Postprocessing

Postprocessing#

# Enable Jupyter Widget support for matplotlib.
%matplotlib widget

import numpy as np
import pandas as pd
import matplotlib.pyplot as plt

Find QMC data#

The variable dirs is a list of all directories containing an ALF results file data.h5.

from py_alf.utils import find_sim_dirs
dirs = find_sim_dirs()
dirs

['./ALF_data/temper_Nematic_Dirac_Model_vers=1_L1=10_L2=10_beta=40.0_xi=0.25_h=2.5/Temp_0',
 './ALF_data/temper_Nematic_Dirac_Model_vers=1_L1=10_L2=10_beta=40.0_xi=0.25_h=2.5/Temp_1',
 './ALF_data/temper_Nematic_Dirac_Model_vers=1_L1=10_L2=10_beta=40.0_xi=0.25_h=2.5/Temp_2',
 './ALF_data/temper_Nematic_Dirac_Model_vers=1_L1=10_L2=10_beta=40.0_xi=0.25_h=2.5/Temp_3',
 './ALF_data/temper_Nematic_Dirac_Model_vers=1_L1=4_L2=4_beta=16.0_xi=0.25_h=2.5/Temp_0',
 './ALF_data/temper_Nematic_Dirac_Model_vers=1_L1=4_L2=4_beta=16.0_xi=0.25_h=2.5/Temp_1',
 './ALF_data/temper_Nematic_Dirac_Model_vers=1_L1=4_L2=4_beta=16.0_xi=0.25_h=2.5/Temp_2',
 './ALF_data/temper_Nematic_Dirac_Model_vers=1_L1=4_L2=4_beta=16.0_xi=0.25_h=2.5/Temp_3',
 './ALF_data/temper_Nematic_Dirac_Model_vers=1_L1=6_L2=6_beta=24.0_xi=0.25_h=2.5/Temp_0',
 './ALF_data/temper_Nematic_Dirac_Model_vers=1_L1=6_L2=6_beta=24.0_xi=0.25_h=2.5/Temp_1',
 './ALF_data/temper_Nematic_Dirac_Model_vers=1_L1=6_L2=6_beta=24.0_xi=0.25_h=2.5/Temp_2',
 './ALF_data/temper_Nematic_Dirac_Model_vers=1_L1=6_L2=6_beta=24.0_xi=0.25_h=2.5/Temp_3',
 './ALF_data/temper_Nematic_Dirac_Model_vers=1_L1=8_L2=8_beta=32.0_xi=0.25_h=2.5/Temp_0',
 './ALF_data/temper_Nematic_Dirac_Model_vers=1_L1=8_L2=8_beta=32.0_xi=0.25_h=2.5/Temp_1',
 './ALF_data/temper_Nematic_Dirac_Model_vers=1_L1=8_L2=8_beta=32.0_xi=0.25_h=2.5/Temp_2',
 './ALF_data/temper_Nematic_Dirac_Model_vers=1_L1=8_L2=8_beta=32.0_xi=0.25_h=2.5/Temp_3']

Custom observables#

Defining some observables derived from measurements done during the simulations (cf. Section 4.2.3.2).

Correlation ratio \(1 - \frac{O(\boldsymbol{k}+\boldsymbol{\delta})}{O(\boldsymbol{k})}\). RG-invariant quantity

def R_k(obs, back, sign, N_orb, N_tau, dtau, latt,
        ks=[(0., 0.)], mat=None, NNs=[(1, 0), (0, 1), (-1, 0), (0, -1)]):
    """RG-invariant quantity derived from a correlation function.
    
    obs.shape = (N_orb, N_orb, N_tau, latt.N)
    back.shape = (N_orb,)
    """
    if mat is None:
        mat = np.identity(N_orb, dtype=np.double)
    out = 0.
    for k in ks:
        n = latt.k_to_n(k)

        J1 = (obs[..., n].sum(axis=-1) * mat).sum()
        J2 = 0
        for NN in NNs:
            i = latt.nnlistk[n, NN[0], NN[1]]
            J2 += (obs[..., i].sum(axis=-1) * mat).sum() / len(NNs)
        out += (1 - J2/J1)

    return out / len(ks)

Binder cumulant \(\left(3 - \frac{\langle s^4 \rangle}{\langle s^2 \rangle^2}\right)/2\). RG-invariant quantity

def binder(obs, sign, N_obs):
    return (3 - obs[2] / obs[1]**2 * sign)/2

Susceptibility \(\int_{0}^{\beta}\!\text{d}\tau\, C\left(\boldsymbol{k},\tau\right)\)

def susceptibility(obs, back, sign, N_orb, N_tau, dtau, latt,
                   ks=[[0., 0.]]):
    """Susceptibility of a time-displaced correlation function.
    
    obs.shape = (N_orb, N_orb, N_tau, latt.N)
    back.shape = (N_orb,)
    """
    out = 0.
    for k in ks:
        n = latt.k_to_n(k)
        out += dtau*obs[..., n].trace().sum() / sign
    return out / len(ks)

Fermionic single particle gap determined through a quick and dirty fit of the time-displaced Green function (cf. Section 2.7.2.1).

from scipy.optimize import curve_fit

def fit_gap_lazy(obs, back, sign, N_orb, N_tau, dtau, latt):
    """Lazily fit Green function to determin gap.
    
    obs.shape = (N_orb, N_orb, N_tau, latt.N)
    """
    i1 = (2*N_tau)//24
    i2 = (4*N_tau)//24
    green = obs[0, 0, i1:i2, :]/sign
    
    def func(x, a, b):
        return a*np.exp(-b*x)
    taus = np.arange(0., N_tau*dtau, dtau)
    res = np.empty((latt.N,), dtype=np.cdouble)
    for n in range(latt.N):
        popt, pcov = curve_fit(func, taus[i1:i2], green[:, n])
        res[n] = popt[1]
    
    return res

custom_obs = {}

# Structure factor correlation ratio
custom_obs['R_S']= {
    'needs': ['IsingZ_eq'],
    'function': R_k,
    'kwargs': {}
}

# Susceptibility correlation ratio
custom_obs['R_chi']= {
    'needs': ['IsingZT_tau'],
    'function': R_k,
    'kwargs': {}
}

# Binder cumulant
custom_obs['B']= {
    'needs': ['m_scal'],
    'function': binder,
    'kwargs': {}
}

# Susceptibility
custom_obs['chi']= {
    'needs': ['IsingZT_tau'],
    'function': susceptibility,
    'kwargs': {}
}

# Susceptibility
custom_obs['gap_lazy']= {
    'needs': ['Green_tau'],
    'function': fit_gap_lazy,
    'kwargs': {}
}

Check warmup and autocorrelation times#

C.f. Section 4.2.3.3.

from py_alf import check_warmup, check_rebin

check_warmup(dirs, ['m_scal', 'ising_x_scal', 'chi', 'R_S', 'B', 'R_chi'],
             custom_obs=custom_obs, gui='ipy')

check_rebin(dirs, ['m_scal', 'ising_x_scal', 'chi', 'R_S', 'B', 'R_chi'],
            custom_obs=custom_obs, gui='ipy')

Error analysis#

The analysis results are saved in each simulation directory, both in plain text in the folder res and as a pickled Python dictionary in the file res.pkl.

