The marginal stability of the metastable TAP states

The existing investigations on the complexity are extended. In addition to the Edward–Anderson parameter q2 the fourth moment q4=1/N∑imi4 of the magnetizations mi is included to the set of constrained variables and the constrained complexity Σ(T; q2, q4) is numerical determined. The maximum of Σ(T; q2, q4) (representing the total complexity) sticks at the boundary for temperatures at and below a new critical temperature. This implies marginal stability for the nearly all metastable states. The temperature dependence of the lowest value of the Gibbs potential consistent with various physical requirements is presented.


Introduction
The Thouless-Anderson-Palmer (TAP) approach [1][2][3] to the Sherrington and Kirkpatrick (SK) model [4] plays a central role in the attempt to understand the physics of spin glasses and related interdisciplinary problems (neural networks, computer science, theoretical biology, econophysics etc). The system exhibits meta-stability below the spin glass temperature.
The well established work of Bray and Moore (BM) [5] leads to a finite complexity below the spin glass temperature, which implies an exponential increase of the number of metastable TAP states with increasing system size. It is essential to count exclusively the physical states and neglect non-physical ones. However, the BM and subsequent work [6][7][8] do not completely satisfy this requirement.
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An alternative method to work out the characteristic properties of the metastable TAP states are numerical investigations [7,[9][10][11][12] based on iteration techniques or phenomenological dynamics for systems of finite size. The numerical determined fix-points are interpreted as metastable TAP states. Extrapolation to infinite size systems results in the conclusion that these states are marginally stable. Such an extrapolation procedure, however, leads generally to some uncertainty. Note in this context, that the numerical investigation are usually performed for systems with just some hundreds spins. Increasing the system size results in a drastic reduce of the rate finding a TAP state.
In this work the existing approaches on the complexity are extended by the additional inclusion of the forth moment of the magnetizations to the set of constrained variables. This procedure enables a complete counting of the physical TAP states. In section 2 the results of the calculation are presented. The total complexity and various averages are discussed to some detail in section 3. Finally conclusions are drawn in section 4.

