Typical correlations and entanglement in random MPS and PEPS

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Typical correlations and entanglement in random MPS and PEPS

Mostly based on joint work with David Pérez-García, available at arXiv:1906.11682

Cécilia Lancien

Institut de Mathématiques de Toulouse & CNRS

Uniba Mathematical Physics seminar – June 29 2021

Cécilia Lancien Typical correlations and entanglement in random MPS and PEPS Uniba Mathematical Physics seminar – June 29 2021 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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The curse of dimensionality when dealing with many-body quantum systems

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

−→Exponential growth of the dimension with the number of subsystems.

However,‘physically relevant’ statesof many-body quantum systems are often well approximated by so-calledtensor network states (TNS), which form a small subset of the global state space.

Tensor network state construction 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|˜χ_Giassociated to a same edge to get a tensor|χ_Gi ∈(C^d)^⊗^N.

degree ofv

pure state on(C^d)^⊗N(up to normalization)

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

• •

• • Gwith 6 vertices and 7 edges

• •

|˜χGi ∈(C^d)^⊗⁶⊗(C^q)^⊗¹⁴

• •

|χGi ∈(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|χ_Giis a matrix product state (MPS)or aprojected entangled pair state (PEPS).

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The curse of dimensionality when dealing with many-body quantum systems

degree ofv

• •

|˜χGi ∈(C^d)^⊗⁶⊗(C^q)^⊗¹⁴

• •

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The curse of dimensionality when dealing with many-body quantum systems

degree ofv

• •

|˜χGi ∈(C^d)^⊗⁶⊗(C^q)^⊗¹⁴

• •

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Motivations behind TNS and related questions

A TNS representation is thought to be amathematically tractabledescription of the state of a system composed ofmany subsystems having a certain geometryand subject to interactions respecting this geometry.

Example:Being a TNS is (conjectured to be) approximately equivalent to being a ground state of a gapped local Hamiltonian(Hastings, Landau/Vazirani/Vidick...)

composed of terms which act only on nearby sites spectral gap lower bounded by a constant independent ofN

−→In condensed-matter physics, TNS are used as Ansatz in computations: optimization over a manageable number of parameters, even for largeN.

General question:Among the variousqualitativeintuitions about TNS, which ones are at least true generically? And canquantitativestatements be made about the generic case?

Idea:Sample a TNS at random and study the characteristics that it exhibits with high probability. In particular, what is itstypical amount of correlations and entanglement?

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.

In particular, what is itstypical amount of correlations and entanglement?

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Outline

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A simple model of random translation-invariant MPS

Nparticles on a circle

• • • • • N

Pick a tensor|χi ∈C^d⊗(C^q)^⊗²whose entries are independent complex Gaussians with mean 0 and variance 1/dq.

Repeat it on all sites and contract neighbouringq-dimensional indices.

−→Obtained tensor|χNi ∈(C^d)^⊗^N:random translation-invariant MPS with periodic boundary conditions(typically almost normalized).

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^q, obtained 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

Nparticles on a circle

• • • • • N

Pick a tensor|χi ∈C^d⊗(C^q)^⊗²whose entries are independent complex Gaussians with mean 0 and variance 1/dq.

−→Obtained tensor|χNi ∈(C^d)^⊗^N:random translation-invariant MPS with periodic boundary conditions(typically almost normalized).

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^q, obtained 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

MNparticles on a torus

• • • • •

• • • • • N

M

Pick a tensor|χi ∈C^d⊗(C^q)^⊗⁴whose entries are independent complex Gaussians with mean 0 and variance 1/dq².

−→Obtained tensor|χMNi ∈(C^d)^⊗^MN:random translation-invariant PEPS with periodic boundary conditions(typically almost normalized).

q d q

q q

|χi= ∑^d

x=1 q

∑

i,j,i⁰,j⁰=1

g_xiji⁰_j⁰|xiji⁰j⁰i

M

|χMi ∈(C^d⊗C^q⊗C^q)^⊗^M

Associatedtransfer operator:T_M: (C^q⊗C^q)^⊗^M→(C^q⊗C^q)^⊗^M, obtained 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

MNparticles on a torus

• • • • •

• • • • • N

M

Pick a tensor|χi ∈C^d⊗(C^q)^⊗⁴whose entries are independent complex Gaussians with mean 0 and variance 1/dq².

