SU2xSU2 package

SU2xSU2.SU2_mat_routines module

SU2xSU2.SU2_mat_routines.alpha_to_a(alpha)

A lattice of SU(2) matrices may be defined through the exponential map with parameters alpha. The exponential map can be explicitly evaluated, resulting in a linear combination of the Pauli matrices and unity: U = exp(i alpha_i sigma_i) = a_0*1 + i*a_i sigma_i This function finds the coefficients a based on alpha.

Parameters:

alpha ((L,L,3) array) – parameters when representing a SU(2) group element via the exponential map at every lattice site

Returns:

a – parameters of matrices at each lattice site when explicitly evaluating the exponential map

Return type:

(L,L,4) array

SU2xSU2.SU2_mat_routines.det(a)

The determinant of an SU(2) matrix is given by the squared length of the parameter vector.

Parameters:

a ((L,L,4) array) – parameters of the SU(2) valued lattice

Returns:

determinant – determinants of the SU(2) valued lattice

Return type:

(L,L) array

SU2xSU2.SU2_mat_routines.dot(a, b)

Computes the elementwise matrix product between two lattices of SU(2) matrices with parameter vectors a and b.

Parameters:

• a ((L,L,4) array) – parameters of first SU(2) valued lattice

• b ((L,L,4) array) – parameters of second SU(2) valued lattice

Returns:

c – parameters of SU(2) valued lattice resulting from the elementwise matrix products

Return type:

(L,L,4) array

SU2xSU2.SU2_mat_routines.hc(a)

Returns the parameter vector of the hermitian conjugate at each lattice site.

Parameters:

a ((L,L,4) array) – parameters of the SU(2) valued lattice

Returns:

a_hc – parameters of hermitian conjugate SU(2) valued lattice

Return type:

(L,L,4) array

SU2xSU2.SU2_mat_routines.make_mats(a)

Constructs explicit matrices corresponding to parameter vector a.

Parameters:

a ((L,L,4) array) – parameters of the SU(2) valued lattice

Returns:

mats – np.matrix instance at every site

Return type:

(L,L) object array

SU2xSU2.SU2_mat_routines.norm2(a)

Returns squared norm of the parameter vector a.

Parameters:

a ((L,L,4) array) – parameters of the SU(2) valued lattice

Returns:

norm_sq – containing the norm at each site

Return type:

(L,L) array

SU2xSU2.SU2_mat_routines.renorm(a)

Renormalises matrix to have det = 1

Parameters:

a ((L,L,4) array) – parameters of the SU(2) valued lattice

Returns:

renormed – renormalised parameters of the SU(2) valued lattice

Return type:

(L,L,4) array

SU2xSU2.SU2_mat_routines.sum(a, b)

Computes the elementwise sum of two SU(2) valued lattices A and B with parameters a and b. Let C = A + B, i.e. c = a + b. Note that the sum of two SU(2) matrices is proportional to an SU(2) matrix with proportionality constant k, meaning D = C/k = 1/k (A + B) is in SU(2). To only having to perform manipulations on SU(2) matrices, the parameters d of the SU(2) valued lattice D and the constant k are returned such that their product gives the parameter vectors of C, the sum of lattice A and B.

Parameters:

• a ((L,L,4) array) – parameters of first SU(2) valued lattice

• b ((L,L,4) array) – parameters of second SU(2) valued lattice

Returns:

• d ((L,L,4) array) – parameters of SU(2) valued lattice proportional to a+b

• k ((L,L,1) array) – proportionality constant between d and a+b

SU2xSU2.SU2_mat_routines.tr(a)

Returns the trace of the matrices at each lattice site.

Parameters:

a ((L,L,4) array) – parameters of the SU(2) valued lattice

Returns:

trace – trace at each site of the SU(2) valued lattice

Return type:

(L,L) array

SU2xSU2.SU2xSU2 module

class SU2xSU2.SU2xSU2.SU2xSU2(L, a, ell, eps, beta, mass=0.1)

Bases: object

Each instance describes a realization of the SU(2)xSU(2) model on a square lattice whose dynamics can be simulated using the Fourier Accelerated Hybrid Monte Carlo algorithm.

Ham(phi, pi)

Computes the Hamiltonian for a lattice configuration phi and momentum configuration pi. The kinetic term is chosen such that the momenta follow a standard Gaussian.

