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In this tutorial, we will apply dynamical mean field theory (DMFT) on Sr2RuO4 on a QSGW stating point that you can find here

Impurity solver

Questaal package does not supply an impurity solver. There are several choices of solver on the market, in this tutorial, the CTQMC solver from the TRIQS package is used (see TRIQS package and its hybridization-expansion solver solver) TRIQS is already installed on the machine used for this workshop.

Prepare Input folder and files

Open dmft_sr2ruo4.tar.bz2

tar xvf dmft_sr2ruo4.tar.bz2
cd dmft_sr2ruo4

If you list your directory now, you should have :

atm.sr2ruo4  basp.sr2ruo4  ctrl.sr2ruo4  rst.sr2ruo4  sigm.sr2ruo4  site.sr2ruo4

Definition of the correlated subspace

At the end of ctrl file add the following :

DMFT    BETA=30 NOMEGA=999 NLOHI=16,42
        BLOCK:
        SITES=3 L= 2
        SIDXD= 1 2 0 2 0

Remark: Make sure you have the right indentation between the token DMFT and BLOCK.

Projection window

The first tokens have means :

  • BETA=30 : The DMFT loop is done at inverse temperature (ev^-1) 30.
  • NOMEGA : Number of Matsubara frequencies for the impurity self energy.
  • NLOHI=16.42 The projector are constructed with the bands from 16 to 42.

    In principle, the window is infinite and all the bands should be included. In practice, for numerical reasons, we choose a large finite windows around the d-state.

    plot the partial dos on Ru atom (see tutorial here

    mpirun -np 24 lmf sr2ruo4 -vnkabc=8 --quit=rho --pdos~mode=2~sites=3~lcut=2
    lmdos sr2ruo4 -vnkabc=8 --quit=rho --pdos~mode=2~sites=3~lcut=2 --dos:npts=1001:window=-1,1
    echo 1.4 5 -20 20 | pldos -fplot~sh~long~open~tmy=.25~dmin=0.40~xl=E -esclxy=13.6 -ef=0 -lst="5;6;7;8;9" dos.sr2ruo4
    

    which prints a file fplot.ps that you can open with evince :

    pdos_sr2ruo4

    Panel 1,2,4 are t2g bands and 3,5 are eg.

    In this tutorial, we will include the t2g bands so we want at least a windows at least of around -8 ev (0.58 Ry) to 8 eV the following command :

    mpirun -np 24 lmf ctrl.sr2ruo4 --minmax --quit=rho
    

    prints an array with the lower and higher energies (in Ry) of each bands over the Brillouin zone.

