Universality in the full counting statistics of trapped fermions
aa r X i v : . [ c ond - m a t . s t a t - m ec h ] A ug Universality in the full counting statistics of trapped fermions
Viktor Eisler
Vienna Center for Quantum Science and Technology, Faculty of Physics,University of Vienna, Boltzmanngasse 5, A-1090 Wien, Austria (Dated: September 27, 2018)We study the distribution of particle number in extended subsystems of a one-dimensional non-interacting Fermi gas confined in a potential well at zero temperature. Universal features areidentified in the scaled bulk and edge regions of the trapped gas where the full counting statisticsare given by the corresponding limits of the eigenvalue statistics in Gaussian unitary random matrixensembles. The universal limiting behavior is confirmed by the bulk and edge scaling of the particlenumber fluctuations and the entanglement entropy.
The techniques for the manipulation of ultracoldatomic gases have undergone a rapid development in thelast decade and have provided experimental access to var-ious interesting aspects of many-body quantum systems[1, 2]. The common feature in the experiments is the pres-ence of trapping potentials that can be tuned to confineparticles into effective one-dimensional geometries [3–5]and theoretical predictions on various properties of 1Dquantum gases [6–8] can directly be tested.The strongly correlated phases of 1D quantum gasesare characterized by the simultaneous presence of ther-mal and quantum noise. At ultra low temperatures, thedominating quantum noise reveals important informationon the non-local character of correlations in the corre-sponding many-body states. In particular, 1D Bose liq-uids can be probed by measuring the full distributionof interference amplitude in experiments [9], showing re-markable agreement with the predictions of the theory[10]. In case of a Fermi gas, an analogous concept isthe full counting statistics (FCS) [11] which encodes thedistribution of particle number in extended subsystems.The FCS shows interesting properties in the ground stateof the Fermi gas and, in the non-interacting case, can alsobe used to extract the entanglement entropy of the sub-system [12, 13].The presence of trapping potentials leaves character-istic signatures on the FCS. For the ground state of thenon-interacting Fermi gas, the FCS was studied in thepresence of a periodic potential [14] and recently the ef-fect of harmonic traps has been analyzed on the particlenumber fluctuations and entanglement [15]. However,the question whether some properties of the FCS holdirrespectively of the details of the potential has not yetbeen addressed.Here we point out a remarkable universality and showthat, for a broad class of trapping potentials, the properscaling limits of the FCS in the bulk and edge regime ofthe trapped gas are given by the corresponding eigenvaluestatistics of Gaussian unitary random matrix ensembles.Physically, the universality can be understood from thegeneric behaviour of the trapping potential, being flat inthe center and approximately linear around the edge ofthe high-density region. The FCS is derived by a semi- classical treatment of the single-particle wavefunctionsand by finding scaling variables for the bulk and edgeregimes through which the details of the potential can becompletely eliminated in the thermodynamical limit.The appearance of random matrix eigenvalue statisticsin the FCS is rooted in the free fermion nature of theproblem. However, using the Fermi-Bose mapping [16]the results immediately carry over to the bosonic Tonks-Girardeau gas. Since the latter one is accessible in cold-atom experiments [3, 4], the measurement of the FCSmight be feasible in the spirit of Ref. [9] where the fulldistribution function of an analogous observable could beextracted for trapped bosons.The FCS is defined through the generating function χ ( λ ) = h exp( iλN A ) i , N A = Z A ρ ( x )d x (1)where N A is the total number of particles in subsystem A , given by the integral of the density ρ ( x ), and theexpectation value is taken with the N -particle groundstate of the system. For the spinless free Fermi gas, theFCS can be expressed as a Fredholm determinant [13, 17] χ ( λ ) = det (cid:2) iλ − K A (cid:3) (2)where K A is an integral operator with the kernel K A ( x, y ) = N − X k =0 ϕ ∗ k ( x ) ϕ k ( y ) (3)given by the