Dimensional reduction

The Parisi–Sourlas construction originally aimed at "dimensional reduction" proposed in 1976 by Amnon Aharony, Yoseph Imry, and Shang-keng Ma^[10] who proved that to all orders in perturbation expansion, the critical exponents in a d-dimensional (4 < d < 6) system with short-range exchange and a random quenched field are the same as those of a (d–2)-dimensional pure system.^[30] Their arguments indicated that the "Feynman diagrams which give the leading singular behavior for the random case are identically equal, apart from combinatorial factors, to the corresponding Feynman diagrams for the pure case in two fewer dimensions."^[30]

... Parisi and Sourlas ... observed that the most infrared divergent diagrams are those with the maximum number of random source insertions, and, if the other diagrams are neglected, one is left with a diagrammatic expansion for a classical field theory in the presence of random sources.
Parisi and Sourlas then pointed out that the underlying phenomenon for the connection between random systems and pure systems in two fewer dimensions is that a classical field theory in the presence of random sources is perturbatively equivalent to the corresponding quantum field theory in two fewer dimensions. Parisi and Sourlas explained this dimensional reduction by a hidden supersymmetry.^[30]

Topological interpretation

Topological aspects of the Parisi–Sourlas construction can be briefly outlined in the following manner.^[22][31] The delta-functional, i.e., the collection of the infinite number of delta-functions, ensures that only solutions of the Langevin SDE contribute to ${\displaystyle {\mathcal {W}}}$. In the context of BRST procedure, these solutions can be viewed as Gribov copies. Each solution contributes either positive or negative unity: ${\textstyle {\mathcal {W}}=\textstyle \left\langle I_{N}(\xi )\right\rangle _{\text{noise}}}$with ${\textstyle I_{N}(\xi )=\sum _{\text{solutions}}\operatorname {sign} J}$ being the index of the so-called Nicolai map, ${\textstyle \xi =({\dot {x}}+\partial U)/(2\Theta )^{1/2}}$, which in this case is the map from the space of closed paths in ${\textstyle X}$ to the space of noise configurations, a map that provides a noise configuration at which a given closed path is a solution of the Langevin SDE. ${\textstyle I_{N}(\xi )}$ can be viewed as a realization of Poincaré–Hopf theorem on the infinite-dimensional space of close paths with the Langevin SDE playing the role of the vector field and with the solutions of Langevin SDE playing the role of the critical points with index ${\displaystyle \operatorname {sign} J}$. ${\textstyle I_{N}(\xi )}$ is independent of the noise configuration because it is of topological character. The same it true for its stochastic average, ${\displaystyle {\mathcal {W}}}$, which is not the partition function of the model but, instead, its Witten index.

Path integral representation

With the help of a standard field theoretic technique that involves introduction of additional field called Lagrange multiplier, ${\displaystyle B}$, and a pair of fermionic fields called Faddeev–Popov ghosts, ${\displaystyle \chi ,{\bar {\chi }}}$, the Witten index can be given the following form,

$${\displaystyle {\mathcal {W}}=\int \dots \int _{p.b.c.}e^{(Q,\Psi )}D\Phi ,}$$

where ${\displaystyle \Phi }$ denotes collection of all the fields, p.b.c. stands for periodic boundary conditions, the so-called gauge fermion, ${\displaystyle \textstyle \Psi =\int _{t'}^{t}d\tau (\imath _{\dot {x}}-{\bar {d}})(\tau )}$, with ${\displaystyle \textstyle \imath _{\dot {x}}=i{\bar {\chi }}_{j}{\dot {x}}^{j}}$ and ${\textstyle \textstyle {\bar {d}}=-i{\bar {\chi }}_{j}\delta ^{jk}(\partial _{k}U+\Theta iB_{k})}$, and the BRST symmetry defined via its action on arbitrary functional ${\displaystyle A(\Phi )}$ as ${\displaystyle (Q,A(\Phi ))=\textstyle \int _{t'}^{t}d\tau (\chi ^{i}(\tau )\delta /\delta x^{i}(\tau )+B_{i}(\tau )\delta /\delta {\bar {\chi }}_{i}(\tau ))A(\Phi )}$. In the BRST formalism, the Q-exact pieces like, ${\displaystyle (Q,\Psi )}$, serve as gauge fixing tools. Therefore, the path integral expression for ${\displaystyle {\mathcal {W}}}$ can be interpreted as a model whose action contains nothing else but the gauge fixing term. This is a definitive feature of Witten-type topological field theories and in this particular case of BRST procedure approach to SDEs, the BRST symmetry can be also recognized as the topological supersymmetry.^[22]

A common way to explain the BRST procedure is to say that the BRST symmetry generates the fermionic version of the gauge transformations, whereas its overall effect on the path integral is to limit the integration only to configurations that satisfy a specified gauge condition. This interpretation also applies to Parisi–Sourlas approach with the deformations of the trajectory and the Langevin SDE playing the roles of the gauge transformations and the gauge condition respectively.

