5.4. Output of DSQSS/DLA¶

5.4.1. 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.1.2.0

P D       =            1
P L       =            8
P BETA    =      10.0000000000000000
P NSET    =           10
P NMCSE   =         1000
P NMCSD   =         1000
P NMCS    =         1000
P SEED    =    198212240
P NSEGMAX =        10000
P NVERMAX =        10000
P NCYC    =            7
P ALGFILE = algorithm.xml
P LATFILE = lattice.xml
P CFINPFILE  = cf.xml
P SFINPFILE  = sf.xml
P CKINPFILE  = sf.xml
P OUTFILE    = res.dat.000
P CFOUTFILE  = cfout.dat.000
P SFOUTFILE  = sfout.dat.000
P CKOUTFILE  = ckout.dat.000
P SIMULATIONTIME   =     0.000000
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 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 amzs = -9.25000000e-04 4.02247160e-03
R bmzs = -2.03918502e-04 2.22828174e-03
R smzs = 8.72503175e-01 8.93939492e-03
R xmzs = 3.00500011e+00 2.99056535e-02
R time = 9.01378000e-08 1.61529255e-09
I [the maximum number of segments]          = 123
I [the maximum number of vertices]          = 66
I [the maximum number of reg. vertex info.] = 3

5.4.1.1. 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.

5.4.2. Main results¶

Main results are written in a file with the name specified by outfile keyword in the input parameter file.

anv

The mean number of the vertices.

\(\displaystyle \frac{\langle N_v \rangle}{N_s}\)

ene

The energy density (energy per site)

\(\displaystyle \epsilon \equiv \frac{1}{N_s}\left(E_0 - T\langle N_v\rangle\right)\)

spe

The specific heat

\(\displaystyle C_V \equiv \frac{\partial \epsilon}{\partial T}\)

len

The mean length of worm

xmx

The transverse susceptibility

amzu

The “magnetization” (uniform, \(\tau=0\)).

\(\left\langle m^z \right\rangle\) , where \(\displaystyle m^z \equiv \frac{1}{N_s} \sum_i^{N_s} M^z_i\)

bmzu

The “magnetization” (uniform, average over \(\tau\)). \(\left\langle \tilde{m}^z \right\rangle\) .

smzu

The structure factor (uniform).

\(\displaystyle S^{zz}(\vec{k}=0) \equiv \frac{1}{N_s} \sum_{i, j} e^{i \vec{k}\cdot(\vec{r}_i-\vec{r}_j)} \left[ \left\langle M^z_i M^z_j\right\rangle - \left\langle M_i^z \right\rangle \left\langle M_j^z \right\rangle \right] \Bigg|_{\vec{k}=0} = N_s \left[ \left\langle (m^z)^2\right\rangle - \left\langle m^z\right\rangle^2 \right]\)

xmzu

The longitudinal susceptibility (uniform).

\(\displaystyle \chi^{zz}(\vec{k}=0, \omega=0) \equiv \frac{\partial \left\langle \tilde{m}^z \right\rangle}{\partial h} = \beta N_s\left[ \left\langle \left(\tilde{m}^z\right)^2\right\rangle - \left\langle \tilde{m}^z\right\rangle^2 \right]\)

amzs

The “magnetization” (“staggered”, \(\tau=0\))

\(\left\langle m_s^z \right\rangle\) , where \(\displaystyle m_s^z \equiv \frac{1}{N_s} \sum_i^{N_s} M_i^z \cos\left( 2\pi \frac{\text{mtype}(i)}{N_\text{mtype}} \right)\) , \(\text{mtype}(i)\) is the kind of measurement of \(i\) site (see lattice file), and \(N_\text{mtype}\) is the number of kinds of measurements.

bmzu

The “magnetization” (“staggered”, average over \(\tau\)). \(\left\langle \tilde{m}_s^z \right\rangle\) .

smzs

The structure factor (“staggered”).

\(\displaystyle S^{zz}(\vec{k}_s) = N_s \left[ \left\langle (m_s^z)^2 \right\rangle - \left\langle m_s^z \right\rangle^2 \right]\)

xmzs

The longitudinal susceptibility (“staggered”).

\(\displaystyle \chi^{zz}(\vec{k}_s, \omega=0) = \beta N_s \left[\left\langle (\tilde{m}_s^z)^2 \right\rangle - \left\langle \tilde{m}_s^z \right\rangle^2 \right]\)

5.4.3. 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:

\[S^{zz}(\vec{k}, \tau) \equiv \left\langle M^z(\vec{k}, \tau) M^z(-\vec{k}, 0) \right\rangle - \left\langle M^z(\vec{k}, \tau)\right\rangle \left\langle M^z(-\vec{k}, 0)\right\rangle\]

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 SF tag) in the structure factor input file and <t> is an index of the discretized imaginary time.

5.4.4. 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:

\[G(\vec{r}_{ij}, \tau) \equiv \left\langle M_i^+(\tau) M_j^- \right\rangle\]

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 CF tag) in the real space temperature Green’s function input file, and <t> is an index of the discretized imaginary time.

5.4.5. 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:

\[G(\vec{k}, \tau) \equiv \left\langle M^+(\vec{k}, \tau) M^-(-\vec{k}, 0) \right\rangle\]

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 SF tag) in the momentum space temperature Green’s function input file, and <t> is an index of the discretized imaginary time.