PT-symmetry¶

These examples illustrate the PT-symmetry and the PT-symmetry breaking.

Note

Throughout the examples, the hoppings $t_{ab}=t_a$ and $t_{ba}=t_b$ are real and the onsite energies are purely imaginary with $\epsilon_a=i\gamma$ and $\epsilon_b=-i\gamma$ . Then, the A sites are amplified (positive imaginary part) and the B sites are absorbed (negative imaginary part).

Dimer¶

The Hamiltian reads

$\left( \begin{array}{cc} i\gamma & t \\ t & i\gamma \end{array} \right)$

Hermitian case¶

In the case $\gamma=0$ , the Hamiltonian is real symmetric, the eigenenergies are real and given by: $E_\pm=\pm t$ .

from chainTB import *
# lattice
tags = [b'a', b'b']
ri = [[0, 0], [1, 0]]
chain_tb = latticeTB(tags=tags, ri=ri, nor=2, ang=0.)
chain_tb.get_lattice(nx=1, ny=1)
# solve eigenvalue problem
eig_chain = eigChain(lat=chain_tb)
t = 1.
eig_chain.set_hop(ho=[t])
eig_chain.get_ham()
eig_chain.get_eig(eigenvec=True)
# plots
plot_chain = plotTB(sys=eig_chain)
lattice  = plot_chain.plt_lattice(plt_index=True, ms=50, fs=20)
fig_spec = plot_chain.plt_spec(ms=20, pola_tag=b'a')
#propagation
prop_chain = propTB(lat=chain_tb, steps=1400, dz=0.02)
psi_init =  [1, 0]
prop_chain.get_prop(ham=eig_chain.ham, psi_init=psi_init, norm=True)
fig_prop = prop_chain.plt_prop1d()
fig_prop_dimer = prop_chain.plt_prop_dimer()
plt.show()
# save figures
save_chain = saveFigTB(sys=eig_chain, params={'t': t}, dir_name='chain', ext='png')
save_chain.save_fig(fig_spec, 'spec')
save_chain.save_fig(fig_prop_dimer, 'prop_dimer')
save_chain.save_fig(fig_prop, 'prop')

_images/prop_dimer_t(1+0j)_ea0j_eb0j.png

Non-Hermitian case¶

In the case $\gamma\neq0$ , the Hamiltonian is complex non-Hermitian, and display the space-time reflection symmetry. The eigenenergies are given by: $E_\pm=\pm \sqrt{t^2-|\gamma|^2}$ . As the result, the energies are real if $t>\gamma$ , degenerated if $t=\gamma$ (PT-symmetric phase), and complex if $t<\gamma$ . $\gamma$ (broken PT-symmetry). $t=\gamma$ . $\gamma$ is the PT-symmetry breaking point.

Note

This example shows that a non-Hermitian matrix can have a real spectrum.

PT-symmetry¶

Below the PT-symmetry breaking point, the eigenenergies are real: $E_\pm=\pm \sqrt{t^2-|\gamma|^2}$ .

from chainTB import *
# lattice
tags = [b'a', b'b']
ri = [[0, 0], [1, 0]]
chain_tb = latticeTB(tags=tags, ri=ri, nor=2, ang=0.)
chain_tb.get_lattice(nx=1, ny=1)
# solve eigenvalue problem
eig_chain = eigChain(lat=chain_tb)
t = 1.
on = [.8j, -.8j]
eig_chain.set_hop(ho=[t])
eig_chain.set_onsite(on=on)
eig_chain.get_ham()
eig_chain.get_eig(eigenvec=True)
# plots
plot_chain = plotTB(sys=eig_chain)
lattice  = plot_chain.plt_lattice(plt_index=True, ms=50, fs=20)
fig_spec = plot_chain.plt_spec(ms=20, pola_tag=b'a')
#propagation
prop_chain = propTB(lat=chain_tb, steps=1400, dz=0.02)
psi_init =  [1, 0]
prop_chain.get_prop(ham=eig_chain.ham, psi_init=psi_init, norm=True)
fig_prop = prop_chain.plt_prop1d()
fig_prop_dimer = prop_chain.plt_prop_dimer()
plt.show()
# save figures
save_chain = saveFigTB(sys=eig_chain, params={'t': t}, dir_name='chain', ext='png')
save_chain.save_fig(fig_spec, 'spec')
save_chain.save_fig(fig_prop_dimer, 'prop_dimer')
save_chain.save_fig(fig_prop, 'prop')

_images/prop_dimer_ea0,8j_eb-0,8j_t(1+0j).png

breaking point¶

At the breaking point, $\gamma=t$ , the eigenenergies coincide: $E_\pm=0$ .

