Single-mode approximation (SMA)

1 Introduction

The single-mode approximation (SMA) can be used to analyse excitation spectrum of Quantum dimer models (QDMs).

1.1 Hamilton

H=−t∑r(|||><=|+|=><|||)+v∑r(|||><|||+|=><=|)

where the first term is the kinetic term, which describes flipping two dimers on every flippable plaquette, and the second term represents a repulsion between two parallel dimers

We focus on a special point, namely, the Rokhsar-Kivelson (RK) point (v=t=1), at which the ground state (|GS>) of the Hamilton is exactly solvable as an equal amplitude superposition of all dimer coverings (|C>).

|GS>=∑CAC|C>

Dimer covering is that each vertex in the lattice touches exactly one dimer under certain constraints.

1.2 SMA

We define a state |q>=Dα(q)|GS>, where the density operator Dα(q)=1N∑rexp(−iq.r)nα(r) and nα(r) is dimer number operator. Energy can be defined as:

ω(Q)=12<GS|[D^x(−Q),[H,D^x(Q)]]|GS>Dx(Q)

2 code

We present SMA dispersion of the square lattice using fortran language.

2.1 Define parameters

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module configuration
save
integer :: lx=64,nbins=10     !system size:lx*ly  
integer :: ly,path
integer :: nn,np      !nn:the number of bond along x-direction   np: the number of plaquette    
integer :: nb         !nb: the number of bond alond x and y direction
real(8) :: beta,qx=0,qy=0     !qx,qy:  the value of momentum space
integer, allocatable :: bond(:),bondold(:)   
integer, allocatable :: bsites(:,:)   
complex(8),parameter:: ipi2=(0,1.5707963268),ipi=2*ipi2
end module configuration
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module measurementdata
 save
 complex :: d_d=0.d0      !<GS|D(-q)D(q)|GS>
 complex :: d_hd=0.d0     !<GS|D(-q)HD(q)|GS>
 complex :: dh_d=0.d0     !<GS|D(q)HD(-q)|GS>
 real(8) :: data1(7)=0.d0 
 real(8) :: data2(7)=0.d0 
 end module measurementdata

2.2 Encode each bond of the lattice

The system is made up of np plaquettes, and the bonds of each plaquette are coded in turn

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subroutine makelattice()
 use configuration; implicit none
 integer :: s,x,y,z,i
 nn=lx*ly
 nb=2*nn
 np=nn   
 
!|-2-|
!3   4  
!|-1-| 

 allocate(bsites(4,np))
 do s=1,nn
    bsites(1,s)=s
    x=s+lx
    if (x>nn) then
        bsites(2,s)=x-nn
    else
        bsites(2,s)=x
    endif

    bsites(3,s)=s+nn
    x=s+nn

    if (mod(x,lx)==0) then
         bsites(4,s)=x-lx+1
    else
         bsites(4,s)=x+1
    endif
 enddo
 end subroutine makelattice

2.3 Initial configuration

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!Generate random numbers
real(8) function ran()
 implicit none
 real(8)    :: dmu64
 integer(8) :: ran64,mul64,add64
 common/bran64/dmu64,ran64,mul64,add64

 ran64=ran64*mul64+add64
 ran=0.5d0+dmu64*dble(ran64)

 end function ran
!---------------------!
 subroutine initran(w)
 implicit none
 integer:: myid, numprocs, namelen, rc,ierr ,comm

 integer(8) :: irmax
 integer(4) :: w
 real(8) :: rdm
 real(8)    :: dmu64
 integer(8) :: ran64,mul64,add64,ii,i
 common/bran64/dmu64,ran64,mul64,add64

 irmax=2_8**31
 irmax=2*(irmax**2-1)+1
 mul64=2862933555777941757_8
 add64=1013904243
 dmu64=0.5d0/dble(irmax)

do ii=1,w
call random_seed()
call random_number(rdm)
enddo
do ii=1,w+3
call random_number(rdm)
enddo
ran64=abs((rdm*mul64)/5+5265361)
 end subroutine initran
!-------------------------------------------------------------------!
!Select the initial configuration (dimer covering) based on random numbers
subroutine initconfig()
 use configuration; implicit none
 integer :: i,b1,b2,b3,b4,ii,next,s1,s2,bs,v0
 real(8),external :: ran
 real(8) :: rann

