Typical correlations in many-body quantum systems
Joint work with David Pérez-García (Universidad Complutense de Madrid) arXiv:1906.11682
Cécilia Lancien
Institut de Mathématiques de Toulouse & CNRS
Journée LPT-IMT, Toulouse – September 26 2019
Outline
1 Background and motivations
2 Random MPS and PEPS : construction and statement of the main questions & results
3 Typical correlation length in a random TNS through typical spectral gap of its transfer operator
4 Typical correlation length in a random TNS through spectral gap of its parent Hamiltonian
Cécilia Lancien Typical correlations in many-body quantum systems Journée LPT-IMT, Toulouse – September 26 2019 2
Mathematical formalism of quantum physics in a nutshell
Physical system composed of 1 particle : Complex Hilbert spaceH.
Physical system composed ofNparticles : Complex Hilbert spaceH1⊗ · · · ⊗HN, tensor product of the spaces corresponding to each particle.
State of systemH:
Purestate : unit vector inH(or rather associated rank 1 projector).
Mixedstate : convex combination of pure states, i.e. positive semidefinite and trace 1 operator onH.
A stateρonH1⊗H2isseparableif it is a convex combination of product states, i.e.ρ=∑
i
piσi1⊗τi2. Otherwise it isentangled.
(A pure state|ϕiis separable iff it is product, i.e.|ϕi=|φi1⊗ |ωi2.) Observable on systemH: Hermitian operator onH.
Valueof an observableAwhen measured on a system in stateρ: Tr(Aρ). (Ifρ=|ϕihϕ|is pure, this is simply equal tohϕ|A|ϕi.)
In this talk :Always finite-dimensional systems, i.e.H≡Cd,H1⊗ · · · ⊗HN≡Cd1⊗ · · · ⊗CdN.
Mathematical formalism of quantum physics in a nutshell
Physical system composed of 1 particle : Complex Hilbert spaceH.
Physical system composed ofNparticles : Complex Hilbert spaceH1⊗ · · · ⊗HN, tensor product of the spaces corresponding to each particle.
State of systemH:
Purestate : unit vector inH(or rather associated rank 1 projector).
Mixedstate : convex combination of pure states, i.e. positive semidefinite and trace 1 operator onH.
A stateρonH1⊗H2isseparableif it is a convex combination of product states, i.e.ρ=∑
i
piσi1⊗τi2. Otherwise it isentangled.
(A pure state|ϕiis separable iff it is product, i.e.|ϕi=|φi1⊗ |ωi2.)
Observable on systemH: Hermitian operator onH.
Valueof an observableAwhen measured on a system in stateρ: Tr(Aρ). (Ifρ=|ϕihϕ|is pure, this is simply equal tohϕ|A|ϕi.)
In this talk :Always finite-dimensional systems, i.e.H≡Cd,H1⊗ · · · ⊗HN≡Cd1⊗ · · · ⊗CdN.
Cécilia Lancien Typical correlations in many-body quantum systems Journée LPT-IMT, Toulouse – September 26 2019 3
Mathematical formalism of quantum physics in a nutshell
Physical system composed of 1 particle : Complex Hilbert spaceH.
Physical system composed ofNparticles : Complex Hilbert spaceH1⊗ · · · ⊗HN, tensor product of the spaces corresponding to each particle.
State of systemH:
Purestate : unit vector inH(or rather associated rank 1 projector).
Mixedstate : convex combination of pure states, i.e. positive semidefinite and trace 1 operator onH.
A stateρonH1⊗H2isseparableif it is a convex combination of product states, i.e.ρ=∑
i
piσi1⊗τi2. Otherwise it isentangled.
(A pure state|ϕiis separable iff it is product, i.e.|ϕi=|φi1⊗ |ωi2.) Observable on systemH: Hermitian operator onH.
Valueof an observableAwhen measured on a system in stateρ: Tr(Aρ). (Ifρ=|ϕihϕ|is pure, this is simply equal tohϕ|A|ϕi.)
In this talk :Always finite-dimensional systems, i.e.H≡Cd,H1⊗ · · · ⊗HN≡Cd1⊗ · · · ⊗CdN.
Mathematical formalism of quantum physics in a nutshell
Physical system composed of 1 particle : Complex Hilbert spaceH.
Physical system composed ofNparticles : Complex Hilbert spaceH1⊗ · · · ⊗HN, tensor product of the spaces corresponding to each particle.
State of systemH:
Purestate : unit vector inH(or rather associated rank 1 projector).
Mixedstate : convex combination of pure states, i.e. positive semidefinite and trace 1 operator onH.
A stateρonH1⊗H2isseparableif it is a convex combination of product states, i.e.ρ=∑
i
piσi1⊗τi2. Otherwise it isentangled.
(A pure state|ϕiis separable iff it is product, i.e.|ϕi=|φi1⊗ |ωi2.) Observable on systemH: Hermitian operator onH.
Valueof an observableAwhen measured on a system in stateρ: Tr(Aρ). (Ifρ=|ϕihϕ|is pure, this is simply equal tohϕ|A|ϕi.)
In this talk :Always finite-dimensional systems, i.e.H≡Cd,H1⊗ · · · ⊗HN≡Cd1⊗ · · · ⊗CdN.