from chainTB import *
# lattice
tags = [b'a', b'b']
ri = [[0, 0], [1, 0]]
chain_tb = latticeTB(tags=tags, ri=ri, nor=2, ang=0.)
chain_tb.get_lattice(nx=1, ny=1)
# solve eigenvalue problem
eig_chain = eigChain(lat=chain_tb)
t = 1.
on = [1j, -1j]
eig_chain.set_hop(ho=[t])
eig_chain.set_onsite(on=on)
eig_chain.get_ham()
eig_chain.get_eig(eigenvec=True)
# plots
plot_chain = plotTB(sys=eig_chain)
lattice  = plot_chain.plt_lattice(plt_index=True, ms=50, fs=20)
fig_spec = plot_chain.plt_spec(ms=20, pola_tag=b'a')
#propagation
prop_chain = propTB(lat=chain_tb, steps=1400, dz=0.02)
psi_init =  [1, 0]
prop_chain.get_prop(ham=eig_chain.ham, psi_init=psi_init, norm=True)
fig_prop = prop_chain.plt_prop1d()
fig_prop_dimer = prop_chain.plt_prop_dimer()
plt.show()
# save figures
save_chain = saveFigTB(sys=eig_chain, params={'t': t}, dir_name='chain', ext='png')
save_chain.save_fig(fig_spec, 'spec')
save_chain.save_fig(fig_prop_dimer, 'prop_dimer')
save_chain.save_fig(fig_prop, 'prop')

_images/prop_dimer_ea1j_eb-1j_t(1+0j).png

broken PT-symmetry¶

Above the PT-symmetry breaking point, $\gamma>t$ , the eigenenergies are complex: $E_\pm= \pm i \sqrt{\gamma^2-t^2}$ .

from chainTB import *
# lattice
tags = [b'a', b'b']
ri = [[0, 0], [1, 0]]
chain_tb = latticeTB(tags=tags, ri=ri, nor=2, ang=0.)
chain_tb.get_lattice(nx=1, ny=1)
# solve eigenvalue problem
eig_chain = eigChain(lat=chain_tb)
t = 1.
on = [1.2j, -1.2j]
eig_chain.set_hop(ho=[t])
eig_chain.set_onsite(on=on)
eig_chain.get_ham()
eig_chain.get_eig(eigenvec=True)
# plots
plot_chain = plotTB(sys=eig_chain)
lattice  = plot_chain.plt_lattice(plt_index=True, ms=50, fs=20)
fig_spec = plot_chain.plt_spec(ms=20, pola_tag=b'a')
#propagation
prop_chain = propTB(lat=chain_tb, steps=1400, dz=0.02)
psi_init =  [1, 0]
prop_chain.get_prop(ham=eig_chain.ham, psi_init=psi_init, norm=True)
fig_prop0 = prop_chain.plt_prop1d()
fig_prop_dimer0 = prop_chain.plt_prop_dimer()
psi_init =  [0, 1]
prop_chain.get_prop(ham=eig_chain.ham, psi_init=psi_init, norm=True)
fig_prop1 = prop_chain.plt_prop1d()
fig_prop_dimer1 = prop_chain.plt_prop_dimer()
plt.show()
# save figures
save_chain = saveFigTB(sys=eig_chain, params={'t': t}, dir_name='chain', ext='png')
save_chain.save_fig(fig_spec, 'spec')
save_chain.save_fig(fig_prop_dimer0, 'prop_dimer0')
save_chain.save_fig(fig_prop0, 'prop0')
save_chain.save_fig(fig_prop_dimer1, 'prop_dimer1')
save_chain.save_fig(fig_prop1, 'prop1')

_images/prop_dimer0_ea1,2j_eb-1,2j_t(1+0j).png

_images/prop_dimer1_ea1,2j_eb-1,2j_t(1+0j).png

_images/prop0_ea1,2j_eb-1,2j_t(1+0j).png

_images/prop1_ea1,2j_eb-1,2j_t(1+0j).png

Dimer chain¶

The chain is composed of $n$ dimers i.e. $n$ sites A and $n$ sites B. The chain starts with a A site and ends with a B site.