Operator representation

Physical fermions in the high-energy physics and condensed matter models have antiperiodic boundary conditions in time. The unconventional periodic boundary conditions for fermions in the path integral expression for the Witten index is the origin of the topological character of this object. These boundary conditions reveal themselves in the operator representation of the Witten index as the alternating sign operator,

$${\displaystyle {\mathcal {W}}=\operatorname {Tr} (-1)^{\hat {n}}{\hat {\mathcal {M}}}_{tt'},}$$

where ${\textstyle {\hat {n}}}$ is the operator of the number of ghosts/fermions and the finite-time stochastic evolution operator (SEO), ${\textstyle {\hat {\mathcal {M}}}_{tt'}=e^{-(t-t'){\hat {H}}}}$, where,

$${\displaystyle {\hat {H}}={\hat {L}}_{-\partial U}-\Theta {\hat {\triangle }}=[{\hat {d}},{\hat {\bar {d}}}],}$$

is the infinitesimal SEO with ${\displaystyle \textstyle {\hat {L}}}$ being the Lie derivative along the subscript vector field, ${\textstyle {\hat {\triangle }}}$ being the Laplacian, ${\displaystyle \textstyle {\hat {d}}=\chi ^{i}\partial /\partial x^{i}}$ being the exterior derivative, which is the operator representative of the topological supersymmetry (TS), and ${\displaystyle \textstyle {\hat {\bar {d}}}=-(i{\hat {\bar {\chi }}}_{i})\delta ^{ij}(\partial _{j}U+\Theta (i{\hat {B}}_{j}))}$, where ${\displaystyle i{\hat {B}}_{j}=\partial /\partial x^{j}}$ and ${\displaystyle i{\hat {\bar {\chi }}}_{j}=\partial /\partial \chi ^{j}}$ are bosonic and fermionic momenta, and with square brackets denoting bi-graded commutator, i.e., it is an anticommutator if both operators are fermionic (contain odd total number of ${\displaystyle \chi }$'s and ${\displaystyle {\hat {\bar {\chi }}}}$'s) and a commutator otherwise. The exterior derivative and ${\displaystyle \textstyle {\hat {\bar {d}}}}$ are supercharges. They are nilpotent, e.g., ${\displaystyle {\hat {d}}^{2}=0}$, and commutative with the SEO. In other words, Langevin SDEs possess N=2 supersymmetry. The fact that ${\displaystyle \textstyle {\hat {\bar {d}}}}$ is a supercharge is accidental. For SDEs of arbitrary form, this is not true.

Hilbert space

The wavefunctions are functions not only of the bosonic variables, ${\displaystyle x\in X}$, but also of the Grassmann numbers or fermions, ${\displaystyle \chi \in TX}$, from the tangent space of ${\displaystyle X}$. The wavefunctions can be viewed as differential forms on ${\displaystyle X}$ with the fermions playing the role of the differentials ${\displaystyle \chi \equiv dx\wedge }$.^[26] The concept of infinitesimal SEO generalizes the Fokker–Planck operator, which is essentially the SEO acting on top differential forms that have the meaning of the total probability distributions. Differential forms of lesser degree can be interpreted, at least locally on ${\displaystyle X}$, as conditional probability distributions.^[32] Viewing the spaces of differential forms of all degrees as wavefunctions of the model is a mathematical necessity. Without it, the Witten index representing the most fundamental object of the model—the partition function of the noise—would not exist and the dynamical partition function would not represent the number of fixed points of the SDE (see below). The most general understanding of the wavefunctions is the coordinate-free objects that contain information not only on trajectories but also on the evolution of the differentials and/or Lyapunov exponents.^[33]