Due to its length, the text printed out during the analysis is hidden, but can be viewed in the website version of this document.

from py_alf.analysis import analysis
for directory in dirs:
    analysis(directory, custom_obs=custom_obs, always=True)

Show code cell output Hide code cell output

### Analyzing ./ALF_data/temper_Nematic_Dirac_Model_vers=1_L1=10_L2=10_beta=40.0_xi=0.25_h=2.5/Temp_0 ###
/home/jonas/dissertation/jb/appendix_nematic/pyalf
Custom observables:
custom R_S ['IsingZ_eq']
custom R_chi ['IsingZT_tau']
custom B ['m_scal']
custom chi ['IsingZT_tau']
custom gap_lazy ['Green_tau']

Scalar observables:
Acc_Temp_scal
Kin_Pot_E_scal
Part_scal
chi2_scal
ising_x_alt_scal
ising_x_scal
ising_z_alt_scal
ising_z_scal
m_alt_scal
m_scal
Histogram observables:
B_hist
X_hist
m_hist
Equal time observables:
Green_eq
IsingX_eq
IsingZ_eq
Time displaced observables:
Green_tau
IsingXT_tau
IsingZT_tau
### Analyzing ./ALF_data/temper_Nematic_Dirac_Model_vers=1_L1=10_L2=10_beta=40.0_xi=0.25_h=2.5/Temp_1 ###
/home/jonas/dissertation/jb/appendix_nematic/pyalf
Custom observables:
custom R_S ['IsingZ_eq']
custom R_chi ['IsingZT_tau']
custom B ['m_scal']
custom chi ['IsingZT_tau']
custom gap_lazy ['Green_tau']
Scalar observables:
Acc_Temp_scal
Kin_Pot_E_scal
Part_scal
chi2_scal
ising_x_alt_scal
ising_x_scal
ising_z_alt_scal
ising_z_scal
m_alt_scal
m_scal
Histogram observables:
B_hist
X_hist
m_hist
Equal time observables:
Green_eq
IsingX_eq
IsingZ_eq
Time displaced observables:
Green_tau
IsingXT_tau
IsingZT_tau
### Analyzing ./ALF_data/temper_Nematic_Dirac_Model_vers=1_L1=10_L2=10_beta=40.0_xi=0.25_h=2.5/Temp_2 ###
/home/jonas/dissertation/jb/appendix_nematic/pyalf
Custom observables:
custom R_S ['IsingZ_eq']
custom R_chi ['IsingZT_tau']
custom B ['m_scal']
custom chi ['IsingZT_tau']
custom gap_lazy ['Green_tau']
Scalar observables:
Acc_Temp_scal
Kin_Pot_E_scal
Part_scal
chi2_scal
ising_x_alt_scal
ising_x_scal
ising_z_alt_scal
ising_z_scal
m_alt_scal
m_scal
Histogram observables:
B_hist
X_hist
m_hist
Equal time observables:
Green_eq
IsingX_eq
IsingZ_eq
Time displaced observables:
Green_tau
IsingXT_tau
IsingZT_tau
### Analyzing ./ALF_data/temper_Nematic_Dirac_Model_vers=1_L1=10_L2=10_beta=40.0_xi=0.25_h=2.5/Temp_3 ###
/home/jonas/dissertation/jb/appendix_nematic/pyalf
Custom observables:
custom R_S ['IsingZ_eq']
custom R_chi ['IsingZT_tau']
custom B ['m_scal']
custom chi ['IsingZT_tau']
custom gap_lazy ['Green_tau']
Scalar observables:
Acc_Temp_scal
Kin_Pot_E_scal
Part_scal
chi2_scal
ising_x_alt_scal
ising_x_scal
ising_z_alt_scal
ising_z_scal
m_alt_scal
m_scal
Histogram observables:
B_hist
X_hist
m_hist
Equal time observables:
Green_eq
IsingX_eq
IsingZ_eq
Time displaced observables:
Green_tau
IsingXT_tau
IsingZT_tau
### Analyzing ./ALF_data/temper_Nematic_Dirac_Model_vers=1_L1=4_L2=4_beta=16.0_xi=0.25_h=2.5/Temp_0 ###
/home/jonas/dissertation/jb/appendix_nematic/pyalf
Custom observables:
custom R_S ['IsingZ_eq']
custom R_chi ['IsingZT_tau']
custom B ['m_scal']
custom chi ['IsingZT_tau']
custom gap_lazy ['Green_tau']
Scalar observables:
Acc_Temp_scal
Kin_Pot_E_scal
Part_scal
chi2_scal
ising_x_alt_scal
ising_x_scal
ising_z_alt_scal
ising_z_scal
m_alt_scal
m_scal
Histogram observables:
B_hist
X_hist
m_hist
Equal time observables:
Green_eq
IsingX_eq
IsingZ_eq
Time displaced observables:
Green_tau
IsingXT_tau
IsingZT_tau
### Analyzing ./ALF_data/temper_Nematic_Dirac_Model_vers=1_L1=4_L2=4_beta=16.0_xi=0.25_h=2.5/Temp_1 ###
/home/jonas/dissertation/jb/appendix_nematic/pyalf
Custom observables:
custom R_S ['IsingZ_eq']
custom R_chi ['IsingZT_tau']
custom B ['m_scal']
custom chi ['IsingZT_tau']
custom gap_lazy ['Green_tau']
Scalar observables:
Acc_Temp_scal
Kin_Pot_E_scal
Part_scal
chi2_scal
ising_x_alt_scal
ising_x_scal
ising_z_alt_scal
ising_z_scal
m_alt_scal
m_scal
Histogram observables:
B_hist
X_hist
m_hist
Equal time observables:
Green_eq
IsingX_eq
IsingZ_eq
Time displaced observables:
Green_tau
IsingXT_tau
IsingZT_tau
### Analyzing ./ALF_data/temper_Nematic_Dirac_Model_vers=1_L1=4_L2=4_beta=16.0_xi=0.25_h=2.5/Temp_2 ###
/home/jonas/dissertation/jb/appendix_nematic/pyalf
Custom observables:
custom R_S ['IsingZ_eq']
custom R_chi ['IsingZT_tau']
custom B ['m_scal']
custom chi ['IsingZT_tau']
custom gap_lazy ['Green_tau']
Scalar observables:
Acc_Temp_scal
Kin_Pot_E_scal
Part_scal
chi2_scal
ising_x_alt_scal
ising_x_scal
ising_z_alt_scal
ising_z_scal
m_alt_scal
m_scal
Histogram observables:
B_hist
X_hist
m_hist
Equal time observables:
Green_eq
IsingX_eq
IsingZ_eq
Time displaced observables:
Green_tau
IsingXT_tau
IsingZT_tau
### Analyzing ./ALF_data/temper_Nematic_Dirac_Model_vers=1_L1=4_L2=4_beta=16.0_xi=0.25_h=2.5/Temp_3 ###
/home/jonas/dissertation/jb/appendix_nematic/pyalf
Custom observables:
custom R_S ['IsingZ_eq']
custom R_chi ['IsingZT_tau']
custom B ['m_scal']
custom chi ['IsingZT_tau']
custom gap_lazy ['Green_tau']
Scalar observables:
Acc_Temp_scal
Kin_Pot_E_scal
Part_scal
chi2_scal
ising_x_alt_scal
ising_x_scal
ising_z_alt_scal
ising_z_scal
m_alt_scal
m_scal