Hermitian case¶

from chainTB import *
# lattice
tags = [b'a', b'b']
ri = [[0, 0], [1, 0]]
nx = 10
chain_tb = latticeTB(tags=tags, ri=ri, nor=2, ang=0.)
chain_tb.get_lattice(nx=nx, ny=1)
# solve eigenvalue problem
eig_chain = eigChain(lat=chain_tb)
t = 1.
eig_chain.set_hop(ho=[t])
eig_chain.get_ham()
eig_chain.get_eig(eigenvec=True)
# plots
plot_chain = plotTB(sys=eig_chain)
fig_lat  = plot_chain.plt_lattice(plt_index=True, ms=15, fs=20, figsize=(10, 2))
fig_spec = plot_chain.plt_spec(ms=20, pola_tag=b'a')
#propagation
prop_chain = propTB(lat=chain_tb, steps=1400, dz=0.05)
psi_init =  np.ones(eig_chain.sites) / np.sqrt(eig_chain.sites)
prop_chain.get_prop(ham=eig_chain.ham, psi_init=psi_init, norm=True)
fig_prop = prop_chain.plt_prop1d()
plt.show()
# save figures
save_chain = saveFigTB(sys=eig_chain, params={'t': t}, dir_name='chain', ext='png')
save_chain.save_fig_lat(fig_lat, 'lattice')
save_chain.save_fig(fig_spec, 'spec')
save_chain.save_fig(fig_prop, 'prop')

Non-Hermitian case¶

PT-symmetry¶

from chainTB import *
# lattice
tags = [b'a', b'b']
ri = [[0, 0], [1, 0]]
nx = 10
chain_tb = latticeTB(tags=tags, ri=ri, nor=2, ang=0.)
chain_tb.get_lattice(nx=nx, ny=1)
# solve eigenvalue problem
eig_chain = eigChain(lat=chain_tb)
t = 1.
on = [.1j, -.1j]
eig_chain.set_hop(ho=[t])
eig_chain.set_onsite(on=on)
eig_chain.get_ham()
eig_chain.get_eig(eigenvec=True)
# plots
plot_chain = plotTB(sys=eig_chain)
fig_lat  = plot_chain.plt_lattice(plt_index=True, ms=15, fs=20)
fig_spec = plot_chain.plt_spec(ms=20, pola_tag=b'a')
fig_stateA = plot_chain.plt_intensity1d(stateA, ms=20)
fig_stateB = plot_chain.plt_intensity1d(stateB, ms=20)
#propagation
prop_chain = propTB(lat=chain_tb, steps=1400, dz=0.05)
psi_init =  np.ones(eig_chain.sites) / np.sqrt(eig_chain.sites)
prop_chain.get_prop(ham=eig_chain.ham, psi_init=psi_init, norm=True)
fig_prop = prop_chain.plt_prop1d()
plt.show()
# save figures
save_chain = saveFigTB(sys=eig_chain, params={'t': t}, dir_name='chain', ext='png')
save_chain.save_fig(fig_spec, 'spec')
save_chain.save_fig(fig_prop, 'prop')

broken PT-symmetry¶

from chainTB import *
# lattice
tags = [b'a', b'b']
ri = [[0, 0], [1, 0]]
nx = 10
chain_tb = latticeTB(tags=tags, ri=ri, nor=2, ang=0.)
chain_tb.get_lattice(nx=nx, ny=1)
# solve eigenvalue problem
eig_chain = eigChain(lat=chain_tb)
t = 1.
on = [.2j, -.2j]
eig_chain.set_hop(ho=[t])
eig_chain.set_onsite(on=on)
eig_chain.get_ham()
eig_chain.get_eig(eigenvec=True)
stateA = eig_chain.get_state_pola(b'a')
stateB = eig_chain.get_state_pola(b'b')
# plots
plot_chain = plotTB(sys=eig_chain)
fig_lat  = plot_chain.plt_lattice(plt_index=True, ms=15, fs=20)
fig_spec = plot_chain.plt_spec(ms=20, pola_tag=b'a')
fig_stateA = plot_chain.plt_intensity1d(stateA, ms=20)
fig_stateB = plot_chain.plt_intensity1d(stateB, ms=20)
#propagation
prop_chain = propTB(lat=chain_tb, steps=1400, dz=0.05)
psi_init =  np.ones(eig_chain.sites) / np.sqrt(eig_chain.sites)
prop_chain.get_prop(ham=eig_chain.ham, psi_init=psi_init, norm=True)
fig_prop = prop_chain.plt_prop1d()
plt.show()
# save figures
save_chain = saveFigTB(sys=eig_chain, params={'t': t}, dir_name='chain', ext='png')
save_chain.save_fig(fig_spec, 'spec')
save_chain.save_fig(fig_prop, 'prop')
save_chain.save_fig(fig_stateA, 'stateA')
save_chain.save_fig(fig_stateB, 'stateB')

_images/stateA_ea0,2j_eb-0,2j_t(1+0j).png

_images/stateB_ea0,2j_eb-0,2j_t(1+0j).png

The state with $\langle A|A \rangle>1/2$ is amplified.