Cécilia Lancien Typical correlations in many-body quantum systems Journée LPT-IMT, Toulouse – September 26 2019 3
Tensor network states on many-body quantum systems
A system composed ofN d-dimensional subsystems has dimensiondN.−→Exponential growth of the dimension with the number of particles.
Known :In several contexts, ‘physically relevant’ states of many-body quantum systems are well approximated by states from a very small subset of the global, exponentially large, state space.
Tensor network state (TNS) on(Cd)⊗N:
Take a graphGwithNvertices andLedges. Put at each vertexva tensor|χvi ∈Cd⊗(Cq)⊗δ(v) to get a tensor|˜χGi ∈(Cd)⊗N⊗(Cq)⊗2L. Contract together the indices of|˜χGicorresponding to a same edge to obtain a tensor|χGi ∈(Cd)⊗N.
−→Ifδ(v)6rfor allv, then such state is described by at mostNqrdparameters (linear rather than exponential inN).
• •
• •
• •
Gwith 6 vertices and 7 edges
• •
• •
• •
|˜χGi ∈(Cd)⊗6⊗(Cq)⊗14
• •
• •
• •
|χGi ∈(Cd)⊗6 d-dimensional indices :physicalindices.q-dimensional indices :bondindices.
In this talk :The underlying graphGis a regular lattice in dimension 1 or 2, so that|χGiis a matrix product state (MPS)or aprojected entangled pair state (PEPS).
Tensor network states on many-body quantum systems
A system composed ofN d-dimensional subsystems has dimensiondN.−→Exponential growth of the dimension with the number of particles.
Known :In several contexts, ‘physically relevant’ states of many-body quantum systems are well approximated by states from a very small subset of the global, exponentially large, state space.
Tensor network state (TNS) on(Cd)⊗N:
Take a graphGwithNvertices andLedges. Put at each vertexva tensor|χvi ∈Cd⊗(Cq)⊗δ(v) to get a tensor|˜χGi ∈(Cd)⊗N⊗(Cq)⊗2L. Contract together the indices of|˜χGicorresponding to a same edge to obtain a tensor|χGi ∈(Cd)⊗N.
−→Ifδ(v)6rfor allv, then such state is described by at mostNqrdparameters (linear rather than exponential inN).
• •
• •
• •
Gwith 6 vertices and 7 edges
• •
• •
• •
|˜χGi ∈(Cd)⊗6⊗(Cq)⊗14
• •
• •
• •
|χGi ∈(Cd)⊗6 d-dimensional indices :physicalindices.q-dimensional indices :bondindices.
In this talk :The underlying graphGis a regular lattice in dimension 1 or 2, so that|χGiis a matrix product state (MPS)or aprojected entangled pair state (PEPS).
Cécilia Lancien Typical correlations in many-body quantum systems Journée LPT-IMT, Toulouse – September 26 2019 4
Tensor network states on many-body quantum systems
A system composed ofN d-dimensional subsystems has dimensiondN.−→Exponential growth of the dimension with the number of particles.
Known :In several contexts, ‘physically relevant’ states of many-body quantum systems are well approximated by states from a very small subset of the global, exponentially large, state space.
Tensor network state (TNS) on(Cd)⊗N:
Take a graphGwithNvertices andLedges. Put at each vertexva tensor|χvi ∈Cd⊗(Cq)⊗δ(v) to get a tensor|˜χGi ∈(Cd)⊗N⊗(Cq)⊗2L. Contract together the indices of|˜χGicorresponding to a same edge to obtain a tensor|χGi ∈(Cd)⊗N.
−→Ifδ(v)6rfor allv, then such state is described by at mostNqrdparameters (linear rather than exponential inN).
• •
• •
• •
Gwith 6 vertices and 7 edges
• •
• •
• •
|˜χGi ∈(Cd)⊗6⊗(Cq)⊗14
• •
• •
• •
|χGi ∈(Cd)⊗6 d-dimensional indices :physicalindices.q-dimensional indices :bondindices.
In this talk :The underlying graphGis a regular lattice in dimension 1 or 2, so that|χGiis a matrix product state (MPS)or aprojected entangled pair state (PEPS).
Motivations behind TNS and related questions
Ideally, a TNS representation should be a (mathematically tractable) way of describing the state of a system composed of many sub-systems having a certain geometry and subject to interactions respecting this geometry.
Example :It is rigorously proven in some cases and conjectured in some others that being a TNS is approximately equivalent to being a ground state of a gapped local Hamiltonian (Hastings, Landau/Vazirani/Vidick...)
(‘gapped’=spectral gap lower bounded by a constant independent ofN, ‘local’=composed of terms which act only on nearby sites)
−→In condensed-matter physics, TNS are used as Ansatz in computations : optimization over a tractable number of parameters, even for largeN.
General question :Are statements of this kind at least true generically ?
−→What are the typical features of TNS ?
For instance : What is the typical amount of entanglement and correlations between two subsets of particles ? Is the parent Hamiltonian typically gapped or gapless ? etc.
Idea :Sample a TNS at random and study the characteristics that it exhibits with high probability. (Regime we are interested in : larged,q, ideally anyN.)
