The free energy of a quantum Sherrington-Kirkpatrick spin-glass model for weak disorder

We extend two rigorous results of Aizenman, Lebowitz, and Ruelle in their pioneering paper of 1987 on the Sherrington-Kirkpatrick spin-glass model without external magnetic field to the quantum case with a"transverse field"of strength $b$. More precisely, if the Gaussian disorder is weak in the sense that its standard deviation $v>0$ is smaller than the temperature $1/\beta$, then the (random) free energy almost surely equals the annealed free energy in the macroscopic limit and there is no spin-glass phase for any $b/v\geq0$. The macroscopic annealed free energy (times $\beta$) turns out to be non-trivial and given, for any $\beta v>0$, by the global minimum of a certain functional of square-integrable functions on the unit square according to a Varadhan large-deviation principle. For $\beta v<1$ we determine this minimum up to the order $(\beta v)^4$ with the Taylor coefficients explicitly given as functions of $\beta b$ and with a remainder not exceeding $(\beta v)^6/16$. As a by-product we prove that the so-called static approximation to the minimization problem yields the wrong $\beta b$-dependence even to lowest order. Our main tool for dealing with the non-commutativity of the spin-operator components is a probabilistic representation of the Boltzmann-Gibbs operator by a Feynman-Kac (path-integral) formula based on an independent collection of Poisson processes in the positive half-line with common rate $\beta b$. Its essence dates back to Kac in 1956, but the formula was published only in 1989 by Gaveau and Schulman.

1 Introduction and definition of the model A spin glass is a spatially disordered material exhibiting at low temperatures a complex magnetic phase without spatial long-range order, in contrast to a ferro-or antiferromagnetic phase [FH91,M93,N01]. To this day most theoretical studies of spin glasses are based on models which go back to the classic(al) SHERRINGTON-KIRKPATRICK (SK) model [SK75]. In this simplified model (LENZ-)ISING spins are pairwise and multiplicatively coupled to each other via independent and identically distributed (GAUSSian) random variables and are possibly subject to an external ("longitudinal") magnetic field. The SK model may be viewed as a generalization of the traditional CURIE-WEISS (CW) model in which the spin coupling is given by a single (non-random) constant of a suitable sign. In both models the pair interaction is of the somewhat unrealistic mean-field type in the sense that it is the same for all spin pairs [NS03]. This neglect of geometric distances requires the effective strength of the pair interaction to decrease sufficiently fast with increasing total number of the spins in order to ensure thermostatic behavior in the limit of macroscopically many of them. The notion "mean field" indicates the comfortable fact that the BRAGG-WILLIAMS mean-field approximation of equilibrium statistical mechanics [H87] yields the exact free energy in this limit [FSV80]. According to standard textbook wisdom it is easy to calculate the macroscopic free energy of the CW model and to show that it provides a simplified but qualitatively correct description of the onset of ferromagnetism at low temperatures [D99]. In contrast, for the SK model the calculation has turned out to be much harder due to the interplay between thermal and disorder fluctuations, in particular for low temperatures. Nevertheless, by an ingenious application of the heuristic replica approach, see [FH91,M93,N01], PARISI found that the macroscopic (quenched) free energy of the SK model is given by the global maximum of a rather complex functional of (probabilistic) distribution functions on the unit interval [P80a, P80b]. The unique maximizing distribution function is interpreted as the (functional) spin-glass "order parameter". The attempt at understanding this PARISI formula became a challenge to mathematical physicists and mathematicians [T98]. Highly gratifying for him and his intuition [P09], the formula was eventually confirmed by a mathematically rigorous proof due to the efforts and insights of GUERRA, TALAGRAND, and others [GT02, G03, ASS03, T06, T11b, P13, AC15].
Since magnetic properties cannot be explained at the (sub)microscopic level of atoms and molecules by classical physics alone, some spin glasses require for fundamental and experimental reasons a quantum-theoretical modelling. Of course, the SK model may be viewed as a simplistic quantum model by interpreting the values of the ISING spins as (twice) the eigenvalues of one and the same component of associated three-component spin operators each of them with (main) quantum number 1/2. But a genuine quantum SK model with quantum fluctuations and inherent dynamics needs the presence of different (non-commuting) components of the spin operators. The theory of such a model was pioneered by BRAY and MOORE [BM80] and by SOMMERS [S81]. More precisely, for a quantum spin-glass model with isotropic (FRENKEL-)HEISENBERG(-DIRAC) spin coupling of mean-field type these authors handled the competition of thermal, disorder, and quantum fluctuations by combining the DYSON-FEYNMAN time-ordering of operator products with the replica approach [BM80] or with the THOULESS-ANDERSON-PALMER (TAP) approach [S81]. For the TAP approach see [FH91,M93,N01]. Since these authors did not aim at rigorous results, they applied the so-called static approximation to simplify the rather complicated equations derived by them. However, this approximation is still insufficiently understood -even for higher temperatures.
A simpler genuine quantum SK model is obtained by considering an extremely anisotro-pic pair interaction where only one component of the spins is coupled which is perpendicular to the direction of the external magnetic field. This model was introduced by ISHII and YAMAMOTO [IY85] and approximately studied within the TAP approach. It is usually called the SK model with (or "in") a transverse field, see [SIC13] and references therein. It is this model to which we devote ourselves in the present paper. It is characterized by the random energy operator or HAMILTONian acting selfadjointly on the N-spin HILBERT space (isometrically isomorphic to) 2 ⊗ · · · ⊗ 2 =: ( 2 ) ⊗N ∼ = 2 N , that is, the N-fold tensor product of the two-dimensional complex HILBERT space 2 for a single spin. Here N ≥ 2 is the total number of a collection of three-component spin-1/2 operators where the selfadjoint spin operator S α i /2 with component α and index (or "site") i is given by the tensor product of N factors according to In this definition the identity operator ½ and the operator S α , as the i-th factor, are understood to act (a priori) on 2 and satisfy the (specialized) DIRAC identities (S α ) 2 = ½, S x S y = iS z , S y S z = iS x , S z S x = iS y (1.2) with i ≡ √ −1 denoting the imaginary unit. With respect to the eigenbasis of S z these four operators are represented by the 2 × 2 unit matrix and the triple of 2 × 2 PAULI matrices according to The first term in (1.1) models an ideal (quantum) paramagnet and represents the energy of the spins due to their individual interactions with a constant magnetic field of strength b ≥ 0 externally applied along the positive x-direction. The second term in (1.1) models disorder in spin glasses and represents the energy of the spins due to random mean-field type pair interactions of their z-components. More precisely, we assume the N(N −1)/2 coupling coefficients (g i j ) 1≤i< j≤N to form a collection of jointly GAUSSian random variables with mean [g i j ] = 0 and covariance [g i j g kl ] = δ ik δ jl (in terms of the KRONECKER delta). The parameter v > 0 is the standard deviation of vg i j and stands for the strength of the disorder. At given b or v quantum fluctuations become more important with increasing v or b, respectively -due to the non-commutativity of S x i and S z i . We proceed by introducing the basic thermostatic quantity of the model (1.1). For any reciprocal (absolute) temperature β ∈ ]0, ∞[, we define the random partition function (or sum) as the trace (1.5) The latter is physically less relevant (for spin glasses with "frozen-in" disorder), but mathematically more accessible and provides a lower bound on the quenched free energy by the concavity of the logarithm and the JENSEN inequality [J06] (see also [K02,Lem. 3.5]), (1.6) Over the years the work [IY85] has stimulated many further approximate and numerical studies devoted to the macroscopic quenched free energy of the quantum SK model (1.1) and the resulting phase diagram in the temperature-field plane, among them [FS86, YI87, US87, K88, RCC89, BU90a, GL90, BU90b, MH93, KK02, T07, Y17,MRC18]. Not surprisingly, this has led to partially conflicting results, especially for low temperatures.
