1/f Noise from Glauber Dynamics: Self-Consistent Interaction and Temperature Dependent Correlations

Low frequency flicker noise has been argued to occur in spatially extendedmetastable systems near a critical point (Bak et al., 1987). An Ising-Glauber model based method is suggested here to systematically obtain temperature dependent nth-order correlation functions for N arbitrary interacting two-level systems (TLSs). This model is fully consistent with existing methods to calculate 1/f noise spectra from TLSs and complements them. However, with as such no a priori assumptions on the typical log normal distribution of fluctuation rates, it is shown that 1/fα noise manifests in two different cases: first in the thermodynamic limit on a 2D lattice with long range antiferromagnetic interactions at low temperatures and second in the case of a statistical ensemble of finite-sized spin clusters representing disorder, but where each cluster is ordered due to ferromagnetic interactions.


Introduction
Low frequency flicker noise or 1/ noise is seen in all sorts of systems such as in solid-state devices, road traffic, and biological, geological, and financial systems.Despite its widespread occurrence, there is no common physical underlying mechanism that gives rise to all the different manifestations of this noise [1].However, it has been argued that this flicker noise will occur in dynamical systems with extended spatial degrees of freedom that are barely stable, which evolve into self-organized critical structures [2].
Ever since its first measurement in the 1920s [3], 1/ noise has been seen in a wide variety of solid state systems [1,4,5].In the case of spin glasses, 1/ noise is related to the magnetic fluctuations [6,7].This is directly related to the low temperature kinetics of glassy systems and is often associated with the onset of a spin-glass phase [8].
In many solid state devices the presence of parasitic twolevel systems (TLSs), possibly due to the presence of defects, generates random-telegraphic noise (RTN) [9,10].Typically, if there are a large number of RTN sources (same as flipflopping Ising spins or fluctuating TLSs), then a log normal distribution of their switching rates gives rise to a 1/ type power spectrum within some range of frequencies.
Various theoretical models have been proposed to explain 1/ noise in SQUIDs.Examples include a model proposed by Koch et al. [47], where unpaired and noninteracting electrons 2 Advances in Condensed Matter Physics randomly hop between traps with different spin orientations and a 1/ distribution of trap energies.1/ noise can also be explained using a dangling bond model [48].Choi et al. [49] recently explained this in terms of metal induced gap states that arise due to the potential disorder at the metalinsulator interface.In general, hopping conductivity models are typically used to explain 1/ noise in electron glasses [50][51][52][53].Sometime a model was proposed, where the amplitude of 1/ noise scaled as 1/ [54]; however, this model did not consider interactions between TLSs.
Interactions are important for understanding 1/ noise spectra in different types of systems such as in spin glasses [8], where sometimes the temperature dependence of the noise can even be conflicting [6,55].Coulomb interactions and correlated hopping are particularly important in the case of Coulomb glasses [53,56].Spatially correlated fluctuations and the temperature dependence are also very important for understanding 1/ noise spectrum in metal films [57].In the case of 1/ noise in SQUIDs, experimental evidence suggests that the TLSs are strongly interacting and that there is a net spin polarization [34,42,58], which cannot be explained by a spin glass phase [59].Another example where both interactions and disorder also play an important role is in the case of metal-insulator transitions in 2D systems [60], where 1/ noise has been observed [61,62].
The purpose of this paper is to introduce a general fully self-consistent model that takes the temperature dependence, TLS-TLS interactions, and spatial correlations into account while calculating the noise spectra.This is done within the framework of Ising-Glauber dynamics.A straightforward method is suggested to systematically extract any th correlation function for  interacting Ising spins.This method is well suited for numerics as well as for analytics in smaller systems.It also allows open quantum systems methods such as the quasi-Hamiltonian formalism [24,40,[63][64][65][66][67] to connect to more complicated underlying temperature dependent noise microscopics.A temperature dependent spin-cluster model with ferromagnetic interactions, based on this method, was proposed recently to explain various puzzling features of 1/ magnetization noise in SQUIDs [68].
In the case of the interaction dependent model presented here, as such no a priori assumptions are needed for a log normal distribution of fluctuation rates in order to obtain 1/ noise.For long range antiferromagnetic interactions, there are signatures of 1/ type noise on an infinite 2D lattice, without making any heuristic assumption for switching rates, whereas, for long range ferromagnetic interactions, 1/ noise is shown to manifest from multiple clusters with a normal distribution of lattice constants.This spin-cluster model represents a disordered system.The overall suggested method is fully self-consistent: in the infinite temperature limit, the model can reduce to usual case with some assumed switching rate distribution.
This paper is organized as follows.In Section 2, the model and the correlation function calculation technique are discussed.An explicit example is given for a pair of interacting TLSs, where analytical expressions are derived for the correlation functions and the noise power spectra.More general results are discussed in Section 3, where this model can be used to show the manifestation of 1/ noise in two different cases.This is followed by Section 4.