       1  -3.1933 -3.1923  |   65   1.8788  2.0501  |  129   4.7309  4.9919
     2  -3.1932 -3.1922  |   66   1.9419  2.0890  |  130   4.8393  5.0922
     3  -3.1661 -3.1658  |   67   1.9731  2.1264  |  131   4.9053  5.1016
     4  -2.6274 -2.6215  |   68   2.0040  2.1605  |  132   4.9440  5.1585
     5  -2.6228 -2.6196  |   69   2.0555  2.2153  |  133   4.9919  5.1952
     6  -1.5204 -1.4365  |   70   2.0900  2.3107  |  134   5.0455  5.2968
     7  -1.4430 -1.4355  |   71   2.1252  2.3360  |  135   5.0776  5.3219
     8  -1.4108 -1.3802  |   72   2.1726  2.3635  |  136   5.1263  5.3914
     9  -1.3998 -1.3629  |   73   2.1764  2.4229  |  137   5.1329  5.4252
    10  -1.3041 -1.2684  |   74   2.2579  2.4491  |  138   5.1656  5.4392
    11  -1.2971 -1.2629  |   75   2.2928  2.5067  |  139   5.1886  5.4739
    12  -1.2803 -1.2585  |   76   2.3301  2.5856  |  140   5.2448  5.5131
    13  -1.2747 -1.2463  |   77   2.3612  2.6184  |  141   5.3528  5.5326
    14  -1.2558 -1.2075  |   78   2.4576  2.6479  |  142   5.3625  5.5800
    15  -1.2405 -1.2064  |   79   2.5070  2.6812  |  143   5.4614  5.6184
    16  -0.5752 -0.3531  |   80   2.5863  2.7187  |  144   5.4849  5.6598
    17  -0.5550 -0.3531  |   81   2.6036  2.7533  |  145   5.5521  5.7134
    18  -0.4820 -0.3084  |   82   2.6140  2.7589  |  146   5.6148  5.8006
    19  -0.3974 -0.2972  |   83   2.6419  2.7795  |  147   5.6710  5.8289
    20  -0.3743 -0.2419  |   84   2.6690  2.8733  |  148   5.6927  5.8528
    21  -0.3121 -0.2289  |   85   2.6750  2.8797  |  149   5.7224  5.9200
    22  -0.2710 -0.1819  |   86   2.6853  2.9270  |  150   5.7843  5.9534
    23  -0.2607 -0.1755  |   87   2.7066  2.9472  |  151   5.8318  6.0230
    24  -0.2219 -0.1618  |   88   2.7538  2.9835  |  152   5.8502  6.0298
    25  -0.2137 -0.1341  |   89   2.8097  3.0046  |  153   5.8637  6.0883
    26  -0.1670 -0.1281  |   90   2.8142  3.0375  |  154   5.9125  6.1391
    27  -0.1506 -0.1118  |   91   2.8543  3.0865  |  155   5.9572  6.1855
    28  -0.1404  0.0948  |   92   2.8951  3.1151  |  156   6.0318  6.2192
    29   0.0028  0.0966  |   93   2.9317  3.1592  |  157   6.0892  6.2885
    30   0.0028  0.1064  |   94   3.0004  3.1984  |  158   6.1307  6.3278
    31   0.1241  0.3452  |   95   3.0451  3.2332  |  159   6.1794  6.4674
    32   0.2157  0.4878  |   96   3.1038  3.2800  |  160   6.2245  6.4991
    33   0.3272  0.5175  |   97   3.1583  3.3322  |  161   6.2727  6.5368
    34   0.3602  0.5757  |   98   3.1991  3.4632  |  162   6.2787  6.6025
    35   0.4733  0.6402  |   99   3.2188  3.5471  |  163   6.3411  6.6706
    36   0.5218  0.6815  |  100   3.2625  3.5473  |  164   6.4144  6.6974
    37   0.5741  0.7266  |  101   3.2988  3.5809  |  165   6.4714  6.7229
    38   0.6189  0.7492  |  102   3.3689  3.6773  |  166   6.5543  6.8148
    39   0.6359  0.8077  |  103   3.4046  3.6939  |  167   6.6090  6.8661
    40   0.6657  0.8407  |  104   3.4186  3.7622  |  168   6.6523  6.9259
    41   0.7294  0.8681  |  105   3.4595  3.8401  |  169   6.6834  7.0081
    42   0.7714  0.8902  |  106   3.5179  3.8615  |  170   6.7564  7.0302
    43   0.7813  0.9216  |  107   3.5787  3.8777  |  171   6.8423  7.1140
    44   0.8269  0.9935  |  108   3.6858  3.9177  |  172   6.9238  7.1336
    45   0.9261  1.0750  |  109   3.7300  3.9602  |  173   6.9394  7.2033
    46   0.9507  1.1757  |  110   3.7610  3.9689  |  174   6.9742  7.2077
    47   1.0236  1.2357  |  111   3.8346  4.0433  |  175   6.9983  7.2563
    48   1.0793  1.2666  |  112   3.8837  4.0809  |  176   7.0628  7.3006
    49   1.1336  1.2692  |  113   3.9219  4.0896  |  177   7.1292  7.4603
    50   1.1579  1.4529  |  114   3.9735  4.1142  |  178   7.2345  7.5412
    51   1.1790  1.4529  |  115   4.0064  4.1494  |  179   7.2853  7.5598
    52   1.2092  1.4693  |  116   4.0548  4.2091  |  180   7.3522  7.7069
    53   1.2931  1.5363  |  117   4.1113  4.2737  |  181   7.4270  7.7541
    54   1.3212  1.6060  |  118   4.1363  4.3192  |  182   7.4274  7.9015
    55   1.3494  1.6485  |  119   4.1976  4.3858  |  183   7.6705  7.9530
    56   1.4468  1.6525  |  120   4.2307  4.4231  |  184   7.8429  8.3109
    57   1.4956  1.7020  |  121   4.2748  4.5355  |  185   7.9566  8.5638
    58   1.5960  1.7095  |  122   4.3247  4.5716  |  186   8.1509  9.1719
    59   1.6300  1.7828  |  123   4.4494  4.6198  |  187   8.3788  9.5613
    60   1.7048  1.8179  |  124   4.5232  4.6966  |  188   8.3801  9.5618
    61   1.7308  1.9218  |  125   4.5748  4.7578  |  189   8.5248  9.7693
    62   1.7599  1.9445  |  126   4.6161  4.7945  |  190   9.8480 10.0328
    63   1.8104  1.9558  |  127   4.6564  4.8879  |  191   9.9069 10.8707
    64   1.8320  1.9920  |  128   4.6564  4.9164  |
    

