are worth to mention here. First, the parity for the local boson basis is the parity of its occupation number. To define the parity for the spin basis one should use the eigenstates of the spin operator in the parity operator. For example for SBM1, the eigenstates of σz are used as the basis and the two parities are assigned to the two states respectively. After having defined the parity of each local state, the states of the same parity are combined, therefore all operators will be 2×2 block matrices. Second, one can define the right vacuum state as even parity then the parity of the wave function is defined by the left vacuum state.
The rest of the implementation is along the lines as described for example in the Section 5.1 of Ref. [57].
The expectation value hσxiof Pz parity eigenstates is always 0. If one wants to recover the expectation value of the non-parity eigenstates in the localized regime, one can simply calculate
hG↑|σx|G↑i = 1
2(hG+|σx|G+i+hG−|σx|G−i+hG+|σx|G−i+hG−|σx|G+i)
= Re(hG+|σx|G−i) (4.37)
4.6.4 U(1) symmetry in SBM2
For SBM2, besides the parity symmetry, there is also an Abelian U(1) symmetry. The generator is
S= 1
2σz+iX
k
(b†y,kbx,k−b†x,kby,k) (4.38) It is straight forward to verify that
[H, S] = 0 (4.39)
At the moment when the thesis was written we were still investigating the influence of U(1) symmetry to the VMPS numerics. We have indications that the numerics breaks the U(1) symmetry in the critical regime where the ground states should preserve the U(1) symmetry.
This results in the nonzero hσxi and hσyi in the critical regime for both non-parity code (Fig. 4.10) and parity code (Fig. 4.11) results of SBM2. Explicitly implementing of U(1) symmetry in the VMPS code appears necessary to get improved results. Work along those lines is currently in progress.
4.7 Energy flow diagrams 67
0 0.5 1 1.5 2
0 0.2 0.4 0.6 0.8
hx=h
y=h
z=0, s=0.8, Λ=2, L=60, D=100, d
opt=36
α
|<σ
x>|
|<σ
z>|
0 0.5 1 1.5 2
0 0.5 1 1.5
2x 10−6
α
maximun of the least sigular values 0 0.5 1 1.5 20
0.01 0.02 0.03 0.04
maximun of boson occupacy
α
0 0.5 1 1.5 2
0 0.2 0.4 0.6 0.8
hx=h
y=h
z=0, s=0.8, Λ=2, L=60, D=100, d
opt=36
α
|<σ
x>|
|<σ
z>|
0 0.5 1 1.5 2
0 0.5 1 1.5
2x 10−6
α
maximun of the least sigular values 0 0.5 1 1.5 20
0.01 0.02 0.03 0.04
maximun of boson occupacy
α
0 0.5 1 1.5 2
0 0.2 0.4 0.6 0.8
hx=h
y=h
z=0, s=0.8, Λ=2, L=60, D=100, d
opt=36
α
|<σ
x>|
|<σ
z>|
0 0.5 1 1.5 2
0 0.5 1 1.5
2x 10−6
α
maximun of the least sigular values 0 0.5 1 1.5 20
0.01 0.02 0.03 0.04
maximun of boson occupacy
α
4.6.3 Implement of the parity symmetry
In the previous section I explained that the ground state obtained from VMPS is not parity eigenstate in the localized regime. However one could implement parity symmetry explicitly in the VMPS algorithm to obtain the ground state which is also the parity eigenstate in the localized regime. This is useful when one want to generate a cleaner flow diagram.
Parity symmetry is a special case of Abelian symmetry, so one can use the same method implementing Abelian symmetry to implement parity symmetry. Two points worth to mention here. First, the parity for the local boson basis is the parity of its occupation number. To define the parity for the spin basis one should use the eigenstates of the spin operator in the parity operator. For example for SBM1, the eigenstates of z are used as the basis and the two parities are assigned to the two states respectively. After defined the parity of each base state, the states of the same parity shall be put together therefore all the operator will be a 2⇥2 block matrices. Second, one can define the right vaccum state as even parity then the parity of the wave function is defined by the left vaccum state. The rest of the implementation is exactly the same as described for example in the Section 5.1 of Ref. [?].
