Solvable matter on 2d Causal Dynamical Triangulation? - QG-Paris2017.pdf · Solvable matter on 2d Causal Dynamical Triangulation? John Wheater University of Oxford arXiv: hep-th/0509191,

Solvable matter on 2d Causal Dynamical Triangulation?

John WheaterUniversity of Oxford

arXiv: hep-th/0509191, math-ph/0607020, 0908.3643 Durhuus, Jonsson, JWarXiv:1201.4322 Atkin & Zohren arXiv:1405.6782 Ambjorn, Durhuus, JW

contents

1. Quantum Gravity

2. Triangulations & metrics

3. Causal triangulation & tree bijection

4. Adding matter: dimers on CTs

5. Phase diagram & geometry

6. Concluding remarks

1. quantum gravity

μνg (x,t)Gravity’s dynamical degree of freedom is the metric

Classically obeys Einstein’s equations:g (x,t)μν

〈g (x), t=T｜g (x), t=0〉~ab

g (x,t)μν

g (x,0)μν

space

time

ga gb∑w(g) g∈Γ

Quantum mechanically consider the probability amplitude for evolution from g to gba

Defining Γ and w(g)…

Generically approaches can be:

1. Not obviously geometrical in an elementary sense, for example some continuum field theoretical methods

2. Explicitly geometrical but directly in continuum space-time

3. Explicitly geometrical but in discretised space-time, based on a relationship between the metric and graphs — C(ausal) D(ynamical) T(riangulations) fall into this category.

So how do we do ∑g (on a fixed topology)?

Note that gμν ⬌ {geodesic (minimal) distances x ⟷ y}



Note that gμν ⬌ {geodesic (minimal) distances x ⟷ y} ⬄ {graph (minimal) distances i ⟷ j}




Triangulate a 2-dim space with spherical topology…


































a

∑g ⬌ ∑triangulations T : a ➞ 0, |T| ➞ ∞, |T|a2 = A

Now we know Γ…

1. Construct a dictionary between discrete and continuum properties (remember all triangles are equilateral) eg at a vertex R∝(6 - nedges)

2. Then construct the weight w(g) out of these quantities, so in 2d for a triangulation T (as curvature term topological)

w(T) = exp(-μ|T|)

Established long ago that, with completely general random triangulations, this model (of planar random graphs) is fascinating, but unfortunately it is not very gravity-like as we don’t get physics that looks causal

3. Causal Triangulation & Tree bijection

The Causal Triangulation introduced by Ambjorn and Loll has microscopic causality built in by only allowing sliced triangulations



Start with the slices up to S

S

v



Start with the slices up to S then add the slice S+1

S

v




S

i) vertex v has Ev forward edges &wv = exp(-μEv) v




S





S


1

2

k




S


1

2

k




S

i) vertex v has Ev forward edges &wv = exp(-μEv)

ii) complete triangulation with space- like edges

S+1

v

1

2

k




S

i) vertex v has Ev forward edges &wv = exp(-μEv)

ii) complete triangulation with space- like edges

S+1

space-like forward time-like

v

Z (S,S0;g) = ∑ ∏ exp(-μEv) T v∈T

We define an amplitude

T= {CTs starting at S0 and ending at S}, and g = exp(-μ)

Radius of convergence of Z is g = gc = 1/2




Radius of convergence of Z is g = gc = 1/2CTs have a bijection with trees: discard spatial and rightmost forward edge at each vertex…









at gc the tree offspring probability pn = (1/2)n+1

critical Galton Watson (GW) tree &⟨n⟩ = 1





GW tree conditioned on survival is an example of the Generic Random Tree (GRT) — ∞ tree, measure μ∞, having a single ∞ branch or spine

Bijection implies μ(∞ CDT) ⇔ μ(URT), Uniform RT ∊ GRT

⟨ |NR(r0)| ⟩GW ∼ R with dH = 2 (actually ν-1)

⟨ |BR(r0) | ⟩URT ∼ R with dh = 2

and in fact dh = 2 a.s. ie with measure 1.

dh

dH

2D pure CDT looks genuinely two-dimensional; in fact the continuum limit is shown to be 2d Horava-Lifshitz gravity

So can we disrupt the geometry by adding matter to act as a source?

Numerical simulations with unitary matter c < 1 indicated no matter - gravity interaction strong enough to affect long distance properties (unlike planar random graph case for example)

Are there any solvable systems?