Model for Noise Dynamics
2.1.Uncorrelated Two-Level System Fluctuations.A fluctuating two-level system is an Ising spin whose spin flips have a random time dependence, which is a stochastic process.Its dynamics is therefore governed by the master equation, W()/ = VW() [69], where V is a matrix of transition rates (such that the sum of each of its columns is zero) and W is the flipping probability matrix for the TLS.If the average occupation of the two states is the same (for unbiased fluctuators), then for spin- V  =   (  −   ) and the corresponding flipping probability matrix W = exp(V) is where   is the th spin's relaxation rate.For a single Ising spin, then the th order autocorrelation function can be obtained from W as follows: where W is the same as in For  uncorrelated fluctuating Ising spins, the probability matrix is simply W = W 1 ⊗ W 2 ⊗ ⋅ ⋅ ⋅ W  and the matrix V = ẆW −1 , which for the case of two fluctuating Ising spins is where   is a 2×2 identity matrix.Alternatively, if one started with the matrix of transition rates, then the explicit flipping probability matrix is explicitly W = exp(−V).

𝑧
=   ⊗   , and the projection vectors As expected, the cross-correlation term is zero as we only considered uncorrelated fluctuating Ising spins.The noise power spectrum can be obtained by taking the Fourier transform of the above correlation functions.
In general for  spins (interacting or noninteracting), any th order correlation function between arbitrary spins can be calculated as follows: and it is implied that And W and  ()  are in the same lexicographically ordered Ising spin basis.The W matrices for interacting Ising spins are discussed in the next section.

Correlated Two-Level System Fluctuations with
Ising-Glauber Dynamics.The method described in the previous section can also be used to obtain arbitrary -point correlation functions for correlated spin fluctuations.The system's overall time development is governed by the master equation Ẇ() = VW() as usual.The nonequilibrium spin dynamics for a system of fluctuating spins can be handled using the Ising-Glauber model [70,71].Single-site Glauber dynamics (also known as the kinetic Ising model) is a Markov process where the new distribution of spins depends only on the current spin configuration.A single spin is flipped at a given site, and the new configuration agrees with the old one everywhere except at the site where the spin has been flipped.In Glauber dynamics, the conditional probability for a single spin to flip is determined by the Boltzmann factor.For correlated spin fluctuations, the matrix elements of the matrix of transition rates are Here, s  is a vector denoting the present spin configuration of the lattice, s denotes the spin configuration of the lattice at an earlier instance of time, and  is the relaxation rate of the spin that is flipped.The nonnegative off-diagonal matrix elements in (10) satisfy the detailed balance condition and the diagonal terms are just the negative sum of the off-diagonal column elements so that all column elements sum up to zero, which ensures the conservation of probability.The systems temporal dynamics is then governed by the flipping probability matrix, which is W = exp(−V).The eigenvalues of V are either zero, which corresponds to the equilibrium distribution, or real-negative which also eventually tend to the equilibrium distribution as  → ∞ [69].
The Hamiltonian in the Boltzmann factor in ( 10) is where  is the magnetic field and  , is the spin-spin interaction between the th and th spins.

