4.6. Output of DSQSS/DLA¶
4.6.1. Format¶
DSQSS/DLA generates the result of a simulation as a plain-text file. The first character stands for the meaning of the line.
P <name> = <value>Parameters read from the input files.
R <name> = <mean> <error>Results of observables.
<mean>denotes the expected value and<error>denotes the statistical error of<mean>.I <text> = <value>Other information.
C <text>Comments.
The following one is a result of an antiferromagnetic Heisenberg chain.
C This is DSQSS ver.v2.0.0
I N_PROC = 1
P D = 1
P L = 8
P BETA = 10.0000000000000000
P NSET = 10
P NPRE = 1000
P NTHERM = 1000
P NDECOR = 1000
P NMCS = 1000
P SEED = 198212240
P NSEGMAX = 10000
P NVERMAX = 10000
P BETA = 10.000000000000
P NTAU = 10
P NCYC = 7
P ALGFILE = algorithm.xml
P LATFILE = lattice.xml
P WVFILE = wv.xml
P DISPFILE = disp.xml
P OUTFILE = sample.log
P CFOUTFILE = cf.dat
P SFOUTFILE = sf.dat
P CKOUTFILE = ck.dat
P SIMULATIONTIME = 0.000000
R sign = 1.00000000e+00 0.00000000e+00
R anv = 3.03805000e+00 1.25395375e-02
R ene = -4.55991910e-01 1.20267537e-03
R spe = -1.76672204e-02 4.09064489e-02
R som = -1.76672204e-01 4.09064489e-01
R len = 1.20014021e+01 4.78403202e-02
R xmx = 3.00035053e-01 1.19600800e-03
R amzu = -2.00000000e-04 1.08972474e-04
R bmzu = -2.00000000e-04 1.08972474e-04
R smzu = 1.32382500e-03 1.40792745e-04
R xmzu = 1.32382500e-02 1.40792745e-03
R ds1 = -1.32954309e-03 7.87178338e-04
R wi2 = 2.31040000e+01 3.83762890e-01
R rhos = 1.44400000e-01 2.39851806e-03
R rhof = inf nan
R comp = 2.43165481e+35 1.71412709e+35
R amzs0 = -2.00000000e-04 1.08972474e-04
R bmzs0 = 1.65625000e-04 1.76161818e-05
R smzs0 = 1.32382500e-03 1.40792745e-04
R xmzs0 = 1.32382500e-02 1.40792745e-03
R amzs1 = -9.25000000e-04 4.02247160e-03
R bmzs1 = 1.09209375e-01 1.12051866e-03
R smzs1 = 8.72503175e-01 8.93939492e-03
R xmzs1 = 3.00500011e+00 2.99056535e-02
R time = 1.03679300e-07 1.22794234e-09
I [the maximum number of segments] = 123
I [the maximum number of vertices] = 66
I [the maximum number of reg. vertex info.] = 3
4.6.1.1. Notations¶
- \(N_s\)
The number of sites.
- \(Q(\vec{k})\)
The Fourier transformation of an arbitrary operator on a site \(i\), \(Q_i\).
\(\displaystyle Q(\vec{k}) \equiv \frac{1}{\sqrt{N_s}} \sum_i^{N_s} Q_i e^{-i\vec{r}_i\cdot\vec{k}}\)
- \(Q(\tau)\)
An arbitrary operator at imaginary time \(\tau\).
\(\displaystyle Q(\tau) \equiv \exp\left[\tau \mathcal{H}\right] Q(\tau=0) \exp\left[-\tau \mathcal{H}\right]\)
- \(\tilde{Q}\)
The average of an arbitrary operator \(Q\) over the imaginary time, \(\displaystyle \frac{1}{\beta}\int_0^\beta \! \mathrm{d} \tau Q(\tau)\)
- \(M^z\)
The component of a local degree of freedom along with the quantized axis. For example, \(z\) component of the local spin operator \(S^z\) for spin systems and the number operator \(n\) for the Bose-Hubbard models.
- \(M^\pm\)
The ladder operator. \(M^{\pm} \equiv S^\pm\) for spin systems, and the creation/annihilation operators \(M^+ \equiv b^\dagger\) , \(M^- \equiv b\) for the Bose-Hubbard models.
- \(M^x\)
The off-diagonal order parameter. \(M^x \equiv (S^+ + S^-)/2\) for spin systems and \(M^x \equiv (b + b^\dagger)\) for the Bose-Hubbard models.
- \(T\)
The temperature.
- \(\beta\)
The inverse temperature.
- \(h\)
The conjugate field to the operator \(M^z\) . The longitudinal magnetic field for spin systems and the chemical potential for the Bose-Hubbard models.
- \(\left\langle Q \right\rangle\)
The expectation value of an arbitrary operator \(Q\) over the grand canonical ensemble.
4.6.2. Main results¶
Main results are written in a file with the name specified by outfile keyword in the input parameter file.
signThe sign of the weights.
\(\sum_i W_i / \sum_i |W_i\)
anvThe mean number of the vertices.