The x expectation value of parity eigenstates are always 0. If one wants to recover the expectation value of the non-parity eigenstates Eqs. (4.34)(4.35) in the delocalized regime one could simple calculate like
hG+| x|G+i = 1
2(hG+| x|G+i+hG | x|G i+hG+| x|G i+hG | x|G+i)
= Re(hG+| x|G i) (4.36)
4.6.4 U(1) symmetry in SBM2
For SBM2 besides the parity symmetry there is also the U(1) symmetry. The generator is S = 1
2 z+iX
k
(b†y,kbx,k b†x,kby,k) (4.37) It is straight to verified that
[H, S] = 0 (4.38)
At the moment when the thesis is written we are still investigating the influence of U(1) symmetry to the VMPS numerics. h xi h yi
4.7 Flow diagrams
In NRG, the flow diagram can provide lots of information about the impurity system. For example we can fix the phase boundary of the quantum phase transition accurately with flow diagram. This is very important when calculating the critical exponents at phase boundary.
4.6.3 Implement of the parity symmetry
In the previous section I explained that the ground state obtained from VMPS is not parity eigenstate in the localized regime. However one could implement parity symmetry explicitly in the VMPS algorithm to obtain the ground state which is also the parity eigenstate in the localized regime. This is useful when one want to generate a cleaner flow diagram.
Parity symmetry is a special case of Abelian symmetry, so one can use the same method implementing Abelian symmetry to implement parity symmetry. Two points worth to mention here. First, the parity for the local boson basis is the parity of its occupation number. To define the parity for the spin basis one should use the eigenstates of the spin operator in the parity operator. For example for SBM1, the eigenstates of z are used as the basis and the two parities are assigned to the two states respectively. After defined the parity of each base state, the states of the same parity shall be put together therefore all the operator will be a 2⇥2 block matrices. Second, one can define the right vaccum state as even parity then the parity of the wave function is defined by the left vaccum state. The rest of the implementation is exactly the same as described for example in the Section 5.1 of Ref. [?].
The x expectation value of parity eigenstates are always 0. If one wants to recover the expectation value of the non-parity eigenstates Eqs. (4.34)(4.35) in the delocalized regime one could simple calculate like
hG+| x|G+i = 1
2(hG+| x|G+i+hG | x|G i+hG+| x|G i+hG | x|G+i)
= Re(hG+| x|G i) (4.36)
4.6.4 U(1) symmetry in SBM2
For SBM2 besides the parity symmetry there is also the U(1) symmetry. The generator is S = 1
2 z +iX
k
(b†y,kbx,k b†x,kby,k) (4.37) It is straight to verified that
[H, S] = 0 (4.38)
At the moment when the thesis is written we are still investigating the influence of U(1) symmetry to the VMPS numerics. h xi h yi
4.7 Flow diagrams
In NRG, the flow diagram can provide lots of information about the impurity system. For example we can fix the phase boundary of the quantum phase transition accurately with flow diagram. This is very important when calculating the critical exponents at phase boundary.
Figure 4.10: hσxi andhσyi calculated with the SBM2 non-parity code. The system is in the critical regime forα <0.8 however we still see small non-zero values which is probably caused by the fact that the VMPS code breaks the U(1) symmetry. (Another possible reason is the finite size effect.)