Calculation
More than four decades ago SK introduced the spin glass Hamiltonian The bonds J ij are independent random variables with zero means and standard deviations N −1/2 (which fixes the spin glass temperature to T sg = 1). According to the TAP approach [1-3] the energy the entropy and consequently the Gibbs potential G(T, m i ) = U − TS are given in terms of the local magnetizations m i and the temperature T, where q k is defined as In general the local magnetic fields h i are determined by the TAP equations h i = ∂G/∂m i . This work is exclusively restricted to the case h i = 0 and the TAP equations reduce to where the definition G i is introduced for later use. As shown by the present author [2] the m i have to satisfy the two convergence criteria Criterion c 1 is generally accepted and is related to the de Almeida Thouless condition [13] for the SK solution. Criterion c 2 is controversial [7,14].
These criteria are of some importance for the present work. They result from an application of a theorem of Pastur [15], which requires the in-dependency of the variables m i from the bonds J ij . For a Gibbs potential, indeed these magnetizations m i are the free and independent variables. Note, that for every thermodynamic stability analysis one has to study the influence of all possible m i values including the J ij independent values. This requirement is also essential for the integration procedure used in this work. Thus the application of the theorem of Pastur is justified (compare 1 ) and c 2 > 0 is a necessary (but probably not sufficient [17]) convergence condition for the expansion [2].
Further support for validity of both criteria result from the fact, that they are necessary to prove the positivity of the entropy S(T, m i ) [16]. Simple examples leading to a negative entropy, if c 2 < 0, can easily be constructed (see 2 ). Consequences of the criteria (5) to the T-dependence of q 2 and q 4 has already been presented in [18].
The present approach is related to the studies [5][6][7] of the complexity which describes the extensive number N of solutions of TAP equations (4) where G i is defined in equation (4) and · · · J denotes the J ij averaging. Constraints are considered in the term C(Ω), which is chosen in this work as and the set of constrained variables is Ω = {q 2 , q 4 , w}. The inclusion of q 4 is new, but is essential to take into account the restrictions due to criteria (5). Note that q 4 is a sum of single particle terms and the modifications due to such terms are simple. The use of w instead of the Gibbs potential is a technical tool simplifying the calculation. The further calculation is lengthy but straightforward due to the existing previous work [5][6][7]. Following these investigations the Fourier representations for the arising δ-functions and the exponential representation of the determinant using anti-commuting Grassmann variables are employed to rewrite Σ(T, Ω) in a form which permits to perform the J ij averaging via Gaussian integrations. The remaining integrations are performed by the steepest decent method which is valid due to the large N limit. Adopting the notation of [6] the calculation of the complexity finally leads to 1 Owen [14] investigates a different question, as he conciders m i , which depend on the J ij via the TAP equations and the Pastur theorem cannot be applied. There is no disagreement to [2], which investigates the Gibbs potential with free and independent m i . Moreover, it is generally impossible to conclude anything on the convergence of a series from a partial re-summation, as done in [14]. 2 Consider a special set with m 2 i = 1 − T, which implies c 1 = 0 and c 2 = T(3T − 2). For T < 2/3 condition c 2 is violated. The entropy is given by S/N = s 0 ( √ 1 − T) − 1/4, which is negative for all temperatures smaller than 0.256. where and with β = 1/T. The new variables λ, μ and v enter via the Fourier representations of the δfunctions and are therefore conjugated to q 2 , q 4 and w, respectively. Similarly Δ results from δ(G i ). The stationary of Σ(T, Ω) with respect to λ, Δ, μ, v finally leads to with The set of equations (9)- (14) correspond to the equations (56)-(61) of [6] with the replace- B → 0 and the additional terms proportional to μ resulting from the inclusion of q 4 . The apparent differences of equation (10) and equation (56) of [6] result from a simplification using equation (13). Setting B = 0 corresponds to an exclusion of a non-physical solution. Equation (15) is obvious and results from the stationary with respect to μ. For more details of the calculation and for the performed approximations it is referred to the previous work [5][6][7].
Note, however, that as long as q 2 and q 4 satisfy the criteria (5) the value of the determinant det ∂G k /∂m l in equation (7) is always positive. All previous work disregards the modulus with not completely satisfying arguments.

