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Statistics of Spin Fluctuations in Quantum Dots with Ising Exchange

We explore the effect of single-particle level fluctuations on the Stoner instability in a QD with a strong spin-orbit coupling in the framework of the universal Hamiltonian with the Ising exchange interaction. We reduce the problem to studying the statistics of extrema of a certain Gaussian process and demonstrate that, in spite of the randomness of the single-particle levels, the longitudinal spin susceptibility and all its moments diverge simultaneously at the point of the Stoner instability which is determined by the standard criterion involving the mean level spacing only.

We explore the effect of single-particle level fluctuations on the Stoner instability in a QD with a strong spin-orbit coupling in the framework of the universal Hamiltonian with the Ising exchange interaction. We reduce the problem to studying the statistics of extrema of a certain Gaussian process and demonstrate that, in spite of the randomness of the single-particle levels, the longitudinal spin susceptibility and all its moments diverge simultaneously at the point of the Stoner instability which is determined by the standard criterion involving the mean level spacing only. Introduction.-In the last two decades the physics of quantum dots (QDs) attracted a lot of interest from both experimentalists and theorists [1][2][3][4][5]. Under the assumption E Th /δ ≫ 1, where E Th and δ denote the Thouless energy and the mean single-particle level spacing respectively, an effective zero-dimensional Hamiltonian has been derived [6]. This so-called universal Hamiltonian (UH) provides a convenient framework for the theoretical description of QDs. In the UH the electron-electron interaction that involves a set of matrix elements in the single-particle basis is reduced to just three parameters: the charging energy (E c ), the ferromagnetic exchange (J > 0) and the interaction in the Cooper channel.
At low temperatures T ≪ E c the additional cost due to the charging energy restricts the probability of real electron tunneling through a QD, a phenomenon known as the Coulomb blockade [7]. This leads to suppression of the tunneling density of states (TDOS) in QDs for T ≪ E c [8,9]. Although typically E c ≫ δ, a small enough exchange interaction J δ/2 is important for a quantitative description of the experiments on transport through QDs at T δ in a two-dimensional electron gas [10]. The exchange interaction of a large QD can be estimated by the Fermi-liquid interaction parameter (F σ 0 ) of the corresponding bulk material: J/δ = −F σ 0 . As well-known for bulk materials, strong enough exchange interaction leads to a Stoner instability at F σ 0 = −1 and a corresponding quantum phase transition between a paramagnet and a ferromagnet. In QDs an intermediate case of the ground state (GS) having a finite value of spin can be realized for δ/2 J < δ in the case of the equidistant single-particle spectrum [6]. As J increases towards δ, the GS spin increases and at J = δ all electrons in a QD become spin polarized. This phenomenon of mesoscopic Stoner instability disappears in the thermodynamic limit δ → 0. Due to the entanglement of the charge and spin degrees of freedom in the UH, the mesoscopic Stoner instability affects the electron transport through a QD. For example, it leads to an additional nonmonotonicity of the energy dependence of the TDOS [11][12][13] and to the enhancement of the shot noise [14]. The Cooper channel interaction in the UH describes the superconducting correlations in QDs [15]. We shall assume that there is no attraction in the Cooper channel and, therefore, disregard it below [6]. We also neglect the corrections to the UH due to the fluctuations in the matrix elements of the interaction [16,17], which are small in the regime δ/E Th ≪ 1 but can lead to interesting physics beyond the UH [5].
In the presence of a spin-orbit coupling the UH description of a QD breaks down. For a large spin-orbit length (weak spin-orbit coupling), λ SO ≫ L, where L is a typical size of the QD, fluctuations of the matrix elements of the interaction cannot be neglected [18,19]. However, for a QD fabricated in a two-dimensional electron gas the orbital degrees of freedom are coupled to inplane components of the spin only. Thus in the regime (λ SO /L) 2 ≫ (E Th /δ)(L/λ SO ) 4 ≫ 1 the UH description is restored but with the Ising exchange interaction [18,20]. Contrary to case of the Heisenberg exchange, there is no mesoscopic Stoner instability within the UH with the Ising exchange (J z > 0) for the equidistant single-particle spectrum [6]. As a consequence, the TDOS is almost independent of J z while the longitudinal spin susceptibility χ zz is independent of T as in a clean Fermi liquid [11,21].
