The stochastic approach for anomalies in supersymmetric theories
We discuss how the stochastic approach for supersymmetric theories leads to new ways of characterizing anomalies in how supersymmetry can be broken.
hep-th↗
arXiv subjects
Stam Nicolis
Publications and source records attributed to Stam Nicolis.
A tale of two SUSYs
There are two approaches towards supersymmetry: The ``conventional approach'', in which the fields appear in the classical action and the ``stochastic approach'', in which they emerge upon introducing in the action the contribution of a certain determinant. The second approach relies, in particular, on the so-called Nicolai map. The relation between the two approaches hasn't been clarified and the subject of this contribution is to spell it out. In particular the interpretation proposed by Parisi and Sourlas has remained quite incomplete. The subject of this contribution is to spell out in what way the two approaches complement each other and how the second can provide insights for the first.
hep-th↗
Path integral spin dynamics with exchange and external field
In this work, we propose a path integral-inspired formalism for computing the quantum thermal expectation values of spin systems, when subject to magnetic fields that can be time-dependent and can accommodate the presence of Heisenberg exchange interactions between the spins. This is done by deriving an effective magnetic field from the quantum partition function of the system to use in classical atomistic spin dynamics simulations and generalises the formalism presented in our previous work [Phys. Rev. Research 5, 043075 (2023)]. In special cases where the effective field can be computed exactly we compare our results with exact/numerical diagonalisation methods for both ferromagnetic and antiferromagnetic coupling. We show that our method works well across a large temperature range and can reproduce quantum expectation values for antiferromagnetic coupling which is usually not possible with classical models.
quant-ph↗
Path integral spin dynamics for quantum paramagnets
A path integral method, combined with atomistic spin dynamics simulations, has been developed to calculate thermal quantum expectation values using a classical approach. In this study, we show how to treat Hamiltonians with non-linear terms, that are relevant for describing uniaxial anisotropies and mechanical constraints. These interactions can be expressed solely through quadratic terms of the spin operator along one axis, that can be identified with the quantisation axis.
quant-ph↗
Flavor and Fluctuations
The Parisi-Sourlas approach to supersymmetry implies that, in spacetime dimensions greater than 1, there is a constraint on the minimal number of flavors, in order for a field theory to define a closed system. In particular, this number is greater than 1. This does not preclude that supersymmetry can be broken, however, and the known ways of breaking supersymmetry can be taken into account from this point of view, by using the so-called Nicolai map. This procedure is well-defined for abelian gauge theories and corresponds to the construction of the so-called trivializing maps for non-abelian gauge theories, that is, still, work in progress.
hep-th↗
Exponential mixing of all orders for Arnol'd cat map lattices
We show that the recently introduced classical Arnol'd cat map lattice field theories, which are chaotic, are exponentially mixing to all orders. Their mixing times are well-defined and are expressed in terms of the Lyapunov exponents, more precisely by the combination that defines the inverse of the Kolmogorov-Sinai entropy of these systems. We prove by an explicit recursive construction of correlation functions, that these exhibit $l-$fold mixing for any $l= 3,4,5,\ldots$. This computation is relevant for Rokhlin's conjecture, which states that 2-fold mixing induces $l-$fold mixing for any $l>2$. Our results show that 2-fold exponential mixing, while being necessary for any $l-$fold mixing to hold it is nevertheless not sufficient for Arnol'd cat map lattice field theories.
hep-th↗
Noisy SUSY
We review the idea, put forward in 1982, by Parisi and Sourlas, that the bath of fluctuations, with which a physical system is in equilibrium, can be resolved by the superpartners of the degrees of freedom, defined by the classical action. This implies, in particular, that fermions can be described in terms of their superpartners, using the Nicolai map. We focus on the question, whether the fluctuations of scalar fields can, in fact, produce the absolute value of the stochastic determinant itself, whose contribution to the action can be identified with the fermionic degrees of freedom and present evidence supporting this idea in two spacetime dimensions. The same idea leads to a new formulation of supersymmetric QED. We also review the obstacles for extending this approach to Yang-Mills theories and report on progress for evading the obstructions for obtaining interacting theories in three and four spacetime dimensions. This implies, in particular, that it is possible to describe the effects of fermions in numerical simulations, through their superpartners.
