Just a fast (fast to write, not fast to run) little jordan wigner string code
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import numpy as np from numpy import kron, identity import numpy.linalg as la # sigma functions sigma_x = np.array([[0, 1],[1, 0]]) sigma_y = np.array([[0, -1j],[1j, 0]]) sigma_z = np.array([[1, 0],[0, -1]]) # standard basis spin_up = np.array([[1],[0]]) spin_down = np.array([[0],[1]]) # spin ladder operators sigma_plus = sigma_x + 1j * sigma_y sigma_minus = sigma_x - 1j * sigma_y # pauli spin N = 3 def chainify(mat, pos): if pos == 0: newmat = mat else: newmat = identity(2) for j in range(1,N): if j == pos: newmat = kron(newmat,mat) else: newmat = kron(newmat,identity(2)) return newmat def sx(i): return chainify(sigma_x,i) def sy(i): return chainify(sigma_y,i) def sz(i): return chainify(sigma_z,i) def sp(i): return chainify(sigma_plus,i) def sm(i): return chainify(sigma_minus,i) #print sz(0) #print sz(1) #print sz(2) #print np.dot(sp(0),sp(0)) # sp sm = 2 + 2 sz #print np.dot(sp(0),sm(0))- 2*identity(2**N) - 2*sz(0) I = identity(2**N) fdag = lambda i: sp(i)/2 f = lambda i: sm(i)/2 def stringify(mat, pos): if pos == 0: newmat = mat else: newmat = sigma_z for j in range(1,N): if j == pos: newmat = kron(newmat,mat) elif j<pos: newmat = kron(newmat,sigma_z) else: newmat = kron(newmat,identity(2)) return newmat def cdag(i): return np.mat(stringify(sigma_plus/2, i)) def c(i): return np.mat(stringify(sigma_minus/2, i)) #print np.dot(cdag(1),c(1)) + np.dot(c(1),cdag(1)) # This is 1 #print np.dot(cdag(1),c(2)) + np.dot(c(2),cdag(1)) # This is 0 #It does appear to work. print cdag(1)*c(1) + c(1)*cdag(1) # This is 1 print cdag(1)*c(2) + c(2)*cdag(1) # This anticommutator is 0. |
What fun!