Documentation:dwa

Directed worm algorithm

Theory

Quantum statistical mechanics at finite temperature

At finite temperature T, the physics is essentially captured by the partition function

$Z = \mathrm{Tr} \, \exp \left(-\beta \hat{H} \right)$

and physical quantities such as the local density

$\langle n_i \rangle = \frac{1}{Z} \, \mathrm{Tr} \, \hat{n}_i \exp \left(-\beta \hat{H} \right) = \frac{1}{Z} \sum_{\mathcal{C}} n_i (\mathcal{C}) Z(\mathcal{C})$

for some configuration $\mathcal{C}$ in the complete configuration space, with inverse temperature $\beta = 1/T$. Here, the units will be cleverly normalized later on.

Feynmann perturbation in the path-integral representation

Decompose $\hat{H} = \hat{H}_0 - \hat{V}$, where $\hat{H}_0$ is purely diagonal in the basis of choice, and $\hat{V}$ being off-diagonal.

Feynmann perturbation in the path-integral representation defines the configuration weight:

$Z(\mathcal{C}) = e^{-\beta\epsilon_1} \left( e^{-\tau_1 \epsilon_1} V_{i_1i_2} e^{\tau_1 \epsilon_2} \right) \cdots \left( e^{-\tau_m \epsilon_m} V_{i_mi_1} e^{\tau_m \epsilon_1} \right)$

for configuration

$\mathcal{C} = \left\{ m ; i_1 \cdots i_m ; \tau_1 \cdots \tau_m \, | \, m \in \mathbf{N} ; 0 < \tau_1 < \cdots < \tau_m < \beta \right\}$

where

$\epsilon_i = <i| \hat{H}_0 |i>$

$V_{ij} = <i| \hat{V} |j>$

The derivation can be found in chapter 2.1-2.2 of my thesis.

Quantum Monte Carlo (Directed Worm Algorithm)

The Quantum Monte Carlo simulation is in fact a Markov chain random walk in the (worldlines) configuration space, importance sampled by the configuration weight $Z(\mathcal{C})$ which is just a positive number assigned to some particular configuration $\mathcal{C}$ for instance shown here. How $Z(\mathcal{C})$ is being assigned depends on the model Hamiltonian as well as the ergodic algorithm that satisfies detailed balance.

For the directed worm algorithm, the configuration is updated with the worm transversing to and from the extended configuration space to ensure ergodicty. In addition, $n_i(\mathcal{C})$ is the number of particles (or state) at site i with time 0.

Each configuration update is known as a Monte Carlo sweep.

The complete step-by-step description of the directed worm algorithm can be found in chapter 2.3 of my thesis, and the code implementation in the following.

The dwa code: options

Monte Carlo options

Option	Default	Remark
THERMALIZATION	0	1) number of Monte Carlo configuration updates (sweeps) needed for thermalization     2) no measurements are performed in the thermalization stage
SWEEPS	1000000	total number of Monte Carlo configuration updates (sweeps) after thermalization
SKIP	1.	number of Monte Carlo configuration updates (sweeps) per measurement $t$

ALPS lattice library options

Option	Default	Remark
LATTICE	--	which lattice do you want?
L	--	length of lattice

A first quick guide to the ALPS Lattice library can be found here.

Boson Hubbard model options

Option	Default	Remark
MODEL	--	set as "boson Hubbard"
Nmax	--	maximum number of bosons allowed per site
t	1.	hopping strength $t$
U	0.	onsite interaction strength $U$
mu	0.	chemical potential $\mu$

Note:
The following definitions for mu are allowed:

• mu=0.5
• mu="0.5 - 0.001*((x-(L-1)/2.)*(x-(L-1)/2.) + (y-(L-1)/2.)*(y-(L-1)/2.) + (z-(L-1)/2.)*(z-(L-1)/2.)"

Other options

Option	Default	Remark
T	0.	temperature $T$
tof_phase	0.	time-of-flight phase $\gamma$
MEASURE	true	shall we measure the common observables?
MEASURE[Simulation Speed]	true	shall we measure the simulation performance?