hep-th↗
Numerical Simulations of a Spin Dynamics Model Based on a Path Integral Approach
Inspired by path integral molecular dynamics, we build a spin model, in terms of spin coherent states, from which we can compute the quantum expectation values of a spin in a constant magnetic field, at finite temperature. This formulation facilitates the description of a discrete quantum spin system in terms of a continuous classical model and recasts the quantum spin effects within the framework of path integrals in a double $1/s$ and $\hbar s$ expansion, where $s$ is the magnitude of the spin. In particular, it allows for a much more direct path to the low- and high-temperature limits of the quantum system and to the definition of effective classical Hamiltonians that describe both thermal and quantum fluctuations. In this formalism, the quantum properties of the spins emerge as an effective anisotropy. We use atomistic spin dynamics to sample the path integral, calculate thermodynamic observables and show that our effective classical models can reproduce the thermal expectation values of the quantum system within temperature ranges relevant for studying magnetic ordering.
cond-mat.mtrl-sci↗
Arnol'd cat map lattices
We construct Arnol'd cat map lattice field theories in phase space and configuration space. In phase space we impose that the evolution operator of the linearly coupled maps be an element of the symplectic group, in direct generalization of the case of one map. To this end we exploit the correspondence between the cat map and the Fibonacci sequence. The chaotic properties of these systems can be, also, understood from the equations of motion in configuration space, where they describe inverted harmonic oscillators, with the runaway behavior of the potential competing with the toroidal compactification of the phase space. We highlight the spatio-temporal chaotic properties of these systems using standard benchmarks for probing deterministic chaos of dynamical systems, namely the complete dense set of unstable periodic orbits, which, for long periods, lead to ergodicity and mixing. The spectrum of the periods exhibits a strong dependence on the strength and the range of the interaction.
hep-th↗
Route to chaos in a branching model of neural network dynamics
Simplified models are a necessary steppingstone for understanding collective neural network dynamics, in particular the transitions between different kinds of behavior, whose universality can be captured by such models, without prejudice. One such model, the cortical branching model (CBM), has previously been used to characterize part of the universal behavior of neural network dynamics and also led to the discovery of a second, chaotic transition which has not yet been fully characterized. Here, we study the properties of this chaotic transition, that occurs in the mean-field approximation to the $k_{\sf in}=1$ CBM by focusing on the constraints the model imposes on initial conditions, parameters, and the imprint thereof on the Lyapunov spectrum. Although the model seems similar to the H\'enon map, we find that the H\'enon map cannot be recovered using orthogonal transformations to decouple the dynamics. Fundamental differences between the two, namely that the CBM is defined on a compact space and features a non-constant Jacobian, indicate that the CBM maps, more generally, represent a class of generalized H\'enon maps which has yet to be fully understood.
nlin.CD↗
The continuum limit of the modular discretization of AdS$_2$
According to the holographic picture of 't Hooft and Susskind, the black hole entropy, $S_{\rm BH}$, is carried by the chaotic microscopic degrees of freedom, that live in the near horizon geometry and have a Hilbert space of states of finite dimension, $d=\exp(S_{\rm BH})$. In previous work we have proposed that the near horizon geometry, when the microscopic degrees of freedom can be resolved, can be described by the discrete, finite, random geometry, AdS$_2[\mathbb{Z}_N]$, where $N$ is proportional to $S_{\rm BH}$. What had remained as an open problem was how the smooth AdS2 geometry can be recovered, in the limit when N goes to infinity. In this contribution we present the salient points of the solution to this problem, which involves embedding AdS$_2[\mathbb{Z}_N]$ in a family of finite geometries, AdS$_2^M[\mathbb{Z}_N]$, where $M$ is another integer, within 2+1 Minkowski spacetime. In this construction $N$ and $M$ can be considered IR and UV cutoffs. The continuum limit, corresponding to the smooth AdS$_2$ geometry, is obtained by taking $N$ and $M$ to infinity in a correlated way, using properties of the Fibonacci and $k$-Fibonacci sequences. This method can be directly applied to higher-dimensional AdS spacetimes, also.