From a rigorous point of view, a solid understanding of the low-temperature regime seems still to be out of reach. The main and modest aim of the present paper is therefore to provide the first rigorous explicit results for the opposite regime characterized by β v < 1. Since in this regime β b ≥ 0 may be arbitrary, we call it the weak-disorder regime. In the following sections we firstly compile some properties of f ann N for arbitrary β v > 0. Next we show that f ann N has a well-defined macroscopic limit f ann ∞ with similar and wellunderstood properties, see Theorem 3.1, Theorem 4.3, and Theorem 5.3 below. In particular, for β v < 1 the limit β f ann ∞ takes a rather explicit form as a function of β v and β b. Then we prove that the more important free energies f N and [ f N ] have both f ann ∞ as its (almost sure) macroscopic limit if β v < 1, see Theorem 6.3 and Corollary 6.6. For β v < 1 we also prove the absence of spin-glass order in the sense that lim N→∞ S z 1 S z 2 2 = 0, almost surely, see Corollary 6.4 and Remark 6.5. Here (·) := e Nβ f N Tr e −β H N (·) denotes the (random) GIBBS expectation induced by H N . These results extend two of the pioneering results of AIZEN-MAN, LEBOWITZ, and RUELLE [ALR87] for the model (1.1) with b = 0 to the quantum case b > 0. Unfortunately, for any β v > 1 we only have the somewhat weak result that the difference between the macroscopic quenched and annealed free energies is strictly positive if the ratio b/v is sufficiently small. To our knowledge, the only other rigorous results for the quantum SK model (1.1) are due to CRAWFORD [C07] and to ADHIKARI and BRENNECKE [AB20]. CRAWFORD has extended key results of GUERRA and TONINELLI [GT02] and CARMONA and HU [CH06] for the model (1.1) with b = 0 to the quantum case b > 0. More precisely, he has proved the existence of the macroscopic (quenched) free energy not only for β v < 1, but for all β v > 0 (without a formula). Furthermore, he has shown that the limit is the same for random variables (g i j ) 1≤i< j≤N which are not necessarily GAUSSian but merely independently and identically distributed with [g 12 ] = 0, [(g 12 ) 2 ] = 1, and [|g 12 | 3 ] < ∞. More recently, ADHIKARI and BRENNECKE have provided a variational formula for the macroscopic (quenched) free energy for all β v > 0. This formula is still somewhat implicit and given by a suitable d → ∞ limit of a PARISI-like functional for a classical d-component vector-spin-glass model, due to PANCHENKO.
2 The annealed free energy and its deviation from the quenched free energy In this section we attend to the annealed free energy f ann N for arbitrary values of N ≥ 2, β v > 0, and β b > 0. According to (1.5) we have to perform the GAUSSian disorder average of the partition function Z N . In order to do so explicitly, we will use the following POISSON-FEYNMAN-KAC (PFK) probabilistic representation of Z N in terms of N copies of a POISSON process with constant rate (or intensity parameter) β b: (2.1) Here, the classical HAMILTONian h N , characterizing the zero-field SK model [SK75], is defined by where s := (s 1 , . . . , s N ) ∈ {−1, 1}×· · · ×{−1, 1} =: {−1, 1} N denotes one of the 2 N classical spin configurations and the notation ∑ s indicates summation over all of them. The integrand in (2.1) is obtained from (2.2) by replacing there each s i by the product defines the spin-flip process with index i, in other words, a "(semi-)random telegraph signal" [K74,KR13]. It is a continuous-time-homogeneous pure jump-type two-state MARKOV process [K02, Ch. 12] steered by a simple POISSON process N i in the positive half-line. * The random variable N i (t) is AE 0 -valued and POISSON distributed with mean β bt ≥ 0 independent of the index i. The N POISSON processes N 1 , . . . , N N are assumed to be (stochastically) mutually independent. The angular brackets (·) β b denote the corresponding joint POIS-SON expectation conditional on σ i (1) = 1 for all i ∈ {1, . . . , N}. In (2.1) and in the following we often write σ (t) := σ 1 (t), . . ., σ N (t) and suppress the N-dependence of (·) β b to keep the notation simple. For the validity of the PFK representation (2.1) we refer to (B.16) in Appendix B. For performing the GAUSSian disorder average of the partition function Z N we start out with the disorder mean β h N (s) = 0 (2.4) and the disorder covariance of the classical HAMILTONian in terms of the dimensionless disorder parameter λ := β 2 v 2 /4 and the overlap between two classical spin configurations. Formula (2.1) then gives Here, the functional Z N : σ → Z N (σ ) is a random variable with respect to the underlying N POISSON processes and defined by (2.11) Equation (2.9) reflects the GAUSSianity of the disorder average with (2.4). Interchanging now the expectation with the (two-fold) time integration according to the FUBINI-TONELLI theorem, using (2.5), and observing (s i ) 2 = 1 yields (2.10). By 0 ≤ Q N (s, s) 2 ≤ 1 the two-fold integral P N is a [0, 1]-valued random variable so that we have the crude estimates (2.12) Somewhat to our surprise, we have not succeeded in calculating f ann N explicitly, not even for N → ∞, see however, Theorem 4.3 and Theorem 5.3 below. For the time being we derive certain estimates and properties of f ann N . For the formulation of the corresponding theorem we introduce some notation. We begin with the function µ : Finally, we introduce two positive sequences by Lemma 2.1 (Inequalities between m and p, and bounds on G N ) For any N ≥ 2, β b > 0, and λ > 0 we have the inequalities Proof The first five inequalities in (2.17) are obvious. The last one is a consequence of the elementary inequalities sinh(x) ≥ x + x 3 /6 and tanh(x) ≥ x − x 3 /3 for x ≥ 0. The first inequality in (2.18) follows from the convexity of the exponential and the JENSEN inequality for a {0, 1}-valued BERNOULLI random variable taking the "success" value 1 with probability p N ∈ [0, 1]. For the second inequality we use 1 + p N (e λ − 1) N ≤ 1 + p N (e Nλ − 1) = exp(G N ) by the convexity of the N-th power x → x N for x ≥ 0 and the JENSEN inequality. The last inequality is an application of The dimensionless quantity β f ann N depends on the disorder strength v only via the variable λ > 0. The function λ → β f ann N is concave, is not increasing, and has the following weak-and strong-disorder limits Proof (a) The claimed inequalities (2.21) are equivalent to For later reference we prove this probabilistically and estimate the POISSON expectation in (2.1) from below by restricting it to the single realization without any spin flip (that is, without any jump) in the time interval [0, 1]. This realization occurs if, and only if, the random variable ∏ N i=1 1(σ i ), defined by 1(σ i ) := 1 if σ i (t) = 1 for all t ∈ [0, 1] and 1(σ i ) := 0 otherwise, takes its maximum value 1. The probability of this event is The first sentence is obvious by (2.27). This equation also shows that β f ann N is concave in λ , because the right-hand side of (2.27) is convex by the HÖLDER inequality. The monotonicity in λ then follows from the concavity and β f ann N ≤ − ln 2 cosh(β b) for all λ > 0 by (2.21) with obvious equality in the limiting case λ = 0. The claim (2.23) follows from (2.21) and lim λ ↓0 G N /(λ N) = p N , see (2.18). The claim (2.24) follows from (2.22) and the lower estimate in (2.21) by using lim λ →∞ G N /(λ N) = 1, see (2.19).  [S05]. (iii) The parameter p on both sides of (2.21) is actually a bijective function of the product β b > 0, see (2.15). It is strictly decreasing, approaches its extreme values 1 and 0 in the limiting cases β b ↓ 0 and β b → ∞, respectively, and attains the value 1/2 at β b = 1.19967 . . . , more precisely, at the solution of β b tanh(β b) = 1, see Figure 2.1. In the first limiting case the estimates (2.21) yield the well-known result for the zero-field SK model, given by the right-hand side of (2.22). They also guarantee that f ann N coincides with the free energy of the ideal paramagnet in the absence of disorder (λ ↓ 0). In the opposite limit of extremely strong disorder (λ → ∞) the result (2.24) shows that the magnetic field becomes irrelevant in agreement with the zero-field SK model and "physical intuition" for the case β b ≪ β v. (iv) The lower estimate in (2.21) may be sharpened by replacing G N with the less explicit bound where w N (x) := N/π exp − Nx 2 defines the centered GAUSSian probability density on the real line Ê with variance 1/(2N). In fact, we have the two inequalities functional of Ê-valued POISSON random variables K N : σ → K N (σ ) is convex by the HÖLDER inequality. The first inequality now follows from (2.26), (2.27), the JENSEN inequality applied to the two-fold integration in (2.11), a GAUSSian linearization using w N , and an explicit calculation, confer the proof of Lemma 6.1 below. The first step in the proof of the second inequality is the same as in the proof of the first inequality. But then, instead of performing the GAUSSian linearization, we use (2.29) with P N replaced by Q N σ (t), σ (t ′ ) 2 , combine this with (2.28), and finally apply again the JENSEN inequality to the two-fold integration, but this time using the concavity of the logarithm. We note that G N (like G N ), for any N ≥ 2, may be viewed to depend on β b only via p because the function β b → p is bijective.
For any λ > 0 we also have the lower bound (2.36) (c) A simple condition implying strict positivity of the lower bound in (2.35) is (2.37) The proof of Theorem 2.4, given below, is based on the lower estimate in (2.21), the estimate (2.22), certain quasi-classical estimates for the (random) free energy β f N ≡ β f N (β b, β v), divided by the temperature, and on the (so-called replica-symmetric) SK approximation k(λ ) − λ − ln(2) to the macroscopic quenched free energy of the (zero-field) SK model [SK75]. This approximation provides a lower bound on β f N (0, β v) for all β v and N ≥ 2 according to GUERRA, see [G01, Ineq. (5.7)], [G03], and also [T11a, Thm. 1.3.7]. Since the quasi-classical estimates are of some independent interest, we firstly compile them in Lemma 2.5 (Quasi-classical estimates for the free energy) is due to the elementary inequalities cosh(y) ≤ exp min{y 2 /2, |y|} for y ∈ Ê. Here the second inequality is obvious and the first one follows by comparing the two associated TAYLOR series' termwise and using n!2 n ≤ (2n)! for n ∈ AE. ⊓ ⊔ Remark 2.6 (i) The estimates (2.38) are quasi-classical, because the quantum fluctuations lurking behind the anti-commutativity S z i S x i = −S x i S z i , equivalently behind the randomness of the POISSON process N i , are neglected by the first estimate and taken somewhat into account by the second one in terms of an effective disorder parameter β pv ≤ β v. The last estimate in (2.38) corresponds to the limiting cases v ↓ 0 and b → ∞. The estimates (2.39) control in a simple way the influence of the transverse magnetic field on the values attainable by the free energy.

Proof (Theorem 2.4)
(a) The first bound in (2.34) is (1.6). For the second bound in (2.34) we take the disorder average of the second estimate in (2.38) and combine the result with the lower estimate in (2.21). This gives Finally, we use the disorder average of the last (crude) estimate in (2.38) The simplified last bound in (2.34) is due to (2.19). (b) We begin by claiming the lower estimate (2.42) It simply follows by combining the disorder average of the second estimate in (2.39) with GUERRA's lower bound on β f N (0, β v) , that is, with (2.42) for b = 0. The claim (2.35) now follows by combining (2.42) with (2.22).
⊓ ⊔ Remark 2.7 (i) The upper bound in (2.34) has an underlying random version, namely It simply follows by combining the lower estimate in (2.21) with the estimate in Remark 2.6 (ii).
(ii) We recall some more or less well-known properties of the function k : The lower bound 0 follows from restricting the maximization in (2.36) to q = 0, the other lower bound has just been derived in the proof of Theorem 2.4 (c), and the upper bound follows from using cosh(y) ≥ 1 in (2.36). Second, we have the equivalence 4λ ≤ 1 ⇔ k(λ ) = 0. It follows by considering the first two derivatives with respect to q of the function to be maximized in (2.36). These can be studied via GAUSSian integration by parts. Third, for 4λ > 1 the function k is strictly increasing according to its first derivative k ′ (λ ) = q(λ ) 2 ≥ 0 where q(λ ) is the unique strictly positive solution of q = tanh(g 12 4λ q) 2 , sometimes called the (zero-field) SK equation. (iii) Obviously, any sharpening of (2.41) or (2.42) improves the bound in (2.34) or (2.35), respectively.
3 The topics of Section 2 in the macroscopic limit From now on we are mainly interested in the macroscopic limit N → ∞. The next theorem is the main result of this section and analogous to Theorem 2.2.
Theorem 3.1 (On the macroscopic annealed free energy) (a) For any λ > 0 the macroscopic limit of the annealed free energy exists, is given by and obeys the three estimates The dimensionless limit β f ann ∞ depends on the disorder strength v only via the dimensionless variable λ > 0. The function λ → β f ann ∞ is concave, is not increasing, and has the following weak-and strong-disorder limits The difference between the (macroscopic) annealed free energy and the ideal paramagnetic free energy vanishes for any v > 0 in the high-field limit according to Remark 3.2 (i) For the lower estimate in (3.2), Lemma 2.1 implies the bounds which follows from (3.9) by choosing M ≥ 2/p and observing p ≤ 1. It is smaller than λ if and only if p < (2/3)(1 − e −2λ ). (ii) The infimum over M ≥ 2 in (3.8) is attained and depends on λ . The smaller λ is, the larger is the minimizing M. In particular, if λ < 1/3 then, and only then, the minimizing M is larger than 2. (iii) Returning to inf N≥2 G N /N itself, the minimizing N ≥ 2 depends on λ and, via p, also on β b. In the weak-and strong-disorder limits we have for g(λ ) := inf N≥2 G N /(λ N). The strong-disorder limit is obvious from (3.9). For the proof of the weak-disorder limit we use (3.8) to obtain lim Taking now the infimum over M ≥ 2 and observing p ≤ g(λ ) from (3.7) completes the proof of (3.11). For intermediate values of λ in the sense that 2λ < ln(1 + 2/p), equivalently G 4 /4 < G 2 /2, the infimum in (3.2) is attained at some N ≥ 3. A numerical approach suggests that the minimizer is N = 2 for all λ ≥ 1 if β b ≤ 1/2. In this context we recall from (2.25) the inequality F 2 ≤ G 2 , for all λ and β b, and from (2.33) and (2.26) that F 2 can be calculated exactly. However, while −F 2 /2 provides a sharper lower bound in (3.2) than −G 2 /2, it is a less explicit function of λ and β b.