Ambiguity of path integral representation and Ito–Stratonovich dilemma

BRST gauge fixing procedure goes along the same lines as in case of Langevin SDEs. The topological interpretation of the BRST procedure is just the same and the path integral representation of the Witten index is defined by the gauge fermion, ${\displaystyle \textstyle \Psi }$, given by the same expression but with the generalized version of ${\textstyle \textstyle {\bar {d}}=\imath _{F}-\Theta \imath _{e_{a}}(Q,\imath _{e_{a}})}$. There is one important subtlety, however, that appears on the way to the operator representation of the model. Unlike for Langevin SDEs, classical mechanics, and other SDEs with additive noises, the path integral representation of the finite-time SEO is an ambiguous object. This ambiguity originates from non-commutativity of momenta and position operators, e.g., ${\displaystyle {\hat {B}}{\hat {x}}\neq {\hat {x}}{\hat {B}}}$. As a result, ${\displaystyle B(\tau )x(\tau )}$ in the path integral representation has a whole one-parameter family of possible interpretations in the operator representation, ${\displaystyle (\alpha {\hat {B}}{\hat {x}}+(1-\alpha ){\hat {x}}{\hat {B}})|\psi (\tau )\rangle }$, where ${\displaystyle |\psi (\tau )\rangle }$ denotes an arbitrary wavefunction. Accordingly, there is a whole ${\displaystyle \alpha }$-family of infinitesimal SEOs,

$${\displaystyle {\hat {H}}_{\alpha }={\hat {L}}_{F}-\Theta {\hat {L}}_{e_{a}}{\hat {L}}_{e_{a}}=[{\hat {d}},{\hat {\bar {d}}}_{\alpha }],}$$

with ${\displaystyle \textstyle {\hat {\bar {d}}}_{\alpha }={\hat {\imath }}_{F_{\alpha }}-\Theta {\hat {\imath }}_{e_{a}}{\hat {L}}_{e_{a}}}$, ${\displaystyle {\hat {\imath }}_{F_{\alpha }}}$ being the interior multiplication by the subscript vector field, and the "shifted" flow vector field being ${\displaystyle \textstyle F_{\alpha }=F-\Theta (2\alpha -1)(e_{a}\cdot \partial )e_{a}}$. Noteworthy, unlike in Langevin SDEs, ${\displaystyle {\hat {\bar {d}}}_{\alpha }}$ is not a supercharge and STS cannot be identified as a N=2 supersymmetric theory in the general case.

The path integral representation of stochastic dynamics is equivalent to the traditional understanding of SDEs as of a continuous time limit of stochastic difference equations where different choices of parameter ${\displaystyle \alpha }$ are called "interpretations" of SDEs. The choice ${\displaystyle \alpha =1/2}$, for which ${\displaystyle \textstyle F_{1/2}=F}$ and which is known in quantum theory as Weyl symmetrization rule, is known as the Stratonovich interpretation, whereas ${\displaystyle \alpha =1}$ as the Ito interpretation. While in quantum theory the Weyl symmetrization is preferred because it guarantees hermiticity of Hamiltonians, in STS the Stratonovich–Weyl approach is preferred because it corresponds to the most natural mathematical meaning of the finite-time SEO discussed below—the stochastically averaged pullback induced by the SDE-defined diffeomorphisms.

Eigensystem of stochastic evolution operator

The three possible types of SEO spectra on a 3D sphere. Each line of triples of graphs represents ${\displaystyle spec{\hat {H}}^{(n)},}$ ${\displaystyle n=0,...3}$. for the three types of spectra. Black dots at the origin for the first and the last lines represent supersymmetric eigenstates from the zeroth and the third cohomologies of the 3 sphere. For types b and c, the (fastest growing) ground (eigen)states are non-supersymmetric because they have non-zero eigenvalues. TS is broken spontaneously. Vertical arrowed lines visualize supersymmetry operator.

As compared to the SEO of Langevin SDEs, the SEO of a general form SDE is pseudo-Hermitian.^[19] As a result, the eigenvalues of non-supersymmetric eigenstates are not restricted to be real positive, whereas the eigenvalues of supersymmetric eigenstates are still exactly zero. Just like for Langevin SDEs and nonlinear sigma model, the structure of the eigensystem of the SEO reestablishes the topological character of the Witten index: the contributions from the non-supersymmetric pairs of eigenstates vanish and only supersymmetric states contribute the Euler characteristic of (closed) ${\displaystyle X}$. Among other properties of the SEO spectra is that ${\displaystyle \textstyle {\hat {H}}^{(\operatorname {dim} X)}}$ and ${\displaystyle \textstyle {\hat {H}}^{(0)}}$ never break TS, i.e., ${\displaystyle \textstyle Re(\operatorname {spec} {\hat {H}}^{(\operatorname {dim} X)})\geq 0}$. As a result, there are three major types of the SEO spectra presented in the figure on the right. The two types that have negative (real parts of) eigenvalues correspond to the spontaneously broken TS. All types of the SEO spectra are realizable as can be established, e.g., from the exact relation between the theory of kinematic dynamo and STS.^[34]

The mathematical meaning of stochastic evolution operator

The finite-time SEO can be obtained in another, more mathematical way based on the idea to study the SDE-induced actions on differential forms directly, without going through the BRST gauge fixing procedure. The so-obtained finite-time SEO is known in dynamical systems theory as the generalized transfer operator^[20][21] and it has also been used in the classical theory of SDEs (see, e.g., Refs.^[35][36] ). The contribution to this construction from STS^[9] is the exposition of the supersymmetric structure underlying it and establishing its relation to the BRST procedure for SDEs.