hep-th↗
The hidden fluxes, that control the fluctuations of scalar fields
The fluctuations of scalar fields, that are invariant under rotations of the worldvolume, in Euclidian signature, can be described by a system of Langevin equations. These equations can be understood as defining a change of variables in the functional integral for the noise, with which the physical degrees of freedom are in equilibrium. The absolute value of the Jacobian of this change of variables therefore repackages the fluctuations. This provides a new way of relating the number and properties of scalar fields with the consistent and complete description of their fluctuations and is another way of understanding the relevance of supersymmetry, which, in this way, determines the minimal number of real scalar fields (e.g. two in two dimensions, four in three dimensions and eight in four dimensions), in order for the system to be consistently closed. The classical action of the scalar fields, obtained in this way, contains a surface term and a remainder, in addition to the canonical kinetic and potential terms. The surface term describes possible flux contributions in the presence of boundaries, while the remainder describes additional interactions, that can't be absorbed in a redefinition of the canonical terms. It is, however, through its combination with the surface term that the noise fields can be recovered, in all cases. However their identities can be subject to anomalies. What is of particular, practical, interest is the identification of the noise fields, as functions of the scalars, whose correlation functions are Gaussian. This implies new identities, between the scalars, that can be probed in real, or computer, experiments.
hep-th↗
Emerging magnetic nutation
Nutation has been recognized as of great significance for spintronics; but justifying its presence has proven to be a hard problem. In this paper we show that nutation can be understood as emerging from a systematic expansion of a kernel that describes the history of the interaction of a magnetic moment with a bath of colored noise. The parameter of the expansion is the ratio of the colored noise timescale to the precession period. In the process we obtain the Gilbert damping from the same expansion. We recover the known results, when the coefficients of the two terms are proportional to one another, in the white noise limit; and show how colored noise leads to situations where this simple relation breaks down, but what replaces it can be understood by the appropriate generalization of the fluctuation--dissipation theorem. Numerical simulations of the stochastic equations support the analytic approach. In particular we find that the equilibration time is about an order of magnitude longer than the timescale set by the colored noise for a wide range of values of the latter and we can identify the presence of nutation in the non-uniform way the magnetization approaches equilibrium.
cond-mat.stat-mech↗
The layered phase of anisotropic gauge theories: A model for topological insulators
Topological insulators are materials where current does not flow through the bulk, but along the boundaries, only. They are of particular practical importance, since it is considerably more difficult, by ``conventional'' means, to affect their transport properties, than for the case of conventional materials. They are, thus, particularly robust to perturbations. One way to accomplish such changes is by engineering defects. The defects that have been the most studied are domain walls; however flux compactifications can, also, work. We recall the domain wall construction and compare it to the construction from flux compactification. A particular way of engineering the presence of such defects is by introducing anisotropic couplings for the gauge fields. In this case a new phase appears, where matter is confined along layers and local degrees of freedom cannot propagate through the bulk. It is, also, possible to take into account the ``backreaction'' of the dynamics of the gauge fields on the defects and find that a new phase, the layered phase, where, while transport of local degrees of freedom is confined to surfaces, the topological properties can propagate through the bulk, constituting an example of anomaly flow. The anisotropy itself can be understood as emerging from a particular Maxwell--dilaton coupling.
cond-mat.str-el↗
Supersymmetry and deterministic chaos
We show that the fluctuations of the periodic orbits of deterministically chaotic systems can be captured by supersymmetry, in the sense that they are repackaged in the contribution of the absolute value of the determinant of the noise fields, defined by the equations of motion.
