Parent Hamiltonians of tensor network states

Parent Hamiltonians of tensor network states

Cécilia Lancien

Institut de Mathématiques de Toulouse & CNRS

IMT-LPT Seminar – June 25 2020

Motivation

General belief:Ground states of gapped local Hamiltonians'Tensor network states.

Ion’s talk:(⇒)in 1D(Hastings, Landau/Vazirani/Vidick...)

Note:(⇒)in 2Dnot known (analogue to 1Dstrategy works only under extra assumptions...) Today’s talk: About(⇐)in 1Dand 2D.

Nice review on what is known or not in 2Dby Cirac/Garre-Rubio/Pérez-García.

Motivation

General belief:Ground states of gapped local Hamiltonians'Tensor network states.

Ion’s talk:(⇒)in 1D(Hastings, Landau/Vazirani/Vidick...)

Note:(⇒)in 2Dnot known (analogue to 1Dstrategy works only under extra assumptions...) Today’s talk: About(⇐)in 1Dand 2D.

Nice review on what is known or not in 2Dby Cirac/Garre-Rubio/Pérez-García.

Motivation

General belief:Ground states of gapped local Hamiltonians'Tensor network states.

Ion’s talk:(⇒)in 1D(Hastings, Landau/Vazirani/Vidick...)

Note:(⇒)in 2Dnot known (analogue to 1Dstrategy works only under extra assumptions...) Today’s talk: About(⇐)in 1Dand 2D.

Nice review on what is known or not in 2Dby Cirac/Garre-Rubio/Pérez-García.

Reminders about tensor network states

Tensor network state (TNS) on(C^d)^⊗N: Take a graphGwithNvertices andLedges.

Put at each vertexva tensorA^v∈C^d⊗(C^q)^⊗δ(v)to get a tensorA^G∈(C^d)^⊗N⊗(C^q)^⊗2L. Contract together the indices ofA^Gassociated to a same edge to get a tensorψ(A^G)∈(C^d)^⊗N.

−→Ifδ(v)6^{rfor all}v, then such state is described by at mostNq^rdparameters (linear rather than exponential inN).

• •

• •

• • Gwith 6 vertices and 7 edges

• •

• •

• • A^G∈(C^d)^⊗6⊗(C^q)^⊗14

• •

• •

• • ψ(A^G)∈(C^d)^⊗6 d-dimensional indices:physicalindices.q-dimensional indices:virtualorbondindices.

Here:The underlying graph is a regular latticeΛin dimension 1 or 2.

1D:Λ ={1, . . . ,N} −→matrix product state (MPS)ψ A^[N]

.

2D:Λ ={1, . . . ,M} × {1, . . . ,N} −→projected entangled pair state (PEPS)ψ A^[M]×[N] .

Reminders about tensor network states

Tensor network state (TNS) on(C^d)^⊗N: Take a graphGwithNvertices andLedges.

Put at each vertexva tensorA^v∈C^d⊗(C^q)^⊗δ(v)to get a tensorA^G∈(C^d)^⊗N⊗(C^q)^⊗2L. Contract together the indices ofA^Gassociated to a same edge to get a tensorψ(A^G)∈(C^d)^⊗N.

−→Ifδ(v)6^{rfor all}v, then such state is described by at mostNq^rdparameters (linear rather than exponential inN).

• •

• •

• • Gwith 6 vertices and 7 edges

• •

• •

• • A^G∈(C^d)^⊗6⊗(C^q)^⊗14

• •

• •

• • ψ(A^G)∈(C^d)^⊗6 d-dimensional indices:physicalindices.q-dimensional indices:virtualorbondindices.

Here:The underlying graph is a regular latticeΛin dimension 1 or 2.

1D:Λ ={1, . . . ,N} −→matrix product state (MPS)ψ A^[N]

.

2D:Λ ={1, . . . ,M} × {1, . . . ,N} −→projected entangled pair state (PEPS)ψ A^[M]×[N] .