Namely, for any configuration of the noise, ${\displaystyle \xi }$, and an initial condition, ${\displaystyle x(t')=x'\in X}$, SDE defines a unique solution/trajectory, ${\displaystyle x(t)\in X}$. Even for noise configurations that are non-differentiable with respect to time, ${\displaystyle t}$, the solution is differentiable with respect to the initial condition, ${\displaystyle x'}$.^[37] In other words, SDE defines the family of the noise-configuration-dependent diffeomorphisms of the phase space to itself, ${\displaystyle M_{tt'}:X\to X}$. This object can be understood as a collection and/or definition of all the noise-configuration-dependent trajectories, ${\displaystyle x(t)=M_{tt'}(x')}$. The diffeomorphisms induce actions or pullbacks, ${\displaystyle \textstyle M_{t't}^{*}:\Omega (X)\to \Omega (X)}$. Unlike, say, trajectories in ${\displaystyle X}$, pullbacks are linear objects even for nonlinear ${\displaystyle X}$. Linear objects can be averaged and averaging ${\displaystyle M_{t't}^{*}}$ over the noise configurations, ${\displaystyle \xi }$, results in the finite-time SEO which is unique and corresponds to the Stratonovich–Weyl interpretation of the BRST procedure approach to SDEs, ${\displaystyle {\hat {\mathcal {M}}}_{tt'}=\langle M_{t't}^{*}\rangle _{\text{noise}}=e^{-(t-t'){\hat {H}}_{1/2}}}$.

Within this definition of the finite-time SEO, the Witten index can be recognized as the sharp trace of the generalized transfer operator.^[20][21] It also links the Witten index to the Lefschetz index,${\textstyle \textstyle I_{L}=\operatorname {Tr} (-1)^{\hat {n}}M_{t't}^{*}=\sum _{x\in \operatorname {fix} M_{tt'}}\operatorname {sign} \operatorname {det} (\delta _{j}^{i}-\partial M_{tt'}^{i}(x)/\partial x^{j})}$, a topological constant that equals the Euler characteristic of the (closed) phase space. Namely, ${\textstyle \textstyle {\mathcal {W}}=\operatorname {Tr} (-1)^{\hat {n}}\langle M_{t't}^{*}\rangle _{\text{noise}}=\langle \operatorname {Tr} (-1)^{\hat {n}}M_{t't}^{*}\rangle _{\text{noise}}=I_{L}}$.

The meaning of supersymmetry and the butterfly effect

The N=2 supersymmetry of Langevin SDEs has been linked to the Onsager principle of microscopic reversibility^[16] and Jarzynski equality.^[15] In classical mechanics, a relation between the corresponding N=2 supersymmetry and ergodicity has been proposed.^[6] In general form SDEs, where physical arguments may not be applicable, a lower level explanation of the TS is available. This explanation is based on understanding of the finite-time SEO as a stochastically averaged pullback of the SDE-defined diffeomorphisms (see subsection above). In this picture, the question of why any SDE has TS is the same as the question of why exterior derivative commutes with the pullback of any diffeomorphism. The answer to this question is differentiability of the corresponding map. In other words, the presence of TS is the algebraic version of the statement that continuous-time flow preserves continuity of ${\displaystyle X}$. Two initially close points will remain close during evolution, which is just yet another way of saying that ${\displaystyle M_{tt'}}$ is a diffeomorphism.

In deterministic chaotic models, initially close points can part in the limit of infinitely long temporal evolution. This is the famous butterfly effect, which is equivalent to the statement that ${\displaystyle M_{tt'}}$ losses differentiability in this limit. In algebraic representation of dynamics, the evolution in the infinitely long time limit is described by the ground state of the SEO and the butterfly effect is equivalent to the spontaneous breakdown of TS, i.e., to the situation when the ground state is not supersymmetric. Noteworthy, unlike traditional understanding of deterministic chaotic dynamics, the spontaneous breakdown of TS works also for stochastic cases. This is the most important generalization because deterministic dynamics is, in fact, a mathematical idealization. Real dynamical systems cannot be isolated from their environments and thus always experience stochastic influence.

