Matrix product approximations to multipoint
functions in two-dimensional conformal field theory
Robert Koenig (TUM) and Volkher B. Scholz (Ghent University)
based on arXiv:1509.07414 and 1601.00470

YIKS 2016

Goal••

Understand the entanglement structure of quantum field
theories using tensor network methods•

Tensor networks model the entanglement properties ofmany body systems and are successfully applied incondensed matter physics•

What about quantum field theories?

States of the quantum field theory and tensor network
states live in different Hilbert spaces: how to measurecloseness?

How to approximate a quantum field theory?
•Focus on physical quantities: correlation functions•If tensor networks can approximately reproduce
correlation functions of quantum field theories, then we
can use them to understand the entanglement structureof quantum field theories.

•This talk: Conformal Field Theories and Matrix productstates

Recap: Matrix product states
Tensor network states for spin chains:maximally entangled pairs of (bond) dimension D are placed
between the physical particles (physical dimension d)and are contracted by a DxDxd dimensional tensor••

Correlation functions can be computed efficiently (in D)
and reduce to the computation of a sequence ofcompletely positive maps on matrices of dimension DMain result
Correlation functions of Conformal Field Theories can bearbitrarily well approximated by correlations functions ofMatrix Product states.
Scaling of Parameters:
number of fields n, UV cutoﬀ d (measured in terms of energy), approximation
error ε, C constant depending on CFT (not necessarily central charge)
scaling of bonddimension

fixed n, UV cutoﬀ dfixed εlog(D)~ log(1/ε) C n/d~ √C n

Achievements & shortcomings•

We obtain a sequence of explicit Tensors describing
Matrix product states, which better and better satisfy theconformal symmetry•

Proof is mathematically rigorous and constructive; holds
for most unitary Conformal Field Theories•

However, the parameter scaling is worse than would
could be expected from entropic arguments[Cardy&Calabrese, Holzhey et. al.,..]•Uses the language of Vertex operator algebras: first
introduced by Borcherds in his proof of the Moonshineconjecture

More Symmetries: Wess-Zumino-Witten models•

In addition to conformal symmetries, WZW models
possess an additional local symmetry given by an affine
Lie algebra based on a simple compact Lie group•

These additional symmetries carry over to the MPSTensors; leads to a group invariant MPS•

Moreover, the interactions (fusion rules) are completely
described already in the lowest level; the higher order
Tensors are only needed to model the conformal andaffine symmetries

Proof sketch: regularization•

identify states with Hilbert-Schmidt operators on thechiral theory; field operators become linear maps: need
to approximate by finite-dimensional ones•a finite UV cutoff regularises the unbounded fieldoperators and turns them into bounded operators<φ1(x1)

φ2(x2) ….

φn(xn)>|x1 - x2| > d•techniques: use results of Wassermann for WZW models(explicit bounds), and the existence of genus-1
correlation functions for general CFTs [Zhu, Huang]Proof sketch:
Bounded�eld Bounded
operator�eld opBounded
Bounded�eldrenormalization
Boundedoperator
�eld operator�eldBounded
operator
canthecanthecan changethei (xi ):thei (xchangei ): cani (xchangi ):i (xi ): cani (xchangei ): cani (xchangei ):Bounded field operator
Precision
Truncatedenergyby an arbitraryenergyby anarbi
byenergy energy
by an arbitraryenergyby an arbitraryby an arbitraryenergy
φ(x): can change thebounded field operator φtr(x)amountamountamountamountamountamountenergy by an arbitrary
can only change the energy
amountφ1(x1)

by a fixed amountφ2(x2)

En
Energy Energy
levelsEnergylevelslevels
trtr
φ1 (x1) φ2 (x2)
Energy levels ofthe conformal
Hamiltonian≈Summary & Outlook•

Correlation functions of CFTs can be approximated bythose from MPS•

Our Approximations are constructive, provide rigorous
error bounds and respect additional symmetries (WZW)•Can be analysed further to understand low energy statesof CFTs in quantum information theoretic terms(connection to quantum error correction?)•Generalisation to MERA (multiscale entanglement
renormalization Ansatz) seems possible and may provide
better parameter scaling