Parameters:

• phi ((L,L,4) array) – parameters of the SU(2) valued field

• pi ((L,L,3) array) – parameter values of the conjugate momenta at each lattice site

Returns:

H – the Hamiltonian as the sum of the action and the kinetic term

Return type:

float

Ham_FA(phi, pi)

Analogous to SU2xSU2.SU2xSU2.Ham() but computes the modified Hamiltonian used to accelerate the dynamics.

Parameters:

• phi ((L,L,4) array) – parameters of the SU(2) valued field

• pi ((L,L,3) array) – parameter values of the conjugate momenta at each lattice site

Returns:

H – the modified Hamiltonian as the sum of the action and the modified kinetic term

Return type:

float

See also

SU2xSU2.SU2xSU2.Ham

__init__(L, a, ell, eps, beta, mass=0.1)

Parameters:

• L (int) – Number of lattice sites along one dimension. Must be even for implementation of Fourier acceleration to work properly

• a (float) – Lattice spacing

• ell (int) – Number of steps to integrate Hamilton’s equation, each of size eps

• eps (float) – Step size for integrating Hamilton’s equations

• beta (float) – Model parameter defined in terms of the nearest neighbor coupling parameter g via beta = 1/(2g^2)

• mass (float (optional)) – Mass parameter used in Fourier Acceleration. The physics is independent of the parameter value which only effects the simulation performance. The most efficient choice depends on beta.

action(phi)

Computes the action for lattice configuration phi.

Parameters:

phi ((L,L,4) array) – parameters of the SU(2) valued field at each lattice site

Returns:

S – the action

Return type:

float

exp_update(pi_dt)

Computes the update matrix of the field for the transition t -> t + dt. The update matrix is given by the exponential of the conjugate momenta pi times i dt. As the momentum field is a linear combination of the SU(2) generators, the update matrix is itself an SU(2) element, described by the parameter vector of pi scaled by dt.

Parameters:

pi_dt ((L,L,4) array) – parameters of the momenta times the integration step size

Returns:

update – parameters of the update matrices

Return type:

(L,L,4) array

internal_energy_density(phi)

Computes the internal energy (action) per site of a lattice configuration.

Parameters:

phi ((L,L,4) array) – lattice configuration to perform computation on

Returns:

e – action per site

Return type:

float

kernel_inv_F()

Finds inverse of the action kernel computed in the Fourier space, here referred to as A.

Returns:

A – inverse action kernel in Fourier space

Return type:

(L,L) array

leapfrog(phi_old, pi_old)

Performs the leapfrog integration of Hamilton’s equations for self.ell steps, each of size self.eps. The passed arguments define the initial conditions.

Parameters:

• phi_old ((L,L,4) array) – last accepted sample in the chain of the SU(2) valued field

• pi_old ((L,L,3) array) – parameters of conjugate momenta associated with phi_old

Returns:

• phi_cur ((L,L,4) array) – SU(2) matrix parameters after simulating dynamics

• pi_cur ((L,L,3) array) – momenta parameters after simulating dynamics

leapfrog_FA(phi_old, pi_old)

Leapfrog integrator analogous to SU2xSU2.SU2xSU2.leapfrog() but using the modified EoMs.

See also

SU2xSU2.SU2xSU2.leapfrog

make_NN_mask()

Makes mask to apply to SU(2) valued field phi or momentum field pi which then gives the matrix parameter values of the nearest neighbors (NN) for each lattice site. Hence phi[self.NN_mask] is of shape (L,L,#neighbors,#parameters) i.e (L,L,4,4).

Returns:

NN_mask – tuple of two (L,L,4,1) arrays, each giving the row and column coordinate for all nearest neighbors

Return type:

tuple

pi_dot(phi)

Computes the derivative of pi with respect to the Hamiltonian time. From Hamilton’s equations, this is given as i times the derivative of the action wrt. phi. pi and its time derivative are linear combinations of the Pauli matrices and hence described by 3 real parameters alpha.

Parameters:

phi ((L,L,4) array) – parameters of the SU(2) valued field

Returns:

pi_t – parameters of the time derivative of the conjugate momenta

Return type:

(L,L,3) array

pi_samples()

Returns real space sample of momenta according to the distribution based on the modified kinetic term in the modified Hamiltonian. The sampling is easiest done in Fourier space and in terms of a real and hermitian object PI from which the momentum samples can be reconstructed (both in Fourier space) The size of the lattice along one dimension L is assumed to be even.