two-point correlation function restricted tothe domain x, y ∈ A . It is constructed from the single-particle eigenfunctions of the Schr¨odinger equation12 d ϕ k ( x )d x + [ E k − V ( x )] ϕ k ( x ) = 0 (4)where we we have set ¯ h = m = 1. For simplicity, we con-sider a symmetric V ( − x ) = V ( x ) and monotonously in-creasing trapping potential such that for x → ∞ one has V ( x ) → ∞ . The spectrum is thus discrete and for each k the solutions ϕ k ( x ) admit two classical turning pointsgiven by the condition V ( ± x k ) = E k . In the followingwe will consider x ≥ ϕ k ( − x ) = ( − k ϕ k ( x ).In the classically allowed region, x < x k , the wave-functions are oscillatory with exactly k nodes whereasfor x > x k they must vanish exponentially. The wave-function which approximates the exact solution on bothsides is known as the uniform Airy approximation andcan be derived from a semi-classical treatment of theSchr¨odinger equation [18]. Up to the normalization fac-tor C k , it is given by ψ k ( x ) = C k p ξ ′ k ( x ) Ai [ ± ξ k ( x )] (5)where the + ( − ) sign applies in the classically forbidden(allowed) region and the argument of the Airy functionis given through ξ k ( x ) = (cid:20) Z x x p k ( z )d z (cid:21) / (6)with x = min( x, x k ) and x = max( x, x k ). Themomentum in the integrand of Eq. (6) is defined as p k ( x ) = p | E k − V ( x ) | and the approximate energyniveaus E k can be obtained from the Bohr-Sommerfeldquantization formula [19] Z x k − x k p k ( z )d z = ( k + 1 / π. (7)In particular, we will be interested in power-law po-tentials of the form V p ( x ) = x p /p with some even integer p . Note, that we have set the characteristic length scaleof the trap to one, which can easily be restored using thearguments of trap size scaling [20]. Then the integral inEq. (7) yields E k ≈ [ N p ( k + 1 / θ , N p = √ π Γ(3 / /p ) √ p /p Γ(1 + 1 /p ) (8)with the exponent given by θ = p/ ( p + 2).In general, the approximate wavefunctions ψ k ( x ) givenby Eq. (5) rely on a semi-classical argument and arethus expected to reproduce the exact ones ϕ k ( x ) only for k ≫
1. In fact, however, the uniform Airy approximationgives very good results even for the eigenfunctions of thelow-lying levels. This is demonstrated on Fig. 1 for thequartic potential V ( x ). The eigenfunctions ϕ k ( x ) arecalculated to a high precision by Numerov’s method [21]and compared to ψ k ( x ) where the integrals in Eq. (6) canbe given through special (hypergeometric and incompletebeta) functions and evaluated numerically. While theoverlap is reasonable for k = 0, the deviations are alreadyvery small for k = 3 and the two functions are essentiallyindistinguishable for k = 10.Far away from the turning point x k , the uniform Airyapproximation (5) reproduces the WKB-approximation -0.6-0.4-0.2 0 0.2 0.4 0.6 0.8 0 0.5 1 1.5 2 2.5 3 3.5 4 ϕ k ( x ) , ψ k ( x ) x k=0k=3k=10 FIG. 1. (color online) Exact wavefunctions ϕ k ( x ) (symbols)and their uniform Airy approximations ψ k ( x ) (lines) for thequartic potential V ( x ). [18] which can be obtained from asymptotic expansionsof Ai [ ± ξ k ( x )]. In particular, in the classically allowedregion one has ψ k ( x ) = C k p πp k ( x ) cos (cid:18)Z x k x p k ( z )d z − π (cid:19) . (9)For simple potential wells with two classical turningpoints, the normalization constant can be fixed by im-posing [22] C k Z x k − x k p − k ( z )d z = 2 π. (10)Differentiating Eq. (7) with respect to k , one arrives tothe simple formula C k = 2 d E k d k and thus the normaliza-tion factor accounts for the spectral density.The WKB-form of the wavefunctions (9) gives a goodapproximation in the entire classically allowed regime nottoo close to the turning points, but depends on the detailsof the potential V ( x ). To find universal features of theFCS, we first focus on the bulk of the trapped gas andwe choose the subsystem as the interval A = [ − ℓ, ℓ ] deepin the high-density region, ℓ ≪ x N . Expanding thefunctions p k ( x ) around x = 0, one obtains ψ k ( x ) ≈ C k p π √ E k cos (cid:16) k π − p E k x (cid:17) (11)which is valid up to linear terms in x . Substituting intoEq. (3), one arrives to the following sum K A ( x, y ) ≈ N − X k =0 C k π √ E k cos p E k ( x − y )+ N − X k =0 C k ( − k π √ E k cos p E k ( x + y ) . (12)We are interested in the N → ∞ limit of the FCS,where the first sum in Eq. (12) diverges. Therefore