Histogram observables:
B_hist
X_hist
m_hist
Equal time observables:
Green_eq
IsingX_eq
IsingZ_eq
Time displaced observables:
Green_tau
IsingXT_tau
IsingZT_tau
### Analyzing ./ALF_data/temper_Nematic_Dirac_Model_vers=1_L1=6_L2=6_beta=24.0_xi=0.25_h=2.5/Temp_0 ###
/home/jonas/dissertation/jb/appendix_nematic/pyalf
Custom observables:
custom R_S ['IsingZ_eq']
custom R_chi ['IsingZT_tau']
custom B ['m_scal']
custom chi ['IsingZT_tau']
custom gap_lazy ['Green_tau']
Scalar observables:
Acc_Temp_scal
Kin_Pot_E_scal
Part_scal
chi2_scal
ising_x_alt_scal
ising_x_scal
ising_z_alt_scal
ising_z_scal
m_alt_scal
m_scal
Histogram observables:
B_hist
X_hist
m_hist
Equal time observables:
Green_eq
IsingX_eq
IsingZ_eq
Time displaced observables:
Green_tau
IsingXT_tau
IsingZT_tau
### Analyzing ./ALF_data/temper_Nematic_Dirac_Model_vers=1_L1=6_L2=6_beta=24.0_xi=0.25_h=2.5/Temp_1 ###
/home/jonas/dissertation/jb/appendix_nematic/pyalf
Custom observables:
custom R_S ['IsingZ_eq']
custom R_chi ['IsingZT_tau']
custom B ['m_scal']
custom chi ['IsingZT_tau']
custom gap_lazy ['Green_tau']
Scalar observables:
Acc_Temp_scal
Kin_Pot_E_scal
Part_scal
chi2_scal
ising_x_alt_scal
ising_x_scal
ising_z_alt_scal
ising_z_scal
m_alt_scal
m_scal
Histogram observables:
B_hist
X_hist
m_hist
Equal time observables:
Green_eq
IsingX_eq
IsingZ_eq
Time displaced observables:
Green_tau
IsingXT_tau
IsingZT_tau
### Analyzing ./ALF_data/temper_Nematic_Dirac_Model_vers=1_L1=6_L2=6_beta=24.0_xi=0.25_h=2.5/Temp_2 ###
/home/jonas/dissertation/jb/appendix_nematic/pyalf
Custom observables:
custom R_S ['IsingZ_eq']
custom R_chi ['IsingZT_tau']
custom B ['m_scal']
custom chi ['IsingZT_tau']
custom gap_lazy ['Green_tau']
Scalar observables:
Acc_Temp_scal
Kin_Pot_E_scal
Part_scal
chi2_scal
ising_x_alt_scal
ising_x_scal
ising_z_alt_scal
ising_z_scal
m_alt_scal
m_scal
Histogram observables:
B_hist
X_hist
m_hist
Equal time observables:
Green_eq
IsingX_eq
IsingZ_eq
Time displaced observables:
Green_tau
IsingXT_tau
IsingZT_tau
### Analyzing ./ALF_data/temper_Nematic_Dirac_Model_vers=1_L1=6_L2=6_beta=24.0_xi=0.25_h=2.5/Temp_3 ###
/home/jonas/dissertation/jb/appendix_nematic/pyalf
Custom observables:
custom R_S ['IsingZ_eq']
custom R_chi ['IsingZT_tau']
custom B ['m_scal']
custom chi ['IsingZT_tau']
custom gap_lazy ['Green_tau']
Scalar observables:
Acc_Temp_scal
Kin_Pot_E_scal
Part_scal
chi2_scal
ising_x_alt_scal
ising_x_scal
ising_z_alt_scal
ising_z_scal
m_alt_scal
m_scal
Histogram observables:
B_hist
X_hist
m_hist
Equal time observables:
Green_eq
IsingX_eq
IsingZ_eq
Time displaced observables:
Green_tau
IsingXT_tau
IsingZT_tau
### Analyzing ./ALF_data/temper_Nematic_Dirac_Model_vers=1_L1=8_L2=8_beta=32.0_xi=0.25_h=2.5/Temp_0 ###
/home/jonas/dissertation/jb/appendix_nematic/pyalf
Custom observables:
custom R_S ['IsingZ_eq']
custom R_chi ['IsingZT_tau']
custom B ['m_scal']
custom chi ['IsingZT_tau']
custom gap_lazy ['Green_tau']
Scalar observables:
Acc_Temp_scal
Kin_Pot_E_scal
Part_scal
chi2_scal
ising_x_alt_scal
ising_x_scal
ising_z_alt_scal
ising_z_scal
m_alt_scal
m_scal
Histogram observables:
B_hist
X_hist
m_hist
Equal time observables:
Green_eq
IsingX_eq
IsingZ_eq
Time displaced observables:
Green_tau
IsingXT_tau
IsingZT_tau
### Analyzing ./ALF_data/temper_Nematic_Dirac_Model_vers=1_L1=8_L2=8_beta=32.0_xi=0.25_h=2.5/Temp_1 ###
/home/jonas/dissertation/jb/appendix_nematic/pyalf
Custom observables:
custom R_S ['IsingZ_eq']
custom R_chi ['IsingZT_tau']
custom B ['m_scal']
custom chi ['IsingZT_tau']
custom gap_lazy ['Green_tau']
Scalar observables:
Acc_Temp_scal
Kin_Pot_E_scal
Part_scal
chi2_scal
ising_x_alt_scal
ising_x_scal
ising_z_alt_scal
ising_z_scal
m_alt_scal
m_scal
Histogram observables:
B_hist
X_hist
m_hist
Equal time observables:
Green_eq
IsingX_eq
IsingZ_eq
Time displaced observables:
Green_tau
IsingXT_tau
IsingZT_tau
### Analyzing ./ALF_data/temper_Nematic_Dirac_Model_vers=1_L1=8_L2=8_beta=32.0_xi=0.25_h=2.5/Temp_2 ###
/home/jonas/dissertation/jb/appendix_nematic/pyalf
Custom observables:
custom R_S ['IsingZ_eq']
custom R_chi ['IsingZT_tau']
custom B ['m_scal']
custom chi ['IsingZT_tau']
custom gap_lazy ['Green_tau']
Scalar observables:
Acc_Temp_scal
Kin_Pot_E_scal
Part_scal
chi2_scal
ising_x_alt_scal
ising_x_scal
ising_z_alt_scal
ising_z_scal
m_alt_scal
m_scal
Histogram observables:
B_hist
X_hist
m_hist
Equal time observables:
Green_eq
IsingX_eq
IsingZ_eq
Time displaced observables:
Green_tau
IsingXT_tau
IsingZT_tau
### Analyzing ./ALF_data/temper_Nematic_Dirac_Model_vers=1_L1=8_L2=8_beta=32.0_xi=0.25_h=2.5/Temp_3 ###
/home/jonas/dissertation/jb/appendix_nematic/pyalf
Custom observables:
custom R_S ['IsingZ_eq']
custom R_chi ['IsingZT_tau']
custom B ['m_scal']
custom chi ['IsingZT_tau']
custom gap_lazy ['Green_tau']
Scalar observables:
Acc_Temp_scal
Kin_Pot_E_scal
Part_scal
chi2_scal
ising_x_alt_scal
ising_x_scal
ising_z_alt_scal
ising_z_scal
m_alt_scal
m_scal
Histogram observables:
B_hist
X_hist
m_hist
Equal time observables:
Green_eq
IsingX_eq
IsingZ_eq
Time displaced observables:
Green_tau
IsingXT_tau
IsingZT_tau