\(\displaystyle \frac{\langle N_v \rangle}{N_s}\)
eneThe energy density (energy per site)
\(\displaystyle \epsilon \equiv \frac{1}{N_s}\left(E_0 - T\langle N_v\rangle\right)\)
speThe specific heat
\(\displaystyle C_V \equiv \frac{\partial \epsilon}{\partial T}\)
somThe ratio of the specific heat and the temperature.
\(\displaystyle \gamma \equiv \frac{C_V}{T} = \beta C_V\)
lenThe mean length of worm
xmxThe transverse susceptibility
amzuThe “magnetization” (uniform, \(\tau=0\)).
\(\left\langle m^z \right\rangle\) , where \(\displaystyle m^z \equiv \frac{1}{N_s} \sum_i^{N_s} M^z_i\)
bmzuThe “magnetization” (uniform, average over \(\tau\)). \(\left\langle \tilde{m}^z \right\rangle\) .
smzuThe structure factor (uniform).
\(\displaystyle S^{zz}(\vec{k}=0) \equiv \frac{1}{N_s} \sum_{i, j} e^{i \vec{k}\cdot(\vec{r}_i-\vec{r}_j)} \left[ \left\langle M^z_i M^z_j\right\rangle - \left\langle M_i^z \right\rangle \left\langle M_j^z \right\rangle \right] \Bigg|_{\vec{k}=0} = N_s \left[ \left\langle (m^z)^2\right\rangle - \left\langle m^z\right\rangle^2 \right]\)
xmzuThe longitudinal susceptibility (uniform).
\(\displaystyle \chi^{zz}(\vec{k}=0, \omega=0) \equiv \frac{\partial \left\langle \tilde{m}^z \right\rangle}{\partial h} = \beta N_s\left[ \left\langle \left(\tilde{m}^z\right)^2\right\rangle - \left\langle \tilde{m}^z\right\rangle^2 \right]\)
amzsKThe “magnetization” (“staggered”, \(\tau=0\))
\(\left\langle m_s^z \right\rangle\) where \(\displaystyle m_K^z \equiv \frac{1}{N_s} \sum_i^{N_s} M_i^z \cos\left( \vec{k}\cdot\vec{r_i} \right)\) .
\(K\) is an index of wavevector \(k\) specified in the wavevector XML file.
bmzuThe “magnetization” (“staggered”, average over \(\tau\)). \(\left\langle \tilde{m}_K^z \right\rangle\) .
smzsThe structure factor (“staggered”).
\(\displaystyle S^{zz}(\vec{k}) = N_s \left[ \left\langle (m_K^z)^2 \right\rangle - \left\langle m_K^z \right\rangle^2 \right]\)
xmzsThe longitudinal susceptibility (“staggered”).
\(\displaystyle \chi^{zz}(\vec{k}, \omega=0) = \beta N_s \left[\left\langle (\tilde{m}_K^z)^2 \right\rangle - \left\langle \tilde{m}_K^z \right\rangle^2 \right]\)
wi2The winding number.
\(\displaystyle W^2 = \sum_{d=1}^D L_d^2 \left\langle W_d^2 \right\rangle\)
rhosThe superfluid density.
\(\displaystyle \rho_s = \frac{W^2}{2DV\beta}\)
rhofThe superfluid fraction.
\(\displaystyle \frac{\rho_s}{\left\langle m^z \right\rangle}\)
compThe compressibility.
\(\displaystyle \frac{\chi^{zz}(\vec{k}=0,\omega=0)} {\left\langle \tilde{m}^z \right\rangle^2}\)
timeThe time in a Monte Carlo sweep (in seconds.)
4.6.3. Structure factor output¶
The structure factor is written into a file with the name specified by sfoutfile keyword in the input file.
The structure factor is defined as the following:
Wave vector \(\vec{k}\) and imaginary time \(\tau\) are specified by the name C<k>t<t> as the following:
R C0t0 = 1.32500000e-03 1.40929454e-04
R C0t1 = 1.32500000e-03 1.40929454e-04
R C1t0 = 7.35281032e-02 3.18028565e-04
where <k> is an index of the wave vector specified by kindex (the last element of each RK tag) in the wavevector XML file
and <t> is an index of the discretized imaginary time.
4.6.4. Real space temperature Green’s function output¶
The real space temperature Green’s function is written into a file with the name specified by cfoutfile keyword in the input file.
The real space temperature Green’s function is defined as the following:
Displacement \(\vec{r}_{ij}\) and imaginary time \(\tau\) are specified by the name C<k>t<t> as the same way of structure factor,
where <k> is an index of the displacement specified by kind (the first element of each R tag) in the relative coordinate XML file,
and <t> is an index of the discretized imaginary time.
4.6.5. Momentum space temperature Green’s function output¶
The momentum space temperature Green’s function is written into a file with the name specified by ckoutfile keyword in the input file.
The momentum space temperature Green’s function is defined as the following:
Wave vector \(\vec{r}_{ij}\) and imaginary time \(\tau\) are specified by the name C<k>t<t> as the same way of structure factor,
where <k> is an index of the displacement specified by kind (the last element of each RK tag) in the wavevector XML file,
and <t> is an index of the discretized imaginary time.