Total complexity
As first application the BM work [5] for the total complexity Σ tot (T) is reanalysed, which describes the total number of TAP states.
Setting v = 0 in equations (9)-(15) the resulting equations determine the complexity Σ(T; q 2 , q 4 , v = 0) for fixed values of q 2 and q 4 . These equations are numerical investigated for all possible values of q 2 , q 4 and for all temperatures T < 1.
As example Σ(T = 0.4; q 2 , q 4 , v = 0) is plotted in the q 2 -q 4 plane in figure 1. The region of allowed q 2 -q 4 values is restricted by q 2 2 q 4 q 2 and by the criteria (5). The cyan and the red boundaries represent the lines c 1 = 0 and c 2 = 0, respectively. The physical relevant region c 2 > 0 is above the red borderline. The region below the red line with c 2 < 0 has no physical significance. The absolute maximum of Σ(T; q 2 , q 4 , v = 0) in the q 2 -q 4 plane represents Σ tot (T) and can generally be located in the interior or on the boundary of the relevant region. For temperatures T T 1 the maximum is within the region and for T T 1 the maximum is located on the boundary c 2 = 0. The numerical value of the critical temperature T 1 is given by The coordinates of the maxima are determined by μ = 0 and by ∂Σ/∂q 2 = 0 or by c 2 = 0. At T 1 the internal maximum coincides with the boundary maximum.
In addition to figure 1, which gives an overview, some details are presented in figure 2 for T = 0.6, for T = 0.37 and for T = 0.075. The internal maxima and the boundary maxima are marked by a green and red dots, respectively. (On the scale of figure 1 these two minima are not separated.) Σ tot (T) has two branches Σ i and Σ b resulting from the two different maxima. Σ i is identical to BM and represents the stable branch for T > T 1 . The quantity characteristic for the transition is c i 2 , the c 2 for the internal maximum, which tends to zero for T → T 1 from above. Below T 1 criterion c i 2 is negative, the border maximum is the physical one and the branch Σ b with c b 2 = 0 is relevant. Figure 3 shows the T dependence of these two branches. Both have continuations from T 1 to the irrelevant temperatures regions. The difference of their extremal values is small (in the order of 10 −4 ) and are plotted in figure 4.
For temperatures at and below T 1 the presented results are new and have the consequence that nearly all TAP solutions are marginally stable for these temperatures. This finding differs from the classical BM work which does not claim any marginal stability for any temperature. For temperatures above T 1 the present results agree with BM. Early claims of marginality [9][10][11] are based on c 1 → 0 and are therefore again different from the present findings, which are a consequence of c 2 → 0. The extension of BM worked out in [7] and the present work have a common origin, namely the projector term P i j = − 2 N βm i m j of the Hessian ∂G i /∂m j = χ −1 i j + P i j . Such a term determines the lowest eigenvalue of ∂G i /∂m j , which is non-negative under the condition c 0 ≡ T 2 − 2T/N ij m i χ ij m j 0 (compare [7]). This condition has to be satisfied for every individual TAP solution and should in principle be used to eliminate non-physical solutions from the complexity counting. The approach [7] does not include this requirements as only an average valuec 0 = 0 of a large number of TAP states is calculated, which contains contributions resulting from c 0 = 0. The elimination of non-physical individual states with c 0 < 0 has  a feedback to the average procedure and consequently to thec 0 value. States with c 0 > 0 do not imply marginality. These objections together with missing influence of condition c 2 raise some doubts on the results of [7]. Recall that the numerical results of the extension [7] agree with BM in contrast to the present work which differs from BM for T < T 1 .

Averages
Next some physical interesting averages for the Gibbs potential g and the energy u per spin are calculated. According to equations (1) and (2) these averages are given by and by Let us first consider the averages over all TAP states with equal weights. These 'white' averages g av and of u av are determined by the extremal values of the parameters of the total complexity Σ tot (T). The temperature dependence for T 1 of g av and of u av is plotted in figures 5 and in 6, respectively. The strange increase of u av with decreasing T results from the fast increasing number of TAP solutions with high energies. These white averages have therefore no physical significance or any relevance for low temperatures.
There is an interesting, alternative averaging, that leads to the lowest value g 0 of the Gibbs potential consistent with both, the existence condition of TAP solutions Σ 0 and the validity of the criteria (5). To attack this problem the complete set of equations (9)-(15), (18) and (19) is needed. Keeping q 2 , q 4 and v constant the parameters λ, Δ, and μ are determined numerically with equations (12), (14) and (15). Repeating this procedure for all possible values of q 2 , q 4 and v the dependence of the complexity Σ, of w and of the Gibbs potential g on these quantities is obtained according to equations (10), (13) and (18). Finally the minimum g 0 of the Gibbs potential is determined in the region of the allowed values of q 2 , q 4 and Σ. The findings are a vanishing complexity Σ = 0 for all temperatures, which ensures at least the presence of one TAP state in the thermodynamic limit N → ∞. The resulting g 0 and the corresponding energy u 0 are plotted in figures 5 and in 6, respectively. Both quantities exhibit the expected temperature dependence. Note that the location of g 0 is again on the boundary c 2 = 0 for T < T 1 (compare figure 2). The low temperature behavior (T ≈ 0.1) of g 0 is similar to the results of the replica approach [19]. This is remarkable as the later approach uses the entire ∞-replica symmetry breaking solution of the SK and sophisticated numerical methods.