−→Obtained tensor|χMNi ∈(C^d)^⊗^MN:random translation-invariant PEPS with periodic boundary conditions(typically almost normalized).

q d q

q q

|χi= ∑^d

x=1 q

∑

i,j,i⁰,j⁰=1

g_xiji⁰_j⁰|xiji⁰j⁰i

M

|χMi ∈(C^d⊗C^q⊗C^q)^⊗^M

Associatedtransfer operator:T_M: (C^q⊗C^q)^⊗^M→(C^q⊗C^q)^⊗^M, obtained 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 column ofMsites for a PEPS.

local observables

A A:C^d→C^d

M

A

A: (C^d)^⊗^M→(C^d)^⊗^M

Compute the value on the TNS|χMNiof the observableA1⊗Ik⊗B1⊗IN−k−2, i.e. v_χ(A,B,k) :=hχ_MN|A₁⊗I_k⊗B₁⊗I_N−k−2|χ_MNi.

by translation-invariance of|χMNi Correlationsin the TNS|χ_MNi:γχ(A,B,k) :=

v_χ(A,B,k)−v_χ(A)v_χ(B) . Question:Do we haveγ_χ(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 column ofMsites for a PEPS.

local observables

A A:C^d→C^d

M

A

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

v_χ(A,B,k) :=hχ_MN|A₁⊗I_k⊗B₁⊗I_N₋_k₋₂|χ_MNi.

Compare it to the product of the values on|χ_MNiofA₁⊗I_N₋₁andI_k+1⊗B₁⊗I_N₋_k₋₂, i.e.

v_χ(A)v_χ(B) :=hχMN|A₁⊗IN−1|χMNihχMN|B₁⊗IN−1|χMNi.

by translation-invariance of|χMNi Correlationsin the TNS|χ_MNi:γχ(A,B,k) :=

v_χ(A,B,k)−v_χ(A)v_χ(B) . Question:Do we haveγ_χ(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(χ),τ(χ)>0 such that, for anykN and any observablesA,Bon(C^d)^⊗^M,

γ_χ(A,B,k)6^C(χ)e^−τ(χ)^kkAk_∞kBk_∞.

−→Correlations between two 1-site or 1-column observables decay exponentially with the distance separating the two sites or columns, at a rateτ(χ).

Correlation lengthin the TNS|χ_MNi:ξ(χ) :=1/τ(χ).

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.

going to 0 asd,qgrow with probability going to 1 (exponentially) asd,qgrow

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

Show that thetransfer operatorassociated to the random TNS generically has alarge (upper) spectral gap.

Show that theparent Hamiltonianof the random TNS generically has alarge (lower) spectral gap.

going to 1 asd,qgrow

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

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

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

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

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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. By reading diagrams representingv_χ(A,B,k),v_χ(A),v_χ(B)

‘horizontally’ instead of ‘vertically’, its correlation function can be re-written as:

γ_χ(A,B,k) =

Tr ˜AT^k_BT˜ ^N⁻^k⁻²

−Tr ˜AT^N⁻¹

Tr ˜BT^N⁻¹ .

A

A:C^d→C^d A

A˜:C^q⊗C^q→C^q⊗C^q

Important consequence:

Denote byλ1(T)andλ2(T)the largest and second largest eigenvalues ofT (with multiplicities). There existsC(T)>0 such that, settingε(T) =|λ2(T)|/|λ1(T)|, we have

γχ(A,B,k)6^C(T)ε(T)^kkAk∞kBk∞.

−→Correlations between two 1-site or 1-column observables decay exponentially with the distance separating the two sites or columns, at a rate|logε(T)|, i.e.ξ(χ) =1/|logε(T)|. Question:For a random transfer operatorTwhat are the typical values of|λ1(T)|and|λ2(T)|?

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Correlation length in a TNS and spectrum of its transfer operator

γ_χ(A,B,k) =

−Tr ˜AT^N⁻¹

Tr ˜BT^N⁻¹ .

A

A:C^d→C^d A

A˜:C^q⊗C^q→C^q⊗C^q Important consequence:

Denote byλ1(T)andλ2(T)the largest and second largest eigenvalues ofT (with multiplicities).

There existsC(T)>0 such that, settingε(T) =|λ2(T)|/|λ1(T)|, we have γχ(A,B,k)6^C(T)ε(T)^kkAk∞kBk∞.

−→Correlations between two 1-site or 1-column observables decay exponentially with the distance separating the two sites or columns, at a rate|logε(T)|, i.e.ξ(χ) =1/|logε(T)|.

Question:For a random transfer operatorTwhat are the typical values of|λ1(T)|and|λ2(T)|?

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Correlation length in a TNS and spectrum of its transfer operator

γ_χ(A,B,k) =

−Tr ˜AT^N⁻¹

Tr ˜BT^N⁻¹ .

A

A:C^d→C^d A

A˜:C^q⊗C^q→C^q⊗C^q Important consequence:

Denote byλ1(T)andλ2(T)the largest and second largest eigenvalues ofT (with multiplicities).

There existsC(T)>0 such that, settingε(T) =|λ2(T)|/|λ1(T)|, we have γχ(A,B,k)6^C(T)ε(T)^kkAk∞kBk∞.

