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Combining Tensor Networks and Monte Carlo for Lattice Gauge Theories Entanglement in Strongly Correlated Systems, Benasque 21st of February 2020 Patrick Emonts , E. Zohar, M. C. Banuls, I. Cirac MPI of Quantum Optics Why do we need Lattice

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SLIDE 1

Combining Tensor Networks and Monte Carlo for Lattice Gauge Theories

Entanglement in Strongly Correlated Systems, Benasque

21st of February 2020 Patrick Emonts, E. Zohar, M. C. Banuls, I. Cirac MPI of Quantum Optics

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SLIDE 2

Why do we need Lattice Gauge Theories?

QED

LQED = iΨγμ𝜖μΨ − eΨγμAμΨ − mΨΨ − 1

4FμνFμν

γ e− e+ e− e+

Small coupling

αQED = e2

4π ≈ 1 137

QCD

Image adapted from Alexandre Deur, Stanley J. Brodsky, and Guy F. de Téramond, 2016, Progress in Particle and Nuclear Physics

Slide 2 Combining Tensor Networks and Monte Carlo for Lattice Gauge Theories | 21st of February 2020 | PE, EZ, MCB, IC

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SLIDE 3

Path integral formalism in QFT

pure QED

SQED[Aμ] = − 1

4 ∫ dxαFμν(xα)Fμν(xα) = ∫ dxα𝜖μAν(xα)𝜖νAμ(xα)

vacuum expectation value

⟨Ω|O[Aμ]|Ω⟩ =

∫ DAO[Aμ]e

iSQED[Aμ]

∫ DAe

iSQED[Aμ]

Problems

 Numerator oscillating  Integration measure ill-defined

Slide 3 Combining Tensor Networks and Monte Carlo for Lattice Gauge Theories | 21st of February 2020 | PE, EZ, MCB, IC

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SLIDE 4

Wick rotation

Shift to imaginary time

t → −iτ

Change of metric from Minkowski to Euclidean

eiSM = ei ∫ dxα

ML(xα M) ⟶ e− ∫ dxα EL(xα E) = e−SE

Problems

 Numerator converging  Integration measure ill-defined

Slide 4 Combining Tensor Networks and Monte Carlo for Lattice Gauge Theories | 21st of February 2020 | PE, EZ, MCB, IC

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SLIDE 5

Discretization: Lattice Gauge Theory

xα a

Aμ → Uμ = eiaAμ

Find the lattice action

̃ SE that agrees with SE in the continuum limit of vanishing a ̃ SE[U] → SE[A](a → 0)

Kenneth G. Wilson, 1974, Physical Review D

Slide 5 Combining Tensor Networks and Monte Carlo for Lattice Gauge Theories | 21st of February 2020 | PE, EZ, MCB, IC

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SLIDE 6

Vacuum expectation value in the action formalism

Vacuum expectation value

⟨O[U]⟩ =

∫ DUO[U]e−SE[U] ∫ DUe−SE[U]

with DU = ∏xα dUμ(xα)

Problems

 Numerator converging  Integration with the Haar measure

Slide 6 Combining Tensor Networks and Monte Carlo for Lattice Gauge Theories | 21st of February 2020 | PE, EZ, MCB, IC

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SLIDE 7

Lattice Systems

Hilbert space

H ⊂ Hgauge fields ⊗ Hfermions

A general state

|Ψ⟩ = ∫ DG |G⟩ ∣ΨF(G)⟩

with DG = ∏x,k dg(x, k)

Erez Zohar and J. Ignacio Cirac, 2018, Physical Review D Patrick Emonts and Erez Zohar, 2020, SciPost Physics Lecture Notes

Slide 7 Combining Tensor Networks and Monte Carlo for Lattice Gauge Theories | 21st of February 2020 | PE, EZ, MCB, IC

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SLIDE 8

Gauss law

Gauss law

∑k (Ek(x) − Ek(x − ei)) |phys⟩ = 0 ∀x

Classical analogue in (cont.) electrodynamics

∇ ⋅ E = 0

E2(x) E1(x) E1(x − e1) E2(x − e2)

Slide 8 Combining Tensor Networks and Monte Carlo for Lattice Gauge Theories | 21st of February 2020 | PE, EZ, MCB, IC

