Tensor Networks for Quantum Field Theory

Tensor Networks for Quantum Field Theory

Antoine Tilloy

Theory Division, Max Planck Institute of Quantum Optics, Garching, Germany

Universit¨at Leipzig

via Zoom, from Munich, Germany December 18th, 2020

My profile

3years PhD (ENS),4years postdoc MPQ 3 main lines of inquiry

Continuous measurement

How to gently measure and control quantum systems?

Gravity and quantum

Could gravity, in principle, not be quantum?

Many-body & QFT

How to efficiently parameterize many-body and QFT states

Quantum field theory: a bit of philosophy

Two ways to attackreal world quantum field theories non-perturbatively

1. Startsimpler so that it becomessimpler [e.g. self interacting scalar fieldφ4 2]

2. Startmore complex so that it becomessimpler[e.g. N=4SYM]

φ4₂- pile of dirt QCD - Everest N=4 SYM - Chrysler building 2. is a good way to make fast progress first, but is limited in what it can achieve

Goal - ideal - philosophy: an apology of the pile of dirt approach

Abandon analytical solutions, but find robust methods that can solve simple QFTs non-perturbatively and, if possible, to machine precision,without cheating.

Quantum field theory: a bit of philosophy

Two ways to attackreal world quantum field theories non-perturbatively

1. Startsimpler so that it becomessimpler [e.g. self interacting scalar fieldφ4 2]

2. Startmore complex so that it becomessimpler[e.g. N=4SYM]

φ4₂- pile of dirt QCD - Everest N=4 SYM - Chrysler building 2. is a good way to make fast progress first, but is limited in what it can achieve

Goal - ideal - philosophy: an apology of the pile of dirt approach

Abandon analytical solutions, but find robust methods that can solve simple QFTs non-perturbatively and, if possible, to machine precision,without cheating.

Quantum field theory: a bit of philosophy

Two ways to attackreal world quantum field theories non-perturbatively

1. Startsimpler so that it becomessimpler [e.g. self interacting scalar fieldφ4 2]

2. Startmore complex so that it becomessimpler[e.g. N=4SYM]

φ4₂- pile of dirt QCD - Everest N=4 SYM - Chrysler building 2. is a good way to make fast progress first, but is limited in what it can achieve

Goal - ideal - philosophy: an apology of the pile of dirt approach

Abandon analytical solutions, but find robust methods that can solve simple QFTs non-perturbatively and, if possible, to machine precision,without cheating.

Quantum field theory: a bit of philosophy

Two ways to attackreal world quantum field theories non-perturbatively

1. Startsimpler so that it becomessimpler [e.g. self interacting scalar fieldφ4 2]

2. Startmore complex so that it becomessimpler[e.g. N=4SYM]

φ4₂- pile of dirt QCD - Everest N=4 SYM - Chrysler building 2. is a good way to make fast progress first, but is limited in what it can achieve

Goal - ideal - philosophy: an apology of the pile of dirt approach

Abandon analytical solutions, but find robust methods that can solve simple QFTs non-perturbatively and, if possible, to machine precision,without cheating.

QFT: simplest non-trivial example

φ4

2 is a relativistic QFT ind =1+1 dimensions defined by the Hamiltonian:

H= Z dx 1 2π^ 2₊1 2(∇ ^ φ)2+ m 2 2 ^ φ2+ g 4 : ^φ 4_:

I It exists as a true QFT (rigorously defined by constructive field theorists) I It is not integrable, has no special structure simplifying computations I It has a phase transition which cannot be located perturbatively

=⇒ the ultimate pile of dirt: non-trivial, no cheating, but easy todefine

S. Rychkov challenge'2015

Compute everything one could want to compute about this model for all values ofg. In particular find the position of the phase transition (approximately atg/m2_'10±20%)

QFT: simplest non-trivial example

φ4

2 is a relativistic QFT ind =1+1 dimensions defined by the Hamiltonian:

H= Z dx 1 2π^ 2₊1 2(∇ ^ φ)2+ m 2 2 ^ φ2+ g 4 : ^φ 4_:

I It exists as a true QFT (rigorously defined by constructive field theorists) I It is not integrable, has no special structure simplifying computations I It has a phase transition which cannot be located perturbatively

=⇒ the ultimate pile of dirt: non-trivial, no cheating, but easy todefine

S. Rychkov challenge'2015

Compute everything one could want to compute about this model for all values ofg. In particular find the position of the phase transition (approximately atg/m2_'10±20%)

