Complex complex landscapes

We study the saddle-points of the $p$-spin model -- the best understood example of a `complex' (rugged) landscape -- when its $N$ variables are complex. These points are the solutions to a system of $N$ random equations of degree $p-1$. We solve for $\overline{\mathcal N}$, the number of solutions averaged over randomness in the $N\to\infty$ limit. We find that it saturates the B\'ezout bound $\log\overline{\mathcal N}\sim N\log(p-1)$. The Hessian of each saddle is given by a random matrix of the form $C^\dagger C$, where $C$ is a complex symmetric Gaussian matrix with a shift to its diagonal. Its spectrum has a transition where a gap develops that generalizes the notion of `threshold level' well-known in the real problem. The results from the real problem are recovered in the limit of real parameters. In this case, only the square-root of the total number of solutions are real. In terms of the complex energy, the solutions are divided into sectors where the saddles have different topological properties.

Spin-glasses have long been considered the paradigm of many variable 'complex landscapes,' a subject that includes neural networks and optimization problems, most notably constraint satisfaction. The most tractable family of these are the mean-field spherical -spin models [1] (for a review see [2]) defined by the energy where is a symmetric tensor whose elements are real Gaussian variables and ∈ R is constrained to the sphere 2 = . This problem has been studied in the algebra [3] and probability literature [4,5]. It has been attacked from several angles: the replica trick to compute the Boltzmann-Gibbs distribution [1], a Kac-Rice [6][7][8] procedure (similar to the Fadeev-Popov integral) to compute the number of saddle-points of the energy function [9], and the gradient-descent-or more generally Langevin-dynamics staring from a high-energy configuration [10]. Thanks to the simplicity of the energy, all these approaches yield analytic results in the large-limit. In this paper we extend the study to complex variables: we shall take ∈ C and to be a symmetric tensor whose elements are complex normal, with | | 2 = !/2 −1 and 2 = | | 2 for complex parameter | | < 1. The constraint remains 2 = .
The motivations for this paper are of two types. On the practical side, there are indeed situations in which complex variables appear naturally in disordered problems: such is the case in which they are phases, as in random laser problems [11]. Quiver Hamiltonians-used to model black hole horizons in the zero-temperature limit-also have a Hamiltonian very close to ours [12].
There is however a more fundamental reason for this study: we know from experience that extending a real problem to the complex plane often uncovers underlying simplicity that is otherwise hidden. Consider, for example, the procedure of starting from a simple, known Hamiltonian 00 and studying 00 +(1− 0 ), evolving adiabatically from = 0 to = 1, as is familiar from quantum annealing. The 00 is a polynomial of degree chosen to have simple, known saddles. Because we are working in complex variables, and the saddles are simple all the way (we shall confirm this), we may follow a single one from = 0 to = 1, while with real variables minima of functions appear and disappear, and this procedure is not possible. The same idea may be implemented by performing diffusion in the s and following the roots, in complete analogy with Dyson's stochastic dynamics [13].
The spherical constraint is enforced using the method of Lagrange multipliers: introducing ∈ C, our energy is We choose to constrain our model by 2 = rather than | | 2 = in order to preserve the analyticity of . The nonholomorphic constraint also has a disturbing lack of critical points nearly everywhere: if were so constrained, then 0 = * = − would only be satisfied for = 0.
The critical points are of given by the solutions to the set of equations for all = {1, . . . , }, which for fixed is a set of equations of degree − 1, to which one must add the constraint. In this sense this study also provides a complement to the work on the distribution of zeroes of random polynomials [14], which are for = 1 and → ∞. We see from (3) that at any critical point, = / , the average energy.
Since is holomorphic, any critical point of Re is also a critical point of Im . The number of critical points of is therefore the same as that of Re . From each saddle emerge gradient lines of Re , which are also ones of constant Im and therefore constant phase. Writing = + , Re can be considered a real-valued function of 2 real variables. Its number of saddle-points is arXiv:2012.06299v2 [cond-mat.stat-mech] 18 Dec 2020 given by the usual Kac-Rice formula: Re . (4) The Cauchy-Riemann equations may be used to write this in a manifestly complex way. With the Wirtinger derivative = 1 2 ( − ), one can write Re = Re and Re = − Im . Carrying these transformations through, we have This gives three equivalent expressions for the determinant of the Hessian: as that of a 2 × 2 real matrix, that of an × Hermitian matrix, i.e. the norm squared of that of an × complex symmetric matrix. These equivalences belie a deeper connection between the spectra of the corresponding matrices. Each positive eigen-value of the real matrix has a negative partner. For each such pair ± , 2 is an eigenvalue of the Hermitian matrix and | | is a singular value of the complex symmetric matrix. The distribution of positive eigenvalues of the Hessian is therefore the same as the distribution of singular values of , or the distribution of square-rooted eigenvalues of ( ) † . The expression (5) is to be averaged over to give the complexity Σ as Σ = log N = ∫ log N , a calculation that involves the replica trick. In most the parameter-space that we shall study here, the annealed approximation Σ ∼ log N = log ∫ N is exact.
A useful property of the Gaussian is that gradient and Hessian at fixed are statistically independent [8,15], so that the -functions and the Hessian may be averaged independently. The -functions are converted to exponentials by the introduction of auxiliary fieldsˆ=ˆ+ˆ. The average of those factors over can then be performed. A generalized Hubbard-Stratonovich allows a change of variables from the 4 original and auxiliary fields to eight bilinears defined by = | | 2 ,ˆ= |ˆ| 2 ,ˆ=ˆ2, =ˆ * , and =ˆ(and their conjugates). The result, to leading order in , is where the argument of the exponential is The integral of the distribution of eigenvalues of comes from the Hessian and is dependant on alone. This function has an extremum inˆ, ,ˆ, and at which its value is where = 1 2 arg and This leaves a single parameter, , which dictates the magnitude of | | 2 , or alternatively the magnitude 2 of the imaginary part. The latter vanishes as → 1, where (as we shall see) one recovers known results for the real -spin.
The Hessian = 0 − is equal to the unconstrained Hessian with a constant added to its diagonal. The eigenvalue distribution is therefore related to the unconstrained distribution 0 by a similar shift: ( ) = 0 ( + ).