Cécilia Lancien Typical correlations in many-body quantum systems Journée LPT-IMT, Toulouse – September 26 2019 5
Motivations behind TNS and related questions
Ideally, a TNS representation should be a (mathematically tractable) way of describing the state of a system composed of many sub-systems having a certain geometry and subject to interactions respecting this geometry.
Example :It is rigorously proven in some cases and conjectured in some others that being a TNS is approximately equivalent to being a ground state of a gapped local Hamiltonian (Hastings, Landau/Vazirani/Vidick...)
(‘gapped’=spectral gap lower bounded by a constant independent ofN, ‘local’=composed of terms which act only on nearby sites)
−→In condensed-matter physics, TNS are used as Ansatz in computations : optimization over a tractable number of parameters, even for largeN.
General question :Are statements of this kind at least true generically ?
−→What are the typical features of TNS ?
For instance : What is the typical amount of entanglement and correlations between two subsets of particles ? Is the parent Hamiltonian typically gapped or gapless ? etc.
Idea :Sample a TNS at random and study the characteristics that it exhibits with high probability. (Regime we are interested in : larged,q, ideally anyN.)
Motivations behind TNS and related questions
Ideally, a TNS representation should be a (mathematically tractable) way of describing the state of a system composed of many sub-systems having a certain geometry and subject to interactions respecting this geometry.
Example :It is rigorously proven in some cases and conjectured in some others that being a TNS is approximately equivalent to being a ground state of a gapped local Hamiltonian (Hastings, Landau/Vazirani/Vidick...)
(‘gapped’=spectral gap lower bounded by a constant independent ofN, ‘local’=composed of terms which act only on nearby sites)
−→In condensed-matter physics, TNS are used as Ansatz in computations : optimization over a tractable number of parameters, even for largeN.
General question :Are statements of this kind at least true generically ?
−→What are the typical features of TNS ?
For instance : What is the typical amount of entanglement and correlations between two subsets of particles ? Is the parent Hamiltonian typically gapped or gapless ? etc.
Idea :Sample a TNS at random and study the characteristics that it exhibits with high probability. (Regime we are interested in : larged,q, ideally anyN.)
Cécilia Lancien Typical correlations in many-body quantum systems Journée LPT-IMT, Toulouse – September 26 2019 5
Motivations behind TNS and related questions
Ideally, a TNS representation should be a (mathematically tractable) way of describing the state of a system composed of many sub-systems having a certain geometry and subject to interactions respecting this geometry.
Example :It is rigorously proven in some cases and conjectured in some others that being a TNS is approximately equivalent to being a ground state of a gapped local Hamiltonian (Hastings, Landau/Vazirani/Vidick...)
(‘gapped’=spectral gap lower bounded by a constant independent ofN, ‘local’=composed of terms which act only on nearby sites)
−→In condensed-matter physics, TNS are used as Ansatz in computations : optimization over a tractable number of parameters, even for largeN.
General question :Are statements of this kind at least true generically ?
−→What are the typical features of TNS ?
For instance : What is the typical amount of entanglement and correlations between two subsets of particles ? Is the parent Hamiltonian typically gapped or gapless ? etc.
Idea :Sample a TNS at random and study the characteristics that it exhibits with high probability.
(Regime we are interested in : larged,q, ideally anyN.)
Outline
1 Background and motivations
2 Random MPS and PEPS : construction and statement of the main questions & results
3 Typical correlation length in a random TNS through typical spectral gap of its transfer operator
4 Typical correlation length in a random TNS through spectral gap of its parent Hamiltonian
Cécilia Lancien Typical correlations in many-body quantum systems Journée LPT-IMT, Toulouse – September 26 2019 6
A simple model of random translation-invariant MPS
Pick a tensor|χi ∈Cd⊗(Cq)⊗2whose entries are independent complex Gaussians with mean 0 and variance 1/dq. Repeat it onNsites disposed on a circle. Contract the neighbouring q-dimensional indices. The obtained tensor|χNi ∈(Cd)⊗Nis arandom translation-invariant MPS with periodic boundary conditions.
d q q
|χi= ∑d
x=1 q
∑
i,j=1
gxij|xiji
N
|χNi= ∑d
x1,...,xN=1
q
∑
i1,...,iN=1
gx1iNi1· · ·gxNiN−1iN
|x1· · ·xNi
Associatedtransfer operator:T:Cq⊗Cq→Cq⊗Cqobtained by contracting thed-dimensional indices of|χiand|¯χi.
T= ∑d
x=1
q
∑
i,j,k,l=1
gxij¯gxkl|ikihjl|
=1d ∑d
x=1
Gx⊗G¯x
TheGx’s are independentq×qmatrices whose entries are independent complex Gaussians with mean 0 and variance 1/q.
A simple model of random translation-invariant MPS
Pick a tensor|χi ∈Cd⊗(Cq)⊗2whose entries are independent complex Gaussians with mean 0 and variance 1/dq. Repeat it onNsites disposed on a circle. Contract the neighbouring q-dimensional indices. The obtained tensor|χNi ∈(Cd)⊗Nis arandom translation-invariant MPS with periodic boundary conditions.
d q q
|χi= ∑d
x=1 q
∑
i,j=1
gxij|xiji
N
|χNi= ∑d
x1,...,xN=1
q
∑
i1,...,iN=1
gx1iNi1· · ·gxNiN−1iN
|x1· · ·xNi
Associatedtransfer operator:T:Cq⊗Cq→Cq⊗Cqobtained by contracting thed-dimensional indices of|χiand|¯χi.