weintroduce new variables and define the scaling limit ofthe kernel as K r ( u, v ) = lim N →∞ √ E N K A (cid:18) u √ E N , v √ E N (cid:19) (13)where the subscript refers to the domain [ − r, r ] of thekernel in the scaled variables with the effective length r = ℓ √ E N . Introducing the variable z = p E k /E N , thefirst sum in (12) can be converted into an integral whilethe second, alternating sum vanishes in the scaling limit.Writing d z = d E k d k / √ E k E N and using the expression of C k in terms of the spectral density, one finds K r ( u, v ) = 1 π Z d z cos z ( u − v ) = sin( u − v ) π ( u − v ) . (14)Hence, in the bulk scaling limit, we recover the sine kernelwhich appears in the theory of GUE random matrices.Indeed, the probability E ( n, t ) of finding n eigenvaluesin an interval [ − r, r ] in the bulk of the GUE spectrum isgiven by [23] E ( n, r ) = ( − n n ! d n d z n det(1 − zK r ) (cid:12)(cid:12)(cid:12)(cid:12) z =1 . (15)Fourier transforming Eq. (15) with respect to n yieldsthe determinant in Eq. (2) with K A = K r and thus thebulk FCS of the Fermi gas is identical to the bulk GUEeigenvalue statistics.The bulk scaling limit can be tested through the cu-mulants κ m = ( − i∂ λ ) m ln χ ( λ ) | λ =0 of the particle num-ber. In particular, we calculated the fluctuations κ asa function of ℓ for the potential V ( x ). The numerics issimplified by considering, instead of the integral operator K A , the overlap matrix C A with elements C A,kl = Z A d x ϕ k ( x ) ϕ l ( x ) (16)and using Tr K nA = Tr C nA [24]. Then the FCS of Eq.(2) can be treated as a regular determinant and one has κ ( ℓ ) = Tr C A ( − C A ). This is then compared to ex-isting results derived using the asymptotics of Fredholmdeterminants with the sine kernel [17] κ ( r ) = Tr K r (1 − K r ) = 1 π (log 4 r + γ + 1) (17)where γ is the Euler constant and we neglected termsvanishing for N → ∞ . The results are shown on Fig. 2 for various N withthe dashed lines representing the curves κ ( r ). One cansee a good agreement for small ℓ but the solid curves κ ( ℓ ) deviate from the scaling prediction as soon as thesegment size exceeds the size of the flat region in thedensities ρ ( x ) = K A ( x, x ), shown on the inset. However, the amplitude of the O ( x ) corrections to Eq. (12) isproportional to V ′′ (0) /E k which vanishes for V p ( x ) with p ≥ p . Note also the strong oscillations in κ ( ℓ ) as well asin ρ ( x ) that are results of the alternating sum in Eq.(12) and diminish for higher N . The bulk scaling wasfurther tested by calculating the entanglement entropy S ( ℓ ) = − Tr [ C A ln C A + (1 − C A ) ln(1 − C A )] and com-paring it to the scaling prediction S ( r ) [24, 25] with sim-ilarly looking results as in Fig. 2. κ N=100N=50N=20 ρ (x) x PSfrag replacements ℓ FIG. 2. (color online) Particle number fluctuations for thequartic potential V ( x ) in the interval [ − ℓ, ℓ ] (solid lines) com-pared to the prediction of Eq. (17) in the bulk scaling limit(dashed lines) for various N . The inset shows the correspond-ing density profiles ρ ( x ). The other regime where universal features are expectedto emerge is near the edge of the high-density region.Close to the classical turning point, the argument ξ k ( x )of the Airy function in (5) can be expanded around x k and yields [18] ψ k ( x ) ≈ C k √ α k Ai [ α k ( x − x k )] (18)with α k = (2 V ′ ( x k )) / giving the inverse of the typicallength scale. The subsystem is now fixed as the interval A = ( x N + s/α N , ∞ ) starting close to the edge of thehigh-density region and extending to infinity. Note, thatthis choice of the interval A strongly limits the terms con-tributing to the sum in Eq. (3) since the Airy functionsin Eq. (18) are shifted gradually to the left for decreasing k and for | x k − x N | ≫ | s | /α N they become exponen-tially small in A . The edge scaling limit of the kernel isthen defined as K s ( u, v ) = lim N →∞ α N K A (cid:18) x N + uα N , x N + vα N (cid:19) (19)where the subscript refers to the domain u, v ∈ ( s, ∞ ) inthe new variables. The factors α k /α N appearing in the ρ ( x ) / α N u=(x-x ) α N ρ (u)N=1000N=100N=50 FIG. 3. (color online) Rescaled density profiles (dashed lines)for V ( x ) near the edge of the high-density region for various N . The uppermost (solid) line shows the N → ∞ scalingfunction ρ ( u ), see text. arguments of the