    For this tutorial NLOHI=16,42

Definition of the impurity

In our example, the correlated subspace is composed of Ru d-state. This information is given by :

        BLOCK:
        SITES=3 L= 2
        SIDXD= 1 2 0 2 0

The DMFT subspace can have several correlated atoms. each correlated atoms is defined with it index in the site file, the orbital caracters of the corraleted orbital (l=2 for d, l=3 for f).

  • SITES : index of correlated site (Here Ru which is the atom 3 in site file
  • L : orbital momentum of correlated subspace (L=2 for d)
  • SIDXD indicates which m =[-l .. +l] are taken into account in the impurity self energy To included off-diagonal term, SIDXD has to be replaced SIDXM and a (2L+1)x(2L+1) matrix has to be supplied. Here we take into accout xy, yz, xz where yz and xz are equivalent. The order in given by Questaal’s ordering

# Autogenerated from init.sr2ruo4 using:
# /users/ms4/bin/blm --gw --express=1 --nk=12 --nkgw=6 --gmax=11.0 --nit=100 sr2ruo4
% const nit=100
% const met=5
% const so=0 nsp=so?2:1
% const lxcf=2 lxcf1=0 lxcf2=0
% const pwmode=0 pwemax=3
% const sig=12 gwemax=2.46 gcutb=3.1 gcutx=2.5
% const nkabc=12 nkgw=6 gmax=11 beta=.3

VERS  LM:7 FP:7 # ASA:7
IO    SHOW=f HELP=f IACTIV=f VERBOS=35,35  OUTPUT=*
EXPRESS
  file=   site
  nit=    {nit}
  mix=    B2,b={beta},k=7
  conv=   1e-5
  convc=  3e-5
  nkabc=  {nkabc}
  metal=  {met}
  nspin=  {nsp}
  so=     {so}
  xcfun=  {lxcf},{lxcf1},{lxcf2}

#SYMGRP i r4z mx
HAM   GMAX={gmax}
      AUTOBAS[PNU=1 LOC=1 MTO=4 LMTO=5 GW=1 PFLOAT=2,1]
      PWMODE={pwmode} PWEMIN=0 PWEMAX={pwemax} OVEPS=0
      XCFUN={lxcf},{lxcf1},{lxcf2}
      FORCES={so==0} ELIND=-0.7 NSPIN={nsp} SO={so}
      RDSIG={sig} SIGP[EMAX={gwemax}]
ITER  MIX=B2,b={beta},k=7  NIT={nit}  CONVC=1e-5
BZ    METAL={met}  NKABC={nkabc}
GW    NKABC={nkgw} GCUTB={gcutb} GCUTX={gcutx} DELRE=.01 .1
      GSMEAR=0.003 PBTOL=3e-4,3e-4,1e-3
STRUC
      SHEAR=1 0 0 1.0
      NL=5  NBAS=7  NSPEC=3
      ALAT= 7.30111097
      PLAT= -0.5  0.5  1.6455593  0.5  -0.5  1.6455593  0.5  0.5  -1.6455593
SPEC
  ATOM=Sr       Z= 38  R= 2.985910  LMX=3 LMXA=4
  ATOM=Ru       Z= 44  R= 2.068587  LMX=3 LMXA=4 PZ=0,0,5.4
  ATOM=O        Z=  8  R= 1.581969  LMX=3 LMXA=4
SITE
  ATOM=Sr         POS=  0.5000000   0.5000000  -0.4851109
  ATOM=Sr         POS= -0.5000000  -0.5000000   0.4851109
  ATOM=Ru         POS=  0.0000000   0.0000000   0.0000000
  ATOM=O          POS=  0.0000000   0.5000000   0.0000000
  ATOM=O          POS=  0.0000000   0.0000000   0.5348068
  ATOM=O          POS= -0.5000000   0.0000000   0.0000000
  ATOM=O          POS=  0.0000000   0.0000000  -0.5348068
DMFT    NLOHI=16,42 BETA=30 NOMEGA=999
        BLOCK:
        SITES=3 L= 2
        SIDXD= 1 2 0 2 0

