-
Notifications
You must be signed in to change notification settings - Fork 0
/
operators.py
285 lines (231 loc) · 7.81 KB
/
operators.py
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
import numpy as np
from scipy.sparse import kron as skron
from scipy.sparse import coo_matrix as coo
from scipy.sparse import csr_matrix as csr
TOL = 1e-10
class quantum_operator:
""" Matrix with symmetries
used to generate the full Hamiltonian of interaction systems """
# initialize as an empty operator
def __init__(self, sym=2, dim=1, datatype=np.float, fparity=None, sym_sum=None):
def sym_sum_n(a, b):
return (a+b+sym)%sym
self.sym = sym # number of total symmetry sectors
self.dim = dim # dimension of each symmetry sector
self.datatype = datatype
if fparity is None: self.fparity = np.zeros(sym, dtype=np.int) # fermion parity
else: self.fparity = fparity
if sym_sum is None: self.sym_sum = sym_sum_n # the rule to sum the symmetries
else: self.sym_sum = sym_sum
self.val = [[csr((dim,dim), dtype=datatype) for x in range(sym)] for y in range(sym)]
self.basis = None
self.L = 1
def _empty_(self, row_sym, col_sym):
x =np.sum( np.abs(self.val[row_sym][col_sym].data) )
if x<TOL: return True
else: return False
def identity(self):
id = quantum_operator(self.sym, self.dim, self.datatype, self.fparity, self.sym_sum)
for i in range(self.sym):
id.val[i][i] = csr( np.diag(np.ones(self.dim, dtype=self.datatype)) )
return id
def copy(self):
sym = self.sym
A = quantum_operator(self.sym, self.dim, self.datatype, self.fparity, self.sym_sum)
for row in range(sym):
for col in range(sym):
if not self._empty_(row, col):
A.val[row][col] = self.val[row][col].copy()
A.basis = self.basis
A.L = self.L
return A
def kron(self, B):
fp = self.fparity # def fp just for convenience
ss = self.sym_sum # def ss just for convenience
if self.sym!=B.sym:
raise Exception('A, B symmetries do not match.')
sym = B.sym
dim = sym * self.dim * B.dim
if not np.array_equal(self.fparity, B.fparity):
raise Exception('A, B fermion types do not match.')
datatype = self.datatype
if np.dtype(datatype) < np.dtype(B.datatype): datatype = B.datatype
ret = quantum_operator(sym, dim, datatype, fp, ss)
for row in range(sym):
for col in range(sym):
for row1 in range(sym):
for col1 in range(sym):
row2 = ss(row, -row1)
col2 = ss(col, -col1)
if not self._empty_(row1, col1) and not B._empty_(row2, col2):
sign = 1 - 2*( fp[col1] * fp[ss(col2, -row2)] )
block = self.dim * B.dim
temp = coo( skron(self.val[row1][col1], B.val[row2][col2]) * sign )
add = coo((temp.data, (temp.row + row1*block, temp.col + col1*block)), (dim, dim), dtype=datatype)
ret.val[row][col] += add
return ret
def plus(self, B):
if self.sym!=B.sym:
raise Exception('A, B symmetries do not match.')
sym = B.sym
if self.dim!=B.dim:
raise Exception('A, B dimensions do not match.')
if not np.array_equal(self.fparity, B.fparity):
raise Exception('A, B fermion types do not match.')
if np.dtype(self.datatype) < np.dtype(B.datatype): self.astype(B.datatype)
for row in range(sym):
for col in range(sym):
self.val[row][col] += B.val[row][col]
return self
def times(self, x):
if np.dtype(self.datatype) < np.dtype(type(x)): self.astype(type(x))
for row in range(self.sym):
for col in range(self.sym):
self.val[row][col] = self.val[row][col] * x
return self
def mult(self, B):
if self.sym!=B.sym:
raise Exception('A, B symmetries do not match.')
sym = B.sym
if self.dim!=B.dim:
raise Exception('A, B dimensions do not match.')
dim = B.dim
if not np.array_equal(self.fparity, B.fparity):
raise Exception('A, B fermion types do not match.')
