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Hatano-Nelson-Kitaev Model: Monte Carlo Simulation  |  |
The Hatano-Nelson-Kitaev model is a generalization of the to include non-Hermitian anisotropic pairing potential.
Represents a fermionic Gaussian state. | |
Represents a non-unitary evolution operator associated with local quadratic non-Hermitian Hamiltonian | |
Represents a set of quantum jump operators | |
Simulates the quantum master equation for a dissipative fermionic system |
Q3 functions for the Monte Carlo simulation of open quantum systems of non-interacting fermions.
Make sure to load the Q3 framework.
In[72]:=
Needs["QuantumMob`Q3`"]
The Model
Consider a system of fermion modes.
In[73]:=
$L=4;
In[74]:=
Let[Fermion,f]ff=f[Range@$L]fff=Join[ff,Dagger@ff]
Out[75]=
{,,,}
f
1
f
2
f
3
f
4
Out[76]=
,,,,,,,
f
1
f
2
f
3
f
4
†
f
1
†
f
2
†
f
3
†
f
4
Here is the Hamiltonian for the Kitaev chain of Majorana modes.
In[6]:=
Let[Real,μ,J,Δ,γ]
In[7]:=
hamOp=μ*Q[ff]-(J/2)PlusDagger[Hop[ff]]+(Δ/2)PlusDagger[Pair[ff]]
Out[7]=
Dissipative Model
Dissipative Model
Consider some auxiliary fermion modes to describe the quantum jump operators.
In[8]:=
Let[Fermion,a,b]aa=a[Range[$L-1]]bb=b[Range[$L-1]]
Out[9]=
{,,}
a
1
a
2
a
3
Out[10]=
{,,}
b
1
b
2
b
3
Here are the quantum jump operators that are linear combinations of the fermion creation and annihilation operators.
In[11]:=
a[k_]:=(f[k]+I*Dagger[f[k+1]])/Sqrt[2]b[k_]:=(f[k]-I*Dagger[f[k+1]])/Sqrt[2]ab=Join[aa,Dagger@bb];ab//TableForm
Out[14]//TableForm=
† f 2 f 1 2 |
† f 3 f 2 2 |
† f 4 f 3 2 |
† f 1 f 2 2 |
† f 2 f 3 2 |
† f 3 f 4 2 |
In the fermionic quantum computation, these quantum jump operators are efficiently described by the coefficients matrix.
In[15]:=
jmp=NambuCoefficients[ab,ff];jmp//MatrixForm
Out[16]//MatrixForm=
1 2 | 0 | 0 | 0 | 0 | 2 | 0 | 0 |
0 | 1 2 | 0 | 0 | 0 | 0 | 2 | 0 |
0 | 0 | 1 2 | 0 | 0 | 0 | 0 | 2 |
0 | 2 | 0 | 0 | 1 2 | 0 | 0 | 0 |
0 | 0 | 2 | 0 | 0 | 1 2 | 0 | 0 |
0 | 0 | 0 | 2 | 0 | 0 | 1 2 | 0 |
Construct the effective non-Hermitian Hamiltonian, which results in the so-called Hatano-Nelson model.