In the interacting electron systems a disorder-induced finite temperature transition between the paramagnetic and the ferromagnetic phases is possible in low dimensions d 2 [22,23]. In d = 3 the Stoner instability can be shifted towards the smaller values of the exchange interaction due to disorder [24]. In the UH description the disorder is translated into randomness of the single-particle levels. The latter is crucial in the case of the Ising exchange since the average χ zz acquires a T -dependent contribution of Curie type due to the level fluctuations [6]. The Curie-type contribution dominates at low enough T and for δ − J z ≪ δ. In this regime the level fluctuations become strong with respect to the distance δ − J z to the average position of the Stoner instability at J z = δ (albeit small when compared to the temperature). This implies that, although for δ − J z ≪ δ the QD is in the paramag-netic phase on average, for a particular realization of the single-particle levels the QD can be fully spin-polarized. Such events should affect the tail of the distribution function for χ zz , but how exactly? Can it be possible that at T = 0 the level fluctuations shift the position of the Stoner instability from J z = δ and lead to the existence of a finite temperature transition between the paramagnetic and the ferromagnetic phases?
In this Letter we address these questions within the UH with the Ising exchange interaction approach. We demonstrate that for δ − J z ≪ δ the statistical properties of the spin susceptibility in the temperature range δ ≪ T ≪ δJ z /(δ − J z ) are determined by the statistics of the extrema of a certain Gaussian process with drift that locally resembles a fractional Brownian motion (FBM) with Hurst exponent H = 1 − ǫ with ǫ → 0 (recall that the FBM with Hurst exponent H is the Gaussian process B H (t) with zero mean and the two-point corre- . We estimate the complementary cumulative distribution function (CCDF) for χ zz and show that all moments of χ zz are finite for J z < δ. Thus our results mean that at T ≫ δ in spite of the presence of strong level fluctuations the Stoner instability still occurs at J z = δ.
The partition function.-We consider the following universal Hamiltonian with direct Coulomb and Ising exchange interactions [6]: For an isolated lateral QD with Rashba and Dresselhaus spin-orbit couplings the statistics of single-particle energies ǫ α is described by the unitary Wigner-Dyson ensemble (class A) [20]. The operators of the total number of particlesN = α,σ a † α,σ a α,σ and the total spin S = (1/2) ασσ ′ a † ασ σ σσ ′ a ασ ′ are given as usual in terms of the single-particle creation (a † α,σ ) and annihilation (a α,σ ) operators and the Pauli matrices σ.
Since the operators for the number of spin-up and spindown electrons commute with H, the grand partition function Z = Tr e −βH+βµN can be written as [21] The integers n ↑ and n ↓ represent the number of spin-up and spin-down electrons respectively. The total number of electrons is n = n ↑ + n ↓ , and m = (n ↑ − n ↓ )/2 is the value of S z . The factor Z n ↑ (Z n ↓ ) is the canonical partition function for n ↑ (n ↓ ) noninteracting spinless electrons. They take into account the contributions due to the single-particle energies and are given by the Darwin-Fowler integral: There is a convenient integral representation for Z which is exactly equivalent to Eq. (2): Here Ω 0 (µ) = −T ln γ 1 + e −β(ǫγ −µ) stands for the thermodynamic potential of free spinless electrons; φ 0 and h are the zero-frequency Matsubara components of the electric potential and the magnetic field that can be used to decouple the direct Coulomb [8] and exchange interaction [21,25] terms respectively. At not very low temperatures T ≫ δ we can perform integration over φ 0 in Eq. (3) in the saddle-point approximation [8,26]. In this case the grand canonical partition function factorizes: describes the effect of the charging energy. The exchange interaction is encoded in Hereμ is the solution of the saddle-point equation: for the thermodynamic density of states for the chemical potentialμ. The thermodynamic potential Ω 0 (μ) depends on a particular realization of the single-particle spectrum via the single-particle density of states where δν 0 (E) stands for the deviation of the density of states ν 0 (E) from its average value 1/δ ≡ 1/∆. The longitudinal spin susceptibility is fully determined by the partition function, χ zz = ∂ ln Z/∂J z . Since Z C is independent of J z and therefore does not affect the spin susceptibility, we will discuss Z S only. We note that the normalization is such that Z S = 1 for J z = 0. According to Eq. (2), Z increases with J z ; it follows that Z S 1. It is useful to write down Z S explicitly: . Level fluctuations.-Although the density of states ν 0 (E) has non-Gaussian statistics, V (h) is an even in h Gaussian random function for max{|h|, T /δ} ≫ 1 [27].