QFT: simplest non-trivial example

φ4

2 is a relativistic QFT ind =1+1 dimensions defined by the Hamiltonian:

H= Z dx 1 2π^ 2₊1 2(∇ ^ φ)2+ m 2 2 ^ φ2+ g 4 : ^φ 4_:

I It exists as a true QFT (rigorously defined by constructive field theorists) I It is not integrable, has no special structure simplifying computations I It has a phase transition which cannot be located perturbatively

=⇒ the ultimate pile of dirt: non-trivial, no cheating, but easy todefine

S. Rychkov challenge'2015

Compute everything one could want to compute about this model for all values ofg. In particular find the position of the phase transition (approximately atg/m2_'10±20%)

Tensor network states: a tool

Applications

I Quantum information theory I Statistical Mechanics I Quantum gravity I Many-body quantum Negative theology I Notcovariant/geometric objectsgµνorRµνκσ

I Nottensormodels [Rivasseau, Gurau, ...]

Tensor network states: a tool

Applications

I Quantum information theory I Statistical Mechanics I Quantum gravity I Many-body quantum Negative theology I Notcovariant/geometric objectsgµνorRµνκσ

I Nottensormodels [Rivasseau, Gurau, ...]

Outline

1. Tensor networks on the lattice 2. Bringing QFT to the lattice

Many-body problem

Problem

Finding low energy states of

^ H= N X k=1 ^ hk

is hardbecausedimH ∝DN

Possible solutions I Perturbation theory I Monte Carlo

I Bootstrap IR fixed point

I Variational optimization (e.g. Mean Field, Hamiltonian truncation, tensor networks)

Variational optimization

Exact variational optimization To find the ground state:

|0i= min

|ψi∈H

hψ|H|ψi hψ|ψi

Variational optimization

Approx. variational optimization To find the ground state:

|0i= min

|ψi∈M

hψ|H|ψi hψ|ψi I dimM ∝Poly(N)or fixed

Interesting states are weakly entangled

Low energy state |ψi=|0i or |1i ... Reduced density matrix ρ=tr_Dc h |ψihψ|i Entanglement entropy S = −tr ρlogρ Area law S ∝|∂D|

Interesting states are weakly entangled

Low energy state |ψi=|0i or |1i ... Reduced density matrix ρ=tr_Dc h |ψihψ|i Entanglement entropy S = −tr ρlogρ Area law S ∝|∂D|

Typical states are strongly entangled

Random state |ψi=UHaar|triviali

Reduced density matrix ρ=tr_Dc h |ψihψ|i Entanglement entropy S = −tr ρlogρ Volume law S∝|D|

Constructing weakly entangled states

1. Put auxiliary

maximally entangled states between sites

=

χ

X

j=1

|ji|ji

2. Map to initial Hilbert space on each site

Constructing weakly entangled states

1. Put auxiliary

maximally entangled states between sites

=

χ

X

j=1

|ji|ji

2. Map to initial Hilbert space on each site

Constructing weakly entangled states

1. Put auxiliary

maximally entangled states between sites

=

χ

X

j=1

|ji|ji

2. Map to initial Hilbert space on each site

Tensor network states: definition

Why “tensor” network?

A:_C4χ→_CD _{−→ A}i j1,j2,j3,j4

|Ai=

with tensor contractions on links

Optimization

Find best Afor fixedχ (D×χ4 coeff.)

E0'min

A

hA|H^|Ai hA|Ai for example go down _∂Ai∂E

Tensor network states: definition

Why “tensor” network?

A:_C4χ→_CD _{−→ A}i j1,j2,j3,j4

|Ai=

with tensor contractions on links Optimization

Find best Afor fixedχ (D×χ4 coeff.)

E0'min

A

hA|H^|Ai hA|Ai for example go down _∂Ai∂E

Some facts

d =1spatial dimension

Theorems (colloquially)

1. For gappedH, tensor network states|Ai approximate well |0iwithχfixed 2. All|Aiare ground states of gappedH

d _>2 spatial dimension

Folklore

1. For gappedH, tensor network states |Ai approximate well |0iwithχfixed 2. Most|Aiare ground states of gappedH

Some facts

d =1spatial dimension

Theorems (colloquially)

1. For gappedH, tensor network states|Ai approximate well |0iwithχfixed 2. All|Aiare ground states of gappedH

d _>2 spatial dimension

Folklore

1. For gappedH, tensor network states|Ai approximate well |0iwithχfixed 2. Most|Aiare ground states of gappedH

From condensed matter to QFT

Tensor network are excellenttheoretically andnumericallybut limited to thelattice