 allocate(bondold(nb))
 allocate(bond(nb))
 bond(:)=-1

rann=ran()
print*,rann
if (rann<0.25) then
 do i=1,nn
	if (mod((i-1)/lx,4)==0) then
	    if (mod(i-1,4)==0) then
		bond(i)=1
		bond(i+lx)=1
	           elseif (mod(i-1,4)==2) then
		bond(i+nn)=1
		bond(i+nn+1)=1
	           endif
	elseif (mod((i-1)/lx,4)==2) then
	           if (mod(i-1,4)==2) then
		bond(i)=1
		bond(i+lx)=1
	           elseif (mod(i-1,4)==0) then
		bond(i+nn)=1
		bond(i+nn+1)=1
	           endif
	endif
 enddo
elseif (rann<0.5) then
 do i=1,nn
	if (mod((i-1)/lx,4)==1) then
	    if (mod(i-1,4)==0) then
		bond(i)=1
		bond(i+lx)=1
	    elseif (mod(i-1,4)==2) then
		bond(i+nn)=1
		bond(i+nn+1)=1
	    endif
	elseif (mod((i-1)/lx,4)==3) then
	    if (mod(i-1,4)==2) then
		bond(i)=1
		ii=i+lx
		if (ii>nn) ii=ii-nn
		bond(ii)=1
	    elseif (mod(i-1,4)==0) then
		bond(i+nn)=1
		bond(i+nn+1)=1
	    endif
	endif
 enddo
elseif (rann<0.75) then
 do i=1,nn
	if (mod((i-1)/lx,4)==0) then
	    if (mod(i-1,4)==1) then
		bond(i)=1
		bond(i+lx)=1
	    elseif (mod(i-1,4)==3) then
		bond(i+nn)=1
		ii=i+nn+1
		if (mod(i,lx)==0) ii=ii-lx
		bond(ii)=1
	    endif
	elseif (mod((i-1)/lx,4)==2) then
	    if (mod(i-1,4)==3) then
		bond(i)=1
		bond(i+lx)=1
	    elseif (mod(i-1,4)==1) then
		bond(i+nn)=1
		bond(i+nn+1)=1
	    endif
	endif
 enddo
else
 do i=1,nn
	if (mod((i-1)/lx,4)==1) then
	    if (mod(i-1,4)==1) then
		bond(i)=1
		bond(i+lx)=1
	    elseif (mod(i-1,4)==3) then
		bond(i+nn)=1
		ii=i+nn+1
		if (mod(i,lx)==0) ii=ii-lx
		bond(ii)=1
	    endif
	elseif (mod((i-1)/lx,4)==3) then
	    if (mod(i-1,4)==3) then
		bond(i)=1
		ii=i+lx
		if (ii>nn) ii=ii-nn
		bond(ii)=1
	    elseif (mod(i-1,4)==1) then
		bond(i+nn)=1
		bond(i+nn+1)=1
	    endif
	endif

 enddo
endif
 end subroutine initconfig
!-------------------------------------------------------------------!
!any initial configuration can be obtained from another initial configuration by update (|| change to = or = change to ||)
subroutine update()
 use configuration; implicit none
 integer :: i,ii
 real(8),external :: ran
 logical,external :: bondcheck
 do i=1,np
	if (bondcheck(i).and.(ran()<0.5)) then
		do ii=1,4
		  	 bond(bsites(ii,i))=-bond(bsites(ii,i))
		enddo
	endif
 enddo
 end subroutine update