Two Correlated Fluctuators.
In order to obtain the power spectrum of a given pair of interacting spins, one has to first obtain the two-point correlation functions.Consider two fluctuating Ising spins that are correlated.If  = 0, then the V matrix in the {11, 11, 11, 11} basis has the following form: where  = 1 + exp(2).From this form, it is apparent that if  = 0, then (4) is recovered.The flipping probability matrix is then W = exp(−V) which can be obtained explicitly.The autocorrelation functions and the cross-correlation functions can be calculated using ( 6) and ( 7): ⟨ 1 (0) 1 ()⟩ = ⟨       (1)   W() (1)   W(0) = ⟨       (2)   W() (1)   W(0) where Θ = / and  = √ ( and The projection vectors corresponding to the zero eigenvalue solution of V are In the infinite temperature limit ( → 0) the results of ( 6) and ( 7) are recovered, whereas, in the limit of where Note that here ∑  ⟨  (0)  ()⟩ = 2 −2Γ −|| , which implies that two interacting TLSs can be expressed as a single TLS with effective flipping rate Γ − .This could lead to a simple way to renormalize and reduce large 2  dimensional V matrices for more conducive lattice structures.The effective flipping rates can then be introduced in the form of an effective .
The temperature dependent autocorrelation functions are shown in Figure 1(a) and the cross-correlation functions are shown for ferromagnetic interactions in Figure 1(b) and antiferromagnetic interactions in Figure 1(c).As expected for antiferromagnetic interactions, the cross-correlation function is anticorrelated for a pair of fluctuators at initial times, whereas the autocorrelation function does not depend on the sign of .Also note that the maximum amplitude of the cross-correlation function is inversely proportional to the temperature as expected, thereby ensuring that the crosscorrelation terms vanish at high temperatures.
The respective power spectra are obtained by taking the Fourier transform of the correlation functions (13),   () = ∫ ∞ 0 ⟨  (0)  ()⟩  , and are where Similarly, expressions for  22 and  21 can be obtained by swapping the indices 1 and 2 in the above expressions for the power spectrum and also for the respective coefficients.The total power spectrum, for the pair of spins, is then just the total sum () = ∑ ,   (), where ,  = 1, 2 and is shown in Figure 2 for various ferromagnetic and antiferromagnetic interactions.It is seen that for the ferromagnetic case,  = 1, the amplitude of the power spectra is inversely proportional to the temperature, whereas for  = −1 it is directly proportional.In general one can solve for the systems dynamics by evaluating the eigenvalues and eigenvectors of the V matrix directly for a large number of spins.However, this quickly becomes a prohibitive as the size of V scales as 2  ( being the total number of TLSs).

Self-Consistent 1/𝑓 𝛼 Noise from Glauber Dynamics: Ferromagnetic and Antiferromagnetic Interactions
Usually for 1/ noise a log normal distribution of the fluctuations rates is assumed.In the presence of several uncorrelated TLSs, the power spectrum from the two-point correlation functions is If the distribution () ∝  −1 in some range  min <  <  max , then Here () ∝  −1 when 2 min <  < 2 max since the term in the square parenthesis is ∼1 in that range.