hep-th↗
Nutation Wave as a Platform for Ultrafast Spin Dynamics in Ferromagnets
At short time scales the inertia term becomes relevant for the magnetization dynamics of ferromagnets and leads to nutation for the magnetization vector. For the case of spatially extended magnetic systems, for instance Heisenberg spin chains with isotropic spin-exchange interaction, this leads to the appearance of a novel collective excitation, the "nutation wave", whose properties are elucidated by analytic arguments and numerical studies. The one--particle excitations can be identified as relativistic massive particles. These particles, the "nutatons", acquire their mass via the Brout-Englert-Higgs mechanism, through the interaction of the wave with an emergent topological gauge field. This spin excitation would appear as a peak in the spectrum of the scattering structure factor in inelastic neutron scattering experiments. The high frequency and speed of the nutation wave can open new paths for realizing ultrafast spin dynamics.
cond-mat.mes-hall↗
The arithmetic geometry of AdS$_2$ and its continuum limit
According to the 't Hooft-Susskind holography, the black hole entropy,$S_\mathrm{BH}$, is carried by the chaotic microscopic degrees of freedom, which live in the near horizon region and have a Hilbert space of states of finite dimension $d=\exp(S_\mathrm{BH})$. In previous work we have proposed that the near horizon geometry, when the microscopic degrees of freedom can be resolved, can be described by the AdS$_2[\mathbb{Z}_N]$ discrete, finite and random geometry, where $N\propto S_\mathrm{BH}$. It has been constructed by purely arithmetic and group theoretical methods in order to explain, in a direct way, the finiteness of the entropy, $S_\mathrm{BH}$. What has been left as an open problem is how the smooth AdS$_2$ geometry can be recovered, in the limit when $N\to\infty$. In the present article we solve this problem, by showing that the discrete and finite AdS$_2[\mathbb{Z}_N]$ geometry can be embedded in a family of finite geometries, AdS$_2^M[\mathbb{Z}_N]$, where $M$ is another integer. This family can be constructed by an appropriate toroidal compactification and discretization of the ambient $(2+1)$-dimensional Minkowski space-time. In this construction $N$ and $M$ can be understood as "infrared" and "ultraviolet" cutoffs respectively. The above construction enables us to obtain the continuum limit of the AdS$_2^M[\mathbb{Z}_N]$ discrete and finite geometry, by taking both $N$ and $M$ to infinity in a specific correlated way, following a reverse process: Firstly, by recovering the continuous, toroidally compactified, AdS$_2[\mathbb{Z}_N]$ geometry by removing the ultraviolet cutoff; secondly, by removing the infrared cutoff in a specific decompactification limit, while keeping the radius of AdS$_2$ finite. It is in this way that we recover the standard non-compact AdS$_2$ continuum space-time. This method can be applied directly to higher-dimensional AdS spacetimes.
hep-th↗
Dynamic magnetostriction for antiferromagnets
In this paper we study the switching properties of the dynamics of magnetic moments, that interact with an elastic medium. To do so we construct a Hamiltonian framework, that can take into account the dynamics in phase space of the variables that describe the magnetic moments in a consistent way. It is convenient to describe the magnetic moments as bilinears of anticommuting variables that are their own conjugates. However, we show how it is possible to avoid having to deal directly with the anticommuting variables themselves, only using them to deduce non-trivial constraints on the magnetoelastic couplings. We construct the appropriate Poisson bracket and a geometric integration scheme, that is symplectic in the extended phase space and that allows us to study the switching properties of the magnetization, that are relevant for applications, for the case of a toy model for antiferromagnetic NiO, under external stresses. In the absence of magnetoelastic coupling, we recover the results reported in the literature and in our previous studies. In the presence of the magnetoelastic coupling, the characteristic oscillations of the mechanical system have repercussions on the N\'eel order parameter dynamics. This is particularly striking for the spin accumulation which is more than doubled by the coupling to the strain ; here as well, the mechanical oscillations are reflected on the magnetic dynamics. As a consequence of such a stress induced strain, the switching time of the magnetization is slightly faster and the amplitude of the magnetization enhanced.
cond-mat.mtrl-sci↗