2.4 Calculate

calculate ω ~ k under SMA

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subroutine measureobservables() 
  use configuration; use measurementdata; implicit none
    integer :: i,w,x,y,z,ii
    complex :: dq,dqh
    logical,external :: bondcheck

      dq=0
      do i=1,nn
	x=mod(i-1,lx)
	y=(i-1)/lx
	dq=dq+exp(ipi*(qx*x+qy*y))*bond(i)
      enddo
       d_d=d_d+dq*conjg(dq)

 	do i=1,np
		if (bondcheck(i)) then
			bondold=bond
			do ii=1,4
		  		 bondold(bsites(ii,i))=-bondold(bsites(ii,i))
			enddo

			dqh=0
      			do ii=1,nn
				x=mod(ii-1,lx)
				y=(ii-1)/lx
				dqh=dqh+exp(ipi*(qx*x+qy*y))*bondold(ii)
  			enddo

			d_hd=-conjg(dqh)*dq+conjg(dq)*dq+d_hd   !<GS|D(-q)HD(q)|GS>
			dh_d=-dqh*conjg(dq)+dq*conjg(dq)+dh_d   !<GS|D(q)HD(-q)|GS>
		endif
	 enddo

 end subroutine measureobservables

2.5 main program

we can calculate ω(k) along the Brillouin zone path (0,0)→(π,0)→(π,π)→(0,0)

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program basic_heisenberg_sse
 use configuration;use measurementdata; implicit none

 integer :: i,j,msteps=1000,system,ix,iy
 character(20)::filename
  real(8) :: dqx,dqy

  beta=0
    ly=lx
     dqx=2.d0/lx
     dqy=2.d0/ly

 call initran(8)
 call makelattice()
 call initconfig()

path=0

do ix=0,lx/2   ! path 1 from (0,0) to (1,0)
     qx=dqx*ix
     qy=0

 do j=1,nbins
    do i=1,msteps
       call update()
       call measureobservables()
    enddo
    call writeresults(msteps,j)
 enddo
print*,path
path=path+1
enddo

do iy=1,ly/2   ! path 2 from (1,2/ly) to (1,1)
     qx=1
     qy=dqy*iy
 do j=1,nbins
    do i=1,msteps
       call update()
       call measureobservables()
    enddo
    call writeresults(msteps,j)
 enddo
print*,path
path=path+1
enddo

do iy=1,ly/2-1   ! path 3 from (1-2/lx,1-2/ly) to (2/lx,2/ly)
     qx=1-dqx*iy
     qy=1-dqy*iy
 do j=1,nbins
    do i=1,msteps
       call update()
       call measureobservables()
    enddo
    call writeresults(msteps,j)
 enddo
print*,path
path=path+1
enddo

 end program basic_heisenberg_sse

2.6 Save

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subroutine writeresults(msteps,bins)
 use configuration; use measurementdata; implicit none

 integer :: i,msteps,bins,j
 real(8) :: wdata1(7),wdata2(7),t,tt
 character(20)::filename

 d_d=d_d/msteps
 d_hd=d_hd/msteps
 dh_d=dh_d/msteps

 data1(1)=data1(1)+real(d_hd+dh_d)         !molecule
 data1(2)=data1(2)+real(d_d)               !denominator
 data1(3)=data1(3)+real((d_hd+dh_d)/d_d)   !$\omega(k)$=molecule/denominator

 data2(1)=data2(1)+real(d_hd+dh_d)**2
 data2(2)=data2(2)+real(d_d)**2
 data2(3)=data2(3)+real((d_hd+dh_d)/d_d)**2

if (bins==nbins) then
 do i=1,3
    wdata1(i)=data1(i)/bins
    wdata2(i)=data2(i)/bins
    wdata2(i)=sqrt(abs(wdata2(i)-wdata1(i)**2)/bins)  !error
 enddo

open(114,file='singlemode.out',position='append')
 write(114,*) path,wdata1(3),wdata2(3)               !$\omega(k)$,error
close(114)