Antiferromagnetic Interactions.
In this section, it is shown that a 1/ type noise spectrum can be obtained using a different approach without making any heuristic assumptions on the distribution of fluctuation rates (e.g., see ( 21)).From here on all  = 1.Based on the formalism discussed in the previous section, the noise power is calculated for a 2D lattice of  Ising spins.Consider the case where every spin on the lattice interacts with every other spin via a Ruderman-Kittel-Kasuya-Yosida (RKKY) type interaction: where  , is the separation between two spins.We consider a model where the spins are equally spaced and their separation is given by .It is thought that spin-spin interaction via RKKY mechanism is responsible for the spin polarization and 1/ noise in SQUIDs [35].The RKKY interaction can give rise to unusual magnetic ordering as the magnetic ordering can oscillate between being ferromagnetic for nearer spins to being antiferromagnetic for more distant ones.This unusual magnetic ordering can also lead to the formation of spin clusters.
In Figure 3, the temperature dependent correlation functions, power spectrum, and the respective slopes are shown for  = 10 spins.There are strong hints of the presence of 1/  type noise.However, the  ∼ 1 part is not sustained over longer .In order to study this behavior in the thermodynamic limit, the power spectrum is calculated for six different cases,  = 6, 7, 8, 9, 10, 11 spins, and then every single point for the correlation functions is extrapolated to the limit 1/ → 0 (or  → ∞).From Figure 4 a clearer 1/  type slope can be seen.The 1/ type noise disappears at higher temperatures, which indicates that the  ∼ 1 noise is a result of the antiferromagnetic RKKY interactions.Very similar behavior is seen if the RKKY interactions are truncated at the nearest neighbor site.In the case of ferromagnetic interactions, this  ∼ 1 noise shoulder is not seen.
To get a better understanding of why this behavior arises even when all the s are set to 1, one can look at the Boltzman factor based conditional probabilities: From the structure of the V matrix (10) and from the twopoint correlation functions for two spins (17), it can be seen that Γ determines the effective transition rate at which a spin configuration s will transition into a new configuration s  .Since the V matrix scales as 2  , it is impossible to exactly examine large spin lattices.However, one can still examine a single column at a time or consider an ensemble of equally spaced columns.In Figure 5 a distribution of the Γs, (Γ), is shown for  = 10, 15, and 20 spins at various temperatures for antiferromagnetic RKKY interactions.(Γ) = (Γ) raw / ∑() is normalized over the total number of entries.Since the distributions for the Ising-Glauber model can vary from column to column, (Γ) is averaged over 256 equally spaced columns, which is representative of the sample space.
It can be seen immediately that, at lower temperatures, the Γs take a 1/Γ type distribution, while at higher temperatures and higher , they assume more of a Poisson type distribution.Eventually, in the infinite temperature limit,  → 0, this distribution centers at 1/2 as expected.A notable feature in the distribution is that there are also a significant number of TLSs that sit at Γ ∼ 1 -although, for  = 20, this is still about a couple of orders of magnitude smaller than the maximum height of the distribution.These TLSs at Γ = 1 are most reluctant to flip and could be thought of as being in an ordered state.The relative number of TLSs at Γ = 1 decreases as  is increased.Above a certain temperature, the number of TLSs sitting at Γ = 1 drops sharply and (Γ) assumes a smoother distribution shape.The 1/ type noise feature shown in Figure 4 at lower temperatures comes from the part where (Γ) assumes a 1/Γ type distribution.The bump in the noise spectra's shoulder right after the  = 2 part comes from the TLSs that sit at Γ = 1.And as the TLSs at Γ = 1 dominate the distribution more, then the noise shoulder's  (for the part representing 1/ noise) keeps reducing to values below 1.This effect is much more pronounced for ferromagnetic long range interactions.

