. In order to extract the variance V N ðLÞ from the rate function, we expand ψ L ðκL Þ around its minimum κL ¼ κ ⋆L . −1 We first Þ ffiffihas pffiffiffi notice that ψ L ðκLp ffi a different shape for N p≪ ffiffiffi L < 2 (bulk) and L > 2 (tail), while the edge L ∼ 2 needs a separate treatment. pffiffiffi The bulk N −1 ≪ L < 2.—In this case, ψ L ðκ L Þ is twosided, and setting κ L ¼ κ ⋆L − δ [19], we find that close to its minimum κ ⋆L , it behaves like ψ L ðκ L ¼ κ ⋆L − δÞ ∼ ðπ 2 =4Þδ2 = ln ðLð2 − L2 Þ3=2 =jδjÞ as δ → 0. Therefore, around the critical value κ ⋆L for “sufficiently large" Lð2 − L2 Þ3=2 ≫ jδj, the rate function is nonanalytic and displays a quadratic behavior modulated by a logarithmic singularity. The physical origin of this nonanalytic behavior is linked to a phase transition in the associated Coulomb gas when κ L crosses the critical value κ⋆L (see Fig. 2). Inserting this behavior (close to κ ⋆L ) into (4), using that δ ¼ κL − κ L ¼ Oð1=NÞ, we find that P N;L ðN L Þ has a Gaussian behavior around κ ⋆L N, with a variance growing as in Eq. (3) (first line), thus recovering Dyson’s bulk behavior away from the edge. This Gaussian limiting distribution is thus valid on a scale pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ∼Oð lnðNLð2 − L2 Þ3=2 ÞÞ around κ⋆L N. However, beyond
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FIG. 3 (color online). Behavior of the rate function ψ L ðκ L Þ as a functionpffiffiof ffi κL ∈ ½0; 1 for two different values pffiffiffi of L: L ¼ 0.6 < 2 (green, two-sided) and L ¼ 1.6 > 2 (red, one-sided). The solid blue line in the plane (L, κ L ) is the critical line κ⋆L , where ψ L ðκ L Þ has a minimum (zero).
this scale, the fluctuations of κ L N are instead described by the full large deviation function in Eq. (4) which has nonGaussian tails [19]. analysis holds in the bulk, pffiffiThis ffi pffiffiffi for a fixed N −1 ≪ L < 2p, ffiffibut breaks down for L ∼ 2 (edge) ffi and in the tail, L p> ffiffiffi 2. The tail, L > 2.—In this regime the widthp offfiffiffithe box is much larger than the semicircle sea, L > 2, and κ ⋆L freezes to the value 1 (see Fig. 2), since on average all the eigenvalues are contained within the box. Therefore the rate function ψ L ðκL Þ is one-sided. Setting κL ¼ 1 − δ and expanding ψ L ðκL Þ to leading order in δ > 0, one obtains a linear behavior [19] ψ L ðκ L ¼ 1 − δÞ ∼ ϕðLÞδ, as δ → 0, where pffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi ϕðLÞ ¼ L L2 − 2=2 þ ln ððL − L2 − 2Þ= 2Þ: ð10Þ This function ϕðLÞ turns out to be identical to the large deviation function describing the right tail of the top eigenvalue λmax , Prob:½λmax > L ≈ exp½−βNϕðLÞ. Here NϕðLÞ is the energy cost to pull out one particle at a distance larger than L away from the Wigner sea, while the density of the rest (N − 1) charges remains of the standard semicircular form [24]. To understand this connection with the right tail of λmax one can extrapolate our Coulomb gas calculation of ψ L ðκ L Þ to the case where there is a discrete number of particles outside the interval ½−L; L. Because our Coulomb gas calculation preserves the symmetry ρ ðxÞ ¼ ρ ð−xÞ, which is always true in the continuum limit, this extrapolation can only be done for an even number of particles, say two of them: one in the interval ð−∞; −L and the other one in the interval ½L; þ∞Þ, hence in this case δ ¼ 2=N. Therefore, the energy cost of such a configuration, given by βN 2 ψ L ðκ L ¼ 1 − 2=NÞ, is
