Typical correlations in many-body quantum systems

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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

GdT Modélisation, Paris 7 – November 14 2019

Cécilia Lancien Typical correlations in many-body quantum systems GdT Modélisation, Paris 7 – November 14 2019 1

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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

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Mathematical formalism of quantum physics in a nutshell

Physical system composed of 1 particle : Complex Hilbert spaceH.

Physical system composed ofNparticles : Complex Hilbert spaceH₁⊗ · · · ⊗H_N, 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≡C^d,H1⊗ · · · ⊗HN≡C^d¹⊗ · · · ⊗C^d^N.

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Mathematical formalism of quantum physics in a nutshell

State of systemH:

i

p_iσⁱ₁⊗τⁱ₂. Otherwise it isentangled.

(A pure state|ϕiis separable iff it is product, i.e.|ϕi=|φi₁⊗ |ωi₂.)

Observable on systemH: Hermitian operator onH.

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Mathematical formalism of quantum physics in a nutshell

State of systemH:

i

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Mathematical formalism of quantum physics in a nutshell

State of systemH:

i

In this talk :Always finite-dimensional systems, i.e.H≡C^d,H₁⊗ · · · ⊗H_N≡C^d¹⊗ · · · ⊗C^d^N.

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Tensor network states on many-body quantum systems

A system composed ofN d-dimensional subsystems has dimensiond^N.

−→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(C^d)^⊗^N:

Take a graphGwithNvertices andLedges. Put at each vertexva tensor|χvi ∈C^d⊗(C^q)^⊗δ(^v⁾ to get a tensor|˜χGi ∈(C^d)^⊗^N⊗(C^q)^⊗^2L. Contract together the indices of|˜χGicorresponding to a same edge to obtain a tensor|χ_Gi ∈(C^d)^⊗^N.

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

• •

Gwith 6 vertices and 7 edges

• •

|˜χGi ∈(C^d)^⊗⁶⊗(C^q)^⊗¹⁴

• •

|χ_Gi ∈(C^d)^⊗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).

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Tensor network states on many-body quantum systems

Take a graphGwithNvertices andLedges. Put at each vertexva tensor|χvi ∈C^d⊗(C^q)^⊗δ(^v⁾ to get a tensor|˜χ_Gi ∈(C^d)^⊗^N⊗(C^q)^⊗^2L. Contract together the indices of|˜χ_Gicorresponding to a same edge to obtain a tensor|χ_Gi ∈(C^d)^⊗^N.

• •

|˜χGi ∈(C^d)^⊗⁶⊗(C^q)^⊗¹⁴

• •

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Tensor network states on many-body quantum systems

Take a graphGwithNvertices andLedges. Put at each vertexva tensor|χvi ∈C^d⊗(C^q)^⊗δ(^v⁾ to get a tensor|˜χ_Gi ∈(C^d)^⊗^N⊗(C^q)^⊗^2L. Contract together the indices of|˜χ_Gicorresponding to a same edge to obtain a tensor|χ_Gi ∈(C^d)^⊗^N.

• •

|˜χGi ∈(C^d)^⊗⁶⊗(C^q)^⊗¹⁴

• •

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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.)

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Motivations behind TNS and related questions

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Motivations behind TNS and related questions

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Motivations behind TNS and related questions

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.)

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Outline

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A simple model of random translation-invariant MPS

Pick a tensor|χi ∈C^d⊗(C^q)^⊗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 ∈(C^d)^⊗^Nis arandom translation-invariant MPS with periodic boundary conditions.

d q q

|χi= ∑^d

x=1 q

∑

i,j=1

g_xij|xiji

N

|χNi= ∑^d

x1,...,xN=1

q

∑

i1,...,iN=1

g_x₁_i_N_i₁· · ·g_x_N_i_N₋₁_i_N

|x₁· · ·x_Ni

Associatedtransfer operator:T:C^q⊗C^q→C^q⊗C^qobtained by contracting thed-dimensional indices of|χiand|¯χi.

