J = 1.0 # spin interaction
g = 0.809 # transverse field
h = 0.9045 # parallel field
Omega = 4.5 # drive frequency
#
##### set up alternating Hamiltonians #####
# define time-reversal symmetric periodic step drive
def drive(t, Omega):
return np.sign(np.cos(Omega * t))
drive_args = [Omega]
# compute basis in the 0-total momentum and +1-parity sector
basis = spin_basis_1d(L=L, a=1, kblock=0, pblock=1)
# define PBC site-coupling lists for operators
x_field_pos = [[+g, i] for i in range(L)]
x_field_neg = [[-g, i] for i in range(L)]
z_field = [[h, i] for i in range(L)]
J_nn = [[J, i, (i + 1) % L] for i in range(L)] # PBC
# static and dynamic lists
static = [["zz", J_nn], ["z", z_field], ["x", x_field_pos]]
dynamic = [["zz", J_nn, drive, drive_args], ["z", z_field, drive, drive_args],
["x", x_field_neg, drive, drive_args]]
# compute Hamiltonians
H = 0.5 * hamiltonian(static, dynamic, dtype=np.float64, basis=basis)
#
##### set up second-order van Vleck Floquet Hamiltonian #####
# zeroth-order term
Heff_0 = 0.5 * hamiltonian(static, [], dtype=np.float64, basis=basis)
##### set up semi-classical Hamiltonian #####
# define operators
dipole_op = [[A, 0]]
# define periodic drive and its parameters
def drive(t, Omega):
return np.cos(Omega * t)
drive_args = [Omega]
# define semi-classical static and dynamic lists
static_sc = [["z", at_energy]]
dynamic_sc = [["x", dipole_op, drive, drive_args]]
# compute semi-classical basis
basis_sc = spin_basis_1d(L=1)
# compute semi-classical Hamiltonian H_{sc}(t)
H_sc = hamiltonian(static_sc, dynamic_sc, dtype=np.float64, basis=basis_sc)
#
##### define initial state #####
# define atom ground state
psi_at_i = np.array([1.0, 0.0]) # spin-down eigenstate of \sigma^z
# define photon coherent state with mean photon number Nph
psi_ph_i = coherent_state(np.sqrt(Nph), Nph_tot + 1)
# compute atom-photon initial state as a tensor product
psi_i = np.kron(psi_at_i, psi_ph_i)
#
##### calculate time evolution #####
# define time vector over 30 driving cycles with 100 points per period
t = Floquet_t_vec(Omega, 30) # t.i = initial time, t.T = driving period
# evolve atom-photon state with Hamiltonian H
##### define model parameters ##### L=10 # system size Jxy=1.0 # xy interaction Jzz_0=1.0 # zz interaction at time t=0 h_MBL=3.9 # MBL disorder strength h_ETH=0.1 # delocalised disorder strength vs=np.logspace(-2.0,0.0,num=20,base=10) # log_2-spaced vector of ramp speeds # ##### set up Heisenberg Hamiltonian with linearly varying zz-interaction ##### # define linear ramp function v = 1.0 # declare ramp speed variable def ramp(t): return (0.5 + v*t) ramp_args=[] # compute basis in the 0-total magnetisation sector (requires L even) basis = spin_basis_1d(L,Nup=L/2,pauli=False) # define operators with OBC using site-coupling lists J_zz = [[Jzz_0,i,i+1] for i in range(L-1)] # OBC J_xy = [[Jxy/2.0,i,i+1] for i in range(L-1)] # OBC # static and dynamic lists static = [["+-",J_xy],["-+",J_xy]] dynamic =[["zz",J_zz,ramp,ramp_args]] # compute the time-dependent Heisenberg Hamiltonian H_XXZ = hamiltonian(static,dynamic,basis=basis,dtype=np.float64) # ##### calculate diagonal and entanglement entropies ##### def realization(vs,H_XXZ,basis,real): """ This function computes the entropies for a single disorder realisation. --- arguments --- vs: vector of ramp speeds
from qspin.operators import hamiltonian # Hamiltonians and operators
from qspin.basis import spin_basis_1d # Hilbert space spin basis
import numpy as np # generic math functions
#
##### define model parameters #####
L = 12 # system size
Jxy = np.sqrt(2.0) # xy interaction
Jzz_0 = 1.0 # zz interaction
hz = 1.0 / np.sqrt(3.0) # z external field
#
##### set up Heisenberg Hamiltonian in an enternal z-field #####
# compute spin-1/2 basis
basis = spin_basis_1d(L, pauli=False)
basis = spin_basis_1d(L, pauli=False, Nup=L / 2) # zero magnetisation sector
basis = spin_basis_1d(L, pauli=False, Nup=L / 2,
pblock=1) # and positive parity sector
# define operators with OBC using site-coupling lists
J_zz = [[Jzz_0, i, i + 1] for i in range(L - 1)] # OBC
J_xy = [[Jxy / 2.0, i, i + 1] for i in range(L - 1)] # OBC
h_z = [[hz, i] for i in range(L - 1)]
# static and dynamic lists
static = [["+-", J_xy], ["-+", J_xy], ["zz", J_zz]]
dynamic = []
# compute the time-dependent Heisenberg Hamiltonian
H_XXZ = hamiltonian(static, dynamic, basis=basis, dtype=np.float64)
#
##### various exact diagonalisation routines #####
# calculate entire spectrum only
E = H_XXZ.eigvalsh()
# calculate full eigensystem
E, V = H_XXZ.eigh()