0.5 0.6 0.7 0.8 0.9 1
0 0.1 0.2 0.3 0.4
Parity Code, h
x=h
y=h
z=0, s=0.8, Λ=2, L=50, D=80, dk=100, d
opt=50
α
|<−|σ
x|+>|
|<−|σ
z|+>|
0.4 0.6 0.8 1
10−8 10−7 10−6 10−5 10−4
α
maximun of the least sigular values
Amat Vmat
0.4 0.6 0.8 1
10−3 10−2 10−1
maximun of boson occupacy
α
64 4. One- and Two-Bath Spin-Boson Models
4.6.3 Implement of the parity symmetry
In the previous section I explained that the ground state obtained from VMPS is not parity eigenstate in the localized regime. However one could implement parity symmetry explicitly in the VMPS algorithm to obtain the ground state which is also the parity eigenstate in the localized regime. This is useful when one want to generate a cleaner flow diagram.
Parity symmetry is a special case of Abelian symmetry, so one can use the same method implementing Abelian symmetry to implement parity symmetry. Two points worth to mention here. First, the parity for the local boson basis is the parity of its occupation number. To define the parity for the spin basis one should use the eigenstates of the spin operator in the parity operator. For example for SBM1, the eigenstates of z are used as the basis and the two parities are assigned to the two states respectively. After defined the parity of each base state, the states of the same parity shall be put together therefore all the operator will be a 2⇥2 block matrices. Second, one can define the right vaccum state as even parity then the parity of the wave function is defined by the left vaccum state. The rest of the implementation is exactly the same as described for example in the Section 5.1 of Ref. [?].
The x expectation value of parity eigenstates are always 0. If one wants to recover the expectation value of the non-parity eigenstates Eqs. (4.34)(4.35) in the delocalized regime one could simple calculate like
hG+| x|G+i = 1
2(hG+| x|G+i+hG | x|G i+hG+| x|G i+hG | x|G+i)
= Re(hG+| x|G i) (4.36)
4.6.4 U(1) symmetry in SBM2
For SBM2 besides the parity symmetry there is also the U(1) symmetry. The generator is S = 1
2 z +iX
k
(b†y,kbx,k b†x,kby,k) (4.37) It is straight to verified that
[H, S] = 0 (4.38)
At the moment when the thesis is written we are still investigating the influence of U(1) symmetry to the VMPS numerics. h xi h yi
4.7 Flow diagrams
In NRG, the flow diagram can provide lots of information about the impurity system. For example we can fix the phase boundary of the quantum phase transition accurately with flow diagram. This is very important when calculating the critical exponents at phase boundary.
64 4. One- and Two-Bath Spin-Boson Models
4.6.3 Implement of the parity symmetry
In the previous section I explained that the ground state obtained from VMPS is not parity eigenstate in the localized regime. However one could implement parity symmetry explicitly in the VMPS algorithm to obtain the ground state which is also the parity eigenstate in the localized regime. This is useful when one want to generate a cleaner flow diagram.
Parity symmetry is a special case of Abelian symmetry, so one can use the same method implementing Abelian symmetry to implement parity symmetry. Two points worth to mention here. First, the parity for the local boson basis is the parity of its occupation number. To define the parity for the spin basis one should use the eigenstates of the spin operator in the parity operator. For example for SBM1, the eigenstates of z are used as the basis and the two parities are assigned to the two states respectively. After defined the parity of each base state, the states of the same parity shall be put together therefore all the operator will be a 2⇥2 block matrices. Second, one can define the right vaccum state as even parity then the parity of the wave function is defined by the left vaccum state. The rest of the implementation is exactly the same as described for example in the Section 5.1 of Ref. [?].