Proof (Theorem 3.1) (a) For the proof of (3.1) we show that the sequence (F N ) N≥2 , as defined in (2.26), is subadditive. Indeed, for two arbitrary natural numbers N 1 , N 2 ≥ 2 and classical spin config- by the convexity of the square and the JENSEN inequality. By combining this with (2.27), (2.10), and (2.6) and by using the independence of the involved POISSON processes we get the claimed sub-additivity, F N 1 +N 2 ≤ F N 1 + F N 2 . According to FEKETE [F23] (see also [K02,Lem. 10.21]) this establishes the convergence result where we have used also (2.25) and (2.16). By (2.26) this gives the claim (3.1). The estimates (3.2) and (3.3) follow from (3.1) applied to (2.21) and (2.22), respectively. (b) The first sentence follows from (3.1) and the corresponding one in Theorem 2.2 (b). The function λ → β f ann ∞ is concave and not increasing, because it is the pointwise limit of a family of such functions according to (3.1) and Theorem 2.2 (b). The claim (3.4) follows from (3.2) and (3.11). The claim (3.5) follows from (3.3), the lower estimate in (3. This follows by combining (3.6) with (1.6) and the disorder average of the estimate in Remark 2.6 (ii) in the limit N → ∞. Here we rely on the fact that lim N→∞ [ f N ] exists for all λ > 0. For the classical limit b = 0 this has been shown by GUERRA and TONINELLI in their seminal paper [GT02]. Its extension to the quantum case b > 0 is due to CRAW-FORD [C07] by building on [GT03]. For λ < 1/4 this extension also follows from our main result ∆ ∞ := lim N→∞ [ f N ] − f ann N = 0 obtained by probabilistic arguments in Theorem 6.3 below. Returning to the high-field relation for the macroscopic quenched free energy, we note that it is consistent with the inequality (3.14) following from the disorder averages of (2.38) and (2.39) for any b > 0. Equality in (3.14), for given v > 0, only holds in the limiting cases b ↓ 0 and b → ∞, as follows from (2.38) and the strict concavity of lim N→∞ [β f N ] in b ∈ Ê. This contrasts the quantum random energy model (QREM) for which equality holds in the analog of (3.14) for all b (and v) according to GOLDSCHMIDT in [G90]. Although the QREM is much simpler than the quantum SK model (1.1), a rigorous proof of his statement was achieved only recently by MANAI and WARZEL in [MW20].
(iii) In the limit N → ∞ the bounds in Theorem 2.4 take the form The upper bound in (3.15) is due to the disorder average of (2.38) and the lower estimate in (3.2). At the expense of weakening this bound for intermediate values of λ , it can be made somewhat more explicit with the help of (3.8) and (3.9) or (3.10) for small and large λ , respectively. In any case, by (3.7) the upper bound in (3.15) is not sharp enough to vanish for λ < 1/4. But it vanishes for any λ ∈ ]0, ∞[ in the high-field limit b → ∞ due to (3.10). (iv) According to the monotonicity mentioned in Remark 2.7 (ii) the lower bound in (3.15) is strictly positive for sufficiently large λ > 1/4, in particular, if β b min{β b/2, 1} < λ − 8λ /π, see (2.37). It follows from (3.15) that the difference ∆ ∞ ≥ 0 between the macroscopic quenched and annealed free energies is strictly positive for any pair Physically more important is the situation of a fixed v > 0. Then strict positivity holds for any b ≥ 0 provided that β > 0 is sufficiently large and, conversely, for any β > 1/v provided that b > 0 is sufficiently small. Nevertheless, the lower bound in (3.15) is not sharp enough to characterize the The latter can be seen by combining the main result in [ALR87] (or Theorem 6.3) with (2.42) for b = 0 and the equivalence in Remark 2.7 (ii). These facts are illustrated in Figure 3.1, where we also have included the result of Theorem 6.3 and a cartoon of the critical line, that is, the border line between the spin-glass and the paramagnetic phase as obtained by approximate arguments and/or numerical methods, for example in [ ≥ 0 is strictly positive (light gray) according to (3.15) and another one where it is zero (heavy gray) according to Theorem 6.3. The (red) dashed line is a cartoon of the critical line between the spin-glass and the paramagnetic phase as obtained by approximate arguments and/or numerical methods, see Remark 3.3 (iv). The region with ∆ ∞ > 0 is larger than the light gray region, but we do not know how large. It should at least contain the critical line.

A dual pair of variational formulas for the macroscopic annealed free energy
In the last section we have seen that the macroscopic annealed free energy f ann ∞ exists and obeys explicitly given lower and upper bounds which become sharp in the limits of weak and strong disorder, λ ↓ 0 and λ → ∞, respectively. Furthermore, the bounds in (3.2) coincide asymptotically also in the limits of low and high field, that is, b ↓ 0 and b → ∞. However, so far we have no formula which makes the λ -dependence of β f ann ∞ more "transparent" for general λ > 0 and b > 0. This will be achieved, to some extent, in the present section. More precisely, we will show that β f ann ∞ may be viewed as the global minimum of a non-linear functional with a simple λ -dependence and defined on the HILBERT space of real-valued functions being square-integrable over the unit square. This follows from an asymptotic evaluation of the right-hand side of (2.27) as N → ∞ by large-deviation techniques due to VARADHAN [V66] and others, see [DS89,DZ10] and also [K02].
Lemma 4.1 (Some properties of the functionals Λ and Λ * ) For any ψ, ϕ ∈ L 2 , with µ ∈ L 2 as defined in (2.13), and with 1 ∈ L 2 denoting the (constant) unit function we have for Λ * the equality and inequalities Moreover, we have the properties: Here, the non-linear mapping Λ ′ : This function is in its argument (t,t ′ ) continuous and exchange symmetric (t ↔ t ′ ). Moreover, for ψ ≥ 0 (pointwise) it satisfies the equality and inequalities and also weakly sequentially continuous, that is, sequentially continuous with respect to the weak topology on L 2 . Furthermore, the functional Λ has at any ψ ∈ L 2 the linear and continuous GÂTEAUX differential The functional Λ * is convex and so is its non-empty set D * . Furthermore, Λ * is weakly lower semi-continuous and its lower-level sets D * r := {ϕ ∈ L 2 : Λ * (ϕ) ≤ r} for r ∈ [0, ∞[ , are non-empty, convex, weakly sequentially compact, and weakly compact.
Eventually we are prepared to present the main result of this section. (a) For any λ > 0 the limit β f ann ∞ of the dimensionless annealed free energy satisfies the following two equivalent variational formulas: and obeying the (pointwise) bounds 2λ µ ≤ ψ λ ≤ 2λ 1 (so that 2λ √ p ≤ ψ λ ≤ 2λ ).
Corresponding properties hold for the minimizer in (4.19) by (b).
