投影对角化自旋激发

def Hamiltonian_H0_SU4(k,N,t=-1):
    """
    t 是最近邻hopping系数
    k 是Bloch波矢量
    N 是宽度
    """
    A = np.matrix([[sqrt(3),0,1,np.exp(-1j*k)],[0, sqrt(3),1,1],[1,1,sqrt(3),0],[np.exp(1j*k),1,0,sqrt(3)]])
    B = np.matrix([[0,0,0,0],[0,0,0,0],[1,0,0,0],[0,-1,0,0]])
    H = tridiag(A,B,B.H,N)
    H[0,0] = 1000
    return H

对角化得到本征值和本征向量

nk = 256
N = 512
k = np.linspace(0,2*pi,nk)
band = np.zeros((4*N, nk))
Ak = np.zeros((4*N,nk),dtype="complex")
for i in range(nk):
    Hk0 = Hamiltonian_H0_SU4(k[i],N)
    E, A = LA.eigh(Hk0)
    band[:,i] = E
    Ak[:,i] = A[:,N-1]

画能带

for i in range(4*N):
    plt.plot(k,band[i,:])
plt.ylim((-1,1))

测试本征态

plt.plot(Ak[:,1].real)

在基态基础上构造有效投影自旋激发哈密顿量

def Hamiltonian_Heff(nk,iq,Ak,U,N):
    H = np.zeros((nk,nk),dtype='complex')
    for j in range(nk):
        j_q = (j - iq + nk)%nk
        for i in range(j,nk):
            i_q = (i-iq + nk)%nk
            H[j,i] = -np.sum(Ak[:,i_q].conj()*Ak[:,i]*Ak[:,j_q]*Ak[:,j].conj())
    
    H = H + H.T
    
    for i in range(nk):
        i_q = (i - iq + nk)%nk
        H[i,i] = np.sum(np.power(np.absolute(Ak),2).sum(axis=1)*np.power(np.absolute(Ak[:,i_q]),2))
    
    return U*H/N

对角化并画能带图

for i in range(nk):
    plt.plot(k,band_mag[i,:])
#plt.xlim((0,2*pi/3))
plt.ylim((0,0.1))

计算谱函数，用到一个矩阵循环移位技巧，实现函数如下：

def circshift(matrix,shiftnum1,shiftnum2):
    """
    """
    h,w=matrix.shape
    matrix=np.vstack((matrix[(h-shiftnum1):,:],matrix[:(h-shiftnum1),:]))
    matrix=np.hstack((matrix[:,(w-shiftnum2):],matrix[:,:(w-shiftnum2)]))
    return matrix

测试：

a=np.arange(1,17).reshape(4,4)
print(a)
b= circshift(a,0,2)
print(b)

delta = 0.002
nw = 256
w = np.linspace(0,0.1,nw)
Spectral_A = np.zeros((nw,nk),dtype='complex')
for i in range(nk):
    A_kq = circshift(Ak,0,i)
    S_plus = np.sum(Ak.conj()*A_kq,axis=0).reshape(nk,1)
    for j in range(nw):
        for p in range(nk):
            Spectral_A[j,i] += np.absolute(np.dot(S_plus.conj().T,psi[:,p,i]))**2/(w[j]-band_mag[p,i]+1j*delta)

画谱函数

X,Y = np.meshgrid(k,w)
plt.figure(figsize=(5,5))
plt.pcolormesh(X, Y, -Spectral_A.imag/pi)
plt.colorbar()

投影对角化自旋激发

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