The x expectation value of parity eigenstates are always 0. If one wants to recover the expectation value of the non-parity eigenstates Eqs. (4.34)(4.35) in the delocalized regime one could simple calculate like
hG+| x|G+i = 1
2(hG+| x|G+i+hG | x|G i+hG+| x|G i+hG | x|G+i)
= Re(hG+| x|G i) (4.36)
4.6.4 U(1) symmetry in SBM2
For SBM2 besides the parity symmetry there is also the U(1) symmetry. The generator is S = 1
2 z +iX
k
(b†y,kbx,k b†x,kby,k) (4.37) It is straight to verified that
[H, S] = 0 (4.38)
At the moment when the thesis is written we are still investigating the influence of U(1) symmetry to the VMPS numerics. h xi h yi
4.7 Flow diagrams
In NRG, the flow diagram can provide lots of information about the impurity system. For example we can fix the phase boundary of the quantum phase transition accurately with flow diagram. This is very important when calculating the critical exponents at phase
boundary. 0.5 0.6 0.7 0.8 0.9 1
0 0.1 0.2 0.3 0.4
Parity Code, h
x=h
y=h
z=0, s=0.8, Λ=2, L=50, D=80, dk=100, d
opt=50
α
|<−|σ
x|+>|
|<−|σ
z|+>|
0.4 0.6 0.8 1
10−8 10−7 10−6 10−5 10−4
α
maximun of the least sigular values
Amat Vmat
0.4 0.6 0.8 1
10−3 10−2 10−1
maximun of boson occupacy
α
Figure 4.11: Same as Fig. 4.10 except using the parity code and a bit largerDand shorter L. We still see the non-zero hσxi and hσyi in the critical regime. This indicates that it is caused by symmetry breaking other than the parity symmetry.
[74, 70]. I call it “the quasi flow diagram”. By comparison, we find the quasi-flow diagram is very close to the flow diagram in the delocalized regime and near the phase boundary, especially at the lower part of the energy spectrum. The larger D we use, the better the quasi-flow diagram we get before we reach the machine precision. To fix the phase boundary a few low lying states are already sufficient.
Fig. 4.12 shows the quasi flow diagram in the delocalized regime, on the phase boundary and in the localized regime. One can compare them with the flow diagram in Ref. [12].
10 20 30 40 50 60 70 80 90 100 10−4
10−3 10−2 10−1
site index k
<b k+b k>
ε=0, ∆=0.1, s=0.3, α=0.0346139, Λ=2, L=100, z=1, D=40, d
opt=12
10 20 30 40 50 60 70 80 90 100
0 2 4 6
site index k
Flow diagram
2 4 6 8 10 12 14 16 18 20
10−20 10−10 100
D dimension index
Amat sigular values
1 2 3 4 5 6
10−20 10−10 100
dopt dimenstion index
Vmat sigular values
10 20 30 40 50 60 70 80 90 100
0 0.02 0.04 0.06 0.08 0.1
site index k
Amat von Neumann entropy
10 20 30 40 50 60 70 80 90 100
0 1 2 3x 10−3
site index k
Vmat von Neumann entropy
10 20 30 40 50 60 70 80 90 100
10−4 10−3 10−2 10−1
site index k
<bk+ bk>
ε=0, ∆=0.1, s=0.3, α=0.0346139, Λ=2, L=100, z=1, D=40, d
opt=12
10 20 30 40 50 60 70 80 90 100
0 2 4 6
site index k
Flow diagram
2 4 6 8 10 12 14 16 18 20
10−20 10−10 100
D dimension index
Amat sigular values
1 2 3 4 5 6
10−20 10−10 100
dopt dimenstion index
Vmat sigular values
10 20 30 40 50 60 70 80 90 100
0 0.02 0.04 0.06 0.08 0.1
site index k
Amat von Neumann entropy
10 20 30 40 50 60 70 80 90 100
0 1 2 3x 10−3
site index k
Vmat von Neumann entropy
10 20 30 40 50 60 70 80 90 100
10−3 10−2 10−1
site index k
<bk+bk>
