hengyueli / fermion_table

A usefull tool for quantum cluster calculation

Fermion_Table

Representation of Manybody-State

In the quantum cluster calculation in the Fermi system, we need to 'code' the manybody-state which is a direct product of single particle states. Assume for a system that contains N site (the orbital information is included), the many body state can be represented as:

|ѱ> =|φ₁>|φ₂>...|φ_N> ⊗ |ζ₁>|ζ₂>...|ζ_N>

where |φ_i> (|ζ_i>) = 0 or 1 represent if there is one particle of spin up(down) at site i. In this way we can represent a manybody state of a N site system by 2N bits. For example, for a system with N=3, a state

|ѱ> =|101011> = |101>⊗|011> = |↑0↑>⊗|0↓↓>

represent a state that: at site 1 there is a spin-up particle, at site 2 there is a spin down particle and at site 3 there is 1 spin-up and spin-down particle respectivily. We can further represent this state by a integer:

|ѱ> =|101011> = |53> because 53 is the Decimal for of 101011(←). One should not be confused here why it is not 43. In our definition the first site is the most left one (I have used a '←' to denote that). But anyway, one who want to use this module need not to consider such kind of boring thing at all.
  So we see that we can have a one-on-one mapping relation between a integer and a (manybody) quantum state. For a system with size N, the physical states range from |0> to |4^N-1>. We represent it as {|i>,i=0,...,4^N-1}. When this representation is used, all the fermion operator(I also have this convenient code) can be represent at a action on these integers. So the Hamiltonian(H) becomes a d X d matrix , where d = 4^N.

Subspace

Basically d is a very large number so the diagonalization of H is difficult. For many cases, we can use the symmetry of H. For example, if we have [H,n] = 0, where n is the total particle operator, H will not change the particle number of the system. We can classify all the basis {|i>} into several categories.
  For example N=3 case. The basis {|i>,i=0,63} can be classified into different subspaces as:

• s₀={|0>} ,
• s₁={|1>,|2>,|4>,|8>,...},
• s₂={|3>,|5>,|6>,|10>,...},
• ...

In the binary form we can see that all the basis in s₁ contains 1 particle only and s₂ contains 2,... . In s₀, there is only one state. Because [H,n] = 0 , for all the matrix element H_ij where i and j are belong to different subspace, H_ij = 0. In other words, H can be separated into different blocks. We need only to diagonalize each block independently.

What this module do

As we have shown, we can simplify H when [H,n] = 0. In some system we can even have [H,S_z]=0 where S_z is the total spin (z component). This module create a table which then offer some method to check where a state is located in subspace and vice versa. In the module we define a variant called "symmetry":

• symmetry = 0 : do not use symetry
• symmetry = 1 : [H,n] = 0
• symmetry = 2 : [H,n] = 0 and [H,S_z]=0

By using this module, it becomes very easy to change the case(symmetry) without changing other interface. When some "bad term" which breaks down the symmetry of H is added to H, we need not to range the basis again. One need only to decrease the value of symmetry.

A simple example of usage:

program main
  use fermion_table
  implicit none

  type(table)::t
  integer::m(2)
  integer*8::psi

  !   Initialization.   This example shows a system with 3 site and  symmetry = 2
  call t%Initialization(ns = 3 ,  symmetry = 2 )



  ! show how many subspace are there
  write(*,*)"The number of subspace is:" , t%get_nsub()

  ! show some information of subspace id = 5 for example
  ! The subspace is marked by ( nup =1 , ndown = 1 )
  write(*,*)t%get_subspace_marker(subid=6)




  ! We can use (two) integer::marker(2) to mark the different subspace
  ! if symmtry = 0 , there is only one subspace. So marker is useless
  ! if symmtry = 1 , marker(1) reperent the total particle marker(2) is not used.
  ! if symmtry = 2 , the two value of marker will represent number of particle that spin up and down respectivily.
  call t%get_sub_mark_value( subid = 6 , res = m )
   write(*,*)"the marker value of subspace id = 6:",  m


   ! For a working basis corresponding to the 4-th state in subid = 6
   psi = t%get_subindex_to_basis(subid=6,index=4)
   write(*,*)"The 4-th state in subspace6 is:",psi


   ! We then try to do the inverse thing. For a given basis psi = 17, we check it id in subspace
   write(*,*)"The subspace index is:",t%get_subid_from_basis(basis=17_8)
   ! This function, in fact, is not so usefull in practice. The most import infor is the index of
   ! of this basis in the subspace, which is shown below
   write(*,*)"The order of psi=17 in the subspace is: ",t%get_basis_to_sub_index(basis=17_8)


   ! some other usefull function can be found in source code directly.
   ! This module should contains all the requirments to built a higher level package.
   ! One should not confuse about that some functions can not be found in this module.
   ! If you really think so, the function you really want, in fact, is not needed at all!


end