Read analysis results#

Read all the res.pkl files and combine them in a single pandas DataFrame, called res.

from py_alf.ana import load_res
res = load_res(dirs)

./ALF_data/temper_Nematic_Dirac_Model_vers=1_L1=10_L2=10_beta=40.0_xi=0.25_h=2.5/Temp_0
No orbital locations saved.
./ALF_data/temper_Nematic_Dirac_Model_vers=1_L1=10_L2=10_beta=40.0_xi=0.25_h=2.5/Temp_1
No orbital locations saved.
./ALF_data/temper_Nematic_Dirac_Model_vers=1_L1=10_L2=10_beta=40.0_xi=0.25_h=2.5/Temp_2
No orbital locations saved.
./ALF_data/temper_Nematic_Dirac_Model_vers=1_L1=10_L2=10_beta=40.0_xi=0.25_h=2.5/Temp_3
No orbital locations saved.
./ALF_data/temper_Nematic_Dirac_Model_vers=1_L1=4_L2=4_beta=16.0_xi=0.25_h=2.5/Temp_0
No orbital locations saved.
./ALF_data/temper_Nematic_Dirac_Model_vers=1_L1=4_L2=4_beta=16.0_xi=0.25_h=2.5/Temp_1
No orbital locations saved.
./ALF_data/temper_Nematic_Dirac_Model_vers=1_L1=4_L2=4_beta=16.0_xi=0.25_h=2.5/Temp_2
No orbital locations saved.
./ALF_data/temper_Nematic_Dirac_Model_vers=1_L1=4_L2=4_beta=16.0_xi=0.25_h=2.5/Temp_3
No orbital locations saved.
./ALF_data/temper_Nematic_Dirac_Model_vers=1_L1=6_L2=6_beta=24.0_xi=0.25_h=2.5/Temp_0
No orbital locations saved.
./ALF_data/temper_Nematic_Dirac_Model_vers=1_L1=6_L2=6_beta=24.0_xi=0.25_h=2.5/Temp_1
No orbital locations saved.
./ALF_data/temper_Nematic_Dirac_Model_vers=1_L1=6_L2=6_beta=24.0_xi=0.25_h=2.5/Temp_2
No orbital locations saved.
./ALF_data/temper_Nematic_Dirac_Model_vers=1_L1=6_L2=6_beta=24.0_xi=0.25_h=2.5/Temp_3
No orbital locations saved.
./ALF_data/temper_Nematic_Dirac_Model_vers=1_L1=8_L2=8_beta=32.0_xi=0.25_h=2.5/Temp_0
No orbital locations saved.
./ALF_data/temper_Nematic_Dirac_Model_vers=1_L1=8_L2=8_beta=32.0_xi=0.25_h=2.5/Temp_1
No orbital locations saved.
./ALF_data/temper_Nematic_Dirac_Model_vers=1_L1=8_L2=8_beta=32.0_xi=0.25_h=2.5/Temp_2
No orbital locations saved.
./ALF_data/temper_Nematic_Dirac_Model_vers=1_L1=8_L2=8_beta=32.0_xi=0.25_h=2.5/Temp_3
No orbital locations saved.

The printout of the following command is again hidden, but can be viewed in the website version of this document.

