Directed Loop Algorithm with SSE

Introduction

The dirloop_sse package provides a full generic implementation of the Quantum Monte Carlo (QMC) method called directed loop algorithm in the Stochastic Series Expansion representation. The dirloop_SSE method was invented and developped by Anders Sandvik and coworkers. It is a powerful and elegant QMC method to study quantum spin or bosonic lattice models.

The current implementation we present here uses the most recent developments of this method published in:

  • A. W. Sandvik, Phys. Rev. B 59, 14157 (1999).
  • F. Alet, S. Wessel, and M. Troyer Phys. Rev. E 71, 036706 (2005).
  • L. Pollet, S. M. A. Rombouts, K. Van Houcke, and K. Heyde, Phys. Rev. E 70, 056705 (2005).

This version allows to simulate on arbitrary lattices:

  • Quantum spin (even frustrated - see remark below) models with arbitrary spin size, magnetic field and anisotropy
  • (Softcore) bosonic models

This release allows to simulate systems with a sign problem (e.g. frustrated spin systems). However, this case was moderately tested so please be careful if your model has a sign problem …

Please note that for frustrated models, some values of the parameter Epsilon might render the algorithm non ergodic (for example, when you have a “pure loop” algorithm). One needs to check this carefully.

Running a simulation

is discussed in the tutorial.

Input parameters

In addition to the common input parameters of the ALPS applications the dirloop_sse application takes the following input parameters for experts (use only if you see what it means!):

ParameterDefaultMeaning
SKIP1the number of Monte Carlo sweeps between each measurement
RESTRICT_MEASUREMENTS[N]if defined this restricts measurements to configurations where the quantum number N (particle number) has the value given as this parameter. Note that the simulation will still be performed in the grand canonical ensemble and the chemical potential needs to be tuned to the right range, to actually sample configurations with the desired particle number.
RESTRICT_MEASUREMENTS[Sz]if defined this restricts measurements to configurations where the quantum number Sz (magnetization) has the value given as this parameter. Note that the simulation will still performed in the grand canonical ensemble and the magnetic field needs to be tuned to the right range, to actually sample configurations with the desiredmagnetization.
NUMBER_OF_WORMS_PER_SWEEPCalculated self consistentlynumber of worms done during the loop update. By default, this number is calculated self-consistently during the thermalization part. Nevertheless, you can force its value during the whole simulation
EPSILON0supplementary diagonal energy shift for all interactions. The value of EPSILON affects the performances of the algorithm with the following tradeoff : the higher it is, the longer the simulation time but the lowest are bounce probabilities. Current wisdom indicates that one should use non-zero values for Epsilon, but not too high (for example S/2 for spin S models). Please note that for frustrated models, some values of Epsilon might render the algorithm non-ergodic. You have to check carefully.
WHICH_LOOP_TYPE“minbounce”string to indicate which type of updates should be used for the scattering at the vertices : ( “heatbath” ) heatbath, see A. W. Sandvik, Phys. Rev. B 59, 14157 (1999). ( “minbounce” ) minimum bounces, see F. Alet, S. Wessel, and M. Troyer Phys. Rev. E 71, 036706 (2005). ( “locopt” ) locally optimal, see L. Pollet, S. M. A. Rombouts, K. Van Houcke, and K. Heyde, Phys. Rev. E 70, 056705 (2005). By default the algorithm uses the “minbounce” updates.
NO_WORMWEIGHT0boolean to indicate whether the worm matrixelement should be set to unity (NO_WORMWEIGHT = true) or to its real value depending on the spin/density configuration (NO_WORMWEIGHT=false). By default, NO_WORMWEIGHT is false.

Measurements

The following observables are measured by the dirloop_sse for any model application:

NameDescription
Energytotal energy of the system
Energy Densityenergy per site

The following observables are measured by the dirloop_sse for spin models, i.e. models that have an Sz quantumnumber defined:

NameDescription
Magnetizationthe z-component of the total magnetization
Magnetization Densitythe z-component of the total magnetization per site
|Magnetization|absolute value of the z-component of the magnetization
|Magnetization Density|absolute value of the z-component of the magnetization per site
Magnetization^2square of the z-component of the total magnetization
Magnetization Density^2square of the z-component of the total magnetization per site
Magnetization^4fourth power of the z-component of the total magnetization
Magnetization Density^4fourth power of the z-component of the total magnetization per site
Susceptibilitythe uniform susceptibility (spin models)

Spin models on bipartite lattices also have a staggered magnetization:

NameDescription
Staggered Magnetizationthe z-component of the staggered magnetization
Staggered Magnetization Densitythe z-component of the staggered magnetization per site
Staggered Magnetization^2square of the z-component of the staggered magnetization
Staggered Magnetization Density^2square of the z-component of the staggered magnetization per site

The following observables are measured by the dirloop_sse for particle models, i.e. models that have an N quantumnumber defined:

NameDescription
Densityparticle density
Density^2square of the particle density

And for all models

NameDescription
Stiffnessstiffness of the system (both for spin and bosonic models)

Other observables might also be available depending on the exact version of the application.

Contributors

The following persons have contributed to the dirloop_sse application:

  • Fabien Alet
  • Matthias Troyer
  • Lode Pollet