It has zero mean and the following two-point correlation function (see the Supplemental Material [28]): Here ψ(z) is the Euler digamma function, γ = −ψ(1) is the Euler-Mascheroni constant, and β = 2 since the energy levels ǫ α in Eq. (1) are described by the unitary Wigner-Dyson ensemble. The asymptotics of L(h) are as follows [12,28]: Average spin susceptibility.-As well known, the level fluctuations are small at T ≫ δ. In this regime one can find from Eq. (8) that the variance of the level spacing is ( [12]. Therefore it seems that in order to find the average spin susceptibility at T ≫ δ it is enough to substitute 1/∆ for 1/δ in the expression for χ zz obtained above for the equidistant spectrum. Performing the expansion to the second order in ∆ − δ, we find the average spin susceptibility This result indicates that the effect of fluctuations is small only at temperatures T ≫J z . For such temperatures the fluctuations of the level spacing are small in comparison with the distance to the average position of the Stoner instability, the renormalized exchange interactionJ z ≫ δ and there is a wide interval of temperaturesJ z ≫ T ≫ δ where the effect of level fluctuations can be strong. We stress that the dependence of χ zz on T appears only due to level fluctuations. The result (9) can be also obtained from Eq. (6) by means of the second order perturbation theory in V using the asymptotic expression (8) at |h| ≪ 1. For temperatures T ≪J z the integral in the r.h.s. of Eq. (6) is dominated by |h| ∼ J z /T ≫ 1. Hence in this case to evaluate the integrals involved in the perturbation theory in V one has to use the asymptotic formula (8) at |h| ≫ 1. Expanding Eq. (6) to the fourth order in V and performing the averaging of ln Z S with the help of Eqs. (7) and (8) we obtain [28] where a 2 ≈ 0.29. From Eq. (10) we see that the perturbation theory in V is justified only for T ≫J z /(π 2 β). Therefore the result (10) is valid in the rangeJ z ≫ T ≫J z /(π 2 β). In this regime the fluctuations of the spin susceptibility around its average value are small, (χ zz − χ zz ) 2 /(χ zz ) 2 ∝J z /(π 2 βT ) ≪ 1.
Tail distribution for ln Z S .-The perturbative result (10) suggests that the spin susceptibility can be strongly affected by level fluctuations atJ z /(π 2 β) ≫ T ≫ δ. Such regime is realized in the close vicinity of the average position of the Stoner instability δ − J z ≪ δ/(π 2 β). However, if the effect of level fluctuations is strong then it is useful to know not only the average spin susceptibility but also all its moments. With this in mind, we investigate the CCDF for ln Z S at temperatures in the intervalJ z /(π 2 β) ≫ T ≫ δ. In this range of temperatures, the integral in the r.h.s. of Eq. (6) is dominated by the large values of |h|. Using asymptotic expression (8), one can check that for |h 1 |, |h 2 | ≫ 1 the two-point correlation function (7) is homogeneous of degree two: We can therefore substitute zv(h) for the random function V (h βJ z ) where z = βJ z /(π 2 β); the Gaussian random process v(h) has zero mean, possesses the property v(h) = v(−h), and its correlation function is given by Since for any H = 1 − ǫ < 1, the trajectories of v(h) are continuous and its increments are strongly positively correlated (see inset in Fig. 1). In fact the process v(h) is in many aspects close to the ballistic caseṽ(h) = ξ|h|, where ξ is a Gaussian random variable (recall thatṽ(h) is the unique process with H = 1). We mention that the process v(h) has arisen before in a seemingly unrelated context [30]. The average moments of ln Z S can be conveniently written as [ln Z S ] k = k ∞ 0 dW W k−1 P(W ) where the function P(W ) is the complementary cumulative distribution function, i.e. the probability for ln Z S to exceed W : P(W ) ≡ Prob{ln Z S > W }. We note that P(0) = 1, P(∞) = 0 and P(W ) is monotonously decreasing. Although we cannot find a closed analytical expression for P(W ), we bound it from above to prove that all moments of ln Z S (and consequently all moments of χ zz ) are finite for J z < δ. We first split the Gaussian weight exp(−h 2 ) in the integral in the r.h.s. of Eq. (6) and obtain (0 < γ < 1 is an arbitrary splitting parameter) The inequality (12) allows us to reduce the problem of finding an upper bound for P(W ) to studying the statistics of the maxima of the Gaussian process Y (h) = −h 2 − (z/ √ γ)v(h) which locally resembles a FBM with drift. Indeed, from Eq. (12) we find To give an upper bound for the probability Prob{max h 0 Y (h) > w} we employ an auxiliary Gaussian is the standard Brownian motion (the Hurst exponent H = 1/2). The processes Y (h) and X(h) satisfy the conditions