res

Show code cell output Hide code cell output

	beta	dtau	global_h	global_j	global_type	ham_h	ham_j	ham_t	ham_xi	...	IsingXT_tauK_err	IsingXT_tauR	IsingXT_tauR_err	IsingXT_tau_lattice	IsingZT_tauK	IsingZT_tauK_err	IsingZT_tauR	IsingZT_tauR_err	IsingZT_tau_lattice	lattice
./ALF_data/temper_Nematic_Dirac_Model_vers=1_L1=10_L2=10_beta=40.0_xi=0.25_h=2.5/Temp_0	40.0	0.1	3.0	1.0	b''	2.5	1.0	1.0	0.25	...	[[0.00846007502342428, 0.011361413792811912, 0...	[[-0.8759651432413821, -0.8773726147907193, -0...	[[0.0021307557348112893, 0.002103012359563489,...	{'L1': [7.071067811865475, 7.071067811865475],...	[[0.3279964514764253, 0.3434765067595519, 0.37...	[[0.0008932302580056016, 0.0011906138182334284...	[[0.5773744618389179, 0.5776504791744279, 0.57...	[[0.0005925670020223004, 0.000599508043614475,...	{'L1': [7.071067811865475, 7.071067811865475],...	{'L1': [7.071067811865475, 7.071067811865475],...
./ALF_data/temper_Nematic_Dirac_Model_vers=1_L1=10_L2=10_beta=40.0_xi=0.25_h=2.5/Temp_1	40.0	0.1	3.0	1.0	b''	3.0	1.0	1.0	0.25	...	[[0.010440213255229968, 0.00971142615215967, 0...	[[-1.3122546627824234, -1.3081659457089554, -1...	[[0.0024669838243135925, 0.0022471387853431036...	{'L1': [7.071067811865475, 7.071067811865475],...	[[0.4648468734781728, 0.4990409266176215, 0.55...	[[0.0022719373940343016, 0.002024507665608468,...	[[0.3164376602178704, 0.3172661300913677, 0.31...	[[0.0007501647357670305, 0.0009271215478391133...	{'L1': [7.071067811865475, 7.071067811865475],...	{'L1': [7.071067811865475, 7.071067811865475],...
./ALF_data/temper_Nematic_Dirac_Model_vers=1_L1=10_L2=10_beta=40.0_xi=0.25_h=2.5/Temp_2	40.0	0.1	3.0	1.0	b''	3.5	1.0	1.0	0.25	...	[[0.00998898503248877, 0.008874897552016436, 0...	[[-1.7439667155439509, -1.7451722383207986, -1...	[[0.0015190314978645058, 0.0019186473723686597...	{'L1': [7.071067811865475, 7.071067811865475],...	[[0.5915424611948749, 0.6433643867380482, 0.72...	[[0.0022566441762264552, 0.0022329291182243984...	[[0.037518564379655055, 0.039690493611752135, ...	[[0.0011222534229489557, 0.0009779394619310773...	{'L1': [7.071067811865475, 7.071067811865475],...	{'L1': [7.071067811865475, 7.071067811865475],...
./ALF_data/temper_Nematic_Dirac_Model_vers=1_L1=10_L2=10_beta=40.0_xi=0.25_h=2.5/Temp_3	40.0	0.1	3.0	1.0	b''	4.0	1.0	1.0	0.25	...	[[0.007634992984452745, 0.005835316441545428, ...	[[-1.847695393284891, -1.8495307484039036, -1....	[[0.0025720663795285484, 0.001939963222210881,...	{'L1': [7.071067811865475, 7.071067811865475],...	[[0.6339044988215092, 0.684312635385893, 0.781...	[[0.0020198355137177985, 0.0027236228176412618...	[[0.0024501348260564074, 0.003458403475288739,...	[[0.0002785584053134153, 0.0003883892011143934...	{'L1': [7.071067811865475, 7.071067811865475],...	{'L1': [7.071067811865475, 7.071067811865475],...
./ALF_data/temper_Nematic_Dirac_Model_vers=1_L1=4_L2=4_beta=16.0_xi=0.25_h=2.5/Temp_0	16.0	0.1	3.0	1.0	b''	2.5	1.0	1.0	0.25	...	[[0.01232601938514963, 0.011702811089570902, 0...	[[-0.8012640883169901, -0.7497033999249376, -0...	[[0.00443155929293343, 0.004831066600579312, 0...	{'L1': [2.82842712474619, 2.82842712474619], '...	[[0.36947834874161917, 0.444780551420884, 0.37...	[[0.002013635050318031, 0.003191020480788499, ...	[[0.6226970872195444, 0.6455383454642287, 0.62...	[[0.0015830720439759531, 0.0014751962346094868...	{'L1': [2.82842712474619, 2.82842712474619], '...	{'L1': [2.82842712474619, 2.82842712474619], '...
./ALF_data/temper_Nematic_Dirac_Model_vers=1_L1=4_L2=4_beta=16.0_xi=0.25_h=2.5/Temp_1	16.0	0.1	3.0	1.0	b''	3.0	1.0	1.0	0.25	...	[[0.007912955862575333, 0.00994876501009697, 0...	[[-1.1615682588479128, -1.1120875956846286, -1...	[[0.006140705833959657, 0.0062178274927627425,...	{'L1': [2.82842712474619, 2.82842712474619], '...	[[0.5448155158716456, 0.6931636903733888, 0.54...	[[0.003800827642495056, 0.003889823170868747, ...	[[0.43933860661117075, 0.48550222790231756, 0....	[[0.0023261562340938266, 0.0020577286896956743...	{'L1': [2.82842712474619, 2.82842712474619], '...	{'L1': [2.82842712474619, 2.82842712474619], '...
./ALF_data/temper_Nematic_Dirac_Model_vers=1_L1=4_L2=4_beta=16.0_xi=0.25_h=2.5/Temp_2	16.0	0.1	3.0	1.0	b''	3.5	1.0	1.0	0.25	...	[[0.01275866498225441, 0.01114143763486667, 0....	[[-1.5031378204697217, -1.4656902611341118, -1...	[[0.003974345112283094, 0.004038572323341327, ...	{'L1': [2.82842712474619, 2.82842712474619], '...	[[0.7175970087389038, 0.9643325273738381, 0.71...	[[0.003001937941593038, 0.005409099425790231, ...	[[0.25480021415495147, 0.3279120583054125, 0.2...	[[0.0019484346393753634, 0.0016425474210216464...	{'L1': [2.82842712474619, 2.82842712474619], '...	{'L1': [2.82842712474619, 2.82842712474619], '...
./ALF_data/temper_Nematic_Dirac_Model_vers=1_L1=4_L2=4_beta=16.0_xi=0.25_h=2.5/Temp_3	16.0	0.1	3.0	1.0	b''	4.0	1.0	1.0	0.25	...	[[0.009315833166549096, 0.008333079691029744, ...	[[-1.7562694593373842, -1.7267783438093052, -1...	[[0.005528932722976044, 0.005731379888512149, ...	{'L1': [2.82842712474619, 2.82842712474619], '...	[[0.8306037908478882, 1.1415797703095765, 0.82...	[[0.003998895038665653, 0.005233624185626729, ...	[[0.13330903740321445, 0.22308798426852733, 0....	[[0.0020202622734657063, 0.0017575079018091037...	{'L1': [2.82842712474619, 2.82842712474619], '...	{'L1': [2.82842712474619, 2.82842712474619], '...
./ALF_data/temper_Nematic_Dirac_Model_vers=1_L1=6_L2=6_beta=24.0_xi=0.25_h=2.5/Temp_0	24.0	0.1	3.0	1.0	b''	2.5	1.0	1.0	0.25	...	[[0.013359096995139893, 0.011575996882320966, ...	[[-0.8531159412572458, -0.8472365106670662, -0...	[[0.003275455019155735, 0.00333441534660581, 0...	{'L1': [4.242640687119285, 4.242640687119285],...	[[0.3418454051286131, 0.3917926888854215, 0.42...	[[0.0017208699477740546, 0.0027020403280339765...	[[0.5932356430351405, 0.5939184448708253, 0.59...	[[0.0010273911192221733, 0.001100783332509444,...	{'L1': [4.242640687119285, 4.242640687119285],...	{'L1': [4.242640687119285, 4.242640687119285],...
./ALF_data/temper_Nematic_Dirac_Model_vers=1_L1=6_L2=6_beta=24.0_xi=0.25_h=2.5/Temp_1	24.0	0.1	3.0	1.0	b''	3.0	1.0	1.0	0.25	...	[[0.012372842872302243, 0.00982592979789681, 0...	[[-1.2568991170257189, -1.254691133928998, -1....	[[0.006223036092397936, 0.004391421891827163, ...	{'L1': [4.242640687119285, 4.242640687119285],...	[[0.4913945419659706, 0.5854088345516923, 0.65...	[[0.0032738627894267646, 0.001614294276006036,...	[[0.3634863972006823, 0.37037504294647144, 0.3...	[[0.001596115479707853, 0.0018508902777070052,...	{'L1': [4.242640687119285, 4.242640687119285],...	{'L1': [4.242640687119285, 4.242640687119285],...
./ALF_data/temper_Nematic_Dirac_Model_vers=1_L1=6_L2=6_beta=24.0_xi=0.25_h=2.5/Temp_2	24.0	0.1	3.0	1.0	b''	3.5	1.0	1.0	0.25	...	[[0.00865089544688345, 0.010880094917081352, 0...	[[-1.6639999051857537, -1.6576803641063333, -1...	[[0.004268781005284443, 0.0033731519086098636,...	{'L1': [4.242640687119285, 4.242640687119285],...	[[0.6299816796959656, 0.7832804761376195, 0.91...	[[0.002488745977358733, 0.0033553108719100422,...	[[0.12214017728303422, 0.13420112162969303, 0....	[[0.001852767924198795, 0.0014429850264620877,...	{'L1': [4.242640687119285, 4.242640687119285],...	{'L1': [4.242640687119285, 4.242640687119285],...
./ALF_data/temper_Nematic_Dirac_Model_vers=1_L1=6_L2=6_beta=24.0_xi=0.25_h=2.5/Temp_3	24.0	0.1	3.0	1.0	b''	4.0	1.0	1.0	0.25	...	[[0.00573535696549232, 0.009217756507456423, 0...	[[-1.8295623750819807, -1.829167413199349, -1....	[[0.00295719286077581, 0.0027860252155041043, ...	{'L1': [4.242640687119285, 4.242640687119285],...	[[0.6978523592809309, 0.8709448118019547, 1.00...	[[0.0028454396065431477, 0.0026638919283964143...	[[0.03186750815322241, 0.04542341256626974, 0....	[[0.0010382408217656288, 0.0011561299211574668...	{'L1': [4.242640687119285, 4.242640687119285],...	{'L1': [4.242640687119285, 4.242640687119285],...
./ALF_data/temper_Nematic_Dirac_Model_vers=1_L1=8_L2=8_beta=32.0_xi=0.25_h=2.5/Temp_0	32.0	0.1	3.0	1.0	b''	2.5	1.0	1.0	0.25	...	[[0.011065646508611636, 0.00966847649737193, 0...	[[-0.8683024558232636, -0.8652411497252245, -0...	[[0.0024470447102413547, 0.0023117226930034814...	{'L1': [5.65685424949238, 5.65685424949238], '...	[[0.33132086465198235, 0.362852436977168, 0.40...	[[0.001073583877656575, 0.0012523584761456176,...	[[0.580762577141888, 0.5807116562396739, 0.580...	[[0.0005767105920373921, 0.0005604277525153794...	{'L1': [5.65685424949238, 5.65685424949238], '...	{'L1': [5.65685424949238, 5.65685424949238], '...
./ALF_data/temper_Nematic_Dirac_Model_vers=1_L1=8_L2=8_beta=32.0_xi=0.25_h=2.5/Temp_1	32.0	0.1	3.0	1.0	b''	3.0	1.0	1.0	0.25	...	[[0.009705957792840473, 0.010404507959166889, ...	[[-1.2904725412229632, -1.2885912675540796, -1...	[[0.0029970225441264438, 0.003336183367224488,...	{'L1': [5.65685424949238, 5.65685424949238], '...	[[0.47106830153441287, 0.5278714673490869, 0.6...	[[0.0022689336820031435, 0.002637842724235969,...	[[0.33448391407443134, 0.33618193707633376, 0....	[[0.0017154281277830467, 0.0018677137367773436...	{'L1': [5.65685424949238, 5.65685424949238], '...	{'L1': [5.65685424949238, 5.65685424949238], '...
./ALF_data/temper_Nematic_Dirac_Model_vers=1_L1=8_L2=8_beta=32.0_xi=0.25_h=2.5/Temp_2	32.0	0.1	3.0	1.0	b''	3.5	1.0	1.0	0.25	...	[[0.009937946150227129, 0.008132034964506419, ...	[[-1.7145102527216323, -1.717048775742574, -1....	[[0.0027164831243305675, 0.002706955073502924,...	{'L1': [5.65685424949238, 5.65685424949238], '...	[[0.6049661596398819, 0.6874276349317301, 0.81...	[[0.0016479953788909247, 0.002595997757649418,...	[[0.06581094217085602, 0.06958924887373168, 0....	[[0.0016636449294150323, 0.0016802902003834619...	{'L1': [5.65685424949238, 5.65685424949238], '...	{'L1': [5.65685424949238, 5.65685424949238], '...
./ALF_data/temper_Nematic_Dirac_Model_vers=1_L1=8_L2=8_beta=32.0_xi=0.25_h=2.5/Temp_3	32.0	0.1	3.0	1.0	b''	4.0	1.0	1.0	0.25	...	[[0.00775267435343496, 0.006439130531618064, 0...	[[-1.8463121373967215, -1.8453149418841837, -1...	[[0.0020212561458271237, 0.0019462609683163882...	{'L1': [5.65685424949238, 5.65685424949238], '...	[[0.6539311815241208, 0.7445460076284947, 0.87...	[[0.0025705487171792706, 0.002465948564277555,...	[[0.00991609307970665, 0.013097558903952613, 0...	[[0.00045949679783958714, 0.000459128937341380...	{'L1': [5.65685424949238, 5.65685424949238], '...	{'L1': [5.65685424949238, 5.65685424949238], '...