4. Adding matter: Dimers on cdt arXiv:1201.4322 Atkin & Zohren arXiv:1405.6782 Ambjorn, Durhuus & Wheater

Introduce dimers - possible configurations

Two dimers may not share a triangle

31D D

2DDDD D

2’

31D D2 DDDD D2’

D4

wG = ∏ gE + 1v∈G

v

Z( 𝛏 ;g) = ∑ wG 𝛏 𝛏 𝛏 𝛏 𝛏 G,D

D1 D2’D2 D4D3

21 32’ 4

Z fn of 𝛏 + 𝛏 so set 𝛏 = 0 2 2’ 2’

If 𝛏 = 0 then Z(𝛏 ;g) = Z(0;g(𝛏 +1)) so still pure CDT2 1 1

➞

Ambjorn, Durhuus & JW: 𝛏4 = 0 still allows bijection to trees with vertices labelled as follows

3

10 2

0

Atkin & Zohren showed 𝛏3 = 𝛏4 = 0 allows bijection to labelled trees, and found some strange properties…

Label allocation L survives bijection —tree 𝛕 inherits a set of labelling rules enforcing dimer rule:

1. if a leaf vertex has label 3, then left spatial neighbour is 0 and right is 0 or 1

2. if a non-leaf vertex is 2, rightmost forward neighbour is not 13. a leaf vertex that is a rightmost or leftmost forward neighbour

is not 3

leaf vertex

3

10 2

0

𝛕 can be decomposed

0Z = g W - 1-1

these are geometric series so F are rational fns of W, also

j

i

k

W( 𝛏 ;g) = ∑ g 𝛏 𝛏 𝛏 𝛕,L:i

L1 L3L2

1 32i

𝛕

= F( W ; 𝛏 ;g)i

So for i = 0,1,2

and W satisfies a cubic — we are in business 0

➞

➞➞

𝛕 can be decomposed

0Z = g W - 1-1

these are geometric series so F are rational fns of W, also

j

i

k

W( 𝛏 ;g) = ∑ g 𝛏 𝛏 𝛏 𝛕,L:i

L1 L3L2

1 32i

𝛕

= F( W ; 𝛏 ;g)i

So for i = 0,1,2

and W satisfies a cubic — we are in business 0

➞

➞➞

5.phase diagram

gc(𝛏) is free energy; setting 𝛏 = 𝛏 = 𝛏 2

find W ( 𝛏 ;g) ∼ W (𝛏) - A(𝛏) (gc(𝛏) - g)0 0c𝛂

1

3

ξ

ξ

0

➞➞

➞

➞➞

gc(𝛏) is analytic except on dimer critical line, wherevariance of dimer density diverges

➞

5.phase diagram


Dimers sub-critical, pure CT 𝛂 = 1/2


1

3

ξ

ξ

0

➞➞

➞

➞➞


➞

5.phase diagram




1

conditionalconvergence

3

ξ

ξ

0

➞➞

➞

➞➞


➞

5.phase diagram



𝛂 = 1/3 Dimer critical line


1

conditionalconvergence

3

ξ

ξ

0

➞➞

➞

➞➞


➞

finite tree N ∼ Rinfinite tree B ∼ R

3

ξ

ξ

0

hdHd

What happens to the geometry?1. Far enough to the right of the line each individual graph has

positive weight after summing over Dimer config2. Close enough to the line, some individual graphs have

negative weight after summing over Dimer config3. But everywhere except in the non-conditional region we get

single spine trees


3

ξ

ξ

0

d = d = 2h H

hdHd




single spine trees


3

ξ

ξ

0

d = d = 2h H

d = 1, d = 3/2h HhdHd




single spine trees


3

ξ

ξ

0

d = d = 2h H

d = 1, d = 3/2h H

d = d = 3h H

hdHd




single spine trees

6.Concluding Remarks I

1. Away from dimer criticality the continuum theory is just (Horava-Lifshitz) 2d gravity

2. The critical line is believed to be critical dimer CFT (c=-22/5) coupled to H-L 2d gravity but no constructive proof exists

3. The AZ point is special, but not that special — in the full 𝛏 space it is a two-parameter family. Presumably it describes a CFT coupled to H-L gravity but we do not know which CFT

d = d = 3h H

d = 1, d = 3/2h H

Concluding Remarks 2

4. Systems with negative weights are intricate and can easily confound out intuition

5. But it is encouraging that systems definitely exist where matter interacts strongly with 2D CDT gravity

6. There has been only rather weak analytic progress on coupling unitary (+ve weight) matter to 2D CDT although there is a significant amount of numerical work on Ising spins and scalar fields

7. It seems difficult to extend the technical trick (labelled trees) to other matter systems in contrast to planar random graphs

Documents

Solvable matter on 2d Causal Dynamical Triangulation? - QG-Paris2017.pdf · Solvable matter on 2d Causal Dynamical Triangulation? John Wheater University of Oxford arXiv: hep-th/0509191,