Ferromagnetic Interactions.
Next consider ferromagnetic RKKY interactions.The distribution of Γ is shown in Figure 6.While the low temperature (Γ) has a line shape that is quite similar to that of the antiferromagnetic case, the number of TLSs at Γ = 1 is now significantly higher.This is expected as the ferromagnetic RKKY interactions induce more TLSs to be in the ordered state.These TLSs are least likely to flip due to strong ferromagnetic interactions between them.The ratio of the number of TLSs that are ordered versus ones that are not increases as  increases.
For ferromagnetic interactions a 1/ noise slope cannot be seen for a single cluster.The calculated correlation functions for ferromagnetic interactions are very long lived and even more so at lower temperatures.This is of course expected for +|  |.Even though at low temperatures the distribution has a 1/Γ type line shape, the dominant TLS density at Γ = 1 dominates and washes out any possibilities of seeing the 1/ noise slope.For example, for  = 20, the TLS density at Γ = 1 is at least 2 orders of magnitude higher than that anywhere else for  −1 = 0.1 and is 3 orders of magnitude higher for  −1 = 0.2 case.A key observation here is that this difference of the peak densities reduces as  decreases.This will next prove to be conducive for obtaining 1/ type noise that spans several decades in frequency.As the temperature is raised, the distribution broadens.Finally, in the  → 0 limit, the distribution centers at Γ = 1/2 as expected.
Although, for ferromagnetic interactions, 1/ noise is not seen for a single spin lattice, 1/  noise can be obtained if one considers a statistical ensemble of various clusters.This is a representation of disorder.The correlation functions averaged over all clusters, the respective noise spectrum, and the respective noise slopes are shown in Figure 7. 20 spin clusters were considered with 9 spins each.Each cluster is assigned a random lattice constant    with a uniform distribution (see Figure 7(d)).The 1/  flux noise ( ∼ 1) appears at an intermediate range of frequencies.At high frequencies, the noise slope of 2 corresponds to the Lorentzian tail.The upper and lower cutoff frequencies for the 1/ noise depend on the distribution of    and   (or T).Eventually, for all T,  → 0 at very low frequencies.Note that, in the absence of interactions, the noise spectrum is just a simple Lorentzian.Similar results can also be obtained with short range interactions, though the temperature dependence will be much higher then [68].
The reason why 1/ type noise can now be obtained in the spin-cluster model is because the interactions assume a 1/ type distribution.If the RKKY   is expanded for small   , then   ∝ 1/  ; hence, a uniform distribution of   results in a 1/  distribution of interaction strengths at a certain crossover length 1/  .This long range ferromagnetic interaction model is particularly helpful for explaining magnetization noise in SQUIDs [68].

Summary
In summary, a general method is suggested for calculating the temperature dependent noise spectra of interacting TLSs.The TLSs are modeled as flip-flopping Ising spins whose temperature dependent fluctuations are governed by Glauber dynamics.A method is suggested to systematically extract any th-order correlation function, from the solutions of the stochastic master equation, for  arbitrary TLSs.This model is fully consistent with the usual existing method of calculating 1/ noise from random telegraphic signals and complements the existing method by adding interactions and temperature dependence.
Analytical expressions are obtained for the autocorrelation functions, cross-correlation functions, and the power spectrum for a pair of interacting spins using this model.Numerical results are obtained for larger spin lattices.In general this method can be used to model a multidimensional lattice with any type of interaction.With no a priori assumptions on a log normal distribution of fluctuation rates, it is shown that 1/ type noise can be obtained, at low temperatures, in two different cases on a 2D lattice: first in the thermodynamic limit on an infinite lattice with antiferromagnetic RKKY interactions and the second case on multiple finite sized spin clusters representing spatial disorder, but where each cluster is magnetically ordered by ferromagnetic RKKY interactions.This behaviour can be understood from the distribution of the conditional probabilities of spin configurations.Overall this provides general insight into different manifestations of 1/ noise in spatially extended metastable dynamical systems.

Figure 3 :
Figure 3: Temperature dependent (a) correlation functions, (b) power spectrum, and (c) slopes of the power spectrum for  = 10 spins.

Figure 4 :
Figure 4: Temperature dependent (a) correlation functions and (b) power spectrum shown in the thermodynamic limit ( → ∞).(c) Some samples of best, mediocre, and worst case fits are shown for  = 6-11 spins.

Figure 7 :
Figure 7: (a) Temperature dependent net correlation function for 20 spin clusters, each with 9 spins with ferromagnetic (+|  |) RKKY interactions.Note that the overall results are independent of cluster size.(b) Corresponding flux noise spectrum showing 1/  noise below ∼0.1 Hz and (c) its respective slope , where  ≲ 1 below about 0.1 Hz.(d) Distribution of    for each cluster, with mean ⟨  ⟩ and standard deviation (Σ) as indicated.