2 options:

I Discretize QFT, solve with best known tensor network algorithms S(φ) =X hi,ji (φi−φj)2 2a2 a 2 (∇φ)2_/2 +X i 1 2µ 2 aφ2i + 1 4λaφ 4 i and takea→0

Lattice

φ

Discretize the action:

S(φ) =X hi,ji (φi−φj)2 2a2 a 2 (∇φ)2_/2 +X i 1 2µ 2 aφ 2 i + 1 4λaφ 4 i

Taking the limit

The right way to get the continuum limit is to take: µ2_a=µ2a2+ 3

2log(a)a

2_λ

λa=λa2 which is equivalent to normal ordering

At first order in perturbation theory, theφ4_{term behaves like aφ}2_{term times a log divergent}

Results with GILT tensor renormalization

With C. Delcamp, we found the critical pointfc =lima→0λ_µa2

a in the continuum limit to the

highest precision everarXiv:2003.12993

0.00 0.02 0.04 0.06 0.08 0.10 λ 10.6 10.7 10.8 10.9 11.0 11.1 Critical coupling fc ( λ ) fcont. c,1 +α1λlogλ+β1λ fcont. c,2 +α2λlogλ+β2λ+γ2λ2 fcont. c,3 +α3λlogλ+β3λ+γ3λ2logλ

Thealoga correction of the critical point position as a function of lattice spacinga was not known before

Directly in the continuum

What was known (since 2010)

Continuous matrix product states for1+1dimensionalnon-relativisticQFT

My contribution

I Define continuous tensor networks for1+d dimensionalnon-relativisticQFT [with I. Cirac]

I Demonstrate that they have the right UV properties and fast convergence [with T. Karanikolaou]

I Define relativistic continuous matrix product states for1+1relativisticQFT [preliminary numerics]

Directly in the continuum

What was known (since 2010)

Continuous matrix product states for1+1dimensionalnon-relativisticQFT My contribution

I Define continuous tensor networks for1+d dimensionalnon-relativisticQFT [with I. Cirac]

I Demonstrate that they have the right UV properties and fast convergence [with T. Karanikolaou]

I Define relativistic continuous matrix product states for1+1relativisticQFT [preliminary numerics]

Continuous Matrix Product states

[Verstraete & Cirac 2010]: continuum limit ofMatrix ProductStates (d=1 tensor networks)

UV

IR

Works for Lieb-Liniger model (boson with contact interactions) Best method on the marketfor1+1 non-relativisticQFT But no version ford+1QFT, even “no-go” theorems

Continuous Tensor Networks: blocking

Uponblocking:

♦ Thephysical Hilbert space dimensionDincreases ♦ Thebond(auxiliary space)

Result

auxiliary field physical field

auxiliary degrees of fredom physical degrees of fredom

AT, J. I. Cirac,Phys. Rev. X 2019

Continuous tensor network state (heuristically)

State|αiofd+1 QFTfrom an auxiliary d dimensional theory of random fieldsφ: |αi= Z Dφexp − Z ddxL[φ(x)] −α[φ(x)]ψ^†(x) creation |Ωi

1. Genuine continuum limit of discrete tensor networks

2. Right UV scaling and exponential convergence to the ground state as the number of auxiliary fieldsφin increasedarXiv:2006.13143

Result

auxiliary field physical field

auxiliary degrees of fredom physical degrees of fredom

AT, J. I. Cirac,Phys. Rev. X 2019

Continuous tensor network state (heuristically)

State|αiofd+1 QFTfrom an auxiliary d dimensional theory of random fieldsφ: |αi= Z Dφexp − Z ddxL[φ(x)] −α[φ(x)]ψ^†(x) creation |Ωi

1. Genuine continuum limit of discrete tensor networks

2. Right UV scaling and exponential convergence to the ground state as the number of auxiliary fieldsφin increasedarXiv:2006.13143

Preliminary continuous relativistic results in

1

+

1

Test of a brand newrelativistic continuous matrix product state ansatzatg =4(deeply non perturbative). NoUVnorIRcutoffs!

Summary of tensor networks in QFT

Tensor networks are promising for non-perturbative QFT:

I They are already the best numerical method for QFT in1+1 dimensions I They can now be applied to (non-relativistic) QFT in 1+d

I They will very soon give rigorousresults for relativistic QFT in1+1dimensions In the near future:

I Push lattice based approach to lattice gauge theory (go beyond scalar) I Push continuous approach to relativistic 1+d