MPS and PEPS

(periodic boundary conditions: circle)

• • • • • N

d q q Aⁱ= ∑^d

x=1 q

∑

α,β=1

Aⁱ_x

αβ|xαβi ∈C^d⊗(C^q)^⊗2

PEPS:MNparticles on a grid (periodic boundary conditions: torus)

• • • • •

• • • • •

• • • • • M

N

q d q

q q A^(ij)= ∑^d

x=1 q

∑

α,β,γ,δ=1

A⁽_x^ij)

αβγδ|xαβγδi ∈C^d⊗(C^q)^⊗4

Explicit expression for an MPS with periodic boundary conditions:

N

ψ A^[N]

=

d

∑

x1,...,xN=1 q

∑

α1,...,αN=1

A¹_x1αNα1· · ·A^N_xNαN−1αN

!

|x₁· · ·x_Ni

=

d

∑

x1,...,xN=1

Tr A¹_x

1· · ·A^N_x

N

|x₁· · ·x_Ni

Aⁱ_x:=

q

∑

α,β=1

Aⁱ_x

αβ|αihβ|,q×qmatrix.

MPS and PEPS

(periodic boundary conditions: circle)

• • • • • N

d q q Aⁱ= ∑^d

x=1 q

∑

α,β=1

Aⁱ_x

αβ|xαβi ∈C^d⊗(C^q)^⊗2 PEPS:MNparticles on a grid

(periodic boundary conditions: torus)

• • • • •

• • • • •

• • • • • M

N

q d q

q q A^(ij)= ∑^d

x=1 q

∑

α,β,γ,δ=1

A⁽_x^ij)

αβγδ|xαβγδi ∈C^d⊗(C^q)^⊗4

Explicit expression for an MPS with periodic boundary conditions:

N

ψ A^[N]

=

d

∑

x1,...,xN=1 q

∑

α1,...,αN=1

A¹_x1αNα1· · ·A^N_xNαN−1αN

!

|x₁· · ·x_Ni

=

d

∑

x1,...,xN=1

Tr A¹_x

1· · ·A^N_x

N

|x₁· · ·x_Ni

Aⁱ_x:=

q

∑

α,β=1

Aⁱ_x

αβ|αihβ|,q×qmatrix.

MPS and PEPS

(periodic boundary conditions: circle)

• • • • • N

d q q Aⁱ= ∑^d

x=1 q

∑

α,β=1

Aⁱ_x

αβ|xαβi ∈C^d⊗(C^q)^⊗2 PEPS:MNparticles on a grid

(periodic boundary conditions: torus)

• • • • •

• • • • •

• • • • • M

N

q d q

q q A^(ij)= ∑^d

x=1 q

∑

α,β,γ,δ=1

A⁽_x^ij)

αβγδ|xαβγδi ∈C^d⊗(C^q)^⊗4

Explicit expression for an MPS with periodic boundary conditions:

N

ψ A^[N]

=

d

∑

x1,...,xN=1 q

∑

α1,...,αN=1

A¹_x1αNα1· · ·A^N_xNαN−1αN

!

|x₁· · ·x_Ni

=

d

∑

x1,...,xN=1

Tr A¹_x

1· · ·A^N_x

N

|x₁· · ·x_Ni

Aⁱ_x:=

q

∑

α,β=1

Aⁱ_x

αβ|αihβ|,q×qmatrix.

Parent Hamiltonian of a TNS

Definition

Aparent Hamiltonianof a TNS is a local Hamiltonian (i.e. composed of terms which act only on a bounded number of nearby sites) which has the TNS as ground state.

Observation:For a large enough sub-latticeRof the latticeΛ(i.e. a segment or a rectangle), we haved^|R|>q^|∂R|(because volume grows faster than area).

physical space (bulk) larger than virtual space (boundary) MPS:d^L>q². PEPS:d^KL>q^2(K+L).

Parent Hamiltonian construction for an MPSψ A^[N]

≡ψ: LetLbe s.t.d^L>q².

For all 16ⁱ6N, define the reduced stateρⁱ:= Tr_{[N]\{i,...,i+L−1}}(|ψihψ|)on(C^d)^⊗L. By construction,rank(ρⁱ)6^q2<d^L. SoΠⁱ, the projector ontoker(ρⁱ), is non-trivial. SetH:= ∑^N

i=1

Πⁱ_{{i,...,i+L−1}}⊗I_{[N]\{i,...,i+L−1}}. By construction,





 H>⁰ hψ|H|ψi= ∑^N

i=1

Tr(Πⁱρⁱ) =0 .

−→His anL-local frustration-free Hamiltonian havingψas ground state. Case of a PEPSψ A^[M]×[N]

:Same construction withK,Ls.t.d^KL>q^2(K+L), and for all 16ⁱ6^M,16^j6N, the reduced stateρ^ijon(C^d)^⊗KL, to get aKL-local frustration-free Hamiltonian havingψas ground state.