Returns:

pi – parameters for the sample of the conjugate momenta in real space

Return type:

(L,L,3) array

prod_A_pi(pi_F)

Computes the elementwise product of the inverse action kernel in Fourier space (denoted as ‘A’) and the momenta in Fourier space.

Parameters:

pi_F ((L,L,3) array) – parameter vector of momenta in Fourier space

Returns:

prod – parameter vector of momenta in Fourier space, each site being weighted by the inverse Fourier space kernel

Return type:

(L,L,3)

run_HMC(M, thin_freq, burnin_frac, accel=True, measurements=[], chain_paths=[], saving_bool=True, partial_save=5000, starting_config_path='', RNG_state_path='', chain_state_dir='data/chain_state/', renorm_freq=10000)

Performs the Hybrid Monte Carlo simulation, by default with Fourier Acceleration.

A total of M trajectories will be simulated and measurements are taken every thin_freq steps after the thermalisation period defined as the first M*burnin_frac samples. Due to accumulating rounding errors, the molecular dynamics cause field to leave the group manifold slowly. This is counteracted by enforcing the unitarity constraint of the matrix parameters every renorm_freq-th step. The Monte Carlo chain is fully defined (and thus reproducible) by the model and simulation parameters as well as the initial configuration of the chain and the state of the random number generator (RNG). By using the last configuration of a previous chain and the associated RNG state, one can continue the chain seamlessly in a new simulation.

Parameters:

• M (int) – number of HMC trajectories and thus total number of generated samples

• thin_freq (int) – frequency at which measurements will be taken

• burin_frac (float) – fraction of total HMC samples which are rejected as burn in

• accel (bool (optional)) – By default True, indicating to use Fourier Acceleration

• measurements (list of callables (optional)) – can select from: internal_energy_density, susceptibility, ww_correlation_func to measure the internal energy density, susceptibility, and wall-to-wall correlation respectively

• chain_paths (list of str (optional)) – Only required if saving_bool=True, otherwise can be left empty. Listing the file paths relative to current working directory to store the measurements. The data will always be saved as a .npy file, allowing to omit the file extension.

• saving_bool (bool (optional)) – save measurement data

• partial_save (int (optional)) – after how many steps preliminary measurements and chain state is saved to disk. Requires saving_bool=True.

• starting_config_path (str (optional)) – path to configuration to initialize the chain (.npy file). If not passed a disordered (i.e hot) start will be used.

• RNG_state_path (str (optional)) – relative path to a .obj file containing the internal state of the random number generator from a previous run. When using the final configuration and the associated RNG state of a previous run as the starting configuration for a new one, the chain is seamlessly continued.

• chain_state_dir (str (optional)) – path to directory in which the RNG state and the last configuration will be saved

• renorm_freq (int (optional)) – after how many trajectories the SU(2) valued fields are projected back to the group manifold. Set to None to never renormalize

susceptibility(phi)

Computes the susceptibility (the average point to point correlation) for the configuration phi which can be obtained by summing up the correlation function.

Parameters:

phi ((L,L,4) array) – configuration to perform computation on

Returns:

chi – susceptibility of phi

Return type:

float

ww_correlation_func(phi)

Computes the non-normalized wall to wall correlation function for the lattice configuration phi. Observing that the correlation of two variables can be written as a convolution, one can apply the cross correlation theorem to efficiently compute the latter using FFTs.

Parameters:

phi ((L,L,4) array) – configuration to perform computation on

Returns:

ww_cor – wall to wall correlation evaluated for wall separation in interval [0, L), not normalized to value at zero separation.

Return type:

(L,) array

SU2xSU2.analysis module

SU2xSU2.analysis.acceleration_mass_search(num_traj, burnin_frac, beta, L, corlength, masses, chain_dir='data/acceleration_mass/', data_path='data/accelertion_mass.txt', plot_path='plots/acceleration_mass.pdf')

Performs a grid search of the mass acceleration parameter for the passed value pair of beta, L. To assess the quality of the canonical choice of M=1/correlation length against other values of the parameter, a plot is produced comparing the simulation cost agaist values of the acceleration mass. The plot is stored at plot_path and the associated data is stored at data_path. Computing the cost function requires measurming the susceptibility integrated autocorrelation time. The susceptibility chain is stored in the directory chain_dir with the file being labeled by the value of beta and L.