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SLIDE 9

Expectation value of an Observable

Assume that O acts only on the gauge field and is diagonal in the group element basis:

⟨O⟩ = ⟨Ψ|O|Ψ⟩ ⟨Ψ|Ψ⟩ = ∫ DG ⟨G| O |G⟩ ⟨ΨF(G)∣ΨF(G)⟩ ∫ DG′ ⟨ΨF(G′)∣ΨF(G′)⟩ = ∫ DGFO(G)p(G)

with p(G) =

⟨ΨF(G)∣ΨF(G)⟩ ∫ DG′⟨ΨF(G′)∣ΨF(G′)⟩ = ⟨ΨF(G)∣ΨF(G)⟩ Z

Slide 9 Combining Tensor Networks and Monte Carlo for Lattice Gauge Theories | 21st of February 2020 | PE, EZ, MCB, IC

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SLIDE 10

The rest of this talk

Expectation value

⟨O⟩ = ∫ DGFO(G)p(G)

with p(G) =

⟨ΨF(G)∣ΨF(G)⟩ Z

TODO List

1 How do we construct ∣ΨF(G)⟩? 2 How do we efficiently calculate p(G)? 3 Are those states useful?

Slide 10 Combining Tensor Networks and Monte Carlo for Lattice Gauge Theories | 21st of February 2020 | PE, EZ, MCB, IC

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SLIDE 11

Creation of the fermionic state

Desirable properties

|Ψ⟩ fulfills the Gauss law ∣ΨF(G)⟩ allows efficient calculations of

the norm expectation values

Definition of Ψ

|Ψ⟩ = ∫ DG |G⟩ ∣ΨF(G)⟩

Choice for ∣ΨF(G)⟩

We construct ∣ΨF(G)⟩ with a tensor network.

Patrick Emonts and Erez Zohar, 2020, SciPost Physics Lecture Notes

Slide 11 Combining Tensor Networks and Monte Carlo for Lattice Gauge Theories | 21st of February 2020 | PE, EZ, MCB, IC

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SLIDE 12

Definition of Modes

Gauss law in terms of our modes

G0 = Er − El + Eu − Ed = r†

+r+ − r† −r− − l† +l+ + l† −l− + u† +u+ − u† −u− − d† +d+ + d† −d−

Ψ r+ r− l+ l− u+ u− d+ d−

Definition of positive and negative modes

a: {l+, r−, u−, d+} (neg. modes) b: {l−, r+, u+, d−} (pos. modes)

Erez Zohar et al., 2015, Annals of Physics

Slide 12 Combining Tensor Networks and Monte Carlo for Lattice Gauge Theories | 21st of February 2020 | PE, EZ, MCB, IC

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SLIDE 13

Creating a fermionic state

The state

∣ψ0⟩ = ⟨ΩV∣ ∏

x,k

ω(x, k) ∏

x

A(x) |Ω⟩

A(x) = exp ⎛ ⎜ ⎝ ∑

ij

Tija†

i (x)b† j (x)⎞

⎟ ⎠ ω(x, k) =ωk(x)Ωk(x)ω†

k(x)

ω0(x) = exp(l†

+(x + e1)r† −(x))

exp(l†

−(x + e1)r† +(x))

Slide 13 Combining Tensor Networks and Monte Carlo for Lattice Gauge Theories | 21st of February 2020 | PE, EZ, MCB, IC

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SLIDE 14

Creating a fermionic state

The state

∣ψ0⟩ = ⟨ΩV∣ ∏

x,k

ω(x, k) ∏

x

A(x) |Ω⟩

x00 x10 x20 x01 x11 x21 A(x) = exp ⎛ ⎜ ⎝ ∑

ij

Tija†

i (x)b† j (x)⎞

⎟ ⎠ ω(x, k) =ωk(x)Ωk(x)ω†

k(x)

ω0(x) = exp(l†

+(x + e1)r† −(x))

exp(l†

−(x + e1)r† +(x))

Slide 13 Combining Tensor Networks and Monte Carlo for Lattice Gauge Theories | 21st of February 2020 | PE, EZ, MCB, IC