Proof (a) At first we note that the infimum in (4.19) is lower bounded by −λ , as follows from Λ * (ϕ) ≥ 0 and ϕ, ϕ ≤ 1 for ϕ ∈ D * , see Lemma 4.1(f). By (4.19) and (3.9) this lower bound may be recognized as a weakened version of the one in (3.2). From this lemma and Remark 4.2 (ii) we also recall that Λ is convex and weakly lower semicontinuous, two properties which are well-known to be shared by the L 2 -norm and its square (due to the SCHWARZ and the JENSEN inequality using (2.11), (2.6), and the empirical averages ξ N introduced above Lemma 4.1. In view of (2.26) and (3.1) we need to show that (4.23) To this end, we observe that the sequence (ξ N ) N≥1 satisfies a large-deviation principle (LDP) with convex (good) rate functional Λ * with respect to the weak topology on L 2 . Equivalently, this LDP is satisfied by the sequence of BOREL § probability mea-  [K02,Thm. 27.10]), as an extension of the classic asymptotic method of LAPLACE, then yields lim N→∞ F N /N = sup ϕ∈B 1 λ ϕ, ϕ − Λ * (ϕ) . Here we have used the norm-continuity of the squared norm ϕ → ϕ, ϕ = ϕ 2 and the fact that the measure N is supported on the closed unit-ball B 1 ⊂ L 2 because ξ N ≤ 1. Since λ ϕ, ϕ − Λ * (ϕ) = −∞ for all ϕ ∈ B 1 \ D * , the desired equality (4.23) follows. (b) At first we show that the infimum I := inf ψ∈L 2 Ω (ψ) > −λ in (4.20) is attained. Here, defines the underlying non-linear functional Ω : L 2 → Ê. We will use two properties of Ω . By Λ (ψ) ≤ ψ from (4.3) we have Ω (ψ) ≥ ψ ψ − 4λ /(4λ ) so that Ω is (super-)coercive, that is, lim ψ →∞ Ω (ψ)/ ψ = ∞. Furthermore, we claim that Ω is w. s. l. s. c. in the sense of Remark 4.2 (ii), because it is the sum of two functionals with this property. For the first functional, the squared norm, this is well-known. Namely, if (ϕ n ) n≥1 ⊂ L 2 converges weakly to ψ ∈ L 2 , then we get ψ 2 ≤ lim inf n→∞ ϕ n 2 from the obvious inequality ψ 2 ≤ ϕ n 2 + 2 ψ 2 − 2 ψ, ϕ n . The second functional, −Λ , is w. s. l. s. c., because Λ is even weakly sequentially continuous by Lemma 4.1 (d). Now we choose an arbitrary sequence (η j ) j≥1 ⊂ L 2 infimizing Ω in the sense that lim j→∞ Ω (η j ) = I. Since the (converging) sequence Ω (η j ) j≥1 ⊂ [−λ , ∞[ is bounded, § The norm and weak topologies on L 2 induce the same BOREL sigma algebra of events [E77]. ¶ This functional LDP is one natural extension from Ê d -to L 2 -valued random variables of the pioneering refinement of the weak law of large numbers due to CRAMÉR (1938) and H. CHERNOFF (1952). the coerciveness of Ω implies the same for the underlying sequence (η j ) j≥1 , see [BC17,Prop. 11.20]. Therefore this has at least one sub-sequence (ϕ n ) n≥1 , ϕ n := η j(n) , weakly converging to some limit in L 2 , see [BC17, Lem. 2.45]. We name this (unknown) limit ψ λ , use the fact that Ω is w. s. l. s. c., and conclude by a well-known beautiful argument going back to BOLZANO and WEIERSTRASS, confer for example [BB92]: (4.25) To summarize, each weak accumulation point of any infimizing sequence is a minimizer.

The macroscopic annealed free energy for weak disorder
Unfortunately, we do not know explicitly a single minimizer in (4.19) or (4.20) if λ > 0. § In this section we therefore compare the global minimum Ω (ψ λ ) = β f ann ∞ + ln 2 cosh(β b) , see Theorem 4.3 (b), to its simple upper bound Ω (2λ µ), that is, to the functional (4.24) evaluated at ψ = 2λ µ. Fortunately, it turns out that Ω (2λ µ) not only shares with Ω (ψ λ ) the properties of convexity and monotonicity in λ , but also constitutes a very good approximation to Ω (ψ λ ) for small λ . More precisely, their respective asymptotic expansions, as λ ↓ 0, coincide up to the second order. ‡ The corresponding second-order coefficient turns out to be a rather complicated function of β b > 0 taking only negative values larger than −0.14, see Figure 5.1 below. The main drawback of Ω (2λ µ) is the fact that it does not yield the true behavior of Ω (ψ λ ) for large λ . However, due to our main result Theorem 6.3 in the next section, it is (presently) only the weak-disorder regime, 4λ < 1, for which Ω (ψ λ ) is known to be physically relevant. We begin with (c) has the second-order TAYLOR formula with the variance  4). (c) We use the expectation ( · ) λ := ( · ) β b,2λ µ , see (4.7). Since q ∈ [0, m], the convex function λ → Λ (2λ µ) = ln e 2λ q 0 is arbitrarily often differentiable and its secondorder TAYLOR formula (at λ = 0 with remainder in LAGRANGE form) affirms that for each λ > 0 there exists some (unknown) number a ∈ ]0, 1[ such that The claim now follows from q 0 = p and an explicit calculation of q 2 0 . The latter is based on (2.13), (A.7) in Appendix A for σ 1 (t 1 )σ 1 (t 2 )σ 1 (t 3 )σ 1 (t 4 ) 0 , and a straightforward but somewhat tedious integration over the four-dimensional unit-cube [0, 1] 4 . (d) The limit (5.5) follows from (5.2).
⊓ ⊔ Remark 5.2 (i) Since the explicit expression (5.4) for the variance c 0 is rather complicated, we mention its simple bounds according to (5.8) In the notation of the proof of Lemma 5.1 the lower bound follows from c 0 = (q − p) 2 0 ≥ (q − p) 2 1(σ 1 ) 0 = (m − p) 2 1(σ 1 ) 0 , confer the proof of (2.22). The upper bound simply follows from q 2 = qq ≤ mq. The lower bound implies the expected strict positivity of c 0 (for β b > 0). The upper bound shows that c 0 vanishes (only) in the limiting cases β b ↓ 0 and β b → ∞, and is smaller than p/4. As a function of β b the variance c 0 is continuous and attains its maximum value 0.0695 . . . at β b = 0.9089 . . .
The rather explicit lower estimate in (5.1) shares with Ω (2λ µ) the properties of concavity and monotonicity. It also has the same leading asymptotic behaviors in the limits of small and large λ and b. But the second-order coefficient of its small-λ TAYLOR series is (already) smaller and given by −2(m − p)p, confer (5.3) and (5.8). Nevertheless, the (positive) difference between Ω (2λ µ) and the lower estimate in (5.1) does not exceed 2(m − p)λ min{mλ , 1} for all λ . This follows from (5.1) combined with (2.20), respectively with p/m ≤ 1. (iii) Obviously, (5.5) does not reflect the true strong-disorder limit (3.5) of Ω (ψ λ ) because 2m − p ≤ m 2 /p < 1. But by generalizing Ω (2λ µ) to the one-parameter variational expression min x∈[0,1] Ω 2λ (µ + x(1 − µ)) ≤ Ω (2λ µ) this limit may be included (for x = 1) without changing the first two terms on the right-hand side of (5.3). The next theorem shows that Ω (2λ µ) is a very good approximation to the global minimum Ω (ψ λ ) = β f ann ∞ + ln 2 cosh(β b) for small λ , see Theorem 4.3 (b) and (4.24). Theorem 5.3 (The macroscopic annealed free energy up to second order in λ ) We have the following error estimates and the two-term asymptotic expansion for weak disorder with c 0 given by (5.4) and the usual understanding of the LANDAU big-Oh notation that O(λ 3 ) stands for some function of λ with lim sup λ ↓0 |O(λ 3 )|/λ 3 < ∞.