Parameters:

• num_traj (int) – number of trajectories in each simulation using a different acceleration mass

• burnin_frac (float) – fraction of trajectories discarded as burn in

• beta (float) – model parameter beta for which the simulations are performed

• L (int) – lattice size along one direction. Must be even.

• corlength (float) – correlation length infered for simulations at beta

• masses (array) – values of the acceleration parameter to include in the grid search

• chain_dir (str (optional)) – path to directory containing susceptibility measurement chain.

• data_path (str (optional)) – path at which the collected data will be stored: masses, cost function and its error, simulation time, acceptance rate, susceptbility IAT and its error. Must be a text file.

• plot_path (str (optional)) – path at which the plot of the cost fucntion vs the acceleration mass is stored

SU2xSU2.analysis.critical_slowingdown(num_traj, burnin_frac, corlengthdata_path='data/corlength.txt', chi_chain_dir='data/slowdown/rawchains/', chi_data_dir='data/slowdown/', slowdownplot_path='plots/slowdown.pdf', costfuncplot_path='plots/costfunc.pdf', xticks=[2, 5, 10, 20, 50, 100], yticks=[1, 5, 10, 25])

Computes the integrated autocorrelation time (IAT) of the susceptibility using beta, L form previous simulations where the correlation length was measured and stored. For example, the data produced in SU2xSU2.analysis.mass_lambda() and stored at its argument corlengthdata_path can be used. Its is assumed that the data has a row wise structure of Ls, betas, correlation length. The computation is done for accelerated and unaccelerated dynamics, with the measurement chains being stored in directories accel/, unaccel/ at chi_chain_dir. Files are labeled by the used value of beta and L. The ensemble average results are stored as text files at chi_data_dir while the associated plot of IAT vs correlation length is stored at slowdownplot_path. Power laws are fitted for either acceleration choice, allowing to quantify the degree of critical slowing down through the fitted value of the critical exponent ‘z’. A further plot, showing the simulation cost of either acceleration choice, is stored at costfuncplot_path.

Parameters:

• num_traj (int) – number of trajectories in each simulation

• burnin_frac (float) – fraction of trajectories discarded as burn in

• corlengthdata_path (str (optional)) – path to the Ls, betas, correlation length data (must be .txt). The default causes to use the result from SU2xSU2.analysis.mass_lambda() when its argument with the same name is also left as default.

• chi_chain_dir (str (optional)) – path of directory where two directories (named accel and unaccel) will be created to store the measurement chains of the susceptibility.

• chi_data_dir (str (optional)) – path of directory where two directories (named accel and unaccel) will be created to store L, beta, the IAT, its error, the ensemble averaged susceptibility, its error, the simulation time, and the acceptance rate (row wise).

• slowdownplot_path (str (optional)) – path to plot showing the power law scaling of the susceptibility IAT with the correlation length

• costfuncplot_path (str (optional)) – path to save the simulation cost function plot at

• xticks (list (optional)) – list of ints or floats specifying the tick labels for the correlation length in the two plots

• yticks (list (optional)) – list of ints or floats specifying the tick labels for the ratio unaccel cost function / accel cost function in the cost function plot

SU2xSU2.analysis.get_avg_error(data, get_IAT=False)

Performs the ensemble average based on a sequence of measurements of a single observable and estimates its error as the standard error on the mean (SEM), corrected by the square root of the observable’s integrated autocorrelation time. Optionally also returns integrated autocorrelation time and its error. Each element of data is refereed to as a data point and can either be a float or a 1D array of length N. The returned average and error will be of the same shape as the data point.

Parameters:

• data ((M,) or (M,N) array) – set of measurements made during the chain

• get_IAT (bool (optional)) – False by default. Set to True to return IAT and its error

Returns:

• avg (float or (N, )) – ensemble average of data

• error (float or (N, )) – autocorrelation corrected SEM

• IAT (float (if get_IAT==True)) – integrated autocorrelation time (IAT)

• IAT_err (float (if get_IAT==True)) – error of IAT

SU2xSU2.analysis.get_corlength(ww_cor, ww_cor_err, data_save_path)

Infers the correlation length based on the ensemble averaged wall to wall correlation function and its error. The correlation function data is processed by normalizing and averaging it about its symmetry axis at L/2 which will be saved at data_save_path.