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SLIDE 15

Creating a fermionic state

The state

∣ψ0⟩ = ⟨ΩV∣ ∏

x,k

ω(x, k) ∏

x

A(x) |Ω⟩

x00 x10 x20 x01 x11 x21

ω00,h ω10,h ω01,h ω11,h ω00,v ω10,v ω20,v

A(x) = exp ⎛ ⎜ ⎝ ∑

ij

Tija†

i (x)b† j (x)⎞

⎟ ⎠ ω(x, k) =ωk(x)Ωk(x)ω†

k(x)

ω0(x) = exp(l†

+(x + e1)r† −(x))

exp(l†

−(x + e1)r† +(x))

Slide 13 Combining Tensor Networks and Monte Carlo for Lattice Gauge Theories | 21st of February 2020 | PE, EZ, MCB, IC

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SLIDE 16

Moving towards local symmetry

Lattice Gauge theory

We demand a local symmetry

∑x G(x) |Ψ⟩ = 0 → G(x) |Ψ⟩ = 0

x00 x10 x20 x01 x11 x21

Erez Zohar et al., 2015, Annals of Physics Erez Zohar and Michele Burrello, 2016, New Journal of Physics

Slide 14 Combining Tensor Networks and Monte Carlo for Lattice Gauge Theories | 21st of February 2020 | PE, EZ, MCB, IC

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SLIDE 17

Local symmetry – The state

Substitution

r†

±(x) → e±iθ(x)r† ±(x)

u†

±(x) → e±iθ(x)u† ±(x)

x00 x10 x20 x01 x11 x21

Slide 15 Combining Tensor Networks and Monte Carlo for Lattice Gauge Theories | 21st of February 2020 | PE, EZ, MCB, IC

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SLIDE 18

Fermionic state

Fermionic state

∣ψ(G)⟩ = ⟨Ωv∣ ∏

x

ω(x) ∏

x

UΦ(x) ∏

x

A(x) |Ω⟩

 Gauge invariance of |Ψ⟩ by constructing Ψ(G)  Obeys all demanded symmetries ? Efficient to calculate with

Slide 16 Combining Tensor Networks and Monte Carlo for Lattice Gauge Theories | 21st of February 2020 | PE, EZ, MCB, IC

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SLIDE 19

Is ∣ΨF(G)⟩ special?

The fermionic state ∣ΨF(G)⟩

∣ΨF(G)⟩ = ⟨Ωv∣ ∏

x

ω(x) ∏

x

UΦ(x) ∏

x

A(x) |Ω⟩ A(x) = exp ⎛ ⎜ ⎝ ∑

ij

Tija†

i (x)b† j (x)⎞

⎟ ⎠ ω(x) = ω0(x)ω1(x)Ω(x)ω†

1(x)ω† 0(x)

ω0(x) = exp(l†

+(x + e1)r† −(x)) exp(l† −(x + e1)r† +(x))

ω1(x) = exp(d†

+(x + e2)u† −(x)) exp(d† −(x + e2)u† +(x))

Slide 17 Combining Tensor Networks and Monte Carlo for Lattice Gauge Theories | 21st of February 2020 | PE, EZ, MCB, IC

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SLIDE 20

Gaussian States

Definition

Fermionic Gaussian states are represented by density operators that are exponentials of a quadratic form in Majorana operators.

ρ = K exp(− i

4γTGγ)

Covariance matrix

Covariance matrix for a state Φ:

Γab = i

2 ⟨[γa, γb]⟩ = i 2 ⟨Φ|[γa,γb]|Φ⟩ ⟨Φ|Φ⟩

Sergey Bravyi, 2005, Quantum Inf. and Comp.