T= ∑d
x=1
q
∑
i,j,k,l=1
gxij¯gxkl|ikihjl|
=1d ∑d
x=1
Gx⊗G¯x
TheGx’s are independentq×qmatrices whose entries are independent complex Gaussians with mean 0 and variance 1/q.
Cécilia Lancien Typical correlations in many-body quantum systems Journée LPT-IMT, Toulouse – September 26 2019 7
A simple model of random translation-invariant PEPS
Pick a tensor|χi ∈Cd⊗(Cq)⊗4whose entries are independent complex Gaussians with mean 0 and variance 1/dq2. Repeat it onMNsites disposed on a torus. Contract the neighbouring q-dimensional indices. The obtained tensor|χMNi ∈(Cd)⊗MNis arandom translation-invariant PEPS with periodic boundary conditions.
q d q
q q
|χi= ∑d
x=1 q
∑
i,j,i0,j0=1
gxiji0j0|xiji0j0i
M
|χMi ∈(Cd⊗Cq⊗Cq)⊗M
Associatedtransfer operator:TM: (Cq⊗Cq)⊗M→(Cq⊗Cq)⊗Mobtained by contracting the d-dimensional indices of|χMiand|¯χMi.
TM=dM1qM d
∑
x1,...,xM=1 q
∑
i1,j1,...,iM,jM=1
Gx1iMi1⊗G¯x
1jMj1⊗ · · · ⊗GxMiM−1iM⊗G¯x
MjM−1jM
TheGxij’s are independentq×qmatrices whose entries are independent complex Gaussians with mean 0 and variance 1/q.
A simple model of random translation-invariant PEPS
Pick a tensor|χi ∈Cd⊗(Cq)⊗4whose entries are independent complex Gaussians with mean 0 and variance 1/dq2. Repeat it onMNsites disposed on a torus. Contract the neighbouring q-dimensional indices. The obtained tensor|χMNi ∈(Cd)⊗MNis arandom translation-invariant PEPS with periodic boundary conditions.
q d q
q q
|χi= ∑d
x=1 q
∑
i,j,i0,j0=1
gxiji0j0|xiji0j0i
M
|χMi ∈(Cd⊗Cq⊗Cq)⊗M
Associatedtransfer operator:TM: (Cq⊗Cq)⊗M→(Cq⊗Cq)⊗Mobtained by contracting the d-dimensional indices of|χMiand|¯χMi.
TM=dM1qM d
∑
x1,...,xM=1 q
∑
i1,j1,...,iM,jM=1
Gx1iMi1⊗G¯x
1jMj1⊗ · · · ⊗GxMiM−1iM⊗G¯x
MjM−1jM
TheGxij’s are independentq×qmatrices whose entries are independent complex Gaussians with mean 0 and variance 1/q.
Cécilia Lancien Typical correlations in many-body quantum systems Journée LPT-IMT, Toulouse – September 26 2019 8
Correlations in a TNS
MPS :M=1. PEPS :M>1.A,Bobservables on(Cd)⊗M, i.e. on 1 site for an MPS and on 1 row ofMsites for a PEPS.
A
A:Cd→Cd
M
A
A: (Cd)⊗M→(Cd)⊗M
Compute the value on the TNS|χMNiof the observableA1⊗Idk⊗B1⊗IdN−k−2, i.e. vχ(A,B,k) : = hχMN|A1⊗Idk⊗B1⊗IdN−k−2|χMNi
hχMN|χMNi .
Compare it to the product of the values on|χMNiofA1⊗IdN−1andIdk+1⊗B1⊗IdN−k−2, i.e. vχ(A)vχ(B) : =hχMN|A1⊗IdN−1|χMNihχMN|B1⊗IdN−1|χMNi
hχMN|χMNi2 . Correlationsin the TNS|χMNi:γχ(A,B,k) : =
vχ(A,B,k)−vχ(A)vχ(B) . Question :Do we haveγχ(A,B,k) −→
kN→∞0 ? And if so, at which speed ?
A B
N
k
A B
'?
Nk1 ×
Correlations in a TNS
MPS :M=1. PEPS :M>1.A,Bobservables on(Cd)⊗M, i.e. on 1 site for an MPS and on 1 row ofMsites for a PEPS.
A
A:Cd→Cd
M
A
A: (Cd)⊗M→(Cd)⊗M
Compute the value on the TNS|χMNiof the observableA1⊗Idk⊗B1⊗IdN−k−2, i.e.
vχ(A,B,k) : = hχMN|A1⊗Idk⊗B1⊗IdN−k−2|χMNi hχMN|χMNi .
Compare it to the product of the values on|χMNiofA1⊗IdN−1andIdk+1⊗B1⊗IdN−k−2, i.e.
vχ(A)vχ(B) : =hχMN|A1⊗IdN−1|χMNihχMN|B1⊗IdN−1|χMNi hχMN|χMNi2 . Correlationsin the TNS|χMNi:γχ(A,B,k) : =
vχ(A,B,k)−vχ(A)vχ(B) . Question :Do we haveγχ(A,B,k) −→
kN→∞0 ? And if so, at which speed ?