T= ∑^d

x=1

q

∑

i,j,k,l=1

g_xij¯g_xkl|ikihjl|

=¹_d ∑^d

x=1

G_x⊗_G¯_x

TheG_x’s are independentq×qmatrices whose entries are independent complex Gaussians with mean 0 and variance 1/q.

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A simple model of random translation-invariant MPS

Pick a tensor|χi ∈C^d⊗(C^q)^⊗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 ∈(C^d)^⊗^Nis arandom translation-invariant MPS with periodic boundary conditions.

d q q

|χi= ∑^d

x=1 q

∑

i,j=1

g_xij|xiji

N

|χNi= ∑^d

x1,...,xN=1

q

∑

i1,...,iN=1

g_x₁_i_N_i₁· · ·g_x_N_i_N₋₁_i_N

|x₁· · ·x_Ni

Associatedtransfer operator:T:C^q⊗C^q→C^q⊗C^qobtained by contracting thed-dimensional indices of|χiand|¯χi.

T= ∑^d

x=1

q

∑

i,j,k,l=1

g_xij¯g_xkl|ikihjl|

=¹_d ∑^d

x=1

G_x⊗_G¯_x

TheG_x’s are independentq×qmatrices whose entries are independent complex Gaussians with mean 0 and variance 1/q.

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A simple model of random translation-invariant PEPS

Pick a tensor|χi ∈C^d⊗(C^q)^⊗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 ∈(C^d)^⊗^MNis arandom translation-invariant PEPS with periodic boundary conditions.

q d q

q q

|χi= ∑^d

x=1 q

∑

i,j,i⁰,j⁰=1

g_xiji⁰_j⁰|xiji⁰j⁰i

M

|χ_Mi ∈(C^d⊗C^q⊗C^q)^⊗^M

Associatedtransfer operator:TM: (C^q⊗C^q)^⊗^M→(C^q⊗C^q)^⊗^Mobtained by contracting the d-dimensional indices of|χ_Miand|¯χMi.

T_M=_dM¹q^M d

∑

x1,...,xM=1 q

∑

i1,j1,...,iM,jM=1

G_x₁_i_M_i₁⊗_G¯_x

1jMj1⊗ · · · ⊗G_x_M_i_M₋₁_i_M⊗_G¯_x

MjM−1jM

TheG_xij’s are independentq×qmatrices whose entries are independent complex Gaussians with mean 0 and variance 1/q.

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A simple model of random translation-invariant PEPS

Pick a tensor|χi ∈C^d⊗(C^q)^⊗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 ∈(C^d)^⊗^MNis arandom translation-invariant PEPS with periodic boundary conditions.

q d q

q q

|χi= ∑^d

x=1 q

∑

i,j,i⁰,j⁰=1

g_xiji⁰_j⁰|xiji⁰j⁰i

M

|χ_Mi ∈(C^d⊗C^q⊗C^q)^⊗^M

Associatedtransfer operator:TM: (C^q⊗C^q)^⊗^M→(C^q⊗C^q)^⊗^Mobtained by contracting the d-dimensional indices of|χ_Miand|¯χMi.

T_M=_dM¹q^M d

∑

x1,...,xM=1 q

∑

i1,j1,...,iM,jM=1

G_x₁_i_M_i₁⊗_G¯_x

1jMj1⊗ · · · ⊗G_x_M_i_M₋₁_i_M⊗_G¯_x

MjM−1jM

TheG_xij’s are independentq×qmatrices whose entries are independent complex Gaussians with mean 0 and variance 1/q.

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Correlations in a TNS

MPS :M=1. PEPS :M>1.

A,Bobservables on(C^d)^⊗^M, i.e. on 1 site for an MPS and on 1 row ofMsites for a PEPS.

A

A:C^d→C^d

M

A

A: (C^d)^⊗^M→(C^d)^⊗^M

Compute the value on the TNS|χMNiof the observableA1⊗Idk⊗B1⊗IdN−k−2, i.e. v_χ(A,B,k) : = hχMN|A₁⊗Idk⊗B₁⊗IdN−k−2|χMNi

hχ_MN|χ_MNi .

hχ_MN|χ_MNi² . Correlationsin the TNS|χMNi:γ_χ(A,B,k) : =

v_χ(A,B,k)−v_χ(A)v_χ(B) . Question :Do we haveγχ(A,B,k) −→

kN→∞0 ? And if so, at which speed ?