In[17]:=
dmpOp=γ*Total[Dagger[ab]**ab]/2//Garner
Out[17]=
3γ
2
1
2
†
f
1
†
f
2
1
2
†
f
2
†
f
3
1
2
†
f
3
†
f
4
1
2
f
2
f
1
1
2
f
3
f
2
1
2
f
4
f
3
In[18]:=
gmm=dmpOp/.Thread[ff0]
Out[18]=
3γ
2
In[19]:=
dmp=NambuHermitian[2*CoefficientTensor[dmpOp,Dagger@fff,fff]]
Out[19]=
NambuHermitian
 |
|
In[20]:=
nonOp=hamOp-I*dmpOp//Garner
Out[20]=
-+(γ-Δ)+μ-J+(γ-Δ)-J+μ-J+(γ-Δ)-J+μ-J-J+μ+(-γ-Δ)+(-γ-Δ)+(-γ-Δ)
3γ
2
1
2
†
f
1
†
f
2
†
f
1
f
1
1
2
†
f
1
f
2
1
2
†
f
2
†
f
3
1
2
†
f
2
f
1
†
f
2
f
2
1
2
†
f
2
f
3
1
2
†
f
3
†
f
4
1
2
†
f
3
f
2
†
f
3
f
3
1
2
†
f
3
f
4
1
2
†
f
4
f
3
†
f
4
f
4
1
2
f
2
f
1
1
2
f
3
f
2
1
2
f
4
f
3
In[21]:=
non=2*CoefficientTensor[nonOp,Dagger@fff,fff]//SimplifyThrough;ArrayShort[-2non,"Size"8]
Out[22]//MatrixForm=
-2μ | J | 0 | 0 | 0 | -γ+Δ | 0 | 0 |
J | -2μ | J | 0 | γ-Δ | 0 | -γ+Δ | 0 |
0 | J | -2μ | J | 0 | γ-Δ | 0 | -γ+Δ |
0 | 0 | J | -2μ | 0 | 0 | γ-Δ | 0 |
0 | -γ-Δ | 0 | 0 | 2μ | -J | 0 | 0 |
γ+Δ | 0 | -γ-Δ | 0 | -J | 2μ | -J | 0 |
0 | γ+Δ | 0 | -γ-Δ | 0 | -J | 2μ | -J |
0 | 0 | γ+Δ | 0 | 0 | 0 | -J | 2μ |
In[23]:=
altOp=MultiplyDot[Dagger@fff,non,fff]/2altOp-nonOp//Garner
Out[23]=
1
2
†
f
1
†
f
2
†
f
1
f
1
†
f
1
f
2
†
f
2
†
f
3
†
f
2
f
1
†
f
2
f
2
†
f
2
f
3
†
f
3
†
f
4
†
f
3
f
2
†
f
3
f
3
†
f
3
f
4
†
f
4
f
3
†
f
4
f
4
f
2
f
1
f
3
f
2
f
4
f
3
Out[24]=
3γ
2
In summary, the dissipative Hatano-Nelson-Kitaev (HNK) model can be written as
In[25]:=
Clear[HNKDissipative]HNKDissipative[{μ_,J_,Δ_},γ_,L_Integer]:=Module[{hrm,del,ham,jmp},hrm=-μ*One[L]-(J/2)*PlusTopple@One[{L,L},1];del=-(Δ/2)*One[{L,L},1];del=del-Transpose[del];ham=NambuHermitian[{hrm,del}];jmp={One[{L-1,L}],I*One[{L-1,L},1]};jmp=ArrayFlatten@{jmp,Reverse@jmp};jmp=Sqrt[γ]*WickCanonicalize[NambuJump@jmp];{ham,jmp}]
Projective Model
Projective Model
In summary, the projective Hatano-Nelson-Kitaev (HNK) model can be written as
Simulation of the Dissipative Model
Set the number of fermion modes.
Set the parameters of the Hatano-Nelson-Kitaev model.
Set the dissipative HNK model.
Set the initial state.
Do the Monte Carlo simulation.
Choose a subsystem.
Calculate the entanglement entropy (EE) between the subsystem and the rest.
Check the entanglement entropy as a function of time.
Construct approximate density matrices by taking averages of the covariance matrices in different trajectories, and calculate the logarithmic negativity (LN) between the subsystem and the rest in the density matrices.
Check the logarithmic negativity as a function of time.
Simulation of the Projective Model
Set the number of fermion modes.
Set the parameters of the Hatano-Nelson-Kitaev model.
Set the projective HNK model.
Set the initial state.
Do the Monte Carlo simulation.
Choose a subsystem.
Calculate the entanglement entropy (EE) between the subsystem and the rest.
Check the entanglement entropy as a function of time.
Construct approximate density matrices by taking averages of the covariance matrices in different trajectories, and calculate the logarithmic negativity (LN) between the subsystem and the rest in the density matrices.
Check the logarithmic negativity as a function of time.