Parent Hamiltonian of a TNS

Definition

Aparent Hamiltonianof a TNS is a local Hamiltonian (i.e. composed of terms which act only on a bounded number of nearby sites) which has the TNS as ground state.

Observation:For a large enough sub-latticeRof the latticeΛ(i.e. a segment or a rectangle), we haved^|R|>q^|∂R|(because volume grows faster than area).

physical space (bulk) larger than virtual space (boundary) MPS:d^L>q². PEPS:d^KL>q^2(K+L).

Parent Hamiltonian construction for an MPSψ A^[N]

≡ψ: LetLbe s.t.d^L>q².

For all 16ⁱ6N, define the reduced stateρⁱ:= Tr_{[N]\{i,...,i+L−1}}(|ψihψ|)on(C^d)^⊗L. By construction,rank(ρⁱ)6^q2<d^L. SoΠⁱ, the projector ontoker(ρⁱ), is non-trivial. SetH:= ∑^N

i=1

Πⁱ_{{i,...,i+L−1}}⊗I_{[N]\{i,...,i+L−1}}. By construction,





 H>⁰ hψ|H|ψi= ∑^N

i=1

Tr(Πⁱρⁱ) =0 .

−→His anL-local frustration-free Hamiltonian havingψas ground state. Case of a PEPSψ A^[M]×[N]

:Same construction withK,Ls.t.d^KL>q^2(K+L), and for all 16ⁱ6^M,16^j6N, the reduced stateρ^ijon(C^d)^⊗KL, to get aKL-local frustration-free Hamiltonian havingψas ground state.

Parent Hamiltonian of a TNS

Definition

Aparent Hamiltonianof a TNS is a local Hamiltonian (i.e. composed of terms which act only on a bounded number of nearby sites) which has the TNS as ground state.

Observation:For a large enough sub-latticeRof the latticeΛ(i.e. a segment or a rectangle), we haved^|R|>q^|∂R|(because volume grows faster than area).

physical space (bulk) larger than virtual space (boundary) MPS:d^L>q². PEPS:d^KL>q^2(K+L).

Parent Hamiltonian construction for an MPSψ A^[N]

≡ψ: LetLbe s.t.d^L>q².

For all 16ⁱ6N, define the reduced stateρⁱ:= Tr_{[N]\{i,...,i+L−1}}(|ψihψ|)on(C^d)^⊗L. By construction,rank(ρⁱ)6^q2<d^L. SoΠⁱ, the projector ontoker(ρⁱ), is non-trivial.

SetH:= ∑^N

i=1

Πⁱ_{{i,...,i+L−1}}⊗I_{[N]\{i,...,i+L−1}}. By construction,





 H>⁰ hψ|H|ψi=

N

∑

i=1

Tr(Πⁱρⁱ) =0 .

−→His anL-local frustration-free Hamiltonian havingψas ground state.

Case of a PEPSψ A^[M]×[N]

:Same construction withK,Ls.t.d^KL>q^2(K+L), and for all 16ⁱ6^M,16^j6N, the reduced stateρ^ijon(C^d)^⊗KL, to get aKL-local frustration-free Hamiltonian havingψas ground state.

Parent Hamiltonian of a TNS

Definition

Aparent Hamiltonianof a TNS is a local Hamiltonian (i.e. composed of terms which act only on a bounded number of nearby sites) which has the TNS as ground state.

Observation:For a large enough sub-latticeRof the latticeΛ(i.e. a segment or a rectangle), we haved^|R|>q^|∂R|(because volume grows faster than area).

physical space (bulk) larger than virtual space (boundary) MPS:d^L>q². PEPS:d^KL>q^2(K+L).

Parent Hamiltonian construction for an MPSψ A^[N]

≡ψ: LetLbe s.t.d^L>q².

For all 16ⁱ6N, define the reduced stateρⁱ:= Tr_{[N]\{i,...,i+L−1}}(|ψihψ|)on(C^d)^⊗L. By construction,rank(ρⁱ)6^q2<d^L. SoΠⁱ, the projector ontoker(ρⁱ), is non-trivial.

SetH:= ∑^N

i=1

Πⁱ_{{i,...,i+L−1}}⊗I_{[N]\{i,...,i+L−1}}. By construction,





 H>⁰ hψ|H|ψi=

N

∑

i=1

Tr(Πⁱρⁱ) =0 .

−→His anL-local frustration-free Hamiltonian havingψas ground state.