for the Slepian's inequality [29], Prob{max probabilities for X(h) are trivial to compute [31], and we find the following upper bound for the CCDF [28]: From Eq. (14) it follows that all moments of ln Z S (and hence all moments of χ zz ) are finite for J z < δ for temperatures in the rangeJ z /(π 2 β) ≫ T ≫ δ. Therefore even in the presence of the strong level fluctuations the Stoner instability occurs at J z = δ. For J z < δ and for temperatures T ≫ δ the QD is in the paramagnetic state. For z ≫ 1 the saddle-point approximation in Eq. (6) becomes exact and the statistics of ln Z S reduces to the statistics of maxima of Y (h) directly. Since local behavior of Y (h) may be compared to that of the FBM with Hurst exponent H = 1 − ǫ, one can adapt the results of Ref. [32] for locally stationary processes and find that for W ≫ 2z 2 ln 2 with logarithmic accuracy [28] This result valid in the temperature rangeJ z /(π 2 β) ≫ T ≫ δ is consistent with the upper bound (14). To illustrate the result (15) we approximate the Gaussian process v(h) by a degenerate oneṽ(h) = ξ|h| where ξ is the Gaussian random variable with zero mean ξ = 0 and variance ξ 2 = 4 ln 2. We estimate the partition function (6) as Z S ≃ J z /J z exp(z 2 ξ 2 /4) 1 − erf(zξ/2) . The large values of Z S correspond to large negative values of ξ such that ln Z S ≈ z 2 ξ 2 /4. Hence we find that for z ≫ 1 the tail of the function P(W ) is given by Eq. (15) without the logarithm in the pre-exponent [28]. As shown in Fig.  1 the overall behavior of P(W ) for z ≫ 1 is well enough approximated by the CCDF for the degenerate process v(h). Also we mention that the behavior of P(W ) for z ≫ 1 is very different from its behavior at z 1. For the later, P(W ) is given by the CCDF for the normal distribution (see Fig. 1).
Average moments of χ zz .-Equation (15) implies that the average moments of ln Z S scale as (ln Z S ) k ∼ z 2k for z ≫ 1. Hence for δ ≪ T ≪J z /(π 2 β) the k-th moment of the spin susceptibility is given by The result (16) can be obtained from the saddle-point analysis of the integral in the r.h.s. of Eq. (6), i.e., in essence, by Larkin-Imry-Ma type arguments [33,34]. The scaling of the average spin susceptibility (Eq. (16) with k = 1) was indeed found in Ref. [6] using arguments of Larkin-Imry-Ma type.
Summary.-To summarize, we have studied the Stoner instability in a QD within the UH with the Ising exchange interaction. We demonstrated that in the regime δ − J z ≪ δ where the level fluctuations are dangerous all moments of the spin susceptibility χ zz (T ) are finite at temperatures T ≫ δ for J z < δ. This means that i) the Stoner instability is not shifted by the level fluctuations away from J z = δ and ii) randomness in the single-particle levels does not lead to transition at finite T ≫ δ between the paramagnetic and ferromagnetic phases. Although we expect that these conclusions hold also for temperatures T δ we cannot argue it within our approach; a separate analysis is needed.
Similar conclusions about the influence of randomness of the single-particle levels on the Stoner instability can be made for QDs with the Heisenberg exchange [35]. Our approach can be extended to the analysis of the effect of level fluctuations on the transverse spin susceptibility and the TDOS for the UH with the Ising exchange.
Our results, in principle, can be checked in QDs made of materials close to the Stoner instability such as Co impurities in a Pd or Pt host, Fe or Mn dissolved in various transition-metal alloys, Ni impurities in a Pd host, and Co in Fe grains, as well as nearly ferromagnetic rare-earth materials [36]. However to test our most interesting results (Eqs. (15) and (16)) one needs to explore the regime (δ − J z )/δ ≪ 1/(π 2 β). At present the closest material to the Stoner instability we are aware of, YFe 2 Zn 20 , has the exchange interaction J ≈ 0.94δ which is near the boarder of the regime with strong level fluctuations at low temperatures.
We acknowledge useful discussions with Y.

ONLINE SUPPLEMENTAL MATERIAL: Statistics of Spin Fluctuations in Quantum Dots with Ising Exchange
We present some details on i) the analysis of the correlation function V (h1)V (h2), ii) the perturbative treatment of the average χzz, and iii) the calculation of the probability Prob{ln ZS > W }.
In this section we present a brief derivation of Eqs. (7) and (8) of the paper. The correlation function of the single-particle density of states is given as [S1] Here the function R(x) depends on the statistics of the ensemble of single-particle energies. Using Eq. (1), the identity ∞ −∞ R(x)dx = π and the definition of V (h) we obtain where .