More measurement options

Option	Default	Boolean control
MEASURE[Total Particle Number^2]	false	measure_number2_
MEASURE[Energy^2]	false	measure_energy2_
MEASURE[Density^2]	false	measure_density2_
MEASURE[Energy Density^2]	false	measure_energy_density2_
MEASURE[Local Kink: Number]	false	measure_local_num_kinks_
MEASURE[Winding Number]	false	measure_winding_number_
MEASURE[Local Density]	false	measure_local_density_
MEASURE[Local Density^2]	false	measure_local_density2_
MEASURE[Green Function]	false	measure_green_function_

The dwa code: starting the simulation

in command line

An example is found here.

in python

An example is found here.

The dwa code: output

List of measurement observables

When the measurement mode is turned on, the following is a list of common observables available to the user.

Observable		Boolean control	Binning analysis	Remark
Total Particle Number	$\langle N \rangle$	—	detailed	measure always
Energy	$\langle E \rangle$	—	detailed	measure always
Energy:Vertex	$\langle E_v \rangle$	—	detailed	measure always
Energy:Onsite	$\langle E_o \rangle$	—	detailed	measure always
Density	$\langle n \rangle$	—	detailed	measure if lattice is homoogeneous
Energy Density	$\langle \epsilon \rangle$	—	detailed	measure if lattice is homoogeneous
Energy Density:Vertex	$\langle E_v \rangle$	—	detailed	measure if lattice is homoogeneous
Energy Density:Onsite	$\langle E_o \rangle$	—	detailed	measure if lattice is homoogeneous
Total Particle Number^2	$\langle N^2 \rangle$	measure_number2_	detailed	—
Energy^2	$\langle E^2 \rangle$	measure_energy2_	detailed	—
Density^2	$\langle N^2 \rangle$	measure_density2_	detailed	measure if lattice is homogeneous
Energy Density^2	$\langle E^2 \rangle$	measure_energy_density2_	detailed	measure if lattice is homogeneous
Winding Number^2	$\langle W_\alpha^2 \rangle$	measure_winding_number_	detailed	measure if lattice is periodic: $\alpha=x,y,z$
Stiffness (superfluid density)	$\langle \rho_s \rangle$	measure_winding_number_	detailed	measure if lattice is periodic: $\alpha=x,y,z$
Local Kink:Number	$\langle n_i^r \rangle$	measure_local_num_kinks_	simple	—
Local Density	$\langle n_i \rangle$	measure_local_density_	simple	—
Local Density^2	$\langle n_i^2 \rangle$	measure_local_density2_	simple	—
Green Function:0	$g_f \left(\alpha=0\right)$	—	detailed	measure always
Green Function:1	$\sum_{i=x,y,z} g_f \left(\alpha_i =1\right)$	—	detailed	measure always
Green Function	$g_f \left(\alpha ; \gamma = 0 \right)$	measure_green_function_	simple	—
Green Function:TOF	$g_f \left(\alpha \right)$	measure_green_function_	simple	measure if tof_phase != 0
Momentum Distribution:0	$\langle n_k \left( 0 ; \gamma = 0 \right) \rangle$	—	detailed	measure if tof_phase == 0
Momentum Distribution:TOF:0	$\langle n_k \left( 0 \right) \rangle$	—	detailed	measure if tof_phase != 0

Evaluating the simulation in Python

An example is found here.

Extracting and visualizing the worldlines configuration in Python

Illustrating from this example, we want to, for instance, extract the worldlines configuration of task 30 after the first run.

import pyalps.dwa;
wl = pyalps.dwa.extract_worldlines('parm1b.task30.out.run1.h5');

We can easily visualise, for instance, the cross-sectional worldlines configuration of this 8x8 lattice at y=4:

pyalps.dwa.show_worldlines(wl, reshape = [8,8], at = '[:,4]'); 

© 2013 by Matthias Troyer, Ping Nang Ma.