Case of a PEPSψ A^[M]×[N]

:Same construction withK,Ls.t.d^KL>q^2(K+L), and for all 16ⁱ6^M,16^j6N, the reduced stateρ^ijon(C^d)^⊗KL, to get aKL-local frustration-free Hamiltonian havingψas ground state.

Injectivity

Given a sub-latticeRof the latticeΛ, consider the restriction of the TNSψ(A^Λ)onR, with non-contracted boundary, i.e.

ψ(˜ A^R)∈(C^d)^⊗|R|⊗(C^q)^⊗|∂R|.

Question:Consideringψ(˜ A^R)as a map from(C^q)^⊗|∂R|to (C^d)^⊗|R|, when is it injective?

Necessary condition:d^|R|>^q|∂R|. Generically it is also sufficient.

ψ˜ A^[3]

: (C^q)^⊗2→(C^d)^⊗3

Definition

An MPSψ(A^[N])isnormalif there existsL₀s.t., for all 16ⁱ6^N,ψ˜ A^{{i,...,i+L0−1}}

is injective.

−→Injectivity indexofψ(A^[N]): smallest suchL0.

A PEPSψ(A^[M]×[N])isnormalif there existK₀,L₀s.t., for all 16ⁱ6^M,16^j6^N, ψ˜ A^{{i,...,i+K0−1}×{j,...,j+L0−1}}

is injective.

−→Injectivity indexofψ(A^[M]×[N]): such(K₀,L₀)so thatK₀L₀is the smallest. Observation:If restrictions of sizeL₀, resp.K₀L₀, of a TNS are all injective, than so are its restrictions of sizeLforL>^L0, resp.KLforK>^KO,L>^L0.

Remark:By blocking, i.e. redefining a sub-lattice of sites as being 1 site, any normal TNS can be seen as having injectivity index 1.

d d d

q q

d³ q q

−→

Injectivity

Given a sub-latticeRof the latticeΛ, consider the restriction of the TNSψ(A^Λ)onR, with non-contracted boundary, i.e.

ψ(˜ A^R)∈(C^d)^⊗|R|⊗(C^q)^⊗|∂R|.

Question:Consideringψ(˜ A^R)as a map from(C^q)^⊗|∂R|to (C^d)^⊗|R|, when is it injective?

Necessary condition:d^|R|>^q|∂R|. Generically it is also sufficient.

ψ˜ A^[3]

: (C^q)^⊗2→(C^d)^⊗3

Definition

An MPSψ(A^[N])isnormalif there existsL₀s.t., for all 16ⁱ6^N,ψ˜ A^{{i,...,i+L0−1}}

is injective.

−→Injectivity indexofψ(A^[N]): smallest suchL0.

A PEPSψ(A^[M]×[N])isnormalif there existK₀,L₀s.t., for all 16ⁱ6^M,16^j6^N, ψ˜ A^{{i,...,i+K0−1}×{j,...,j+L0−1}}

is injective.

−→Injectivity indexofψ(A^[M]×[N]): such(K₀,L₀)so thatK₀L₀is the smallest.

Observation:If restrictions of sizeL₀, resp.K₀L₀, of a TNS are all injective, than so are its restrictions of sizeLforL>^L0, resp.KLforK>^KO,L>^L0.

Remark:By blocking, i.e. redefining a sub-lattice of sites as being 1 site, any normal TNS can be seen as having injectivity index 1.

d d d

q q

d³ q q

−→

Injectivity

Given a sub-latticeRof the latticeΛ, consider the restriction of the TNSψ(A^Λ)onR, with non-contracted boundary, i.e.

ψ(˜ A^R)∈(C^d)^⊗|R|⊗(C^q)^⊗|∂R|.

Question:Consideringψ(˜ A^R)as a map from(C^q)^⊗|∂R|to (C^d)^⊗|R|, when is it injective?

Necessary condition:d^|R|>^q|∂R|. Generically it is also sufficient.

ψ˜ A^[3]

: (C^q)^⊗2→(C^d)^⊗3

Definition

An MPSψ(A^[N])isnormalif there existsL₀s.t., for all 16ⁱ6^N,ψ˜ A^{{i,...,i+L0−1}}

is injective.

−→Injectivity indexofψ(A^[N]): smallest suchL0.

A PEPSψ(A^[M]×[N])isnormalif there existK₀,L₀s.t., for all 16ⁱ6^M,16^j6^N, ψ˜ A^{{i,...,i+K0−1}×{j,...,j+L0−1}}

is injective.