The function g(E, h) has the following Fourier transform with respect to variable E: Since the function g(E, h) is even in E, the function g(t, h) is even in t. The function under the integral sign in the r.h.s. of Eq. (4) has poles at E = π(2n + 1)i, ±h + π(2m + 1)i where n and m are integers. Computation of the residues yields Substitution into Eq. (2) leads to At x ≫ 1 the function R(x) has the following asymptotic behavior [S1]: Recall that β = 1 for the orthogonal Wigner-Dyson ensemble, β = 2 for the unitary Wigner-Dyson ensemble and β = 4 for the simplectic Wigner-Dyson ensemble. Then at max{|h|, T /δ} ≫ 1 we find where is even in h. Next, for h > 0 This is the Eq. (7) of the paper. Using the well-known asymptotic expressions for the Euler digamma function ψ(x) at small and large values of its argument one arrives at Eq. (8).

II. PERTURBATION THEORY FOR THE SPIN SUSCEPTIBILITY
In this section we present the derivation of the perturbative results (9) and (10) of the paper. We start the from expansion of the average ln Z S to the fourth order in V : Here we introduced A. Second order in V The contribution of the second order in V is given by F 2 and F 1,1 . We find It is instructive to compare the second order contribution (13) with the second order contribution to the variance of ln Z S : In the regime T ≫J z the arguments of L in the r.h.s. of Eqs. (13) and (14) are small. Using the asymptotic expression for L(h) at |h| ≪ 1, we obtain In the regime T ≫J z the fourth order contributions are proportional to (J z /T ) 4 and therefore negligible. For low temperatures T ≪J z the contributions of the fourth order in V are listed below: (|r| 1/2). Hence we are in position to apply the Slepian's inequality [S2] to the pair (X(t), Y (t)) and claim that for all real w A well-known result for the Brownian motion with linear drift (see, e.g., [S3]) reads (w > 0) Combining Eq. (30) and Eq. (31), we obtain the inequality (14) of the paper. The above reasoning has the advantages of being rigorous and self-contained, but it provides only an upper bound for the CCDF tail; one can in fact go further. As it can be seen from rescaling the time axis, the probability that the maximum of Y (t) = −t 2 − z γ v(t) exceeds w equals the probability that the maximum of Y (s) = v(s)/(1 + s 2 ) defined on s 0 exceeds w 1/2 /z γ . From the results of Hüsler and Piterbarg [S4] it follows that the large-w tail of Prob{max t 0 Y (t) > w} is determined by a small vicinity of the point s * = 1 where the variance of Y (s) attains its maximum ln 2. Furthermore, should we have a finite limit for some function K(x) regularly varying at 0 with index α ∈ (0, 1), the precise asymptotics would read where K −1 (x) stands for the functional inverse of K(x). In our case [v(h + u) − v(h)] 2 → −2u 2 ln |u|+O(u 2 ) translates into K(x) = x ln(1/x) which is regularly varying with index α = 1 (recall that a function f (x) is regular varying at 0 with index α if lim t→0 f (at)/f (t) = a α for any a > 0). The result of Ref. [S4] is therefore not directly applicable, but we believe this to be a technicality. In analogy with a similar situation for fractional Brownian motion, we expect the asymptotics (33) to hold with only the W -independent factor const(α) modified. Note that the exponential part can be tracked to be the tail of a normal distribution with variance ln 2 taken at W 1/2 /z, and that it had been correctly reproduced by our initial estimate. For z ≫ 1 the saddle-point approximation in Eq. (6) of the paper becomes exact and the statistics of ln Z S reduces to the statistics of maxima of Y (h) directly. For x ≪ 1 the inverse of K(x) is K −1 (x) ≈ x/ ln 1/x 2 (with logarithmic accuracy), and we arrive at Eq. (15) of the paper.
Large values of Z S correspond to large negative values of ξ such that ln Z S ≈ z 2 ξ 2 /4. On the other hand, since Z S 1 Eq. (34) limits variable ξ to be in the interval −∞ < ξ < ξ 0 where ξ 0 is given as the solution of equation Z S (ξ 0 ) = 1. It can be estimated as ξ 0 ≈ (2/z) J z /(πJ z ) = πβT /J z ≫ 1. Then, within the degenerate case, we find the following result for the complementary cumulative distribution function: where ξ W is the solution of the equation ln Z S (ξ W ) = W . At z ≫ 1, for W ≫ z 2 one finds ξ W ≈ 2 √ W /z and, hence, that the tail of the function P deg (W ) is given by Eq. (15) of the paper without the logarithm in the pre-exponent. The function P deg (W ) for T = 3δ and J z /δ = 0.9997 is shown as the red dashed curve in Fig. 1 of the paper.