16 rows × 128 columns

Plot order parameter#

fig, ax = plt.subplots(constrained_layout=True)
for L in [4, 6, 8, 10]:
    df = res[res.l1 == L].sort_values(by='ham_h')
    ax.errorbar(df.ham_h, df.m_scal0, df.m_scal0_err, label=f'L={L}')
ax.legend()
ax.set_xlabel('Transverse field $h$')
ax.set_ylabel('$S(\boldsymbol{k}=0)$');

Plot RG-invariant quantities#

RG-invariant quantities behave at critical point as:

\[ R = f(L^z/\beta, (h-h_\mathrm{c})L^{1/\nu}, L^{-\omega}) \]

Dismissing dependence on \(\beta\) and finite size corrections:

(6)#\[ R = f((h-h_\mathrm{c})L^{1/\nu}) \]

Therefore, they should cross for different system sizes at \(h = h_{\rm c}\).

fig, axs = plt.subplots(1, 3, constrained_layout=True, figsize=(7, 2.5))
for L in [4, 6, 8, 10]:
    df = res[res.l1 == L].sort_values(by='ham_h')
    for obs_name, ylabel, ax in zip(['R_S', 'R_chi', 'B'],
                                    [r'$R_S$', r'$R_\chi$', r'$B$'],
                                    axs):
        ax.errorbar(
            df.ham_h, df[obs_name], df[obs_name+'_err'], label=f'L={L}')
        ax.set_xlabel('Transverse field $h$')
        ax.set_ylabel(ylabel)
ax.legend();

Data collapse#

I demonstrate here briefly how one can perform a data collapse. The amount of data is too little and the system sizes too small to get meaningful results, but the overall approach can still be presented well.

Manual data collapse#

The first step is to manually vary the parameters, in this case \(h_{\rm c}\) and \(a = 1/\nu\), to collapse the data. The results can then be used as starting parameters of the automatic data collapse in the next step.

hc = 3.15
a = 1.4  # a=1/nu
fig, axs = plt.subplots(1, 3, constrained_layout=True, figsize=(7, 2.5))
for L in [4, 6, 8, 10]:
    df = res[res.l1 == L].sort_values(by='ham_h')
    for obs_name, ylabel, ax in zip(['R_S', 'R_chi', 'B'],
                                    [r'$R_S$', r'$R_\chi$', r'$B$'],
                                    axs):
        ax.errorbar((df.ham_h-hc)*L**a, df[obs_name],
                    df[obs_name+'_err'], label=f'L={L}')
        ax.set_xlabel(r'$(h-h_{\rm c})*L^{1/\nu}$')
        ax.set_ylabel(ylabel)
ax.legend();

Data collapse fit#

See Appendix A.3 for the source code of collapse.

from collapse import collapse

The function func defines the scaling assumption of Eq. (6).

def func(L, x, dx, y, dy, par):
    """Scaling assumption of RG-invariant quantities
    without corrections to scaling."""
    xc = par[0]
    a = par[1]  # a=1/nu 
    x_scaled = (x-xc) * L**a
    if dx is None:
        dx_scaled = None
    else:
        dx_scaled = (x-xc) * L**a
    y_scaled = y
    dy_scaled = dy
    return x_scaled, dx_scaled, y_scaled, dy_scaled