The Hessian of the unconstrained Hamiltonian is
which makes its ensemble that of Gaussian complex symmetric matrices, when the direction along the constraint is neglected. Given its variances where = 1 2 arg [16]. The eigenvalue spectrum of is therefore constant inside the same ellipse translated so that its center lies at − . Examples of these distributions are shown in the insets of Fig. 1.
The eigenvalue spectrum of the Hessian of the real part is different from the spectrum ( ) of , but rather equivalent to the square-root eigenvalue spectrum of ( ) † ; in other words, the singular value spectrum ( ) of . When = 0 and the elements of are standard complex normal, this is a complex Wishart distribution. For ≠ 0 the problem changes, and to our knowledge a closed form is not in the literature. We have worked out an implicit form for this spectrum using the replica method.
Introducing replicas to bring the partition function into the numerator of the Green function [17] gives with sums taken over repeated Latin indices. The average is then made over and Hubbard-Stratonovich is used to change variables to the replica matrices = ( ( ) ) * · ( ) and = ( ) · ( ) and a series of replica vectors. The replicasymmetric ansatz leaves all off-diagonal elements and vectors zero, and = 0 , = 0 . The result is The argument of the exponential has several saddles. The solutions 0 are the roots of a sixth-order polynomial, and the root with the smallest value of Re 0 in all the cases we studied gives the correct solution. A detailed analysis of the saddle point integration is needed to understand why this is so. Given such 0 , the density of singular values follows from the jump across the cut, or Examples can be seen in Fig. 1 compared with numeric experiments. The transition from a one-cut to two-cut singular value spectrum naturally corresponds to the origin leaving the support of the eigenvalue spectrum. Weyl's theorem requires that the product over the norm of all eigenvalues must not be greater than the product over all singular values [18]. Therefore, the absence of zero eigenvalues implies the absence of zero singular values. The determination of the threshold energy -the energy at which the distribution of singular values becomes gapped -is then reduced to a geometry problem, and yields for = −( −2) . Given , the integral in (8) may be preformed for arbitrary . The resulting expression is maximized for = ∞ for all values of and . Taking this saddle gives log N ( , ) = log( − 1). (15) This is, to this order, precisely the Bézout bound, the maximum number of solutions that equations of degree − 1 may have [19]. That we saturate this bound is perhaps not surprising, since the coefficients of our polynomial equations (3) are complex and have no symmetries. Reaching Bézout in (15) is not our main result, but it provides a good check.
Analogous asymptotic scaling has been found for the number of pure Higgs states in supersymmetric quiver theories [20]. More insight is gained by looking at the count as a function of , defined by N ( , , ) = ( ) . In the large-limit, this is the cumulative number of critical points, or the number of critical points for which | | 2 ≤ . We likewise define the -dependant complexity Σ( , , ) = log N ( , , ) Everything is analytically tractable for = 0, giving Notice that the limit of this expression as → ∞ corresponds with (15), as expected. This is plotted as a function of for several values of in Fig. 2. For any | | < 1, the complexity goes to negative infinity as → 1, i.e., as the spins are restricted to the reals. This is natural, given that the contribution to the volume shrinks to zero as that of an -dimensional sphere 2 = ( −1) with volume ∼ ( −1) . However, when the result is analytically continued to = 1 (which corresponds to real ) something novel occurs: the complexity has a finite value at = 1. Since the -dependence gives a cumulative count, this implies a -function density of critical points along the line = 0. The number of critical points contained within is half of (15) and corresponding precisely to the number of critical points of the real -spin model (note the role of conjugation symmetry, already underlined in [14]). The full -dependence of the real -spin is recovered by this limit as is varied. These qualitative features carry over to nonzero . In Fig. 3 we show that for < 1 there is always a gap of close to one for which there are no solutions. When = 1-the analytic continuation to the real computation-the situation  is more interesting. In the range of energies where there are real solutions this gap closes, which is only possible if the density of solutions diverges at = 1. Another remarkable feature of this limit is that there is still a gap without solutions around 'deep' real energies where there is no real solution. A moment's thought tells us that this is a necessity: otherwise a small perturbation of the s could produce an unusually deep solution to the real problem, in a region where this should not happen.
The relationship between the threshold and ground, or extremal, state energies is richer than in the real case. In Fig. 4 these are shown in the complex-plane for several examples. Depending on the parameters, the threshold might always come at smaller magnitude than the extremal state, or always come at larger magnitude, or cross as a function of complex argument. For sufficiently large the threshold always comes at larger magnitude than the extremal state. If this were to happen in the real case, it would likely imply our replica symmetric computation is unstable, since having a ground state above the threshold implies a ground state Hessian with many negative eigenvalues, a contradiction. However, this is not an obvious contradiction in the complex case. The relationship between the threshold, i.e., where the gap appears, and the dynamics of, e.g., a minimization algorithm or physical dynamics, are a problem we hope to address in future work.
This paper provides a first step for the study of a complex landscape with complex variables. The next obvious one is to study the topology of the critical points and gradient lines of constant phase. We anticipate that the threshold level, where the system develops a mid-spectrum gap, will play a crucial role as it does in the real case.
We wish to thank Alexander Altland, Satya Majumdar and Gregory Schehr for a useful suggestions. JK-D and JK are supported by the Simons Foundation Grant No. 454943.