Double Counting

We will use a static double counting given by the fully localized limit (FLL), with U = 4.5 eV, J = 1 eV, n = 4

Edc = U(n-1/2)-J(n-1)/2 = 14.25 eV

echo "14.25 14.25" > dc.sr2ruo4  # there is one value for each independant active orbital

Solver parameters

The CTQMC solver is a Monte Carlo solver. The parameters needed by the solver are given in para_triqs.txt. In this tutorial, we just need one params file. An example is given below :

U 4.5
J 1
n_warmup_cycles 100000
n_cycles 300000
length_cycle 30
n_l 50
  • U : Effective interation
  • J : Hund coupling
  • n_warmup_cycles : number of warmup steps.
  • n_cycles : number of measures per core
  • length_cycle : number of steps between two measures. If this number is too small, the data will be noisy due to correlated sampling. If this number is too large, the calculation will be longer uselessly.
  • n_l : number of Legendre coefficients (see [3])

Put the parameters above in a file called para_triqs.txt

Note: The total number of measures is n_cycles*number_of_cores. So increasing the number of core will increase the quality of the result.

First iteration

In general, it is adviced to perform the first iteration and check the quality of the result in order to be sure than the number set in para_triqs.txt are ok.

Create a new folder for the first iteration

mkdir it1
cp -f * it1/.
cd it1/

In it1, run lmfdmft to create a self energy file sig.inp

lmfdmft sr2ruo4 --ldadc~fn=dc --job=1

this command creates a file sig.inp

sig.inp is a text file which contains the self energy.

there is NOMEGA lines Then the structure of the collun is

  w Re(S_1) Im(S_1) ... Re(S_i) Im(S_i)

where i is the index given in the token SIDXD or SIDXM

Once the file sig.inp is created, run again

mpirun -np 24 lmfdmft sr2ruo4 --ldadc~fn=dc --job=1

It will create the file solver_input.h5 which is a hdf5 file format.

not fixed yet.

Let try to run the ctqmc solver now.

mpirun -np 24 lmtriqs sr2ruo4

The calculation takes about 10 minutes. In the plot show below, we run the calculation on 24 cores.

The first thing to check is that the number of Legendre coefficient for the Green function is correct. it has to be high enough so the coefficient goes to zeros. If it is too high, the high coefficient just contained Monte Carlo noise which pollutes the final result.

The file gl.inp contains g_l with the following format :
  nb_of_legendre_coef Re(G_l_1_up) ... Re(G_l_i_up) .... Re(G_l_1_down) ...

A simple python script to plot it could be :

import numpy as np
import matplotlib.pyplot as plt
import sys
Gl = np.loadtxt(sys.argv[1]).T
wl = Gl[0]
Gl = Gl[1:]
#plot just G_l_up
for i in range(Gl.shape[0]) :
    plt.plot(wl,Gl[i])
plt.xlabel('n_l')
plt.ylabel('Gl')
plt.savefig('gl.png')
plt.show()

put this script in a file plot_gl.py and run

python plot_gl.py path_to/gl.inp

You should obtain :

G_l All the coefficients go to 0.

Then, the Green function on the Matsubara axis is deduced from G_l and the self energy is obtained from the Dyson equation.

The self energy is contained in the file sig.inp

A simple python script to plot sig.inp could be :

import numpy as np
import matplotlib.pyplot as plt
import sys
S = np.loadtxt(sys.argv[1]).T
w = S[0]
Sig = S[1::2] + 1j* S[2::2]
#plot just Sig_up
plt.figure(figsize=(8,8))
for i in range(Sig.shape[0]) :
    plt.plot(w,Sig[i].imag,label=i+1)
plt.xlabel(r'$\Omega_n$', fontsize = 20)
plt.ylabel(r'$\Sigma$', fontsize = 20)
plt.xlim(0,200)
plt.legend()
plt.savefig('sig_sr2ruo4.png')
plt.show()

put this script in a file plot_giw.py and run

python plot_giw.py path_to/sig.inp

You should obtain :

Self energy

The self energy of the second orbital is noiser than the first one because the second one is the average of yz and xz.

