Parent Hamiltonianof a TNS: local Hamiltonian which has the TNS as ground state (unique ground state in theinjectivity regime, by Cirac/Pérez-García/Verstraete/Wolf).
Fact:By the Lieb-Robinson (LR) bound (Hastings/Koma...) or the detectability lemma (Aharonov/Arad/Landau/Vazirani...), if the parent Hamiltonian of a TNS is gapped above its ground state energy then the correlations between two observables on sub-regions decay exponentially with the distance separating the two sub-regions.
Question:Is a random parent Hamiltonian typically gapped or gapless?
Construction for our random MPS:Assumed>q2, so that|χNiis almost surely injective. DefineV: =Span
( d
∑
x1,x2=1
Tr(Gx1Gx2M)|x1x2i:M q×qmatrix )
⊂Cd⊗Cd, where theGx’s are theq×qmatrices appearing in the transfer operatorT.
Denoting byΠthe projector onV, the parent HamiltonianHof|χNiis then defined as H: =
N
∑
i=1
Π⊥i,i+1⊗I1,...,i−1,i+2,...,N.
By constructionHis a 2-local frustration-free Hamiltonian, that has|χNias unique ground state. Construction for our random PEPS:Assumed>q4, so that|χMNiis almost surely injective.
Cécilia Lancien Correlation length in random MPS and PEPS UCL Quantum Information seminar – June 18 2020 17
Parent Hamiltonian of a TNS
Parent Hamiltonianof a TNS: local Hamiltonian which has the TNS as ground state (unique ground state in theinjectivity regime, by Cirac/Pérez-García/Verstraete/Wolf).
Fact:By the Lieb-Robinson (LR) bound (Hastings/Koma...) or the detectability lemma (Aharonov/Arad/Landau/Vazirani...), if the parent Hamiltonian of a TNS is gapped above its ground state energy then the correlations between two observables on sub-regions decay exponentially with the distance separating the two sub-regions.
Question:Is a random parent Hamiltonian typically gapped or gapless?
Construction for our random MPS:Assumed>q2, so that|χNiis almost surely injective. DefineV: =Span
( d
∑
x1,x2=1
Tr(Gx1Gx2M)|x1x2i:M q×qmatrix )
⊂Cd⊗Cd, where theGx’s are theq×qmatrices appearing in the transfer operatorT.
Denoting byΠthe projector onV, the parent HamiltonianHof|χNiis then defined as H: =
N
∑
i=1
Π⊥i,i+1⊗I1,...,i−1,i+2,...,N.
By constructionHis a 2-local frustration-free Hamiltonian, that has|χNias unique ground state. Construction for our random PEPS:Assumed>q4, so that|χMNiis almost surely injective.
Cécilia Lancien Correlation length in random MPS and PEPS UCL Quantum Information seminar – June 18 2020 17
Parent Hamiltonian of a TNS
Parent Hamiltonianof a TNS: local Hamiltonian which has the TNS as ground state (unique ground state in theinjectivity regime, by Cirac/Pérez-García/Verstraete/Wolf).
Fact:By the Lieb-Robinson (LR) bound (Hastings/Koma...) or the detectability lemma (Aharonov/Arad/Landau/Vazirani...), if the parent Hamiltonian of a TNS is gapped above its ground state energy then the correlations between two observables on sub-regions decay exponentially with the distance separating the two sub-regions.
Question:Is a random parent Hamiltonian typically gapped or gapless?
Construction for our random MPS:Assumed>q2, so that|χNiis almost surely injective. DefineV: =Span
( d
∑
x1,x2=1
Tr(Gx1Gx2M)|x1x2i:M q×qmatrix )
⊂Cd⊗Cd, where theGx’s are theq×qmatrices appearing in the transfer operatorT.
Denoting byΠthe projector onV, the parent HamiltonianHof|χNiis then defined as H: =
N
∑
i=1
Π⊥i,i+1⊗I1,...,i−1,i+2,...,N.
By constructionHis a 2-local frustration-free Hamiltonian, that has|χNias unique ground state. Construction for our random PEPS:Assumed>q4, so that|χMNiis almost surely injective.
Cécilia Lancien Correlation length in random MPS and PEPS UCL Quantum Information seminar – June 18 2020 17
Parent Hamiltonian of a TNS
Parent Hamiltonianof a TNS: local Hamiltonian which has the TNS as ground state (unique ground state in theinjectivity regime, by Cirac/Pérez-García/Verstraete/Wolf).
Fact:By the Lieb-Robinson (LR) bound (Hastings/Koma...) or the detectability lemma (Aharonov/Arad/Landau/Vazirani...), if the parent Hamiltonian of a TNS is gapped above its ground state energy then the correlations between two observables on sub-regions decay exponentially with the distance separating the two sub-regions.
Question:Is a random parent Hamiltonian typically gapped or gapless?
Construction for our random MPS:Assumed>q2, so that|χNiis almost surely injective.
DefineV: =Span ( d
∑
x1,x2=1
Tr(Gx1Gx2M)|x1x2i:M q×qmatrix )
⊂Cd⊗Cd, where theGx’s are theq×qmatrices appearing in the transfer operatorT.
Denoting byΠthe projector onV, the parent HamiltonianHof|χNiis then defined as H: =
N
∑
i=1
Π⊥i,i+1⊗I1,...,i−1,i+2,...,N.
By constructionHis a 2-local frustration-free Hamiltonian, that has|χNias unique ground state.
Construction for our random PEPS:Assumed>q4, so that|χMNiis almost surely injective.
Cécilia Lancien Correlation length in random MPS and PEPS UCL Quantum Information seminar – June 18 2020 17
Parent Hamiltonian of a TNS
Parent Hamiltonianof a TNS: local Hamiltonian which has the TNS as ground state (unique ground state in theinjectivity regime, by Cirac/Pérez-García/Verstraete/Wolf).
Fact:By the Lieb-Robinson (LR) bound (Hastings/Koma...) or the detectability lemma (Aharonov/Arad/Landau/Vazirani...), if the parent Hamiltonian of a TNS is gapped above its ground state energy then the correlations between two observables on sub-regions decay exponentially with the distance separating the two sub-regions.
Question:Is a random parent Hamiltonian typically gapped or gapless?
Construction for our random MPS:Assumed>q2, so that|χNiis almost surely injective.
DefineV: =Span ( d
∑
x1,x2=1
Tr(Gx1Gx2M)|x1x2i:M q×qmatrix )
⊂Cd⊗Cd, where theGx’s are theq×qmatrices appearing in the transfer operatorT.
Denoting byΠthe projector onV, the parent HamiltonianHof|χNiis then defined as H: =
N
∑
i=1
Π⊥i,i+1⊗I1,...,i−1,i+2,...,N.
By constructionHis a 2-local frustration-free Hamiltonian, that has|χNias unique ground state.
Construction for our random PEPS:Assumed>q4, so that|χMNiis almost surely injective.
Cécilia Lancien Correlation length in random MPS and PEPS UCL Quantum Information seminar – June 18 2020 17