(iii) In view of (4.21) combined with (4.12) the (unique) minimizer ψ λ of Ω for 2λ < 1 may be determined (numerically) with arbitrary precision by the successive approximations (5.13) The (norm-)convergence of this minimizing sequence is exponentially fast according to (5.14) Here, the first inequality follows from (4.12) by mathematical induction. For the next inequalities see the proof of Theorem 5.3. || Using ϕ = ψ (n) λ in (4.26) combined with (5.14) yields an approximation to the macroscopic annealed free energy with an error not exceeding (2λ ) 2n+1 /2. For n = 1 we get back to (5.9). (iv) By Theorem 5.3 we know that Ω ψ (1) λ coincides with Ω (ψ λ ) up to the order λ 2 , as λ ↓ 0. By (5.14) we see that ψ (2) λ coincides with the minimizer ψ λ up to the same order. Therefore it is of interest to determine ψ (2) λ up to that order. To this end, we recall that for each non-zero η ∈ L 2 the mapping ψ → η, ψ η defines a positive rank-one operator on L 2 , which we denote by |η η| following DIRAC. The POISSON average E := |ξ 1 ξ 1 | β b of the projection |ξ 1 ξ 1 | is a positive integral operator with a continuous [0, 1]-valued kernel given by By an extension of (5.7) we now have for any ϕ ∈ L 2 with the derivative Hence, we arrive at with 1 denoting the identity operator on L 2 . Like the first-order term in (5.17) also the second-order term is a continuous, exchange symmetric, and positive L 2 -function, in agreement with Theorem 4.3 (c). The first two properties are directly inherited from the integral kernel of the operator E. The positivity follows from the (pointwise) inequality Eψ ≥ µ, ψ µ for all ψ ≥ 0 due to (4.13). The function Eµ can be calculated explicitly. By (5.4) we have, in particular, µ, Eµ = c 0 + p 2 . Further properties of E are given by the operator inequalities 0 ≤ |µ µ| ≤ E ≤ 1 and the equality tr E = 1 for its trace. Consequently, the uniform norm of the operator difference A := E − |µ µ| ≥ 0 obeys A ≤ tr A = 1 − p.
Sometimes variational problems in function spaces like (4.20) are drastically simplified by restricting the set of all allowed variational functions to the one-parameter subset of functions of the form ψ = y1 where 1 is the constant unit function (in L 2 for the present case) and y ∈ Ê is arbitrary. This is often called, for an obvious reason, the static approximation, behavior for small λ , not even up to the first order in λ . As opposed to that, Ω (2λ 1) has the same strong-disorder limit as Ω (ψ λ ) which, however, does not reflect the true behavior of the macroscopic (quenched) free energy in this limit. The main properties of the static approximation to the macroscopic annealed free energy are compiled in for all y ∈ Ê. The second inequality follows from (2.20). This gives The proof of (5.21) is completed by completing the square in (5.24) and dividing by 4λ . (d) The strong-disorder limit follows from (5.20), the lower estimate in (5.19), and (3.11).
For the weak-disorder limit we start from m 2 /(r −1) ≤ lim inf λ ↓0 J(λ )/λ by (5.21), take the supremum over r ∈ ]0, 1[, and observe the upper estimate in (5.19). ⊓ ⊔ Remark 5.6 (i) The small-λ estimate (5.21) is not only useful for the proof of the weakdisorder limit in (5.22), which differs from the true limit in (5.10) since m 2 < p, but it also implies that J(λ ) is strictly larger than the minimum Ω (ψ λ ) in (4.20) for sufficiently small λ > 0. More precisely, by choosing r < 1 − (m 2 /p) we get from (3.2), (4.20) with (4.19), and (5.21) that It follows from a GAUSSian linearization and a consequence of the PFK formula, see Remark B.2 (iii) in Appendix B. The restriction to y ≥ 0 in (5.26) causes no problem, because one may restrict to x ≥ 0 in (5.18) without losing generality. This follows from Λ (ψ) ≤ Λ (|ψ|) in (4.3). (iii) For related models without disorder one may restrict to constant variational functions without losing generality as has been shown in [D09,CCIL08]. In particular, for the quantum CW model (defined by (1.1) with non-random g i j = 1/ √ N) this observation provides one, but not the simplest, rigorous approach to its well-known macroscopic free energy and to the equation tanh(β b) = b/v of its critical line [BMT66,S86].
6 The macroscopic free energy and absence of spin-glass order for weak disorder In this section we are going to prove that for weak disorder, more precisely for any 4λ (= β 2 v 2 ) in the open unit interval ]0, 1[ and any β b > 0, the free energy f N coincides almost surely with the annealed free energy f ann N in the macroscopic limit N → ∞. We begin by comparing the first and the second moment of the partition function with respect to the GAUSSian disorder average. By the positivity of general variances we know that [Z N ] 2 ≤ (Z N ) 2 . In the present case of (1.1) we also have provided that 4λ < 1. This is a special case of the following lemma, which in its turn is an extension of [T11b, Lem. 11.2.3] for the zero-field SK model to the present (quantum) case with a transverse field. For its formulation we recall definition (2.6) and introduce three "tensor expectations". We write (·) ⊗ β b for the joint (conditional) expectation with respect to the given set {N 1 , . . . , N N } of POISSON processes and an independent copy (or replica) { N 1 , . . ., N N } thereof. The joint GIBBS expectation (·) ⊗ corresponding to the duplicated quantum SK model with HILBERT space 2 N ⊗ 2 N and HAMILTONian defined as the sum of H N ⊗ ½, see (1.1), and a copy ½ ⊗ H N thereof (with spin operators S α i , but the same random variables (g i j ) 1≤i< j≤N and parameters b, v) then, in the spin-flip process representation, takes the form Two simple examples for the expectation (·) ⊗ β b , revealing the (dynamical) independence and symmetry between the original SK model and its copy, are given by Lemma 6.1 (Controlling a generalized second moment of Z N by its first moment) For any N ≥ 2, λ > 0, and a ≥ 0 with 4aλ < 1 we have with the [0, 1]-valued random variable R N := 1 0 dt 1 0 dt ′ Q N sσ (t), s σ (t ′ ) 2 , see (2.6).