Performing automatic data collapse of \(R_S\) for system sizes \(L \in \{4, 6, 8, 10\}\) and starting parameters \(h_{\rm c} = 3.15\), \(1/\nu = 1.4\) from the manual data collapse.

fig = plt.figure(constrained_layout=True, figsize=(4, 2.7))
plt.xlabel(r'$(h-h_{\rm c})L^{1/\nu}$')
plt.ylabel(r'$R_S$')
collapse(func, 'ham_h', 'R_S', res, Ls=[4, 6, 8, 10] , par0=[3.15, 1.3])

{'Ls': [4, 6, 8, 10],
 'L0': 4,
 'NL': 4,
 'popt': array([ 3.19239096e+00,  1.28013715e+00,  9.08264521e-01, -1.67682563e-02,
        -1.24065522e-03, -3.29972113e-05]),
 'perr': array([8.89780198e-03, 4.91240345e-03, 2.45520108e-03, 6.27747079e-04,
        5.22775205e-05, 1.42967090e-06]),
 'S': 1136.7062136799598}

The resulting data collapse does not look very good and the quality of fit function \(S \approx 1500\), which should be of order 1, is much too big.

Some likely reasons for this are:

Too small system sizes.
Too few data points.
Too large fitting range, meaning to big values of \(\left|(h-h_{\rm c})L^{1/\nu}\right|\).

We dismiss system sizes \(L=4\) and restrict the data to \((h-h_{{\rm c},0})L^{1/\nu_0} \in [-5, 10]\). This leads to a much more agreeable, but still too large \(S \approx 24\). Furthermore, there are only 6 data points left and the result \(1/\nu = 1.2 \pm 0.1\) is not in agreement with Section 2.8. Nevertheless, this might be the best that can be done with the available data in terms of data collapses.

hc0 = 3.15
a0 = 1.3
df = res[(res['ham_h']-hc)*res.l1**a > -5]
df = df[(df['ham_h']-hc)*df.l1**a < 10]
fig = plt.figure(constrained_layout=True, figsize=(4, 2.7))
plt.xlabel(r'$(h-h_{\rm c})L^{1/\nu}$')
plt.ylabel(r'$R_S$')
collapse(func, 'ham_h', 'R_S', df, Ls=[6, 8, 10] , par0=[hc0, a0])

{'Ls': [6, 8, 10],
 'L0': 6,
 'NL': 3,
 'popt': array([ 3.22588756e+00,  1.22585025e+00,  8.74205610e-01, -3.71090125e-02,
        -3.65818772e-03,  2.59712699e-05]),
 'perr': array([0.04638255, 0.10616724, 0.0102565 , 0.00745803, 0.00196077,
        0.00023591]),
 'S': 26.630914312767914}

Plot correlation#

Accessing elements of the dataframe#

res.columns

Index(['beta', 'dtau', 'global_h', 'global_j', 'global_type', 'ham_chem',
       'ham_h', 'ham_j', 'ham_t', 'ham_xi',
       ...
       'IsingXT_tauK_err', 'IsingXT_tauR', 'IsingXT_tauR_err',
       'IsingXT_tau_lattice', 'IsingZT_tauK', 'IsingZT_tauK_err',
       'IsingZT_tauR', 'IsingZT_tauR_err', 'IsingZT_tau_lattice', 'lattice'],
      dtype='object', length=128)

res.index

Index(['./ALF_data/temper_Nematic_Dirac_Model_vers=1_L1=10_L2=10_beta=40.0_xi=0.25_h=2.5/Temp_0',
       './ALF_data/temper_Nematic_Dirac_Model_vers=1_L1=10_L2=10_beta=40.0_xi=0.25_h=2.5/Temp_1',
       './ALF_data/temper_Nematic_Dirac_Model_vers=1_L1=10_L2=10_beta=40.0_xi=0.25_h=2.5/Temp_2',
       './ALF_data/temper_Nematic_Dirac_Model_vers=1_L1=10_L2=10_beta=40.0_xi=0.25_h=2.5/Temp_3',
       './ALF_data/temper_Nematic_Dirac_Model_vers=1_L1=4_L2=4_beta=16.0_xi=0.25_h=2.5/Temp_0',
       './ALF_data/temper_Nematic_Dirac_Model_vers=1_L1=4_L2=4_beta=16.0_xi=0.25_h=2.5/Temp_1',
       './ALF_data/temper_Nematic_Dirac_Model_vers=1_L1=4_L2=4_beta=16.0_xi=0.25_h=2.5/Temp_2',
       './ALF_data/temper_Nematic_Dirac_Model_vers=1_L1=4_L2=4_beta=16.0_xi=0.25_h=2.5/Temp_3',
       './ALF_data/temper_Nematic_Dirac_Model_vers=1_L1=6_L2=6_beta=24.0_xi=0.25_h=2.5/Temp_0',
       './ALF_data/temper_Nematic_Dirac_Model_vers=1_L1=6_L2=6_beta=24.0_xi=0.25_h=2.5/Temp_1',
       './ALF_data/temper_Nematic_Dirac_Model_vers=1_L1=6_L2=6_beta=24.0_xi=0.25_h=2.5/Temp_2',
       './ALF_data/temper_Nematic_Dirac_Model_vers=1_L1=6_L2=6_beta=24.0_xi=0.25_h=2.5/Temp_3',
       './ALF_data/temper_Nematic_Dirac_Model_vers=1_L1=8_L2=8_beta=32.0_xi=0.25_h=2.5/Temp_0',
       './ALF_data/temper_Nematic_Dirac_Model_vers=1_L1=8_L2=8_beta=32.0_xi=0.25_h=2.5/Temp_1',
       './ALF_data/temper_Nematic_Dirac_Model_vers=1_L1=8_L2=8_beta=32.0_xi=0.25_h=2.5/Temp_2',
       './ALF_data/temper_Nematic_Dirac_Model_vers=1_L1=8_L2=8_beta=32.0_xi=0.25_h=2.5/Temp_3'],
      dtype='object')

item = res.loc['./ALF_data/temper_Nematic_Dirac_Model_vers=1_L1=10_L2=10_beta=40.0_xi=0.25_h=2.5/Temp_0']
item

beta                                                                40.0
dtau                                                                 0.1
global_h                                                             3.0
global_j                                                             1.0
global_type                                                          b''
                                             ...                        
IsingZT_tauK_err       [[0.0008932302580056016, 0.0011906138182334284...
IsingZT_tauR           [[0.5773744618389179, 0.5776504791744279, 0.57...
IsingZT_tauR_err       [[0.0005925670020223004, 0.000599508043614475,...
IsingZT_tau_lattice    {'L1': [7.071067811865475, 7.071067811865475],...
lattice                {'L1': [7.071067811865475, 7.071067811865475],...
Name: ./ALF_data/temper_Nematic_Dirac_Model_vers=1_L1=10_L2=10_beta=40.0_xi=0.25_h=2.5/Temp_0, Length: 128, dtype: object

item['IsingZ_eqK']