ε=0, ∆=0.1, s=0.3, α=0.0346139, Λ=2, L=100, z=1, D=40, d
opt=12
10 20 30 40 50 60 70 80 90 100
0 2 4 6
site index k
Flow diagram
2 4 6 8 10 12 14 16 18 20
10−15 10−10 10−5 100
D dimension index
Amat sigular values
1 2 3 4 5 6
10−20 10−10 100
dopt dimenstion index
Vmat sigular values
10 20 30 40 50 60 70 80 90 100
0 0.02 0.04 0.06 0.08 0.1
site index k
Amat von Neumann entropy
10 20 30 40 50 60 70 80 90 100
0 1 2 3x 10−3
site index k
Vmat von Neumann entropy
10 20 30 40 50 60 70 80 90 100
10−3 10−2 10−1
site index k
<b k+b k>
ε=0, ∆=0.1, s=0.3, α=0.0346139, Λ=2, L=100, z=1, D=40, d
opt=12
10 20 30 40 50 60 70 80 90 100
0 2 4 6
site index k
Flow diagram
2 4 6 8 10 12 14 16 18 20
10−15 10−10 10−5 100
D dimension index
Amat sigular values
1 2 3 4 5 6
10−20 10−10 100
dopt dimenstion index
Vmat sigular values
10 20 30 40 50 60 70 80 90 100
0 0.02 0.04 0.06 0.08 0.1
site index k
Amat von Neumann entropy
10 20 30 40 50 60 70 80 90 100
0 1 2 3x 10−3
site index k
Vmat von Neumann entropy
10 20 30 40 50 60 70 80 90 100
10−2 100 102
site index k
<b k+b k>
ε=0, ∆=0.1, s=0.3, α=0.0346139, Λ=2, L=100, z=1, D=40, d
opt=12
10 20 30 40 50 60 70 80 90 100
0 2 4 6
site index k
Flow diagram
2 4 6 8 10 12 14 16 18 20
10−15 10−10 10−5 100
D dimension index
Amat sigular values
1 2 3 4 5 6
10−15 10−10 10−5 100
dopt dimenstion index
Vmat sigular values
10 20 30 40 50 60 70 80 90 100
0 0.02 0.04 0.06 0.08 0.1
site index k
Amat von Neumann entropy
10 20 30 40 50 60 70 80 90 100
0 0.02 0.04 0.06 0.08
site index k
Vmat von Neumann entropy
10 20 30 40 50 60 70 80 90 100
10−2 100 102
site index k
<bk+bk>
ε=0, ∆=0.1, s=0.3, α=0.0346139, Λ=2, L=100, z=1, D=40, d
opt=12
10 20 30 40 50 60 70 80 90 100
0 2 4 6
site index k
Flow diagram
2 4 6 8 10 12 14 16 18 20
10−15 10−10 10−5 100
D dimension index
Amat sigular values
1 2 3 4 5 6
10−15 10−10 10−5 100
dopt dimenstion index
Vmat sigular values
10 20 30 40 50 60 70 80 90 100
0 0.02 0.04 0.06 0.08 0.1
site index k
Amat von Neumann entropy 10 20 30 40 50 60 70 80 90 1000
0.02 0.04 0.06 0.08
site index k
Vmat von Neumann entropy
(a)
(c)
(b)
(d)
(e) (f)
Figure 4.12: The boson occupation number (a),(c), (e) and flow diagrams (b),(d),(f) generated after we implemented the parity symmetry in our program. The results are cal-culated ats= 0.3 for SBM1. (a), (b) In the delocalized regime close to the phase boundary where α = 0.0346138680; (c), (d) On the phase boundary α=αc= 0.0346138682; (e),(f) In the localized regime close to the phase boundary where α= 0.0346138683.
With these quasi flow diagrams we can fix the critical αc for a given s with a relative accuracy better than 10−8. If one wants to improve the accuracy one needs a longer Wilson chain as explained in Sec.4.2.
There is another very tricky problem to use VMPS to generate flow diagram, due to the degeneracy from the parity symmetry of the spin-boson model. Unlike NRG, the variational nature of VMPS does not preserve parity symmetry, and the degeneracy of the ground states will sometimes cause unstable “jumps” in the flow diagram. To eliminate this instability, one has to implement parity symmetry explicitly in the VMPS program as described in the previous section.