A B

N

k

A B

'?

Nk1 ×

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Correlations in a TNS

MPS :M=1. PEPS :M>1.

A,Bobservables on(C^d)^⊗^M, i.e. on 1 site for an MPS and on 1 row ofMsites for a PEPS.

A

A:C^d→C^d

M

A

A: (C^d)^⊗^M→(C^d)^⊗^M Compute the value on the TNS|χMNiof the observableA1⊗Idk⊗B1⊗IdN−k−2, i.e.

v_χ(A,B,k) : = hχMN|A₁⊗Idk⊗B₁⊗IdN−k−2|χMNi hχ_MN|χ_MNi .

Compare it to the product of the values on|χ_MNiofA₁⊗Id_N−1andId_k+1⊗B₁⊗Id_N−k−2, i.e.

v_χ(A,B,k)−v_χ(A)v_χ(B) . Question :Do we haveγχ(A,B,k) −→

kN→∞0 ? And if so, at which speed ?

A B

N

k

A B

'?

Nk1 ×

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Main result : random TNS typically exhibit fast exponential decay of correlations

Intuition :Given a TNS|χMNi ∈(C^d)^⊗^MN, there existC,τ>0 such that, for anykNand any observablesA,Bon(C^d)^⊗^M,

γχ(A,B,k)6^Ce^−τ^kkAk∞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)

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Main result : random TNS typically exhibit fast exponential decay of correlations

Correlation lengthin the TNS|χMNi:ξ(χ) : =1/τ. Our main result (informal)

(‘generically’=with probability going to 1 asd,qgrow, ‘short’=going to 0 asd,qgrow)

Two ways of proving this :(suited to different dimensional regimes)

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Main result : random TNS typically exhibit fast exponential decay of correlations

Correlation lengthin the TNS|χMNi:ξ(χ) : =1/τ. Our main result (informal)

(‘generically’=with probability going to 1 asd,qgrow, ‘short’=going to 0 asd,qgrow) Two ways of proving this :(suited to different dimensional regimes)

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Outline

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Correlation length in a TNS and spectrum of its transfer operator

|χMNi ∈(C^d)^⊗^MNan MPS (M=1) or a PEPS (M>1).

Tits associated transfer operator on(C^q⊗C^q)^⊗^M. Its correlation function can be re-written as : γχ(A,B,k) =

Tr _AT˜ ^k˜_BT^N⁻^k⁻²

Tr(T^N) −^Tr _AT˜ ^N⁻¹

Tr _BT˜ ^N⁻¹ (Tr(T^N))²

.

A

A:C^d→C^d A

A˜:C^q⊗C^q→C^q⊗C^q

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)|.

γ_χ(A,B,k)6^C _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)|?

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Correlation length in a TNS and spectrum of its transfer operator

Tr _BT˜ ^N⁻¹ (Tr(T^N))²

.

A

A:C^d→C^d A

A˜:C^q⊗C^q→C^q⊗C^q 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)|.

γ_χ(A,B,k)6^C _Tr

(T)

∆(T) 2

Question :For a randomTwhat are typically the values of|λ1(T)|and|λ2(T)|?

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Correlation length in a TNS and spectrum of its transfer operator

Tr _BT˜ ^N⁻¹ (Tr(T^N))²

.

A

A:C^d→C^d A

A˜:C^q⊗C^q→C^q⊗C^q 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)|.

γ_χ(A,B,k)6^C _Tr

(T)

∆(T) 2

Question :For a randomTwhat are typically the values of|λ₁(T)|and|λ₂(T)|?

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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|χ_Ni) 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 ∑^d

x=1

G_xXG^∗_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).

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Typical spectral gap of the transfer operator of a random MPS

P

|λ1(T)|>¹−√^C

>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 ∑^d

x=1

T

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Typical spectral gap of the transfer operator of a random MPS

P

|λ1(T)|>¹−√^C

>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 ∑^d

x=1

T

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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 ∈(C^q)^⊗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: =^√¹_q

q

∑

i=1

|iii ∈C^q⊗C^q. Intuition :Recall thatT=¹_d ∑^d

x=1

G_x⊗_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∞6^C/√ 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).