A B
N
k
A B
'?
Nk1 ×
Cécilia Lancien Typical correlations in many-body quantum systems Journée LPT-IMT, Toulouse – September 26 2019 9
Main result : random TNS typically exhibit fast exponential decay of correlations
Intuition :Given a TNS|χMNi ∈(Cd)⊗MN, there existC,τ>0 such that, for anykNand any observablesA,Bon(Cd)⊗M,γχ(A,B,k)6Ce−τkkAk∞kBk∞.
−→Correlations between two 1-site or 1-row observables decay exponentially with the distance separating the two sites or rows, at a rateτ.
Correlation lengthin the TNS|χMNi:ξ(χ) : =1/τ.
Our main result (informal)
For random translation-invariant MPS and PEPS with periodic boundary conditions this intuition is generically true, with a short correlation length.
(‘generically’=with probability going to 1 asd,qgrow, ‘short’=going to 0 asd,qgrow) Two ways of proving this :(suited to different dimensional regimes)
Show that the transfer operator associated to the random TNS generically has a large (upper) spectral gap.
Show that the parent Hamiltonian of the random TNS generically has a large (lower) spectral gap.
(‘large’=going to 1 asd,qgrow)
Main result : random TNS typically exhibit fast exponential decay of correlations
Intuition :Given a TNS|χMNi ∈(Cd)⊗MN, there existC,τ>0 such that, for anykNand any observablesA,Bon(Cd)⊗M,γχ(A,B,k)6Ce−τkkAk∞kBk∞.
−→Correlations between two 1-site or 1-row observables decay exponentially with the distance separating the two sites or rows, at a rateτ.
Correlation lengthin the TNS|χMNi:ξ(χ) : =1/τ. Our main result (informal)
For random translation-invariant MPS and PEPS with periodic boundary conditions this intuition is generically true, with a short correlation length.
(‘generically’=with probability going to 1 asd,qgrow, ‘short’=going to 0 asd,qgrow)
Two ways of proving this :(suited to different dimensional regimes)
Show that the transfer operator associated to the random TNS generically has a large (upper) spectral gap.
Show that the parent Hamiltonian of the random TNS generically has a large (lower) spectral gap.
(‘large’=going to 1 asd,qgrow)
Cécilia Lancien Typical correlations in many-body quantum systems Journée LPT-IMT, Toulouse – September 26 2019 10
Main result : random TNS typically exhibit fast exponential decay of correlations
Intuition :Given a TNS|χMNi ∈(Cd)⊗MN, there existC,τ>0 such that, for anykNand any observablesA,Bon(Cd)⊗M,γχ(A,B,k)6Ce−τkkAk∞kBk∞.
−→Correlations between two 1-site or 1-row observables decay exponentially with the distance separating the two sites or rows, at a rateτ.
Correlation lengthin the TNS|χMNi:ξ(χ) : =1/τ. Our main result (informal)
For random translation-invariant MPS and PEPS with periodic boundary conditions this intuition is generically true, with a short correlation length.
(‘generically’=with probability going to 1 asd,qgrow, ‘short’=going to 0 asd,qgrow) Two ways of proving this :(suited to different dimensional regimes)
Show that the transfer operator associated to the random TNS generically has a large (upper) spectral gap.
Show that the parent Hamiltonian of the random TNS generically has a large (lower) spectral gap.
(‘large’=going to 1 asd,qgrow)
Outline
1 Background and motivations
2 Random MPS and PEPS : construction and statement of the main questions & results
3 Typical correlation length in a random TNS through typical spectral gap of its transfer operator
4 Typical correlation length in a random TNS through spectral gap of its parent Hamiltonian
Cécilia Lancien Typical correlations in many-body quantum systems Journée LPT-IMT, Toulouse – September 26 2019 11
Correlation length in a TNS and spectrum of its transfer operator
|χMNi ∈(Cd)⊗MNan MPS (M=1) or a PEPS (M>1).
Tits associated transfer operator on(Cq⊗Cq)⊗M. Its correlation function can be re-written as : γχ(A,B,k) =
Tr AT˜ k˜BTN−k−2
Tr(TN) −Tr AT˜ N−1
Tr BT˜ N−1 (Tr(TN))2
.
A
A:Cd→Cd
A
A˜:Cq⊗Cq→Cq⊗Cq
Important consequence :
Denote byλ1(T),λ2(T), . . .the eigenvalues ofT, ordered such that|λ1(T)|>|λ2(T)|>· · ·. Set∆(T) =|λ1(T)| − |λ2(T)|andε(T) =|λ2(T)|/|λ1(T)|.
γχ(A,B,k)6C Tr
(T)
∆(T) 2
ε(T)kkAk∞kBk∞.
−→Correlations between two 1-site or 1-row observables decay exponentially with the distance separating the two sites or rows, at a rate|logε(T)|, i.e.ξ(χ) =1/|logε(T)|.
Question :For a randomTwhat are typically the values of|λ1(T)|and|λ2(T)|?