−→Correlations between two 1-site or 1-column observables decay exponentially with the distance separating the two sites or columns, at a rate|logε(T)|, i.e.ξ(χ) =1/|logε(T)|. Question:For a random transfer operatorTwhat are the typical 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 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 constantsc,C>0 such that, for anyd,q,N∈N,

P

ξ(χ)6 ^C logd

>¹−e⁻^cq.

Remark:Similar results obtained on slightly different models, withT composed of independent Haar unitaries(Hastings, Pisier)or blocks of a Haar unitary(González-Guillén/Junge/Nechita). Motivation: Construction ofquantum expanders. This Gaussian model provides a new example. Proof idea:Approximate first eigenvector forT: maximally entangled state|ψi ∈C^q⊗C^q. Observing that|λ1(T)|>|hψ|T|ψi|and|λ2(T)|6kT(I− |ψihψ|)k∞, we show that:

1 E|hψ|T|ψi|=1 andEkT(I− |ψihψ|)k∞6^C/√

d(Gaussian moment computations).

2 These two quantities have a small probability of deviating from their average (local version of Gaussian concentration inequality).

by minimax principle for singular values, doing as ifTwere Hermitian

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Typical spectral gap of the transfer operator of a random MPS

P

|λ₁(T)|>¹−√^C

>¹−e⁻^cq.

P

ξ(χ)6 ^C logd

>¹−e⁻^cq.

Remark:Similar results obtained on slightly different models, withT composed of independent Haar unitaries(Hastings, Pisier)or blocks of a Haar unitary(González-Guillén/Junge/Nechita). Motivation: Construction ofquantum expanders. This Gaussian model provides a new example. Proof idea:Approximate first eigenvector forT: maximally entangled state|ψi ∈C^q⊗C^q. Observing that|λ1(T)|>|hψ|T|ψi|and|λ2(T)|6kT(I− |ψihψ|)k∞, we show that:

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Typical spectral gap of the transfer operator of a random MPS

P

|λ₁(T)|>¹−√^C

>¹−e⁻^cq.

P

ξ(χ)6 ^C logd

>¹−e⁻^cq.

Remark:Similar results obtained on slightly different models, withT composed of independent Haar unitaries(Hastings, Pisier)or blocks of a Haar unitary(González-Guillén/Junge/Nechita). Motivation: Construction ofquantum expanders. This Gaussian model provides a new example.

Proof idea:Approximate first eigenvector forT: maximally entangled state|ψi ∈C^q⊗C^q. Observing that|λ1(T)|>|hψ|T|ψi|and|λ2(T)|6kT(I− |ψihψ|)k∞, we show that:

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Typical spectral gap of the transfer operator of a random MPS

P

|λ₁(T)|>¹−√^C

>¹−e⁻^cq.

P

ξ(χ)6 ^C logd

>¹−e⁻^cq.

Remark:Similar results obtained on slightly different models, withT composed of independent Haar unitaries(Hastings, Pisier)or blocks of a Haar unitary(González-Guillén/Junge/Nechita). Motivation: Construction ofquantum expanders. This Gaussian model provides a new example.

Proof idea:Approximate first eigenvector forT: maximally entangled state|ψi ∈C^q⊗C^q. Observing that|λ1(T)|>|hψ|T|ψi|and|λ2(T)|6kT(I− |ψihψ|)k∞, we show that:

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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(I− |ψihψ|)k∞6²kT− |ψihψ|k∞.

Now, since|ψihψ|=ET (computations as above):T− |ψihψ|=_d¹ ∑^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

d

∑

x=1

G_x⊗_G¯_x−H_x⊗_H¯_x ∞

6²^E

d

∑

x=1

G_x⊗_H¯_x ∞

. For anyp∈N,E

d

∑

x=1

G_x⊗_H¯_x ∞

6

ETr

d

∑

x=1

G_x⊗_H¯_x

p1/p

6√

d(ETr|G|^p)²^/^p. Choosingp=2bq²^/³c:E

d

∑

x=1

Gx⊗_H¯_x ∞

6⁴√ d

_q7/3

4

1/q²^/³

6¹⁰√ d.

Conclusion:EkT(I− |ψihψ|)k_∞6^√⁴⁰_d^.

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Concentration around average

Gaussian concentration

f:Cⁿ→RL-Lipschitz (for the Euclidean norm),g∈CⁿGaussian with mean 0 and varianceσ². For allε>0,P(f(g)≷^Ef±ε)6^e^−ε²^/σ²^L²^.

In our case:f: (A₁, . . . ,A_d)∈(C^q^×^q)^d7→

1 d

d

∑

x=1

A_x⊗_A¯_x

P ∞

∈R,Pany fixed projector.

Problem: The Lipschitz constant offon the whole set(C^q^×^q)^d is not good enough...