Airy functions can be approximated as α k α N ≈ V ′′ ( x N )3 V ′ ( x N ) ( x k − x N ) (20)and since | x k − x N | ∼ | s | /α N , the second term vanishesin the limit N → ∞ for any well behaved potential. Wethus set α k = α N in evaluating (19) and introduce z = α N ( x N − x k ). Using x k = V − ( E k ), one finds d z ≈− C k /α N and consequently K s ( u, v ) = Z ∞ d z Ai( u + z )Ai( v + z )= Ai( u )Ai ′ ( v ) − Ai ′ ( u )Ai( v ) u − v . (21)Thus we recover the Airy kernel in the edge scaling limit.The FCS is then identical to the GUE edge eigenvaluestatistics [26], which follows immediately from a formulaanalogous to (15) by replacing r with s .As a first test of the scaling limit, we calculated theedge density profiles ρ ( x ) in the potential V ( x ) for vari-ous N and compared them to the density scaling function ρ ( u ) = K s ( u, u ) = (Ai ′ ( u )) − u Ai ( u ), with the resultshown on Fig. 3. To reach larger particle numbers N , weused, instead of the exact wavefunctions ϕ k ( x ), the ψ k ( x )in Eq. (5), that gives excellent results for the profile inthe edge region. The finite-size scaling of the data can beinferred from the correction term in Eq. (20). For power-law potentials one has V ′′ p ( x N ) /V ′ p ( x N ) = ( p − /x N and multiplying by ( x k − x N ) ∼ α − N , we obtain a N − / scaling of the finite-size corrections, which is consistentwith the data in Fig. 3. Note, that the exponent of N is independent of p , however, the prefactor p − p which weindeed observed in the numerics for p = 6. κ , S s κ (s) S(s) N=500N=100N=50 FIG. 4. (color online) Particle number fluctuations κ andentropy S in the scaled edge region for various N . The solidlines show the respective scaling functions κ ( s ) and S ( s ). The edge limit of the FCS is verified by calculatingthe particle number fluctuations κ and entanglement en-tropy S of the interval ( s, ∞ ) in the scaled coordinates.They are shown in Fig. 4 for various N and compared tothe scaling prediction, calculated using a powerful numer-ical toolbox for the evaluation of Fredholm determinantsarising in random matrix theory [27]. Both κ and S converge slowly to their respective scaling functions.In conclusion, we have shown the universality of theFCS for a noninteracting trapped Fermi gas in the bulkand edge regions, given by the respective eigenvaluestatistics of GUE random matrices. Interestingly, thesame universal limits emerge for non-Gaussian unitaryrandom matrix ensembles, where the potential V ( x ) ap-pears in the exponential weight function and the semi-classical asymptotics of the corresponding orthogonalpolynomials has to be analyzed [28, 29]. This leads to re-sults that are surprisingly similar to Eq. (5) even thoughthe two problems coincide only in the trivial Gaussiancase, V ( x ) being the harmonic potential.One expects that the same universality would emergefor trapped fermions on a lattice and might even gener-alize to other potentials. In fact, the connection betweenFCS and GUE statistics has recently been pointed outfor the time evolution of lattice fermions from a step-initial condition [30]. However, in this case the correla-tion matrix is unitarily equivalent to the one describingthe ground state of a chain with a linear potential [31]and hence the universality of the FCS carries over to thegradient problem. Note that without the lattice, the Airyfunctions in (18) are the exact eigenstates and the spec-trum is continuous, thus the edge limit (21) describesthe FCS in the entire high-density region while the bulkregime is missing.It would also be interesting to consider the dynam-ical FCS after releasing the gas from the trap. Inthe harmonic case, the corresponding time-dependentSchr¨odinger equation can be solved exactly and the ker-nel has, up to an irrelevant phase factor, the equilibriumform in appropriately rescaled variables [32]. Thus, thelimiting scaling forms of the FCS are also unchanged.However, for general V ( x ) the situation is more com-plicated and requires a careful analysis. Finally, in thisnon-equilibrium context one could study the waiting timedistribution between counting events where connectionsto random matrix theory have recently been revealed [33].The author thanks Z. R´acz for interesting discussionsand acknowledges support by the ERC grant QUERG. [1] M. 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