Parameters:

• ww_cor ((L,) array) – ensemble average of correlation function on a lattice of length L

• ww_cor_err ((L,) array) – associated error for all possible wall separations

• data_save_path (str) – relative path to cwd at which the wall separations, the processed correlation function and its error will be stored row wise as an .npy file (extension not needed in path)

Returns:

• cor_length (float) – fitted correlation length in units of the lattice spacing

• cor_length_err (float) – error in the fitted correlation length

• reduced_chi2 (float) – chi-square per degree of freedom as a goodness of fit proxy

SU2xSU2.analysis.internal_energy_coupling_exp(betas, Ls, num_traj, burnin_frac, accel=True, chaindata_pathbase='data/energy_density/', simdata_path='data/energy_density.txt', plot_path='plots/coupling_expansion.pdf')

Computes and stores the internal energy per site for the passed value pairs of betas, Ls. As a density is computed, the value of the lattice size is not crucial and finite size effects are often negligible. For each value pair, num_traj trajectories are simulated, with burnin_frac specifying the fraction of these rejected as burn in. By default the Fourier accelerated Hybrid Monte Carlo algorithm is used. For each simulation, the chain of internal energy measurements is stored at chaindata_pathbase and files are labeled by the used value of beta and L. The ensemble average values of the internal energy densities are stored at simdata_path as a text file. A plot, stored at plot_path, is produced comparing the simulation result with the weak and strong coupling expansions (w.c. and s.c.).

Parameters:

• betas ((n,) array) – values of the model parameter beta for which simulations are performed in ascending order

• Ls (int or (n,) array) – size of the lattice along one dimension for each simulation at different beta. When an integer is passed, the size will be assumed for all values of beta.

• num_traj (int) – number of trajectories in each simulation

• burnin_frac (float) – fraction of trajectories discarded as burn in

• accel (bool (optional)) – using Fourier Acceleration by default

• chaindata_pathbase (str) – path of the directory where the internal energy measurement chains for all value pairs of beta, L will be saved

• simdata_path (str (optional)) – path of .txt file (relative to the current working directory) to store the ensemble averaged simulation results for the internal energy density and its error (with file extension)

• plot_path (str) – path of final plot file (with file extension)

SU2xSU2.analysis.mass_lambda(betas, Ls, num_traj, burnin_frac, accel=True, corlengthdata_path='data/corlength.txt', corfuncs_chain_dir='data/corfuncs/rawchains/', corfuncs_dir='data/corfuncs/', corfuncs_plot_dir='plots/corfuncs/', plot_path='plots/asymptotic_scaling.pdf')

Computes the mass over lambda parameter ratio for for the passed value pairs of betas, Ls (lattice size along one dimension). The lattice size must be chosen sufficiently large (meaning a multiple of the correlation length) to avoid finite size effects. In general, the required lattice size increases quickly with beta.

Computing the mass over lambda ratio requires the correlation length for each beta,L value pair and will be stored in the text file at corlengthdata_path. For each of the considered beta,L value pairs, the raw measurement chain for the correlation function is saved in the directory corfuncs_chain_dir, while the the processed correlation function data (normalized as well as ensemble averaged and averaged across the symmetry axis at L/2) is stored at corfuncs_dir. Plots of the correlation functions, with the fitted analytical expectation are stored at corfuncs_plot_dir. In all cases the file names are given by considered values of beta,L.

A plot of mass over lambda ratio is produced and stored at plot_path, allowing to assess the convergence of the simulation data to the continuum mass gap prediction as beta gets large.

Parameters:

• betas ((n,) array) – values of the model parameter beta for which simulations are performed in ascending order

• Ls ((n,) array) – size of the lattice along one dimension for each simulation at different beta. Must be even.

• num_traj (int) – number of trajectories in each simulation

• burnin_frac (float) – fraction of trajectories discarded as burn in

• accel (bool (optional)) – using Fourier Acceleration by default

• couplingdata_path (str (optional)) – path of .txt file (relative to the current working directory and with file extension) to store the correlation length, its error and the associated chi-squared value for all betas,Ls.

• chaindata_pathbase (str (optional)) – path of the directory where the correlation function measurement chains for all value pairs of beta,L will be saved

• corfuncs_dir (str (optional)) – path of the directory where the processed correlation function data for all value pairs of beta,L will be saved

• corfuncs_plot_dir (str (optional)) – path of the directory where the plots of the processed correlation function for all value pairs of beta,L will be saved

• plot_path (str (optional)) – path (with file extension) of the plot showing the numerical mass over lambda ratio against beta.