Correlation length in a TNS and spectrum of its transfer operator
|χMNi ∈(Cd)⊗MNan MPS (M=1) or a PEPS (M>1).
Tits associated transfer operator on(Cq⊗Cq)⊗M. Its correlation function can be re-written as : γχ(A,B,k) =
Tr AT˜ k˜BTN−k−2
Tr(TN) −Tr AT˜ N−1
Tr BT˜ N−1 (Tr(TN))2
.
A
A:Cd→Cd
A
A˜:Cq⊗Cq→Cq⊗Cq
Important consequence :
Denote byλ1(T),λ2(T), . . .the eigenvalues ofT, ordered such that|λ1(T)|>|λ2(T)|>· · ·. Set∆(T) =|λ1(T)| − |λ2(T)|andε(T) =|λ2(T)|/|λ1(T)|.
γχ(A,B,k)6C Tr
(T)
∆(T) 2
ε(T)kkAk∞kBk∞.
−→Correlations between two 1-site or 1-row observables decay exponentially with the distance separating the two sites or rows, at a rate|logε(T)|, i.e.ξ(χ) =1/|logε(T)|.
Question :For a randomTwhat are typically the values of|λ1(T)|and|λ2(T)|?
Cécilia Lancien Typical correlations in many-body quantum systems Journée LPT-IMT, Toulouse – September 26 2019 12
Correlation length in a TNS and spectrum of its transfer operator
|χMNi ∈(Cd)⊗MNan MPS (M=1) or a PEPS (M>1).
Tits associated transfer operator on(Cq⊗Cq)⊗M. Its correlation function can be re-written as : γχ(A,B,k) =
Tr AT˜ k˜BTN−k−2
Tr(TN) −Tr AT˜ N−1
Tr BT˜ N−1 (Tr(TN))2
.
A
A:Cd→Cd
A
A˜:Cq⊗Cq→Cq⊗Cq
Important consequence :
Denote byλ1(T),λ2(T), . . .the eigenvalues ofT, ordered such that|λ1(T)|>|λ2(T)|>· · ·. Set∆(T) =|λ1(T)| − |λ2(T)|andε(T) =|λ2(T)|/|λ1(T)|.
γχ(A,B,k)6C Tr
(T)
∆(T) 2
ε(T)kkAk∞kBk∞.
−→Correlations between two 1-site or 1-row observables decay exponentially with the distance separating the two sites or rows, at a rate|logε(T)|, i.e.ξ(χ) =1/|logε(T)|.
Question :For a randomTwhat are typically the values of|λ1(T)|and|λ2(T)|?
Typical spectral gap of the transfer operator of a random MPS
Theorem (Typical spectral gap of the random MPS transfer operatorT) There exist universal constantsc,C>0 such that, for anyd,q∈N,
P
|λ1(T)|>1−√C
d and|λ2(T)|6√C d
>1−e−cq.
Corollary (Typical correlation length in the random MPS|χNi) There exist universal constantsc,C>0 such that, for anyd,q,N∈N,
P
ξ(χ)6 C logd
>1−e−cmin(q,d1/3).
Remarks :
Comparable results obtained on slightly different models : by Hastings and Pisier with unitary operators, by González-Guillén/Junge/Nechita with truncated unitary operators. Other interpretation :
T
:X7→d1 ∑dx=1
GxXG∗x, random completely positive map.
T
is with high probability a goodquantum expander(regularity≡high entropy approximate fixed point, low degree≡few Kraus operators, high expansion≡large spectral gap).Cécilia Lancien Typical correlations in many-body quantum systems Journée LPT-IMT, Toulouse – September 26 2019 13
Typical spectral gap of the transfer operator of a random MPS
Theorem (Typical spectral gap of the random MPS transfer operatorT) There exist universal constantsc,C>0 such that, for anyd,q∈N,
P
|λ1(T)|>1−√C
d and|λ2(T)|6√C d
>1−e−cq.
Corollary (Typical correlation length in the random MPS|χNi) There exist universal constantsc,C>0 such that, for anyd,q,N∈N,
P
ξ(χ)6 C logd
>1−e−cmin(q,d1/3).
Remarks :
Comparable results obtained on slightly different models : by Hastings and Pisier with unitary operators, by González-Guillén/Junge/Nechita with truncated unitary operators. Other interpretation :
T
:X7→d1 ∑dx=1
GxXG∗x, random completely positive map.
T
is with high probability a goodquantum expander(regularity≡high entropy approximate fixed point, low degree≡few Kraus operators, high expansion≡large spectral gap).Typical spectral gap of the transfer operator of a random MPS
Theorem (Typical spectral gap of the random MPS transfer operatorT) There exist universal constantsc,C>0 such that, for anyd,q∈N,
P
|λ1(T)|>1−√C
d and|λ2(T)|6√C d
>1−e−cq.
Corollary (Typical correlation length in the random MPS|χNi) There exist universal constantsc,C>0 such that, for anyd,q,N∈N,
P
ξ(χ)6 C logd
>1−e−cmin(q,d1/3).
Remarks :
Comparable results obtained on slightly different models : by Hastings and Pisier with unitary operators, by González-Guillén/Junge/Nechita with truncated unitary operators.
