In [1]:

    

from sympy import *
import sympy


    

In [2]:

    

init_printing()


    

In [3]:

    

m,th,ph,kx,ky=symbols('m theta phi kx ky',positive=True)
t,t1,t2,t3,mz=symbols('t t1 t2 t3 mz',positive=True)


    

In [4]:

    

def mataf(h):
    'Array flatten a matrix list of appropriate dimensions'
    H=Matrix.hstack(*h[0])
    for sor in range(1,len(h)):
        H=Matrix.vstack(H,Matrix.hstack(*h[sor]))
    return H


    

In [5]:

    

S1=sympy.physics.matrices.msigma(1)
S2=sympy.physics.matrices.msigma(2)
S3=sympy.physics.matrices.msigma(3)
S0=S1*S1
N=0*S0
S=Matrix([[S1],[S2],[S3]])


    

In [6]:

    

a1=Matrix([[sqrt(3)],
           [1]])/2
a2=Matrix([[-sqrt(3)],
           [1]])/2
k=Matrix([[kx,ky]])


    

In [7]:

    

(k*a1)[0]


    

    
Out[7]:





$$\frac{\sqrt{3} kx}{2} + \frac{ky}{2}$$

In [13]:

    

U=mataf([[N,N,N,N,t*S0,t*S0],
         [N,N,N,t*S0,t*S0,N],
         [N,N,N,t*S0,N,t*S0],
         [N,t*S0,t*S0,N,N,N],
         [t*S0,t*S0,N,N,N,N],
         [t*S0,N,t*S0,N,N,N]])
T1=mataf([[N,N,N,N,N,N],
          [N,N,N,N,N,t*S0+I*mz*S3],
          [N,N,N,N,N,N],
          [N,N,N,N,N,N],
          [N,N,N,N,N,N],
          [N,N,N,N,N,N]])
T2=mataf([[N,N,N,t*S0+I*mz*S3,N,N],
          [N,N,N,N,N,N],
          [N,N,N,N,N,N],
          [N,N,N,N,N,N],
          [N,N,N,N,N,N],
          [N,N,N,N,N,N]])
T3=mataf([[N,N,N,N,N,N],
          [N,N,N,N,N,N],
          [N,N,N,N,N,N],
          [N,N,N,N,N,N],
          [N,N,t*S0+I*mz*S3,N,N,N],
          [N,N,N,N,N,N]])


    

In [14]:

    

H=U+exp(I*(k*a1)[0])*T1+exp(I*(k*a2)[0])*T2+exp(I*(k*(a1+a2))[0])*T3+\
   (exp(I*(k*a1)[0])*T1+exp(I*(k*a2)[0])*T2+exp(I*(k*(a1+a2))[0])*T3).H


    

In [15]:

    

HG=(((H).subs(kx,0)).subs(ky,0))
HM=(((H).subs(kx,2*pi/sqrt(3))).subs(ky,0))


    

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