Local version of Gaussian concentration (Aubrun/Szarek/Werner)

f:Cⁿ→RL-Lipschitz onΩ⊂Cⁿ,g∈CⁿGaussian with mean 0 and varianceσ². For allε>0,P(f(g)≷^Ef±ε)6^e^−ε²^/σ²^L²+P(g∈/Ω).

In our case:Ω = (

(A₁, . . . ,A_d)∈(C^q^×^q)^d: _d

∑

x=1

kA_xk²_∞ 1/2

6³√ d

) . f is 6√

2/√

d-Lipschitz onΩandP((G₁, . . . ,G_d)∈/Ω)6ê⁻^qd^. So for allε>0,P(f(G₁, . . . ,G_d)≷Êf±ε)6ê⁻^qd^ε²^/⁷²+e⁻^qd.

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Putting everything together

We have just proved: For allε>0,P(kTPk_∞≷^EkTPk_∞±ε)6^e⁻^qd^ε²^/⁷²+e⁻^qd(?). We had also proved before:E|hψ|T|ψi|=1 andEkT(I− |ψihψ|)k_∞6⁴⁰/√

d.

So takingε=1/√

dandP=|ψihψ|orP=I− |ψihψ|in(?), we get:

P

|hψ|T|ψi|<1−√¹ d

6^e⁻^q^/⁷²+e⁻^qd 6^2e⁻^q^/⁷², P

kT(I− |ψihψ|)k_∞> √⁴¹ d

6^e⁻^q^/⁷²+e⁻^qd6^2e⁻^q^/⁷².

Since|λ1(T)|>|hψ|T|ψi|and|λ2(T)|6kT(I− |ψihψ|)k∞, this implies, as claimed:

P

|λ1(T)|>¹−√¹

d and|λ1(T)|6√⁴¹ d

>¹−4e⁻^q^/⁷².

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Typical spectral gap of the transfer operator of a random PEPS

Assumption(?):The dimensionsdandqgrow polynomially with the number of sitesM.

More precisely:d,qsatisfyd'M^αandq'M^βwithα>8 and(α+1)/3<β<(α−2)/2.

Theorem [Typical spectral gap of the random PEPS transfer operatorT_M] There exist constantsc,C>0 such that, for anyM∈Nandd,q∈Nsatisfying(?),

P

|λ1(T_M)|>¹− ^C

M^α/²⁻¹^−β and|λ2(T_M)|6 ^C M^α/²⁻¹^−β

>¹−e⁻^cM³^β−α.

Corollary [Typical correlation length in the random PEPS|χMNi]

There exist constantsc,C>0 such that, for anyN>^M∈Nandd,q∈Nsatisfying(?), P

ξ(χ)6 ^C log(M^α/²⁻¹^−β)

>¹−e⁻^cM³^β−α.

Main issue:Results are valid only in the regime whered,qgrow polynomially withM.

−→Enforce this scaling byblocking: Redefine 1 site as being a sublattice oflogMsites. Proof idea:Some kind of recursion procedure that uses the MPS results as building blocks. Approximate first eigenvector forT_M:|ψi^⊗^M∈(C^q⊗C^q)^⊗^M.

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Typical spectral gap of the transfer operator of a random PEPS

P

|λ1(T_M)|>¹− ^C

M^α/²⁻¹^−β and|λ2(T_M)|6 ^C M^α/²⁻¹^−β

>¹−e⁻^cM³^β−α.

Corollary [Typical correlation length in the random PEPS|χMNi]

ξ(χ)6 ^C log(M^α/²⁻¹^−β)

>¹−e⁻^cM³^β−α.

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Typical spectral gap of the transfer operator of a random PEPS

P

|λ1(T_M)|>¹− ^C

M^α/²⁻¹^−β and|λ2(T_M)|6 ^C M^α/²⁻¹^−β

>¹−e⁻^cM³^β−α.

Corollary [Typical correlation length in the random PEPS|χ_MNi]

ξ(χ)6 ^C log(M^α/²⁻¹^−β)

>¹−e⁻^cM³^β−α.

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Typical spectral gap of the transfer operator of a random PEPS

P

|λ1(T_M)|>¹− ^C

M^α/²⁻¹^−β and|λ2(T_M)|6 ^C M^α/²⁻¹^−β

>¹−e⁻^cM³^β−α.

Corollary [Typical correlation length in the random PEPS|χ_MNi]

ξ(χ)6 ^C log(M^α/²⁻¹^−β)

>¹−e⁻^cM³^β−α.

−→Enforce this scaling byblocking: Redefine 1 site as being a sublattice oflogMsites.

Proof idea:Some kind of recursion procedure that uses the MPS results as building blocks. Approximate first eigenvector forT_M:|ψi^⊗^M∈(C^q⊗C^q)^⊗^M.