SU2xSU2.calibrate_paras module

SU2xSU2.calibrate_paras.calibrate(model_paras, accel=False, sim_paras=None)

For a model, specified by the dictionary model_paras, this function calibrates the number of leapfrog integration steps and their size under the constraint that the trajectory length is 1 and that the acceptance rate is with in desireable range between 60 and 75%.

The calibration is done by performing short simulation runs (500 trajectories with 50% burn in unless overridden by passing sim_paras), extracting the acceptance rate and, if the acceptance rate is outside this range, adjusting the number of steps according to the difference to the ideal acceptance rate of 65% for the next run. The number of steps is inferred from constraining the trajectory length to unity. It is not guaranteed that the calibration is successful for all possible model specification given the fixed trajectory length. Hence the calibration is limited to 10 iterations. An indicator that longer trajectories are required is when the calibration algorithm tries to reduce the number of steps below one.

Recommended to use the returned calibrated model_paras to define the model for the production run.

Parameters:

• model_paras (dict) – {L, a, ell, eps, beta} denoting lattice size, lattice spacing, number of integration steps, integration step size and the SU(2)xSU(2) model parameter beta respectively The values of ell, eps are used as guesses to start the calibration and their product must be 1.

• accel (bool, optional) – use Fourier Acceleration or not

• sim_paras (dict, optional) – {M, thin_freq, burnin_frac, accel=True, measurements=[], chain_paths=[], saving_bool=True, partial_save=5000, starting_config=None, RGN_state=None, renorm_freq=10000} Specifying the simulation parameters for the calibration run by calling SU2xSU2.SU2xSU2.run_HMC(). Consider the associated docstring for definitions of these parameters.

Returns:

model_paras – calibrated model parameters and of the same form as model_paras

Return type:

dict

SU2xSU2.correlations module

SU2xSU2.correlations.auto_window(IATs, c)

Windowing procedure of Caracciolo and Sokal, Dynamic critical exponent of some Monte Carlo algorithms for the self-avoiding walk, J. Phys. A: Math. Gen. 19 (1986) 797 to truncate the sum for the IAT.

Parameters:

• IATs (array) – integrated autocorrelation time with increasingly late termination of the sum

• c (float) – defines the window width. For correlations that decay exponentially, a value of 4 or 5 is conventional

Returns:

idx – index for array IATs, representing the IAT estimate from the windowing procedure

Return type:

int

SU2xSU2.correlations.autocorr_func_1d(x)

Computes the autocorrelation of a 1D array x using FFT and the cross correlation theorem. As FFTs yield circular convolutions and work most efficiently when the number of elements is a power of 2, pad the data with zeros to the next power of 2.

Parameters:

x ((N,) array) – 1 dimensional data series of N elements to find the autocorrelation function for

Returns:

acf – the autocorrelation function

Return type:

(N,) array

SU2xSU2.correlations.autocorrelator(data, c=4.0)

Alias for SU2xSU2.correlations.autocorrelator_repeats() when the data has been observed only once, meaning data is of shape (M,).

See also

correlations.autocorrelator_repeats

SU2xSU2.correlations.autocorrelator_repeats(data, c=4.0)

Based on the implementation in emcee. Computes the autocorrelation function and integrated autocorrelation time (IAT) for passed data which is a 2D array such that each row represents a sample of observations. The correlations are computed between rows of the data by finding the 1D autocorrelation function for each column of the data. The overall ACF between rows is estimated as the average of correlations across columns. This also allows to get an estimate of the error of the ACF. An alternative is to average along rows first and to estimate the ACF as the autocorrelation of the final column (Goodman, Weare 2010).

Parameters:

• data ((M,L) array) – each row contains L samples of the same observable while different rows correspond to different positions in the chain of M observations. The correlation is computed across axis 1 i.e. gives the AFC for samples from different positions in the chain.