Other interpretation :
T
:X7→d1 ∑dx=1
GxXG∗x, random completely positive map.
T
is with high probability a goodquantum expander(regularity≡high entropy approximate fixed point, low degree≡few Kraus operators, high expansion≡large spectral gap).Cécilia Lancien Typical correlations in many-body quantum systems Journée LPT-IMT, Toulouse – September 26 2019 13
Proof strategy
Let’s do as ifTwere Hermitian and thus forget about all the subtleties eigen vs singular values...
By the variational characterization of eigenvalues : for any unit vector|ϕi ∈(Cq)⊗2,
|λ1(T)|>|hϕ|T|ϕi|and|λ2(T)|6kT(Id− |ϕihϕ|)k∞.
−→Goal :Find a|ϕiwhich gives good such lower and upper bounds.
Candidate :Maximally entangledunit vector|ψi: =√1q
q
∑
i=1
|iii ∈Cq⊗Cq. Intuition :Recall thatT=1d ∑d
x=1
Gx⊗G¯x.
First,ET =|ψihψ|. SoTis expected to have largest eigenvalue close to 1 and corresponding eigenvector close to|ψi. Second, the singular value distribution of√
d(T− |ψihψ|)converges in moments to the quarter-circle distribution (Aubrun/Nechita). So asymptotically the singular values ofT− |ψihψ|are almost all of order at most 1/√
d(but what about potential isolated ones ?) Two steps in the proof :
Show that :E|hψ|T|ψi|=1 andEkT(Id− |ψihψ|)k∞6C/√ d.
Show that :f: (G1, . . . ,Gd)7→ kTPk∞, forPany fixed projector (e.g.P=|ψihψ|or P=Id− |ψihψ|), has a small probability of deviating from its average (local version of Gaussian concentration inequality by Aubrun/Szarek/Werner).
Proof strategy
Let’s do as ifTwere Hermitian and thus forget about all the subtleties eigen vs singular values...
By the variational characterization of eigenvalues : for any unit vector|ϕi ∈(Cq)⊗2,
|λ1(T)|>|hϕ|T|ϕi|and|λ2(T)|6kT(Id− |ϕihϕ|)k∞.
−→Goal :Find a|ϕiwhich gives good such lower and upper bounds.
Candidate :Maximally entangledunit vector|ψi: =√1q
q
∑
i=1
|iii ∈Cq⊗Cq. Intuition :Recall thatT=1d ∑d
x=1
Gx⊗G¯x.
First,ET =|ψihψ|. SoTis expected to have largest eigenvalue close to 1 and corresponding eigenvector close to|ψi. Second, the singular value distribution of√
d(T− |ψihψ|)converges in moments to the quarter-circle distribution (Aubrun/Nechita). So asymptotically the singular values ofT− |ψihψ|are almost all of order at most 1/√
d(but what about potential isolated ones ?)
Two steps in the proof :
Show that :E|hψ|T|ψi|=1 andEkT(Id− |ψihψ|)k∞6C/√ d.
Show that :f: (G1, . . . ,Gd)7→ kTPk∞, forPany fixed projector (e.g.P=|ψihψ|or P=Id− |ψihψ|), has a small probability of deviating from its average (local version of Gaussian concentration inequality by Aubrun/Szarek/Werner).
Cécilia Lancien Typical correlations in many-body quantum systems Journée LPT-IMT, Toulouse – September 26 2019 14
Proof strategy
Let’s do as ifTwere Hermitian and thus forget about all the subtleties eigen vs singular values...
By the variational characterization of eigenvalues : for any unit vector|ϕi ∈(Cq)⊗2,
|λ1(T)|>|hϕ|T|ϕi|and|λ2(T)|6kT(Id− |ϕihϕ|)k∞.
−→Goal :Find a|ϕiwhich gives good such lower and upper bounds.
Candidate :Maximally entangledunit vector|ψi: =√1q
q
∑
i=1
|iii ∈Cq⊗Cq. Intuition :Recall thatT=1d ∑d
x=1
Gx⊗G¯x.
First,ET =|ψihψ|. SoTis expected to have largest eigenvalue close to 1 and corresponding eigenvector close to|ψi. Second, the singular value distribution of√
d(T− |ψihψ|)converges in moments to the quarter-circle distribution (Aubrun/Nechita). So asymptotically the singular values ofT− |ψihψ|are almost all of order at most 1/√
d(but what about potential isolated ones ?) Two steps in the proof :
Show that :E|hψ|T|ψi|=1 andEkT(Id− |ψihψ|)k∞6C/√ d.
Show that :f : (G1, . . . ,Gd)7→ kTPk∞, forPany fixed projector (e.g.P=|ψihψ|or P=Id− |ψihψ|), has a small probability of deviating from its average (local version of Gaussian concentration inequality by Aubrun/Szarek/Werner).
Typical spectral gap of the transfer operator of a random PEPS
Theorem (Typical spectral gap of the random PEPS transfer operatorTM)
There exist universal constantsc,C>0 such that, for anyd,q,M∈Nsatisfyingd'Mαand q'Mβwithα>8 and(α+1)/3<β<(α−2)/2,
P
|λ1(TM)|>1− C
Mα/2−1−β and|λ2(TM)|6 C Mα/2−1−β
>1−e−cM3β−α.