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Proof strategy

q

∑

i=1

x=1

G_x⊗_G¯_x_.

d(but what about potential isolated ones ?)

Two steps in the proof :

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).

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Proof strategy

q

∑

i=1

x=1

G_x⊗_G¯_x_.

d(but what about potential isolated ones ?) Two steps in the proof :

Show that :f : (G₁, . . . ,G_d)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).

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Average estimates

•Lower bounding the expected largest eigenvalue : E|hψ|T|ψi|=E

1 dq

d

∑

x=1 q

∑

i,j=1

hi|G_x|jihi|_G¯_x|ji

| {z }

=|hi|Gx|ji|²

=_dq¹ ∑^d

x=1 q

∑

i,j=1

E|hi|G_x|ji|²

| {z }

=1/q

=1.

•Upper bounding the expected second largest eigenvalue : First observation :kT(Id− |ψihψ|)k_∞6²kT− |ψihψ|k_∞.

Now, since|ψihψ|=ET (computations as above) :T− |ψihψ|=¹_d

d

∑

x=1

G_x⊗_G¯_x−EH_x⊗_H¯_x . So by Jensen inequality :EkT− |ψihψ|k_∞6^E

1 d

d

∑

x=1

G_x⊗_G¯_x−H_x⊗_H¯_x ∞

. Symmetrization trick :E

1 d

d

∑

x=1

G_x⊗_G¯_x−H_x⊗_H¯_x ∞

6^2E

1 d

d

∑

x=1

G_x⊗_H¯_x ∞

. For anyp∈Neven,E

1 d

d

∑

x=1

G_x⊗_H¯_x ∞

6

ETr

1 d

d

∑

x=1

G_x⊗_H¯_x

p1/p

6

p>3q²^/⁵ p6^2q²^/³

√4 d

_p5

128q

2/p

.

Choosingp=2bq²^/³c:E

1 d

d

∑

x=1

G_x⊗_H¯_x ∞

6^√⁴_d

q⁷^/³ 4

1/q²^/³

6 ^√¹⁰_d^. Conclusion :EkT(Id− |ψihψ|)k_∞6^√⁴⁰_d^.

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Average estimates

•Lower bounding the expected largest eigenvalue : E|hψ|T|ψi|=E

1 dq

d

∑

x=1 q

∑

i,j=1

hi|G_x|jihi|_G¯_x|ji

| {z }

=|hi|Gx|ji|²

=_dq¹ ∑^d

x=1 q

∑

i,j=1

E|hi|G_x|ji|²

| {z }

=1/q

=1.

•Upper bounding the expected second largest eigenvalue : First observation :kT(Id− |ψihψ|)k_∞6²kT− |ψihψ|k_∞.

Now, since|ψihψ|=ET (computations as above) :T− |ψihψ|=¹_d

d

∑

x=1

G_x⊗_G¯_x−EH_x⊗_H¯_x . So by Jensen inequality :EkT− |ψihψ|k_∞6^E

1 d

d

∑

x=1

G_x⊗_G¯_x−H_x⊗_H¯_x ∞

. Symmetrization trick :E

1 d

d

∑

x=1

G_x⊗_G¯_x−H_x⊗_H¯_x ∞

6^2E

1 d

d

∑

x=1

G_x⊗_H¯_x ∞

. For anyp∈Neven,E

1 d

d

∑

x=1

G_x⊗_H¯_x ∞

6

ETr

1 d

d

∑

x=1

G_x⊗_H¯_x

p1/p

6

p>3q²^/⁵ p6^2q²^/³

√4 d

_p5

128q

2/p

.

Choosingp=2bq²^/³c:E

1 d

d

∑

x=1

G_x⊗_H¯_x ∞

6^√⁴_d

q⁷^/³ 4

1/q²^/³

6 ^√¹⁰_d^. Conclusion :EkT(Id− |ψihψ|)k∞6^√⁴⁰_d^.