precisely twice (as there are two particles) the energy pffiffiffi NϕðLÞ to pull out one charge, at a distance L > 2, outside the Wigner sea (ignoring correlation effects between the two particles). Hence N 2 ψ L ðκ L ¼ 1 − 2=NÞ∼ 2NϕðLÞ, in agreement with ψ L ðκ L ¼ 1 − δÞ ∼ ϕðLÞδ, as δ → 0. By using a similar energetic argument [24], one can further show that P N;L ðN L ¼ N − kÞ ∼ Ae−kNβϕðLÞ , with A ¼ ð1 − e−NβϕðLÞ Þ [19], which is valid for k ≪ N. The number variance V N ðLÞ can be easily computed from this (discrete) exponential distribution, yielding in Eq. (3). V N ðLÞ ∼ e−NβϕðLÞ , as announced pffiffiffi The edge, jL − 2j ∼ OðN −2=3 Þ.—This regime smoothly connects the other two ones. The number variance suddenly drops down to p zero ffiffiffi when L approaches the edge of the semicircle L ∼ 2. In this regime, the probability distribution P N;L ðN L ¼ N − kÞ can be expressed, for β ¼ 1; 2 and 4 [30,31], in terms of Fredholm determinants, or equivalently as integrals involving a special solution of the Painlevé II equation. Computing the number variance from these expressions is however quite difficult. A simpler way to compute V N ðLÞ in this regime is to resort to a finite N calculation and then take the large N limit in the edge scaling limit. We illustrate this approach in the case of GUE (β ¼ 2)—but it could be extended to β ¼ 1 and 4. For β ¼ 2, V N ðLÞ is given by [10,11,32] Z L Z L Z L V N ðLÞ ¼ dxK N ðx;xÞ − dx dy½K N ðx;yÞ2 : ð11Þ −L
−L
−L
Here K N ðx; yÞ is the GUE kernel, expressed in terms of Hermite polynomials [19]. in the vicinity of the edge, setting L ¼ pffiffiWe ffi now pffiffizoom ffi [11] and find that when N → ∞, 2þ 2Np2=3 pffiffis=ð ffi ffiffiffi Þ 2=3 V N ð 2 þ s=ð 2N ÞÞ → V~ 2 ðsÞ where [19] Z ∞ ZZ dxK Ai ðx; xÞ − 2 dxdy½K Ai ðx; yÞ2 : V~ 2 ðsÞ ¼ 2 s
½s;∞2
ð12Þ Here, K Ai ðx; yÞ is the Airy kernel given by K Ai ðx; yÞ ¼ ½AiðxÞAi0 ðyÞ − AiðyÞAi0 ðxÞ=ðx − yÞ where AiðxÞ is the Airy function and, at coinciding points, K Ai ðx; xÞ ¼ ðAi0 ðxÞÞ2 − xAi2 ðxÞ. One can show that V~ 2 ðsÞ behaves asymptotically as V~ 2 ðs → −∞Þ ∼ 3=ð2π 2 Þ ln jsj [19,33] 3=2 and V~ 2 ðs → ∞Þ ∼ ð8πÞ−1 s−ð3=2Þ e−ð4=3Þs [19]. These asymptotic behaviors ensure a perfect matching with p theffiffiffi behaviors of V N ðLÞ on both sides of p theffiffiffi edge for L≷ 2 in pffiffi(3). ffi For instance, when L approaches 2 from above, L → 2þ one can substitute of (3) the behavior pffiffiin ffiþ the last line 7=4 pffiffiffi of ϕðLÞ when L → 2 , ϕðLÞ ∼ ð2 =3ÞðL −pffiffi2ffi Þ3=2 . Hence, for β ¼ 2, V N ðLÞ ∼ exp ½−Nð211=4 =3ÞðL − 2Þ3=2 , which after a rearrangement of the argument coincides ~ with pffiffithe ffi 2=3asymptotic pffiffiffi behavior of V 2 ðs → ∞Þ, with s ¼ 2N ðL − 2Þ. We can similarly show that the
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pffiffiffi matching also holds when L approaches 2 from below. Assuming that this matching holds for all values of β, one expects the asymptotic behaviors (3 ln jsj; s → −∞ βπ 2 ð13Þ V~ β ðsÞ ∼ 3=2 exp ð− 2β s Þ; s → ∞: 3 In Fig. 1 we show a plot of V~ 2 ðsÞ together with a comparison with numerical simulations, showing an excellent agreement. In conclusion, our exact large N results in Eq. (3) for GUE random matrix (with β ¼ 2) provide an analytical description of V N ðLÞ for 1D spinless fermions in the ground state, which was computed before only numerically [4,7]. They may also be relevant to the statistics of entanglement entropy in such systems [4]. Even though most experiments in cold atoms are performed in bosonic systems (rather than fermionic), our results are expected to apply to bosonic systems in presence of very strong repulsive interactions between bosons [34]. We also expect a similar “dropoff” effect (see Fig. 1) to occur in the statistics of the so-called index, i.e., the number of positive eigenvalues in a given interval. The index is relevant for instance to describe the energy landscape of complex and glassy systems [35,36] and was thus recently studied for different ensembles [36–39]. Here, we have shown that the presence of the edge induces the “dropoff” effect. Hence it would be interesting to compute the number variance for random matrix ensembles without an edge, such as the Cauchy ensemble (see for instance [40] and references therein). We acknowledge valuable correspondence with Viktor Eisler and Ettore Vicari. S. N. M. and G. S. acknowledge support by ANR Grant No. 2011-BS04-013-01 WALKMAT and in part by the Indo-French Centre for the Promotion of Advanced Research under Project No. 4604-3. This work is supported by “Investissements d’Avenir” LabEx PALM (ANR-10-LABX-0039-PALM) (G. S. and P. V.).
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