Classification of stochastic dynamics

Stochastic dynamical systems can be classified according to whether TS is spontaneously broken or not (ordered/symmetric) and whether the flow vector field is integrable or non-integrable (sometimes chaotic). Symmetric phase can be identified as thermal equilibrium (T). Ordered non-integrable phase can be called chaos (C) because it hosts conventional deterministic chaos. Ordered integrable phase can be called noise-induced chaos (N) because TS breaking involves anti-instantons that disappear in the deterministic limit. The N-phase is also known in the literature as self-organized criticality.

STS provides classification for stochastic models depending on whether TS is broken and integrability of flow vector field. In can be exemplified as a part of the general phase diagram at the border of chaos (see figure on the right). The phase diagram has the following properties:

• For physical models, TS gets restored eventually with the increase of noise intensity.
• Symmetric phase can be called thermal equilibrium or T-phase because the ground state is the supersymmetric state of steady-state total probability distribution.
• In the deterministic limit, ordered phase is equivalent to deterministic chaotic dynamics with non-integrable flow.
• Ordered non-integrable phase can be called chaos or C-phase because ordinary deterministic chaos belongs to it.
• Ordered integrable phase can be called noise-induced chaos or N-phase because it disappears in the deterministic limit. TS is broken by the condensation of (anti-)instantons (see below).
• At stronger noises, the sharp N-C boundary must smear out into a crossover because (anti-)instantons lose their individuality and it is hard for an external observer to tell one tunneling process from another.

Demystification of self-organized criticality

Many sudden (or instantonic) processes in nature, such as, e.g., crackling noise, exhibit scale-free statistics often called the Zipf's law. As an explanation for this peculiar spontaneous dynamical behavior, it was proposed to believe that some stochastic dynamical systems have a tendency to self-tune themselves into a critical point, the phenomenological approach known as self-organized criticality (SOC).^[40] STS offers an alternative perspective on this phenomenon.^[41] Within STS, SOC is nothing more than dynamics in the N-phase. Specifically, the definitive feature of the N-phase is the peculiar mechanism of the TS breaking. Unlike in the C-phase, where the TS is broken by the non-integrability of the flow, in the N-phase, the TS is spontaneously broken due to the condensation of the configurations of instantons and noise-induced antiinstantons, i.e., time-reversed instantons. These processes can be roughly interpreted as the noise-induced tunneling events between, e.g., different attractors. Qualitatively, the dynamics in the N-phase appears to an external observer as a sequence of sudden jumps or "avalanches" that must exhibit a scale-free behavior/statistics as a result of the Goldstone theorem. This picture of dynamics in the N-phase is exactly the dynamical behavior that the concept of SOC was designed to explain. In contrast with the original understanding of SOC,^[42] its STS interpretation has little to do with the traditional critical phenomena theory where scale-free behavior is associated with unstable fixed points of the renormalization group flow.

Kinematic dynamo theory

Magnetohydrodynamical phenomenon of kinematic dynamo can also be identified as the spontaneous breakdown of TS.^[34] This result follows from equivalence between the evolution operator of the magnetic field and the SEO of the corresponding SDE describing the flow of the background matter. The so emerged STS-kinematic dynamo correspondence proves, in particular, that both types of TS breaking spectra are possible, with the real and complex ground state eigenvalues, because kinematic dynamo with both types of the fastest growing eigenmodes are known.^[43]

Transient dynamics

It is well known that various types of transient dynamics, such as quenches, exhibit spontaneous long-range behavior. In case of quenches across phase transitions, this behavior is often attributed to the proximity of criticality. Quenches that do not exhibit a phase transition are also known to exhibit long-range characteristics, with the best known examples being the Barkhausen effect and the various realizations of the concept of crackling noise. It is intuitively appealing that theoretical explanations for the scale-free behavior in quenches must be the same for all quenches, regardless of whether or not it produces a phase transition; STS offers such an explanation. Namely, transient dynamics is essentially a composite instanton and TS is intrinsically broken within instantons. Even though TS breaking within instantons is not exactly due to the phenomenon of the spontaneous breakdown of a symmetry by a global ground state, this effective TS breaking must also result in a scale-free behavior. This understanding is supported by the fact that condensed instantons lead to appearance of logarithms in the correlation functions.^[44] This picture of transient dynamics explains computational efficiency of the digital memcomputing machines.^[45]