Slide 18 Combining Tensor Networks and Monte Carlo for Lattice Gauge Theories | 21st of February 2020 | PE, EZ, MCB, IC

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SLIDE 21

Calculating the Norm and the Observables

∣ψ(G)⟩ = ⟨Ωv∣ ∏

x

ω(x) ∏

x

UΦ(x) ⏟⏟ ⏟ ⏟⏟⏟ ⏟ ⏟⏟

❀Γin(G)

∏

x

A(x) ⏟ ⏟ ⏟ ⏟ ⏟

❀ΓM

|Ω⟩ ΓM

i,j = ( A

B −BT D)

A Physical-Physical correlations B Physical-Virtual correlations C Virtual-Virtual correlations

Norm

⟨ψ(G)∣ψ(G)⟩ = √det (1 − Γin(G)MD 2 )

Slide 19 Combining Tensor Networks and Monte Carlo for Lattice Gauge Theories | 21st of February 2020 | PE, EZ, MCB, IC

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SLIDE 22

The whole framework

Draw new gauge field configuration ∣G′⟩ Build the state ∣Ψ(G′)⟩ Calculate the acceptance probability by computing ⟨Ψ(G′)∣Ψ(G′)⟩ Accept or decline the new configuration G′ [Measure observables]

Slide 20 Combining Tensor Networks and Monte Carlo for Lattice Gauge Theories | 21st of February 2020 | PE, EZ, MCB, IC

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SLIDE 23

Results for Z3

Wilson loop Polyakov loop Transfer matrix Calculation

Erez Zohar et al., 2015, Annals of Physics

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 y 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 z 0.00 0.15 0.30 0.45 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 y 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 z 0.5 0.4 0.3 0.2 0.1 0.0

Different phases

We can model different phases with our variational Ansatz for the state.

Slide 21 Combining Tensor Networks and Monte Carlo for Lattice Gauge Theories | 21st of February 2020 | PE, EZ, MCB, IC

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SLIDE 24

Conclusion and Outlook

We need Lattice Gauge Theories A Hamiltonian approach shows promising possibilities (time evolution, finite μ) The GGPEPS Ansatz shows confined and non-confined phases Formulation of a variational minimization procedure for the energy Optimization of the Monte Carlo procedure for the sampling

x00 x10 x20 x01 x11 x21

Slide 22 Combining Tensor Networks and Monte Carlo for Lattice Gauge Theories | 21st of February 2020 | PE, EZ, MCB, IC

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SLIDE 25

Combining Tensor Networks and Monte Carlo for Lattice Gauge Theories

Entanglement in Strongly Correlated Systems, Benasque

21st of February 2020 Patrick Emonts, E. Zohar, M. C. Banuls, I. Cirac MPI of Quantum Optics

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SLIDE 26

References I

Sergey Bravyi. “Lagrangian Representation for Fermionic Linear Optics”. In: Quantum Inf. and

  • Comp. 5.3 (2005), pp. 216–238.

Alexandre Deur, Stanley J. Brodsky, and Guy F. de Téramond. “The QCD Running Coupling”. In: Progress in Particle and Nuclear Physics 90 (Sept. 2016), pp. 1–74. DOI: 10.1016/j.ppnp.2016.04.003. Patrick Emonts and Erez Zohar. “Gauss Law, Minimal Coupling and Fermionic PEPS for Lattice Gauge Theories”. In: SciPost Physics Lecture Notes (Jan. 17, 2020), p. 12. DOI: 10.21468/SciPostPhysLectNotes.12. Kenneth G. Wilson. “Confinement of Quarks”. In: Physical Review D 10.8 (Oct. 15, 1974),

  • pp. 2445–2459. DOI: 10.1103/PhysRevD.10.2445.

Slide 24 Combining Tensor Networks and Monte Carlo for Lattice Gauge Theories | 21st of February 2020 | PE, EZ, MCB, IC

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SLIDE 27

References II

Erez Zohar and Michele Burrello. “Building Projected Entangled Pair States with a Local Gauge Symmetry”. In: New Journal of Physics 18.4 (Apr. 8, 2016), p. 043008. DOI: 10.1088/1367-2630/18/4/043008. Erez Zohar and J. Ignacio Cirac. “Combining Tensor Networks with Monte Carlo Methods for Lattice Gauge Theories”. In: Physical Review D 97.3 (Feb. 23, 2018). DOI: 10.1103/PhysRevD.97.034510. Erez Zohar et al. “Fermionic Projected Entangled Pair States and Local U(1) Gauge Theories”. In: Annals of Physics 363 (Dec. 2015), pp. 385–439. DOI: 10.1016/j.aop.2015.10.009.

Slide 25 Combining Tensor Networks and Monte Carlo for Lattice Gauge Theories | 21st of February 2020 | PE, EZ, MCB, IC