−→Injectivity indexofψ(A^[M]×[N]): such(K₀,L₀)so thatK₀L₀is the smallest.

Observation:If restrictions of sizeL₀, resp.K₀L₀, of a TNS are all injective, than so are its restrictions of sizeLforL>^L0, resp.KLforK>^KO,L>^L0.

Remark:By blocking, i.e. redefining a sub-lattice of sites as being 1 site, any normal TNS can be seen as having injectivity index 1.

d d d

q q

d³ q q

−→

Injectivity

Given a sub-latticeRof the latticeΛ, consider the restriction of the TNSψ(A^Λ)onR, with non-contracted boundary, i.e.

ψ(˜ A^R)∈(C^d)^⊗|R|⊗(C^q)^⊗|∂R|.

Question:Consideringψ(˜ A^R)as a map from(C^q)^⊗|∂R|to (C^d)^⊗|R|, when is it injective?

Necessary condition:d^|R|>^q|∂R|. Generically it is also sufficient.

ψ˜ A^[3]

: (C^q)^⊗2→(C^d)^⊗3

Definition

An MPSψ(A^[N])isnormalif there existsL₀s.t., for all 16ⁱ6^N,ψ˜ A^{{i,...,i+L0−1}}

is injective.

−→Injectivity indexofψ(A^[N]): smallest suchL0.

A PEPSψ(A^[M]×[N])isnormalif there existK₀,L₀s.t., for all 16ⁱ6^M,16^j6^N, ψ˜ A^{{i,...,i+K0−1}×{j,...,j+L0−1}}

is injective.

−→Injectivity indexofψ(A^[M]×[N]): such(K₀,L₀)so thatK₀L₀is the smallest.

Observation:If restrictions of sizeL₀, resp.K₀L₀, of a TNS are all injective, than so are its restrictions of sizeLforL>^L0, resp.KLforK>^KO,L>^L0.

Remark:By blocking, i.e. redefining a sub-lattice of sites as being 1 site, any normal TNS can be seen as having injectivity index 1.

d d d

q q

d³ q q

−→

Parent Hamiltonian of a normal TNS

Unicity of the ground state of the parent Hamiltonian for a normal MPSψ A^[N] : LetL₀be the injectivity index ofψ A^[N]

. Then, for anyL>^L0+1, the parent HamiltonianH obtained from the previous construction hasψ A^[N]

as unique ground state.

Indeed in this case, setting for allK>^L0and 16ⁱ6^N,ψ˜^i,K:= ˜ψ A^{{i,...,i+K−1}}

, we have:

ψ˜^i,Linjective ⇒supp(ρⁱ) =range(˜ψ^i,L) =: Γ^i,L.

ψ˜^i+1,L−1injective ⇒Γ^i,L⊗(C^d)^⊗(N−L)∩Γ^i+1,L⊗(C^d)^⊗(N−L)= Γ^i,L+1⊗(C^d)^{⊗(N−L−1)}. Hence, iterating:

N

T

i=1

Γ^i,L⊗(C^d)^⊗(N−L)=range ψ˜ A^[N]

=span ψ A^[N] .

:=range(˜ψ^i,L+1)

ground state space ofH Explicitly:range(˜ψ^i,L) =span

( d

∑

x1,...,xL=1

Tr Aⁱ_x

1· · ·Aⁱ_x⁺_L^L−1T

|x₁· · ·x_Li:T q×qmatrix )

. L-site MPS with any boundary condition

Case of a PEPSψ A^[M]×[N]

:Same result forK>^K0+1,L>^L0+1.

Alternative construction: Instead ofMNterms which are(K0+1)(L0+1)-local, 2MNterms which are eitherK₀(L₀+1)-local or(K₀+1)L₀-local.

Parent Hamiltonian of a normal TNS

Unicity of the ground state of the parent Hamiltonian for a normal MPSψ A^[N] : LetL₀be the injectivity index ofψ A^[N]

. Then, for anyL>^L0+1, the parent HamiltonianH obtained from the previous construction hasψ A^[N]

as unique ground state.

Indeed in this case, setting for allK>^L0and 16ⁱ6^N,ψ˜^i,K:= ˜ψ A^{{i,...,i+K−1}}

, we have:

ψ˜^i,Linjective ⇒supp(ρⁱ) =range(˜ψ^i,L) =: Γ^i,L.