array([[[ 0.32638655,  0.34595176,  0.37585751,  0.40512372,
          0.41848529,  0.40560553,  0.3758137 ,  0.34585338,
          0.32686855,  0.32000094,  0.34582087,  0.36959162,
          0.40548   ,  0.44284267,  0.4599152 ,  0.44257476,
          0.40533533,  0.36922645,  0.3460411 ,  0.33724708,
          0.37373131,  0.40447458,  0.45249519,  0.50672501,
          0.53239187,  0.50668498,  0.4533181 ,  0.4044087 ,
          0.37437927,  0.36392956,  0.40205248,  0.43922089,
          0.50464484,  0.58204502,  0.62283904,  0.58161175,
          0.50320508,  0.44007743,  0.40266863,  0.38967683,
          0.41550708,  0.45576157,  0.52808854,  0.6214426 ,
         58.38260793,  0.6214426 ,  0.52808854,  0.45576157,
          0.41550708,  0.40164649,  0.40266863,  0.44007743,
          0.50320508,  0.58161175,  0.62283904,  0.58204502,
          0.50464484,  0.43922089,  0.40205248,  0.38967683,
          0.37437927,  0.4044087 ,  0.4533181 ,  0.50668498,
          0.53239187,  0.50672501,  0.45249519,  0.40447458,
          0.37373131,  0.36392956,  0.3460411 ,  0.36922645,
          0.40533533,  0.44257476,  0.4599152 ,  0.44284267,
          0.40548   ,  0.36959162,  0.34582087,  0.33724708,
          0.32686855,  0.34585338,  0.3758137 ,  0.40560553,
          0.41848529,  0.40512372,  0.37585751,  0.34595176,
          0.32638655,  0.32000094,  0.31925968,  0.33890868,
          0.36577437,  0.3931343 ,  0.40544295,  0.3931343 ,
          0.36577437,  0.33890868,  0.31925968,  0.31334392]]])

Creating Lattice object#

from py_alf import Lattice

latt = Lattice(item['IsingZ_eq_lattice'])

Spin-Spin correlation deep in ordered phase#

latt.plot_k(item['IsingZ_eqK'][0,0])

Spin-Spin correlation in disordered phase#

latt.plot_k(res.loc[
    './ALF_data/temper_Nematic_Dirac_Model_vers=1_L1=10_L2=10_beta=40.0_xi=0.25_h=2.5/Temp_3',
    'IsingZ_eqK'][0,0])

Fermionic dispersion#

See Appendix A.4 for source code of fit_green_tau.

from fit_green_tau import fit_green_tau

dic = {}
for i in res.index:
    print(i)
    dtau = res.loc[i, 'dtau']
    Green = res.loc[i, 'Green_tauK']
    dGreen = res.loc[i, 'Green_tauK_err']
    (N_tau, N) = Green.shape
    taus = np.arange(0., N_tau*dtau, dtau)
    dic[i] = np.empty((N, 2))
    for n in range(N):
        # print(f'{n} out of {N}')
        G = Green[:, n]
        dG = dGreen[:, n]
        dic[i][n] = fit_green_tau(taus, G, dG, plot=False)
res['gap'] = pd.Series(dic)

./ALF_data/temper_Nematic_Dirac_Model_vers=1_L1=10_L2=10_beta=40.0_xi=0.25_h=2.5/Temp_0

./ALF_data/temper_Nematic_Dirac_Model_vers=1_L1=10_L2=10_beta=40.0_xi=0.25_h=2.5/Temp_1
./ALF_data/temper_Nematic_Dirac_Model_vers=1_L1=10_L2=10_beta=40.0_xi=0.25_h=2.5/Temp_2

./ALF_data/temper_Nematic_Dirac_Model_vers=1_L1=10_L2=10_beta=40.0_xi=0.25_h=2.5/Temp_3

./ALF_data/temper_Nematic_Dirac_Model_vers=1_L1=4_L2=4_beta=16.0_xi=0.25_h=2.5/Temp_0
./ALF_data/temper_Nematic_Dirac_Model_vers=1_L1=4_L2=4_beta=16.0_xi=0.25_h=2.5/Temp_1
./ALF_data/temper_Nematic_Dirac_Model_vers=1_L1=4_L2=4_beta=16.0_xi=0.25_h=2.5/Temp_2
./ALF_data/temper_Nematic_Dirac_Model_vers=1_L1=4_L2=4_beta=16.0_xi=0.25_h=2.5/Temp_3
./ALF_data/temper_Nematic_Dirac_Model_vers=1_L1=6_L2=6_beta=24.0_xi=0.25_h=2.5/Temp_0
./ALF_data/temper_Nematic_Dirac_Model_vers=1_L1=6_L2=6_beta=24.0_xi=0.25_h=2.5/Temp_1
./ALF_data/temper_Nematic_Dirac_Model_vers=1_L1=6_L2=6_beta=24.0_xi=0.25_h=2.5/Temp_2
./ALF_data/temper_Nematic_Dirac_Model_vers=1_L1=6_L2=6_beta=24.0_xi=0.25_h=2.5/Temp_3
./ALF_data/temper_Nematic_Dirac_Model_vers=1_L1=8_L2=8_beta=32.0_xi=0.25_h=2.5/Temp_0
./ALF_data/temper_Nematic_Dirac_Model_vers=1_L1=8_L2=8_beta=32.0_xi=0.25_h=2.5/Temp_1
./ALF_data/temper_Nematic_Dirac_Model_vers=1_L1=8_L2=8_beta=32.0_xi=0.25_h=2.5/Temp_2
./ALF_data/temper_Nematic_Dirac_Model_vers=1_L1=8_L2=8_beta=32.0_xi=0.25_h=2.5/Temp_3

We plot the determined dispersion for system sizes \(L=10\) and mark the points \(\boldsymbol{k} = (\pi/2, \pi/2)\) and \(\boldsymbol{k} = (\pi/2, -\pi/2)\), which are the locations of the Dirac points in the disordered phase. One can see how the Dirac cones are displaced in the ordered phase at \(h = 2.5, 3.0\), while they remain in place in the ordered phase at \(h = 3.5, 4.0\). Notably, the simulation at \(h = 2.5\) features \(\langle \hat{s}^z \rangle < 0\) and \(h = 3.0\) features \(\langle \hat{s}^z \rangle > 0\) (cf. Fig. 2.1(a1)), which is random. The displaced Dirac cones are only observable because the simulation is not fully ergodic and randomly “chooses” one of the two symmetry broken phases.

df = res[res.l1 == 10]
latt = Lattice([10, 10], [10, -10], [1, 1], [1, -1])
for i in df.index:
    latt.plot_k(res.loc[i, 'gap_lazy'])
    plt.title(f'$h = {res.loc[i, "ham_h"]}$')
    p = np.pi/2
    plt.plot([p, p], [p, -p], 'o', color='red')

df = res[res.l1 == 10]
latt = Lattice([10, 10], [10, -10], [1, 1], [1, -1])
for i in df.index:
    latt.plot_k(res.loc[i, 'gap'][:, 0])
    plt.title(f'$h = {res.loc[i, "ham_h"]}$')
    p = np.pi/2
    plt.plot([p, p], [p, -p], 'o', color='red')