Postulated marginality
Early numerical investigations [7,[9][10][11] claim marginal metastable states based on c 1 → 0 in the thermodynamic limit. The recent work of Aspelmeier and Moore [12] have numerically studied the N-dependence of the two lowest eigenvalues of the Hessian. They found that both eigenvalues tend to zero in the thermodynamic limit. Thus there is no isolated eigenvalue caused by the projector term, which implies again c 1 → 0 in the thermodynamic limit (compare subsection 3.1). Motivated by all these findings c 1 = 0 is postulated a priory in this subsection and the resulting consequences for the present approach are worked out.
The first quantity of interest is the complexity Σ m , which determines the total number of TAP states with the constraint c 1 = 0. This complexity Σ m is given by the absolute maximum of Σ(T; q 2 , q 4 = 2q 2 − 1 + T 2 , v = 0), as function of q 2 . Again two temperature regimes exist, which are separated by a critical temperature T m av = 0.459. For T > T m av the maximum is located within the allowed q 2 -interval (1 − 1.5 T 2 ) < q 2 < 1 and for T T m av the maximum sticks at the endpoint q 2 = (1 − 1.5 T 2 ). The numerical results for Σ m are consistent to 0 Σ m Σ tot for all temperatures, which ensures the existence of at least one TAP state with the postulated marginality. Note that this last conclusion is a priory not obvious. Figure 7 shows the temperature dependence of Σ m for T < 0.6. Analogue to subsection 3.2 the set of extremal parameters of the maximum determine the averages performed with all TAP states satisfying c 1 = 0. Together with equations (18) and (19) this leads directly to the averages of the Gibbs potential g m av and the energy u m av performed with these states. Both quantities g m av and u m av are plotted in figure 8. The results g m av and u m av are not very useful for low temperatures similar to the above findings for g av and u av .
The lowest value of the Gibbs potential g m 0 consistent with c 1 = 0, c 2 0 and Σ 0 is again determined analogue to procedure of subsection 3.2. As before a vanishing complexity and two temperature regions are found with a sticking below a different critical temperature  [7], to [12] and to [11], respectively.
T m 0 = 0.417. The temperature dependence of g m 0 together with the corresponding energy u m 0 is plotted in figure 8.
For T = 0.2 the numerical value is for the Gibbs potential is g m 0 = −0.7644. This value is in remarkable agreement with the value ofĝ m 0 = −0.076 19 found by the dynamical investigation [11] with c 1 0 and with a vanishing complexity. The resultĝ m 0 is added to figure 8 as point C together with the points A and B found by further investigations [7,12]. Even though c 1 0 is satisfied no results for the complexity are available for the points A and B. Nevertheless these results are compatible with the present work.
According to the above constructions g 0 < g m 0 holds for 0 < T < 1. The numerical differences, however, are small and would not clearly be resolved on a scale of figures 5 or 8. The questions why the states with lowest Gibbs potentials g 0 are not reached by dynamical relaxation or if there are alternative dynamical paths to g 0 are interesting, but are beyond the scope of this work.

Conclusion
The presented investigation of the complexity is based the inclusion of q 4 to the set of constrained variables. This extension together with a strict regard of the validity criteria for the TAP equations leads to new results in low temperatures regime. Numerically the differences to the existing approaches are rather small, the interpretations and conclusions, however, differ considerably. At and below a critical temperature T 1 nearly all TAP states are marginal stable with c 2 = 0, a property not found in previous theories on the total complexity. Marginal stability implies a vanishing eigenvalue of the Hessian and the divergence of a mode of the susceptibility matrix [2]. The system is critical for all temperatures below T 1 and shows critical slowing down effects.
In addition to this findings consequences for averages over all metastable TAP states and averages over states with the lowest value of the Gibbs potential have been worked out for all temperatures. Moreover the influence of a different kind of marginality c 1 = 0, as found by supplementary numerical investigations [7,[9][10][11][12], has been worked out.