To complete the dmft loop, we need to run again lmfdmft to get a updated hybridisation function and impurity energy level and then run again the impurity solver to get a new self energy. It has to be repeated until convergence.

DMFT loop

To perform the DMFT loop, we proposed a bash script which could be adapted for your machine. It’s assumed that you have set correctly the ctrl file and the params file

#!/bin/bash -l
set -xe


x=sr2ruo4
mxit=20 # nb of iteration
np_lmfdmft=24 # nb of cores available for lmfdmft
np_ctqmc=24 # nb of cores available for lmfdmft





if [ ! -e it1 ]; then
    mkdir -p it1
    cp -f * it1/.
fi

for i in `seq $mxit`;
do
    [ ! -e "it$((i-1))" ] || [ ! -e "it${i}"  ] || continue # if some iterations have already be done ...
    [ ! -e "it$((i-1))" ] || [ -e "it${i}"  ] || cp -av "it$((i-1))" "it${i}"
    pushd "it${i}"

    mpirun -n $np_ctqmc  lmfdmft --ldadc~fn=dc -job=1 ctrl.$x
    rm -f evec* proj*
    mpirun -np $np_ctqmc lmtriqs $x
    popd
done

It is adviced to check the convergence by plotting the difference sig.inp in each itX folder. It is also adviced to perform the average over the last iteration.

You can find a converged result in

extra/sig_converge.inp

For the rest of the tutorial replace your sig.inp by the converge one:

cp extra/sig_converge.inp sig.inp

This has been converged using 30 iterations and average on the 10 last interation (with n_cycles=1000000) on a 24 cores machine.

Analyse of the result

Analytic continuation

In this tutorial we use lmfdmft’s built in Pade extrapolator to carry out the analytic continuation.
Note: analytic continuation is a tricky business. Pade works reliably only when the polynomial order is not too large, which means in can operate reliably only over a small energy window. Also the self-energy should be reasonably smooth.

lmfdmft sr2ruo4  --rs=1,0 --ldadc~fn=dc -job=1 --pade~nw=121~window=-6/13.606,6/13.606~icut=30,75

it creates a file sig2.sr2ruo4 containing the self energy in the real axis which has the same format than sig.inp.

Spectral function

In this section, the spectral function

To do so, execute the following command.

 cp extra/syml.sr2ruo4 . # copy the symetry lines.
 mpirun -np 24 lmfdmft sr2ruo4  --rs=1,0 --ldadc~fn=dc -job=1 --gprt~band,fn=syml~rdsigr=sig2~mode=19
 lmfgws sr2ruo4 '--sfuned~units eV~readsek@useef@irrmesh@minmax~eps 0.02~se band@fn=syml nw=2 isp=1 range=-4,4'
 plbnds -sp~atop=10~window=-4,4 spq2.sr2ruo4
 gnuplot gnu.plt

Spectral function

Spectrum DOS

QSGW noninteracting DOS

Making the QSGW noninteracting DOS using lmf in the usual manner, using the --dos switch

lmf sr2ruo4 -vnkabc=12 --dos@npts=721@ef0@rdm@window=-6/13.606,6/13.606 --quit=dos

Note: you could speed up the calculation by using mpi.

It creates file dos.sr2ruo4. --quit=dos tells lmf to stop after the DOS has been written, and the modifies to --dos have the following meanings:

  • npts=721  specifies number of energy points (the spacing between points is the same as for the spectrum dos)
  • window=−6/13.606,6/13.606  energy window for DOS, in Ry. It is 3/2 wider than that generated for the spectrum DOS
  • ef0     Energy window to be defined relative to Fermi level
  • rdm    Write DOS in standard Questaal format for 2D arrays.