Proof Throughout the proof we will, without mention, repeatedly interchange the order of various integrations according to the FUBINI-TONELLI theorem. In a first step, we observe the following identity for the GAUSSian disorder average Here we have used (2.4), (2.5), (2.10), and (2.8). The left hand-side (LHS) of (6.5) can therefore be written as In a second step, we use the JENSEN inequality and the linearization formula with the GAUSSian probability density w N given by w N (x) = N/π exp − Nx 2 , as in Remark 2.3(iv). By combining (6.7), (6.8), and (6.9) we get (6.10) By observing the identities and (2.7), the inequality (6.10) takes the simpler form By the crude inequalities (cosh(y)) N ≤ exp(N|y|) ≤ exp(Ny) + exp(−Ny) for y ∈ Ê the last integral is seen to be bounded from above by 2 exp(2Naλ ) for arbitrary a ≥ 0. If 4aλ < 1, then it even has the N-independent upper bound 1/ √ 1 − 4aλ as claimed in (6.5). It is due to the inequality cosh(y) ≤ exp y 2 /2 mentioned at the end of the proof of Lemma 2.5. ⊓ ⊔ Remark 6.2 For the zero-field SK model equality holds in (6.8) and hence in (6.12), because b = 0 implies σ i (t) = 1 for all t ∈ [0, 1] and all i ∈ {1, . . . , N}, see Remark 2.3 (i). Inequality (6.5) will be applied with a suitable a > 1 in the proof of Corollary 6.4 below. The choice a = 0 leads to equality in (6.5), see (6.7) and/or (6.8).The special case a = 1, see (6.1), is the main ingredient for the proof of the next theorem. This theorem and its two corollaries extend two of the pioneering results of AIZENMAN, LEBOWITZ, and RUELLE in [ALR87] for the zero-field SK model, see also [FZ87,CN95] and [T11b,Ch. 11], to the present (quantum) model with a transverse field of arbitrary strength b > 0.
Theorem 6.3 (The macroscopic quenched free energy for weak disorder) If 4λ < 1, then the macroscopic limit of the quenched free energy exists and is given by that of the annealed free energy, in symbols (6.14) Proof By Theorem 3.1 it is sufficient to show that the (positive) difference ∆ N := [ f N ] − f ann N tends to 0 as N → ∞. In order to do so we adopt the so-called second-moment method as applied in [T11b,Ch. 11] to the zero-field SK model. For this method to work we build on the large-deviation estimate of Lemma C.2 in Appendix C and on the elementary PALEY-ZYGMUND inequality [PZ32] (see also [K02,Lem. 4 We begin by rewriting the given (non-random) difference as the sum of two random differences (6.16) Next we show that there exist constants ε > 0 (independent of N) and γ N > 0 (with γ N ↓ 0 as N → ∞) such that the probability of finding the right-hand side of (6.16) to be smaller than γ N , is larger than ε. This then yields ∆ N ≤ γ N and hence lim N→∞ ∆ N = 0. In fact, with an (initially) arbitrary energy δ > 0 we estimate as follows: Here (6.17) is due to (6.15) with X = Z N and q = 1/2, combined with (6.1), and due to the large-deviation estimate (C.2) in Appendix C using N − 1 < N and replacing δ by β δ / √ N. For (6.18) we have used the inclusion-exclusion formula for two sets/events and the fact that probabilities do not exceed the value 1. Inequality (6.19) is just the monotonicity of (probability) measures. Finally we choose δ so large that ε := 1/(4c) − 2 exp − δ 2 /(2v 2 ) > 0 and put γ N := ln(2)/(β N) + δ / √ N. ⊓ ⊔ Theorem 6.3 has two important consequences. Proof By applying the JENSEN inequality to the left-hand side of (6.5) with respect to the joint expectation [ (·) ⊗ β b ] we obtain (for any N ≥ 2) (6.23) For (6.22) we have used spin-index symmetry and for (6.23) we refer to the example (6.4). By combining this with (6.5) we get under the assumption 4aλ ∈ [0, 1[ of Lemma 6.1. For given 4λ ∈ ]0, 1[ we now choose an arbitrary a ∈ ]1, 1/(4λ )[ = / 0. Then the claim (6.20) follows from (6.24) by observing S z 1 S z 2 2 ≥ 0 and Theorem 6.3. ⊓ ⊔ Remark 6.5 Following [ALR87], see also [PS91,WB04], the left-hand side of (6.20) is the mean of the spin-glass order parameter in the macroscopic limit, because the pre-limit S z 1 S z 2 2 is, by spin-index symmetry, identical to the disorder average of the [0, 1]valued random variable using the (squared) quantum analog of the overlap (2.6) and its GIBBS expectation ( · ) ⊗ induced by the model (1.1) upon duplication. In the spin-flip-process representation this identity takes the form (6.22) (for a = 1). By q 2 N ≤ q N and Corollary 6.4 also the variance By Theorem 6.3 the first difference on the right-hand side tends to 0 as N → ∞. Moreover, the large-deviation estimate (C.2) in Appendix C implies the summability for any δ > 0. A simple and standard application [B96, §11, Example 1] of the easy part of the BOREL-CANTELLI lemma (see [B96,Lem. 11.1] or [K02,Thm. 3.18]) now shows that also the second difference in (6.27) tends to zero, È-almost surely.
⊓ ⊔ Remark 6.7 In the above proof we have used the fact that the summability (6.28) implies the almost-sure relation lim N→∞ ( f N − [ f N ]) = 0. Clearly, the summability and hence the relation hold for arbitrary λ > 0. It may be dubbed as "self-averaging in the mean" of the sequence ( f N ) N≥2 . The physically indispensable self-averaging (or ergodicity) in the sense of the almost-sure relation lim N→∞ f N = lim N→∞ [ f N ] additionally requires the existence of one of the latter limits. Until now the model (1.1) seems to be the only quantum mean-field spin-glass model for which the second limit is known to exist. For arbitrary λ > 0 this is due to CRAWFORD [C07]. Theorem 6.3 above provides for 4λ < 1 a (variational) formula for the limit and therefore its existence for the weak-disorder regime as a by-product, similarly as in [ALR87] for the case b = 0. In view of the complexity of the PARISI formula [P80a, P80b, D81, T06, P09, T11b, P13, AC15] ** , even for vanishing longitudinal field, we conjecture a more complicated (variational) formula to hold for 4λ ≥ 1 and b > 0. ** The last three references also contain results for classical SK models with multi-spin interactions.

Concluding remarks and open problems
The present paper contains the first rigorous explicit results on the thermostatics of the quantum SHERRINGTON-KIRKPATRICK spin-glass model (1.1) for the regime β v < 1. Unfortunately, the opposite (and more important) regime remains not nearly as well understood as in the "classical limit" b ↓ 0, at least from a rigorous point of view. Over the 35 years of research several investigators have provided stimulating and possibly correct results by approximate arguments and/or numerical methods. But for low temperatures these results are typically less reliable, for example due to the unjustified interchange of various limits and/or because of too small "LIE-TROTTER numbers". Therefore one should find rigorous arguments for the actual shape of the (red) dashed line in Figure 3.1. In view of Remark 4.4 (v) it is tempting to conjecture that the assertions (6.14), (6.20), and (6.26) remain true under the (b-dependent) condition β v < 1/m. This would enlarge the heavy gray region in Figure 3.1 slightly beyond the vertical line β v = 1 and help to "localize" the critical line somewhat further. In any case, the precise determination of this line is a demanding problem. A similar challenge is to aspire after the analog of the PARISI formula for the macroscopic (quenched) free energy of the quantum SK model (1.1). A first step in this direction has been achieved recently by ADHIKARI and BRENNECKE [AB20], see the end of Section 1. At present we do not know, how their variational formula reduces to our (4.19) if β v < 1. Unfortunately, for β v ≥ 1 we only have the inequalities (2.38) which may be used to bound the free energy of (1.1) from below and above in terms of the zero-field PARISI formula. Nevertheless, it could be that the quantum analog of the PARISI formula is in certain respects simpler than the classical one because of quantum fluctuations, confer [RCC89,BU90b,MRC18].