ψ˜^i+1,L−1injective ⇒Γ^i,L⊗(C^d)^⊗(N−L)∩Γ^i+1,L⊗(C^d)^⊗(N−L)= Γ^i,L+1⊗(C^d)^{⊗(N−L−1)}. Hence, iterating:

N

T

i=1

Γ^i,L⊗(C^d)^⊗(N−L)=range ψ˜ A^[N]

=span ψ A^[N] .

:=range(˜ψ^i,L+1)

ground state space ofH Explicitly:range(˜ψ^i,L) =span

( d

∑

x1,...,xL=1

Tr Aⁱ_x

1· · ·Aⁱ_x⁺_L^L−1T

|x₁· · ·x_Li:T q×qmatrix )

. L-site MPS with any boundary condition

Case of a PEPSψ A^[M]×[N]

:Same result forK>^K0+1,L>^L0+1.

Alternative construction: Instead ofMNterms which are(K₀+1)(L₀+1)-local, 2MNterms which are eitherK₀(L₀+1)-local or(K₀+1)L₀-local.

Spectral gap of the parent Hamiltonian?

We have shown:Any normal TNS is the unique ground state of a local Hamiltonian.

This Hamiltonian is additionally frustration-free and its range of interaction is related to the injectivity index of the TNS.

Question:What about the spectral gap of this Hamiltonian?

1D: The parent Hamiltonian is always gapped (Kastoryano/Lucia/Pérez-García).

2D: The parent Hamiltonian being gapped or not is an undecidable problem (Scarpa et al.), and it remains an open problem even for simple models.

Conjecture:The parent Hamiltonian of a PEPS being gapped is equivalent to its boundary state being a Gibbs state of a (1D) local Hamiltonian (Cirac/Poilblanc/Schuch/Verstraete).

Boundary state: Consider the PEPSψ(˜ A^[M]×[N])∈(C^d)^⊗MN⊗(C^q)^⊗2(M+N), with

non-contracted boundary. Take its partial trace over the bulk(C^d)^⊗MNto obtain a (mixed) state on the boundary(C^q)^⊗2(M+N).

What next?Understand classification of phases for TNS (cf. Schuch/Pérez-García/Cirac). Without symmetries: All normal MPS are in the same phase but it is not as simple for PEPS. With symmetries...

Spectral gap of the parent Hamiltonian?

We have shown:Any normal TNS is the unique ground state of a local Hamiltonian.

This Hamiltonian is additionally frustration-free and its range of interaction is related to the injectivity index of the TNS.

Question:What about the spectral gap of this Hamiltonian?

1D: The parent Hamiltonian is always gapped (Kastoryano/Lucia/Pérez-García).

2D: The parent Hamiltonian being gapped or not is an undecidable problem (Scarpa et al.), and it remains an open problem even for simple models.

Conjecture:The parent Hamiltonian of a PEPS being gapped is equivalent to its boundary state being a Gibbs state of a (1D) local Hamiltonian (Cirac/Poilblanc/Schuch/Verstraete).

Boundary state: Consider the PEPSψ(˜ A^[M]×[N])∈(C^d)^⊗MN⊗(C^q)^⊗2(M+N), with

non-contracted boundary. Take its partial trace over the bulk(C^d)^⊗MNto obtain a (mixed) state on the boundary(C^q)^⊗2(M+N).

What next?Understand classification of phases for TNS (cf. Schuch/Pérez-García/Cirac). Without symmetries: All normal MPS are in the same phase but it is not as simple for PEPS. With symmetries...

Spectral gap of the parent Hamiltonian?

We have shown:Any normal TNS is the unique ground state of a local Hamiltonian.

This Hamiltonian is additionally frustration-free and its range of interaction is related to the injectivity index of the TNS.

Question:What about the spectral gap of this Hamiltonian?

1D: The parent Hamiltonian is always gapped (Kastoryano/Lucia/Pérez-García).

2D: The parent Hamiltonian being gapped or not is an undecidable problem (Scarpa et al.), and it remains an open problem even for simple models.

Conjecture:The parent Hamiltonian of a PEPS being gapped is equivalent to its boundary state being a Gibbs state of a (1D) local Hamiltonian (Cirac/Poilblanc/Schuch/Verstraete).

Boundary state: Consider the PEPSψ(˜ A^[M]×[N])∈(C^d)^⊗MN⊗(C^q)^⊗2(M+N), with

non-contracted boundary. Take its partial trace over the bulk(C^d)^⊗MNto obtain a (mixed) state on the boundary(C^q)^⊗2(M+N).