• c (float, optional) – parameter used in determining when to truncate the sum for the IAT

Returns:

• ts ((M,) array) – array of considered separations between two samples

• ACF ((M,) array) – autocorrelation function

• ACF_err ((M,) array) – error of the autocorrelation function

• IAT (float) – integrated autocorrelation time, showing how many rows in the data lie between uncorrelated samples

• IAT_err (float) – error of the autocorrelation time

• delta_t (float) – time needed to compute the ACF and the IAT

SU2xSU2.correlations.corr_func_1D(x, y)

Computes the correlation between the equal length 1D arrays x, y using the cross correlation theorem. FFTs yield circular convolutions, such that x and y are assumed to have periodic boundary conditions. When the data is not circular, need to pad it with zeros as done in SU2xSU2.correlations.autocorr_func_1d().

Parameters:

• x ((N,) array) – first data series to correlate

• y ((N,) array) – second data series to correlate

Returns:

cf – correlation function between the data in x and y and of the same length as either array

Return type:

(N,) array

SU2xSU2.correlations.correlator(xs, ys)

Alias for SU2xSU2.correlations.correlator_repeats() when the xs and ys have been observed only once, i.e. are both of shape (L,) This implies that the error of the correlation function cannot be estimated.

See also

correlations.correlator_repeats

SU2xSU2.correlations.correlator_repeats(xs, ys)

Computes the correlation function (CF) using FFTs between two equally sized 1D arrays for which several measurements exists. Each column presents a new observation and correlations are computed along axis 0. The final CF is the average along axis 1.

Parameters:

• xs ((L,M) array) – M measurements of first data vector of length L

• ys ((L,M) array) – M measurements of second data vector of length L

Returns:

• CF ((L,) array) – average correlation function based on the M measurements

• CF_err ((L,) array) – uncertainty in the CF, estimated as the standard error on the mean corrected by the square root of the integrated autocorrelation time

SU2xSU2.plotting module

SU2xSU2.plotting.correlation_func_plot(data_path, plot_path, fit_upper=None, show_plot=True, ybottom=None, xright=None)

Produces a plot (with a logarithmic y axis) of the correlation function data (stored at data_path) and the fitted, analytical expectation. The plot is saved at plot_path with the fitting range and other plot parameters can be adjusted through the arguments.

Allows manual adjustment of fitting the correlation length to the processed correlation function data (normalized and mirror averaged). A plot with the new fitting is produced and the inferred correlation length, its error and the associated chi2 are printed. These can then be manually added to (for example) a data/corlen_data.txt file.

Parameters:

• data_path (str) – file path to the correlation function data that will be plotted. The file is assumed to be .npy and contain the rows separation [0,L/2], correlation function, correlation function error respectively

• plot_path (str) – file path (including file extension) to store the plot at

• fit_upper (int (optional)) – largest separation (in units of the lattice spacing) to be included in the fit. If left as None, only all non-zero values of the correlation function will be included in the fit

• show_plot (bool (optional)) – shows produced plot before saving it

• ybottom (float (optional)) – lower limit for y axis

• xright (float (optional)) – upper limit for x axis

SU2xSU2.plotting.effective_mass_plot(data_path, xright=None, ytop=None, ybottom=None, show_plot=True, plot_path='plots/effective_mass.pdf')

Produces an effective mass plot using the processed correlation function data stored at data_path. The data is expected to contain the separation, the correlation function value and its error row wise. The effective mass will be computed based on the assumption that the correlation function follows the shape of a cosh as analytically expected due to periodic boundary conditions. As the effective mass becomes noisy for large separations, the plot range can be adjusted using the remaining keywords.

Parameters:

• data_path (str) – path to .npy file containing the averaged (ensemble and across the symmetry axis at L/2) and normalized correlation function

• xright (float) – upper limit of the separation shown in the plot

• ytop (float) – upper limit for the effective mass

• ybottom (float) – lower limit for the effective mass

SU2xSU2.plotting.plot_chain(data_chain_path, ylabel, start_idx=0)

Plots raw measurement chain against computer time to gauge when convergence has been achieved. Observables of particular interest are the susceptibility (as sufficient burn is needed for the critical slowing down analysis) and the correlation function at a fixed separation (ideally chosen close to the correlation length as this is a slowly converging quantity and thus gives a lower bound for burn in). If the stored chain has already had some burn in removed, the starting value of the computer time can be adjusted by start_idx.

Parameters:

• data_chain_path (str) – path to measurement chain file (must be .npy)

• ylabel (str) – label for the plotted measurement. When Latex commands are contained, pass as raw string i.e. r’$chi$’

• start_idx (int (optional)) – amount of burn in already removed from the measurement chain