datatype = self.datatype
if np.dtype(datatype) < np.dtype(B.datatype): datatype = B.datatype
ret = quantum_operator(sym, dim, datatype, B.fparity, B.sym_sum)
for row in range(sym):
for col in range(sym):
for k in range(sym):
ret.val[row][col] += self.val[row][k].dot( B.val[k][col] )
return ret
def transpose(self):
ret = quantum_operator(self.sym, self.dim, self.datatype, self.fparity, self.sym_sum)
sym = self.sym
for row in range(sym):
for col in range(sym):
if not self._empty_(row, col):
ret.val[col][row] = self.val[row][col].T
return ret
def hermitian(self):
ret = quantum_operator(self.sym, self.dim, self.datatype, self.fparity, self.sym_sum)
sym = self.sym
for row in range(sym):
for col in range(sym):
if not self._empty_(row, col):
ret.val[col][row] = self.val[row][col].T.conjugate()
return ret
def todense(self):
sym = self.sym
dim = self.dim
ret = np.zeros((sym*dim, sym*dim), dtype=self.datatype)
for row in range(sym):
for col in range(sym):
if not self._empty_(row, col):
ret[row*dim:(row+1)*dim, col*dim:(col+1)*dim] = self.val[row][col].todense()
return ret
def astype(self, dtype):
self.datatype = dtype
sym = self.sym
for row in range(sym):
for col in range(sym):
if not self._empty_(row, col):
self.val[row][col] = self.val[row][col].astype(dtype)
return self
def basis(n):
""" # the basis array has the form basis[sector, state, L] """
return np.arange(n, dtype=int).reshape(n,1,1)
def spinful_fermion_basis():
return np.array([ [[0],[3]], [[1],[2]] ], dtype=int)
def sigma_x():
Sx = quantum_operator(2, 1)
Sx.val[1][0] = csr( np.array([[1.]]) )
Sx.val[0][1] = csr( np.array([[1.]]) )
Sx.basis = basis(2)
return Sx
def sigma_y():
Sy = quantum_operator(2, 1, datatype=np.complex)
Sy.val[0][1] = csr( np.array([[-1.j]]) )
Sy.val[1][0] = csr( np.array([[1.j]]) )
Sy.basis = basis(2)
return Sy
def sigma_z():
Sz = quantum_operator(2, 1)
Sz.val[0][0] = csr( np.array([[1.]]) )
Sz.val[1][1] = csr( np.array([[-1.]]) )
Sz.basis = basis(2)
return Sz
def tau_n(n=3):
w = np.cos(2*np.pi/n) + 1.j * np.sin(2*np.pi/n)
tau = quantum_operator(n, 1, datatype=np.complex)
for i in range(n):
tau.val[i][i] = csr( np.array([[w**i]]) )
tau.basis = basis(n)
return tau
def sigma_n(n=3):
sigma = quantum_operator(n, 1)
for i in range(n):
sigma.val[i][(i+n+1)%n] = csr( np.array([[1.]]) )
sigma.basis = basis(n)
return sigma
def spin1_x():
Sx = quantum_operator(3, 1)
Sx.val[0][1] = csr( np.array([[1.]]) )
Sx.val[1][0] = csr( np.array([[1.]]) )
Sx.val[1][2] = csr( np.array([[1.]]) )
Sx.val[2][1] = csr( np.array([[1.]]) )
Sx.times(1./np.sqrt(2))
Sx.basis = basis(3)
return Sx
def spin1_y():
Sy = quantum_operator(3, 1)
Sy.val[0][1] = csr( np.array([[-1.]]) )
Sy.val[1][0] = csr( np.array([[1.]]) )
Sy.val[1][2] = csr( np.array([[-1.]]) )
Sy.val[2][1] = csr( np.array([[1.]]) )
Sy.times(1.j/np.sqrt(2))
Sy.basis = basis(3)
return Sy
def spin1_z():
Sz = quantum_operator(3, 1)
Sz.val[0][0] = csr( np.array([[1.]]) )
Sz.val[2][2] = csr( np.array([[-1.]]) )
Sz.basis = basis(3)
return Sz
def fermion_c():
f = quantum_operator(2, 1, fparity=np.array([0,1], dtype=np.int))
f.val[0][1] = csr( np.array([[1.]]) )
f.basis = basis(2)
return f
def fermion_up():
f = quantum_operator(2, 2, fparity=np.array([0,1], dtype=np.int))
f.val[0][1] = csr( np.array([[1.,0.],[0.,0.]]) )
f.val[1][0] = csr( np.array([[0.,0.],[0.,1.]]) )
f.basis = spinful_fermion_basis()
return f
def fermion_down():
f = quantum_operator(2, 2, fparity=np.array([0,1], dtype=np.int))
f.val[0][1] = csr( np.array([[0.,1.],[0.,0.]]) )
f.val[1][0] = csr( np.array([[0.,-1.],[0.,0.]]) )
f.basis = spinful_fermion_basis()
return f
def sym_sum4(a, b):
pp = np.array([ [0,1,2,3],
[1,0,3,2],
[2,3,0,1],
[3,2,1,0]])
if b<0: return pp[a,-b]
return pp[a,b]
def fermion_up4():
f = quantum_operator(4, 1, fparity=np.array([0,1,1,0], dtype=np.int), sym_sum=sym_sum4)
f.val[0][1] = csr( np.array([[1.]]) )
f.val[2][3] = csr( np.array([[1.]]) )
f.basis = basis(4)
return f
def fermion_down4():
f = quantum_operator(4, 1, fparity=np.array([0,1,1,0], dtype=np.int), sym_sum=sym_sum4)
f.val[0][2] = csr( np.array([[1.]]) )
f.val[1][3] = csr( np.array([[-1.]]) )
f.basis = basis(4)
return f