A The positivity of certain POISSON-process covariances
For the proofs of (2.13), (4.13), and related facts it is convenient to consider POISSON (point) processes being more general than the one used in the main text (see, for example, [K02,LP18,K93]). A POISSON process in a general sigma-finite measure space (Γ ,A,ρ) is a random measure ν on (Γ ,A). The distribution of ν is uniquely defined, in terms of the (positive) measure ρ, by the elegant and powerful formula for its LAPLACE functional, which dates back to CAMPBELL [C09]. Here and in Appendix B the angular brackets (·) denote the expectation with respect to the probability measure steering the randomness of ν in terms of the {−1,1}-valued products σ ( A) := ∏ m j=1 σ (A j ) and τ A := ∏ m j=1 1 − 2χ A j . In particular, we have σ (A j ) = exp −2ρ(A j ) by choosing A k = / 0 for all k = j. If B := {B 1 ,... ,B n } ⊂ A is another arbitrary collection of n ∈ AE such sets, we obtain the positive covariance by (A.2), the pointwise inequality τ A τ B ≥ τ A + τ B − 1, and the functional equation of the exponential. A simple consequence of (A.3) by iteration is As in the main text we are going to introduce a conditional POISSON expectation. For a fixed Λ ⊆ Γ with Λ ∈ A and ρ(Λ ) < ∞ we write the two KRONECKER deltas δ σ(Λ ),±1 as δ σ(Λ ),±1 = 1 ± σ (Λ ) /2. The POISSON expectation conditional on σ (Λ ) = 1, equivalently on even ν(Λ ), can therefore be written as By (A.5) and (A.3) we immediately see that σ ( A) Λ ≥ σ ( A) . The "conditional analog" of (A.3) is Proof From (A.2) and (A.5) we get the "conditional analog" of (A.2) . Corresponding formulas hold for σ ( B) Λ and σ ( A)σ ( B) Λ . In the latter case τ A has to be replaced with the product τ A τ B and , respectively, for any t ∈ [0,∞[. We also write ( ·) β b instead of ( ·) [0,1] . It is well-known that the stochastic process N (t) : t ∈ [0,∞[ has independent and, in distribution, time-homogeneous increments N (t + u) − N (u) = ν ]t,t + u] for u ≥ 0. This implies that it is a MARKOV process, more precisely, a continuoustime homogeneous MARKOV chain with transition probabilities Also the spin-flip process σ (t) : t ∈ [0,∞[ is such a MARKOV process. Its transition probabilities are

B The POISSON-FEYNMAN-KAC formula
In this appendix we consider an independent collection of N ∈ AE POISSON processes in the positive half-line [0,∞[ with common rate β b > 0 in the sense and notation of Remark A.2 (ii). We begin with the case of a single spin. Here g ∈ Ê is an arbitrary parameter. The rest of the notation has been introduced in Section 1.
Lemma B.1 (Operator-valued PFK formula for a single spin) For a single spin we have the operator identity Proof It is enough to prove (B.1) on the single-spin HILBERT space 2 . So we suppress the spin index i. For the auxiliary operator K g (u,t) and define the operator The last integrand can be rewritten as follows Here we have moved S x N (t) to the utmost right by using N (t) times the relation S  we see that the mapping u → T 0 (u) = exp uβ b(S x − ½) is the "free" or "unperturbed" semigroup on 2 (corresponding to g = 0 and up to the factor e uβ b ). Consequently, the combination of (B.5) and (B.6) implies that u → T g (u) satisfies the same DUHAMEL-DYSON-PHILLIPS integral equation as the "perturbed" semigroup u → exp uβ b(S x − ½) + uβ gS z . Actually, this equation is equivalent to the differential equation ∂ ∂ u T g (u) = T g (u) β b(S x − ½) + β gS z with the initial condition T g (0) = ½. Since the solution is unique, the proof is completed by considering T g (1) . ⊓ ⊔ Remark B.2 (i) Formula (B.1) dates back to KAC [K74]. There he has not written down it explicitly, but it is the backbone of his PFK formula for the solution of the telegraph (or damped-wave) equation. For a modern account of this genre see [KR13] and also [CD06]. (ii) We learned the PFK formula (B.1) for a single-spin semigroup from GAVEAU and SCHULMAN [GS89] who proved it by a suitable LIE-TROTTER(-P. R. CHERNOFF) product formula. Our proof avoids timeslicing and is in the spirit of SIMON's "second proof" of the (WIENER-)FEYNMAN-KAC formula for SCHRÖDINGER semigroups [S05, Thm. 6.1], see also [R94, Sec. 2.2]. It easily extends to time-dependent integrable g : [0,1] → Ê. So does the alternative, slightly more direct, proof at the end of this appendix. that is, the operator S z i in the "imaginary-time" HEISENBERG picture. Then we have for the DUHAMEL-KUBO auto-correlation function of the z-component of a single spin the formula It is due to the (positive) product S z i (t)S z i (t ′ ) = exp 2β b(t ′ − t)S x i and (A.8) for the special case used in the main text. If t < t ′ , then the factor order of the two spin operators in (B.13) has to be reversed. It also follows that (4.13) may be viewed as an inequality for multi-time correlation functions of the z-component of a single spin which interacts with a transverse field only. For an odd number of instants these functions vanish by S z i -reversal symmetry, confer the argument immediately above Lemma 6.1. Proof Using the auxiliary operator K g (u,t) introduced in the first proof we want to show that is an operator semigroup on 2 with generator β b(S x − ½) + β gS z . In the first step, we pick u,t ≥ 0 and get T g (u + t) = S x N (u+t) K g (u + t,0) = S x N (u+t)−N (t) S x N (t) K g (u + t,t)K g (t,0) (B.18) = S x N (u+t)−N (t) K gσ(t) (u + t,t) T g (t) = T g (u)T g (t) .
for any total number of spins N ≥ 2 and any δ > 0.
Proof We interprete the coefficients (g i j ) 1≤i< j≤N in the quantum HAMILTONian H N , defined in (1.1), as the components of a non-random vector g ∈ Ê d with d = N(N − 1)/2, and write more explicitly H N (g) and f N (g) for its (specific) free energy. In view of Proposition C.1 we then only have to show that the function . To this end, we introduce the GIBBS expectation ( ·) g := e Nβ f N (g) Tr e −β H N (g) ( ·) induced by H N (g). Then the JENSEN-PEIERLS-BOGOLYUBOV inequality, see for example [S05], gives For (C.4) we have used the triangle inequality, the operator inequalities −½ ≤ S z i S z j ≤ ½, and the (JENSEN) inequality |x| 1 ≤ √ d |x| between the 1-norm and the 2-norm of x = (x 1 ,... ,x d ) ∈ Ê d . By considering the last chain of inequalities also with g and g ′ interchanged we get the desired LIPSCHITZ continuity. ⊓ ⊔ Remark C.3 A similar result was already given by CRAWFORD [C07]. We include the lemma for two reasons. First, it serves to make the present paper reasonably self-contained. Second, the above proof is simpler than the one in [C07]. It does not need the PFK spin-flip representation and can easily be extended to quantum spin-glass models with additional mean-field type interactions between the spins, for example to the quantum mean-field HEISENBERG spin-glass model with or without an external magnetic field [BM80,S81].