What next?Understand classification of phases for TNS (cf. Schuch/Pérez-García/Cirac).

Without symmetries: All normal MPS are in the same phase but it is not as simple for PEPS.

With symmetries...

Correlations in a TNS

LetR,R⁰⊂Λand letO,O⁰be observables on(C^d)^⊗|R|,(C^d)^⊗|R0|.

Compute the value on the TNSψof the observableO_R⊗O_R⁰0⊗I_Λ\R∪R⁰, i.e.

v_ψ(O,O⁰,R,R⁰) :=hψ|OR⊗O_R⁰0⊗I_Λ\R∪R0|ψi.

Compare it to the product of the values onψofO_R⊗I_Λ\RandO_R⁰0⊗I_Λ\R0, i.e.

v_ψ(O,R)v_ψ(O⁰,R⁰) :=hψ|O_R⊗I_Λ\R|ψihψ|O_R⁰0⊗I_Λ\R⁰|ψi.

Correlationsin the TNSψ:γψ(O,O⁰,R,R⁰) : =

v_ψ(O,O⁰,R,R⁰)−v_ψ(O,R)v_ψ(O⁰,R⁰) . Question:Do we haveγ_ψ(O,O⁰,R,R⁰) '

d(R,R⁰)1

0? And if so, at which speed?

O O’

N

d(R,R⁰)

O O’

'?

d(R,R⁰)1 ×

Exponential decay of correlations in a TNS and spectral gap of its parent Hamiltonian

Intuition:Given a TNSψ A^Λ

, there existC,τ>0 s.t., for anyR,R⁰⊂Λand any observables O,O⁰on(C^d)^⊗|R|,(C^d)^⊗|R0|,

γ_ψ(O,O⁰,R,R⁰)6^{Ce−τd(R,R0)}kOk_∞kO⁰k_∞.

−→Correlations between two local observables decay exponentially with the distance separating them, at a rateτ.Correlation lengthofψ A^Λ

:ξ(ψ) =1/τ.

Theorem Letψ A^Λ

be an MPS, resp. PEPS, with injectivity indexL₀, resp.(K₀,L₀). LetHbe its(L₀+1)-local, resp.(K₀+1)×(L₀+1)-local, parent Hamiltonian. Assume thatHhas a spectral gap∆>0 above its ground state energy. Then, for anyR,R⁰⊂Λand any observablesO,O⁰on(C^d)^⊗|R|,(C^d)^⊗|R0|,

γ_ψ(O,O⁰,R,R⁰)6^{Ce−c∆d(R,R0)}kOk_∞kO⁰k_∞,

whereC>0 is an absolute constant andc>0 is a constant depending only onL₀, resp.K₀,L₀.

Exponential decay of correlations in a TNS and spectral gap of its parent Hamiltonian

Intuition:Given a TNSψ A^Λ

, there existC,τ>0 s.t., for anyR,R⁰⊂Λand any observables O,O⁰on(C^d)^⊗|R|,(C^d)^⊗|R0|,

γ_ψ(O,O⁰,R,R⁰)6^{Ce−τd(R,R0)}kOk_∞kO⁰k_∞.

−→Correlations between two local observables decay exponentially with the distance separating them, at a rateτ.Correlation lengthofψ A^Λ

:ξ(ψ) =1/τ. Theorem

Letψ A^Λ

be an MPS, resp. PEPS, with injectivity indexL₀, resp.(K₀,L₀). LetHbe its(L₀+1)-local, resp.(K₀+1)×(L₀+1)-local, parent Hamiltonian.

Assume thatHhas a spectral gap∆>0 above its ground state energy.

Then, for anyR,R⁰⊂Λand any observablesO,O⁰on(C^d)^⊗|R|,(C^d)^⊗|R0|, γ_ψ(O,O⁰,R,R⁰)6^{Ce−c∆d(R,R0)}kOk_∞kO⁰k_∞,

whereC>0 is an absolute constant andc>0 is a constant depending only onL₀, resp.K₀,L₀.

Detectability lemma

Tool to prove decay of correlations in a TNS from the spectral gap of its parent Hamiltonian:Detectability lemma (Aharonov/Arad/Landau/Vazirani, Gosset/Huang...) Interest: Technically simpler than proofs using Lieb-Robinson bounds (Hastings/Koma...) Goal: Construct an approximation of the ground state projector of a gapped local frustration-free Hamiltonian which leaves the ground state invariant and has some ‘locality structure’.