DMFT interaction DOS

lmfdmft sr2ruo4 -vnkabc=8 --ldadc~fn=dc --job=1 --gprt~rdsigr=sig2~mode=19
lmfgws sr2ruo4 -vnkgw=8 '--sfuned~units eV~readsek@useef@ib=1:27~eps 0.02~dos isp=1 range=-4,4 getev=12 nw=4~savesea~q'

Note: you can speed up the calculation by using mpi for lmfdmft but mpi does not work with lmfg

To compare the two DOS, you can use the fplot utility :

fplot -vef0=0.00037   sdos.sr2ruo4 -ab '(x1-ef0)*13.6' -ord x2/13.6 -lt 2,col=1,0,0 dos.sr2ruo4

The noninteracting DOS are written in Ry, on an absolute energy scale (i.e. not relative to the Fermi level).
-ab ‘(x1-ef0)*13.6’ shifts by ef0 and scales the abscissa from Ry to eV. -ord x2/13.6 scales the ordinate from Ry−1 to eV−1.

Dos sr2ruo4

Spin susceptibility

The spin susceptibility is obtained from the local vertex :

The DMFT vertex is obtained from the local susceptibility :

impurity susceptibility

The local susceptibility is obtained from the two particle green function :

The two-particles Green function has 1 bosonic frequency, 2 fermionic frequencies and 2 orbital indicies (formally 4 but we do not take into account off-diagonal terms in this tutorial). This quantity is computed by the TRIQS cthyb solver.

The first step is to modify the line with DMFT token on the ctrl file by :

DMFT    NLOHI=16,42 BETA=30 NOMEGA=999 NOMF=50 NOMB=1
  • NOMF=50 : Number of fermionic frequencies to included in the local susceptibility
  • NOMB=1 : Number of bosinic frequencies to included in the local susceptibility

The computation of the two particle Green function severaly slows the solver. In this case, it is good to increase length_cycle in the para_triqs file and reduce n_cycles.

For the purpose of this tutorial, the local susceptibility has already be computed on a 240 cores machine with the following para_triqs.txt file :

U 4.5
J 1.0
n_warmup_cycles 100000
n_cycles 100000
length_cycle 500
n_l 50

and run

mpirun -np 24 lmfdmft sr2ruo4  --ldadc~fn=dc --job=1 # makes sure that solver_input.h5 are update
mpirun -np 24 lmtriqs sr2ruo4 --vertex

it takes around 9 hours.

In your folder, do :

cp extra/chiloc_1_example.h5 chiloc_1.h5

The is computed by lmfdmft.

can be computed on the 1 Brillouin zone or in a given q mesh. This q mesh has be given as follow : the first term is the size of the k mesh (should be equal to nkabc in the ctrl file (this option works for nkabc(1)=nkabs(2)=nkabs(3)). then on each line, q is given by :

where Q_1, Q_2, Q_3 are the reciprocal lattice vector and

For instance, the following q list describes a q line along for to on 9x9x9 q-mesh.

9
0 0 0
0 0 1
0 0 2
0 0 3
0 0 4
0 0 5
0 0 6
0 0 7
0 0 8

put the q-list in a file called “qlist” and then run :

mpirun -np 24 lmfdmft sr2ruo4 -vnkabc=9 --ldadc~fn=dc --job=1 --chiloc --qlist

Remark: To make and the full BZ, just remove the –qlist flag.

Finally the Bethe Salpeter equation is done by running :

susceptibility

it stores the susceptibility in a h5 format and in a txt file called chi_spin.txt :

#iq iom real(chi)    imag(chi)
 0  0  13.3551597356 4.90070946787e-13
 1  0  14.8699746786 2.31342722756e-12
 2  0  17.6681324135 8.23262199153e-12
 3  0  12.0345500361 8.8882197898e-12
 4  0  7.24252562004 3.83866087065e-12
 5  0  7.24252562004 4.06060740588e-12
 6  0  12.0345500361 8.91117180046e-12
 7  0  17.6681324135 8.94648059339e-12
 8  0  14.8699746786 2.30697313105e-12

You can then plot it with your favorite program.

References


[1] K. Haule, C.H. Yee, and K. Kim. Dynamical mean field theory within the full-potential methods: Electronic structure of ceirin5, cecoin5, and cerhin5. Phys. Rev. B, 81:195107, 2010.

[2] Paolo Pisanti. A novel QSGW+DMFT method for the study of strongly correlated materials. PhD thesis, King’s College London, 2016.

[3] Lewin Boehnke et al. “Orthogonal polynomial representation of imaginary-time Green’s functions”. Physical Review B 84.7 (2011).