Gauge/Gravity Duality 2013Max Planck Institute for Physics, 29 July to 2 Aug 2013

Coset approach to the Luttinger Liquid

Ingo Kirsch DESY Hamburg, Germany

`M. Isachenkov, I.K., V. Schomerus, arXiv: 1308.XXXX

• Electron transport in conductors is usually well-described by Fermi-liquid theory (d>1)

• BUT d=1: Electrons in a one-dimensional system form a quantum liquid which can be described as a Luttinger liquid rather than by Landau's Fermi-liquid theory

• Fermi surface but no weakly-coupled quasi-particles above FS

• Experimentally realized e.g. in quantum wires/carbon nanotubes

Luttinger Liquid

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carbon nanotube (source: wiki)

Coset approach to the Luttinger Liquid

Vertically aligned Carbon Nanotubes by using a photolithography method (source: Dept. of Electronics U. York)

Luttinger relation

Gopakumar-Hashimoto-Klebanov-Sachdev-Schoutens (2012):

• UV: 2d SU(N) gauge theory coupled to Dirac fermions

• IR: effective low-energy theory flows to 2d coset CFT:

emergent SUSY in the IR - not present in the UV theory!

• coset studied only for N=2, 3: equivalence to minimal models:

• What can we do (no relation to minimal models)?

Luttinger Liquid and matrix cosets

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Coset approach to the Luttinger Liquid

Gauge/Gravity Duality, MPI Munich 2013 -- Ingo Kirsch

Outline:

I. Motivation: Luttinger Liquid

II. Matrix coset theories

III. Partition function ZN (for higher N)

IV. Spectrum of primary fields

V. Chiral ring of chiral primaries

Conclusions

Overview

ETH Zurich, 30 June 2010

Coset approach to the Luttinger Liquid

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Numerator partition function:

Denominator partition function:

D-type modular invariant of the type

Coset partition function:

Example: The partition function Z3 (N=3)

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Coset approach to the Luttinger Liquid

Gauge/Gravity Duality, MPI Munich 2013 -- Ingo Kirsch

It is possible to write

Two successive decompositions:

i) decomposition of SO(16)1 characters into characters of SO(8)1 x SO(8)1:

ii) decomposition of SO(8)1 characters into characters of SU(3)3

branching the characters of SO(16)1 into those of SU(3)6:

Example: The partition function Z3 (N=3)

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Coset approach to the Luttinger Liquid

Gauge/Gravity Duality, MPI Munich 2013 -- Ingo Kirsch

Result for partition function Z3 :

where

and

Example: The partition function Z3 (N=3)

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Coset approach to the Luttinger Liquid

Gauge/Gravity Duality, MPI Munich 2013 -- Ingo Kirsch

We constructed an expression for the coset partition function ZN :

are branching functions:

Partition function ZN (general N)

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Coset approach to the Luttinger Liquid

Gauge/Gravity Duality, MPI Munich 2013 -- Ingo Kirsch

The branching functions are computed using a formula for diagonal cosets.

This gives the q-expansion of .

Spectrum:

Bouwknegt-McCarthy-Pilch formula

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Coset approach to the Luttinger Liquid

Gauge/Gravity Duality, MPI Munich 2013 -- Ingo Kirsch

Bouwknegt-McCarthy-Pilch (1991)

Goal: Compute the coset partition function ZN in terms of the branching functions and then rewrite it in terms of characters.

Example: N=2

cf. w/ characters

N=2,3: Partition function ZN and supersymmetry

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Coset approach to the Luttinger Liquid

Gauge/Gravity Duality, MPI Munich 2013 -- Ingo Kirsch

For N>3 it is difficult to rewrite ZN in terms of characters.

BUT: Still possible to write ZN = ZN (q) using BMP formula

Example: N=3

similarly

Likewise, we found the q-expansions of Z4 = Z4 (q), Z5 = Z5 (q) by computing

700 (N=4) and 10292 (N=5) branching functions…

N=4, 5: The partition function ZN(q)

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Coset approach to the Luttinger Liquid

Gauge/Gravity Duality, MPI Munich 2013 -- Ingo Kirsch

Spectrum: coset elements and their conformal weights

Parallel computing on DESY’s theory and HPC clusters

N=2 N=3N=4

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Coset approach to the Luttinger Liquid

we also have N=5 ...

Gauge/Gravity Duality, MPI Munich 2013 -- Ingo Kirsch

A particular feature of superconformal field theories is the chiral ring of NS sector chiral primary fields. These fields form a closed algebra under fusion.

Let’s identify the chiral primary fields (h=Q) by introducing charge into the branching functions (i.e. make them z-dependent)

Example: N=3

Chiral Ring

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Coset approach to the Luttinger Liquid

Gauge/Gravity Duality, MPI Munich 2013 -- Ingo Kirsch

Do the chiral primaries form a closed algebra under fusion? - Yes.

For instance, for N=3:

Generator of the chiral ring (h=Q=1/6):

Claim: Repeatedly act with x on the identity. This generates the chiral ring of

NS chiral primary fields.

Chiral Ring

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Coset approach to the Luttinger Liquid

Gauge/Gravity Duality, MPI Munich 2013 -- Ingo Kirsch

Visualization of the chiral ring by tree diagrams:

An arrow represents the action of x on a field, e.g. OPE (N=3)

Chiral Ring

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Coset approach to the Luttinger Liquid

Gauge/Gravity Duality, MPI Munich 2013 -- Ingo Kirsch

Chiral Ring

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Coset approach to the Luttinger Liquid

Gauge/Gravity Duality, MPI Munich 2013 -- Ingo Kirsch

N=4, 5:

In the large N limit, the number of chiral primaries is governed by the partition function p(6h).

I discussed diagonal coset theories of the type

Gopakumar et al. studied this space for N=2, 3 (by relating it to minimal models)

Our method works in principle for general N:

• general N: we derived the partition function ZN=ZN (b(q)), the branching functions b(q) can be computed using the BPM formula

(needs a lot of computer power for higher N though) • N=2, 3: we rewrote ZN in terms of characters• N=4, 5: - we explicitly derived the q-expansion of ZN (up to some order) - we identified the chiral primary fields and

- showed that they form a chiral ring under fusion

Outlook (work in progress):

• Large N limit + AdS dual description

Conclusions

Coset approach to the Luttinger Liquid

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