Detectability lemma LetH= ∑^N

i=1

H_ibe a frustration-free Hamiltonian on(C^d)^⊗N, where theH_i’s areL-local projectors, with spectral gap∆>0. LetΠbe its ground state projector.

Given a permutation ofNelementsσ, define the ordered productP_σ=∏^N

i=1

I−H_σ(i) . Then, for any permutation ofNelementsσ,

kΠ−P_σk_∞6p ¹ 1+ ∆/L².

Strategy:Choose a smart orderingσ... Partition the sites into sets so that sitesi’s in a given set do not interact, and thus the correspondingH_i’s commute.

Detectability lemma

Tool to prove decay of correlations in a TNS from the spectral gap of its parent Hamiltonian:Detectability lemma (Aharonov/Arad/Landau/Vazirani, Gosset/Huang...) Interest: Technically simpler than proofs using Lieb-Robinson bounds (Hastings/Koma...) Goal: Construct an approximation of the ground state projector of a gapped local frustration-free Hamiltonian which leaves the ground state invariant and has some ‘locality structure’.

Detectability lemma LetH= ∑^N

i=1

H_ibe a frustration-free Hamiltonian on(C^d)^⊗N, where theH_i’s areL-local projectors, with spectral gap∆>0. LetΠbe its ground state projector.

Given a permutation ofNelementsσ, define the ordered productP_σ=∏^N

i=1

I−H_σ(i) . Then, for any permutation ofNelementsσ,

kΠ−P_σk_∞6p ¹ 1+ ∆/L².

Strategy:Choose a smart orderingσ... Partition the sites into sets so that sitesi’s in a given set do not interact, and thus the correspondingH_i’s commute.

Detectability lemma

Tool to prove decay of correlations in a TNS from the spectral gap of its parent Hamiltonian:Detectability lemma (Aharonov/Arad/Landau/Vazirani, Gosset/Huang...) Interest: Technically simpler than proofs using Lieb-Robinson bounds (Hastings/Koma...) Goal: Construct an approximation of the ground state projector of a gapped local frustration-free Hamiltonian which leaves the ground state invariant and has some ‘locality structure’.

Detectability lemma LetH= ∑^N

i=1

H_ibe a frustration-free Hamiltonian on(C^d)^⊗N, where theH_i’s areL-local projectors, with spectral gap∆>0. LetΠbe its ground state projector.

Given a permutation ofNelementsσ, define the ordered productP_σ=∏^N

i=1

I−H_σ(i) . Then, for any permutation ofNelementsσ,

kΠ−P_σk_∞6p ¹ 1+ ∆/L².

Strategy:Choose a smart orderingσ... Partition the sites into sets so that sitesi’s in a given set

References

D. Aharonov, I. Arad, Z. Landau, U. Vazirani.The detectability lemma and quantum gap amplification.

2009.

D. Aharonov, I. Arad, Z. Landau, U. Vazirani.The detectability lemma and its applications to quantum Hamiltonian complexity. 2011.

J.I. Cirac, J. Garre-Rubio D. Pérez-García.Mathematical open problems in projected entangled pair states. 2019.

J.I. Cirac, D. Pérez-García, F. Verstraete, M.M. Wolf.PEPS as unique ground states of local Hamiltonians. 2008.

J.I. Cirac, D. Poilblanc, N. Schuch, F. Verstraete.Entanglement spectrum and boundary theories with projected entangled-pair states. 2011.

D. Gosset, Y. Huang.Correlation length versus gap in frustration-free systems. 2016.

M.B. Hastings.Solving gapped Hamiltonians locally. 2006.

M.B. Hastings, T. Koma.Spectral gap and exponential decay of correlations. 2006.

M.J. Kastoryano, A. Lucia, D. Pérez-García.Locality at the boundary implies gap in the bulk for 2D PEPS. 2019.

Z. Landau, U. Vazirani, T. Vidick.A polynomial-time algorithm for the ground state of 1D gapped local Hamiltonians. 2015.

G. Scarpa, A. Molnar, Y. Ge, J.J. García-Ripoll, N. Schuch, D. Pérez-García, S. Iblisdir.

Computational complexity of PEPS zero testing. 2018.

N. Schuch, D. Pérez-García, J.I. Cirac.Classifying quantum phases using matrix product states and projected entangled pair states. 2011.