Difference between revisions of "Documentation:dwa"

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(→‎Feynmann perturbation in the path-integral representation)
(→‎Feynmann perturbation in the path-integral representation)
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Decompose <math> \hat{H} = \hat{H}_0 - \hat{V} </math>, where <math>\hat{H}_0 </math> is purely diagonal in the basis of choice, and <math>\hat{V} </math> being off-diagonal.
 
Decompose <math> \hat{H} = \hat{H}_0 - \hat{V} </math>, where <math>\hat{H}_0 </math> is purely diagonal in the basis of choice, and <math>\hat{V} </math> being off-diagonal.
  
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Feynmann perturbation in the path-integral representation defines the configuration weight
+
Feynmann perturbation in the path-integral representation defines the configuration weight:<br/>
 
<math>  
 
<math>  
 
Z(\mathcal{C}) = e^{-\beta\epsilon_1}  
 
Z(\mathcal{C}) = e^{-\beta\epsilon_1}  

Revision as of 20:49, 13 September 2013

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)



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 here, and the code implementation here.

The following table summarizes the basic options for the DWA simulation.

  Option     Default     Remark  
  SWEEPS   1000000   total number of Monte Carlo configuration updates (sweeps)  
  t   1.   hopping strength t  
  U   0.   onsite interaction strength U  
  mu   0.   chemical potential \mu  
  K   0.   parabolic trapping strength K  
  T   —   temperature T  

Basic options

The following table summarizes the basic options for measurements available to the user of our DWA code. When unspecified in the parameter list, they assume the default values.

  Option     Default     Remark  
  SKIP   1   number of Monte Carlo configuration updates (sweeps) per measurement  
  MEASURE   true   turns on/ off measurements
  MEASURE[Simulation Speed]   true   observe time taken per checkpoint

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_number2_   simple     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  

More options

The following table summarizes more options for measurements available to the user of our DWA code. When unspecified in the parameter list, they assume the default values.

  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_  


  Option     Default     Remark  
  tof_phase   0.   time-of-flight phase \gamma  
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