Corollary (Typical correlation length in the random PEPS|χMNi)
There exist universal constantsC0,c,C>0 such that, for anyd,q,M,N∈NsatisfyingN>C0M, d'Mαandq'Mβwithα>11 and(α+1)/3<β<(α−3)/2,
P
ξ(χ)6 C log(Mα/2−1−β)
>1−e−cM1/2.
Main issue :Results are valid only in the regime whered,qgrow polynomially withM,N.
−→Enforce this scaling byblocking: Start from a lattice ofMˆ×Nˆsites, where Mˆ=M√
logM,Nˆ=N√
logM, with physical and bond dimensionsˆd,ˆq= ˇq
√logM. Redefine 1 site as being√
logM×√
logMsites, to obtain a lattice ofM×Nsites, with physical and bond dimensionsd= ˆdlogM,q= ˆq
√logM= ˇqlogM(i.e.d=Mα,q=Mβforα=logˆd,β: =logˇq).
Cécilia Lancien Typical correlations in many-body quantum systems Journée LPT-IMT, Toulouse – September 26 2019 15
Typical spectral gap of the transfer operator of a random PEPS
Theorem (Typical spectral gap of the random PEPS transfer operatorTM)
There exist universal constantsc,C>0 such that, for anyd,q,M∈Nsatisfyingd'Mαand q'Mβwithα>8 and(α+1)/3<β<(α−2)/2,
P
|λ1(TM)|>1− C
Mα/2−1−β and|λ2(TM)|6 C Mα/2−1−β
>1−e−cM3β−α.
Corollary (Typical correlation length in the random PEPS|χMNi)
There exist universal constantsC0,c,C>0 such that, for anyd,q,M,N∈NsatisfyingN>C0M, d'Mαandq'Mβwithα>11 and(α+1)/3<β<(α−3)/2,
P
ξ(χ)6 C log(Mα/2−1−β)
>1−e−cM1/2.
Main issue :Results are valid only in the regime whered,qgrow polynomially withM,N.
−→Enforce this scaling byblocking: Start from a lattice ofMˆ×Nˆsites, where Mˆ=M√
logM,Nˆ=N√
logM, with physical and bond dimensionsˆd,ˆq= ˇq
√logM. Redefine 1 site as being√
logM×√
logMsites, to obtain a lattice ofM×Nsites, with physical and bond dimensionsd= ˆdlogM,q= ˆq
√logM= ˇqlogM(i.e.d=Mα,q=Mβforα=logˆd,β: =logˇq).
Typical spectral gap of the transfer operator of a random PEPS
Theorem (Typical spectral gap of the random PEPS transfer operatorTM)
There exist universal constantsc,C>0 such that, for anyd,q,M∈Nsatisfyingd'Mαand q'Mβwithα>8 and(α+1)/3<β<(α−2)/2,
P
|λ1(TM)|>1− C
Mα/2−1−β and|λ2(TM)|6 C Mα/2−1−β
>1−e−cM3β−α.
Corollary (Typical correlation length in the random PEPS|χMNi)
There exist universal constantsC0,c,C>0 such that, for anyd,q,M,N∈NsatisfyingN>C0M, d'Mαandq'Mβwithα>11 and(α+1)/3<β<(α−3)/2,
P
ξ(χ)6 C log(Mα/2−1−β)
>1−e−cM1/2.
Main issue :Results are valid only in the regime whered,qgrow polynomially withM,N.
−→Enforce this scaling byblocking: Start from a lattice ofMˆ×Nˆsites, where Mˆ=M√
logM,Nˆ=N√
logM, with physical and bond dimensionsˆd,ˆq= ˇq
√logM. Redefine 1 site as being√
logM×√
logMsites, to obtain a lattice ofM×Nsites, with physical and bond dimensionsd= ˆdlogM,q= ˆq
√logM= ˇqlogM(i.e.d=Mα,q=Mβforα=logˆd,β: =logˇq).
Cécilia Lancien Typical correlations in many-body quantum systems Journée LPT-IMT, Toulouse – September 26 2019 15
Idea of the proof and extension of the result
Recall thatTM=dM1qM d
∑
x1,...,xM=1 q
∑
i1,j1,...,iM,jM=1
Gx1iMi1⊗G¯x
1jMj1⊗ · · · ⊗GxMiM−1iM⊗G¯x
MjM−1jM. Proof idea :Some kind of recursion procedure that uses the MPS results as building blocks.
Approximate first eigenvector :|ψi⊗M∈(Cq⊗Cq)⊗M.
Extension :The result on the typical spectral gap ofTMremains unchanged in the non translation-invariant case (1-site random tensors picked independently identically distributed).
−→Potential application : Let
T
Mbe the corresponding random completely positive map, i.e.T
M:X7→ 1 dMqMd
∑
x1,...,xM=1 q
∑
i1,j1,...,iM,jM=1
Gx1
1iMi1⊗ · · · ⊗GMx
MiM−1iM
X G1x∗
1jMj1⊗ · · · ⊗GxM∗
MjM−1jM
.
It is a model of random local dissipative evolution, in analogy to random local circuits in the reversible setting.
Main question : How close to the maximally mixed state is the approximate fixed point of