 d_d=0.d0
 d_hd=0.d0
 dh_d=0.d0 
  data1=0
  data2=0
endif

 d_d=0.d0
 d_hd=0.d0
 dh_d=0.d0 

 end subroutine writeresults

2.7 result

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           0   2.3109267291147261E-003   1.6455131249704898E-003
           1   6.2533550739288328        2.8371740216404522E-002
           2   6.2338413238525394        1.9360616586750323E-002
           3   6.1650729179382324        3.1613608837618649E-002
           4   6.0500032424926760        1.3714428714990519E-002
           5   6.0037383079528812        1.9249298396639518E-002
           6   5.9165374279022220        2.1233933146458623E-002
           7   5.7899007320404055        2.8688790663723224E-002
           8   5.6288352012634277        1.8761953328172567E-002
           9   5.4782701969146732        2.0462955105951773E-002
          10   5.2886437892913820        1.8608940897921072E-002
          11   5.1159195423126222        2.9567157318989505E-002
          12   4.9549531936645508        3.1430773070696548E-002
          13   4.7688477993011471        2.2452612557261262E-002
          14   4.5479550838470457        3.4032958196832191E-002
          15   4.3106336593627930        2.7898038113185218E-002
          16   4.0874701976776127        1.9144380955909032E-002
          17   3.8585180997848512        1.8609396875997147E-002
          18   3.6879651308059693        2.8949933454666853E-002
          19   3.4377574205398558        1.3578068743458328E-002
          20   3.2064547300338746        2.3255134860630623E-002
          21   3.0808235883712767        1.7470583451202289E-002
          22   2.8069951295852662        2.7647364421329677E-002
          23   2.6070729732513427        3.1670710734098995E-002
          24   2.3810294151306151        2.9796689027380193E-002
          25   2.1672029972076414        1.5971279337898391E-002
          26   1.9511149644851684        2.4473548559234566E-002
          27   1.7901781082153321        2.7286673718311588E-002
          28   1.5865640401840211        3.0836123899655121E-002
          29   1.4178456902503966        8.3789057141076002E-003
          30   1.2026528716087341        2.2951941343578548E-002
          31   1.0390071272850037        3.5212021284243736E-002
          32   1.0437846541404725        5.8329403870753194E-002
          33   1.1121289670467376        3.9334604709095188E-002
          34   1.1652176260948182        3.1558839284690707E-002
          35   1.3745294094085694        2.6431431765275463E-002
          36   1.4345147013664246        2.1629957788136263E-002
          37   1.5875914335250854        2.3702880111387638E-002
          38   1.6517681121826171        1.9230563668579375E-002
          39   1.7615051031112672        1.4488433511898375E-002
          40   1.7648772358894349        1.6782384033197116E-002
          41   1.7525395989418029        1.0795956488940417E-002
          42   1.7853079319000245        1.5847920523460547E-002
          43   1.7468797206878661        7.6477434141953996E-003
          44   1.7073505759239196        8.0664708338772963E-003
          45   1.6498860597610474        1.0242478225224066E-002
          46   1.5684979915618897        8.8734704527437969E-003
          47   1.4898424744606018        5.6957569787166405E-003
          48   1.3665797352790832        7.4285902427473982E-003
          49   1.2612158417701722        8.3717839929171971E-003
          50   1.1421930313110351        6.8612852659099147E-003
          51   1.0288167774677277        7.4561234512739369E-003
          52  0.91099560260772705        5.7387275971854494E-003
          53  0.78420571088790891        4.1084180922130774E-003
          54  0.66706750988960262        4.9506690982011343E-003
          55  0.55218040943145752        5.5888892459151265E-003
          56  0.43570946753025053        3.7497197571224258E-003
          57  0.34641086459159853        2.3768882259739979E-003
          58  0.25858314335346222        3.4487059424092325E-003
          59  0.18344641476869583        1.6080926161550495E-003
          60  0.11753480210900306        1.8062113973221030E-003
          61   6.6618058830499649E-002   5.4123314049076877E-004
          62   3.0812994204461576E-002   7.8286098196079113E-004
          63   7.0903261657804251E-003   5.8438091546916360E-004
          64   0.0000000000000000        0.0000000000000000     
          65   1.5889596939086915E-002   5.2551528107296812E-004
          66   6.1861097440123559E-002   1.4868592619685179E-003
          67  0.13374220281839372        1.5077039254814428E-003
!!! do not forget 1/2 (SMA)

References

[1] Läuchli, Andreas M., Sylvain Capponi, and Fakher F. Assaad. “Dynamical dimer correlations at bipartite and non-bipartite Rokhsar–Kivelson points.” Journal of Statistical Mechanics: Theory and Experiment 2008.01 (2008): P01010.

[2] Yan, Zheng, et al. “Widely existing mixed phase structure of the quantum dimer model on a square lattice.” Physical Review B 103.9 (2021): 094421.