TU Wien, April 2014

TU Wien, April 2014

Chiral Primaries in Strange Metals

Ingo Kirsch DESY Hamburg, Germany

`M. Isachenkov, I.K., V. Schomerus, arXiv: 1403.6857

Based on work with

V. Schomerus, M. Isachenkov

A compressible quantum matter is a translationally-invariant quantum system with a globally conserved U(1) charge Q, i.e. [H, Q]=0. The ground state of the Hamiltonian H-mQ is compressible if <Q> changes smoothly as a function of the chemical potential m. (excludes: solids, charge density waves and superfluids)

Options:

i) Fermi liquids (d>1): quasi-particles above Fermi surface which is given by a pole in the fermion Green’s function

ii) Non-Fermi liquids:

• Luttinger liquid (d=1): Fermi surface but no weakly-coupled quasi-particles above FS

Any other realization is referred to as …

• Strange metals: Fermi surface is hidden (since Green’s function not gauge invariant), and characterized by singular, non-quasi-particle low-energy excitations

Dispersion relation:

Motivation: Compressible quantum matter at T=0

2 Technische Universität Wien -- Ingo Kirsch


EtMe3Sb[Pd(dmit)2]2 Yamashita et al, Science (2010)

Triangular lattice of S=½ spins

beyond nearest-neighbor interactions destroy the antiferromagnetic order of the ground state

charge transport is that of an insulator

But: thermal conductivity is that of a metal!

Þ thermal transport of fermions near a Fermi : surface

Þ ground state: spinons (carry spin half but no charge)

An example of a strange metal at T=0



Outline:

I. Motivation: Strange metals at T=0

II. Strange metal model in d=1 spatial dimensions: Coset CFT

III. Partition function ZN (for higher N)

IV. The characters of the coset theory

V. Chiral ring of chiral primaries

Conclusions

Overview

ETH Zurich, 30 June 2010



A very promising candidate of a strange metal is a model of fermions at non-zero density coupled to an Abelian or non-Abelian gauge field.

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

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

• strongly-coupled high density regime:

• approximate the excitations near the zero-dimensional Fermi surface by two sets of relativistic fermions:

• currents generate an SO(2N2-2)1 affine algebra

Strange metal model in d=1 spatial dimensions

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Technische Universität Wien -- Ingo Kirsch

• effective low-energy theory Lagrangian:

• integrate out gauge fields A:

generate an SU(N) at level 2N, SU(N)2N.

• low-energy coset CFT:

• emergent SUSY in the IR - not present in the UV theory (with an emergent U(1) x U(1) global (R-)symmetry rotating the left- and right-moving ferminons separately!)

• central charge:

Strange metal model in d=1 spatial dimensions (cont.)

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• coset studied only for N=2, 3:

• equivalence to minimal models:

• barrier:

For the coset CFT cannot be related to

a supersymmetric minimal model anymore.

New techniques required to study the coset for higher N!

Strange metal model in d=1 spatial dimensions (cont.)

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Part III: Partition function ZN

4 University of Chicago, 23 April 2007Technische Universität Wien -- Ingo Kirsch


GKO construction: The partition function of the coset theory follows from the numerator and denominator partition functions, and .

Numerator:

group:

representations: A = id, v, sp, c

numerator partition function:

with

Numerator partition function ZN

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group:

representations:

conformal weights:

identification current:

monodromy charge:

denominator partition function (D-type):

Denominator partition function ZD

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Example: N=2

So,

and

Denominator partition function ZD (cont.)

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Total partition function:

Substituting the matrices and , we find

Ex: N=2

Problem: The modular invariant possesses non-integer coefficients. This can be fixed by a procedure known as fixed-point resolution (Schellekens, Yankielowicz).

Modular invariant partition function

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Part IV: The characters



The coset characters are defined by

can be computed from

and are known, e.g.

For orbits {a} of maximal length, the branching functions are identical to the characters . For short orbits, they split into a sum of characters.

Branching functions and characters

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The partition function is given by

and the branching functions are

N=2: Characters

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After fixed-point resolution, the partition function becomes

with

and similarly, sp and c.

This can be rewritten as (the partition function of a compactified free boson)

N=2: Characters (cont.)

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Fixed-point resolutione.g. for x=1:

The partition function is given by

and the branching functions are

N=3: Characters

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Part V: Chiral Primaries and Chiral Ring



Chiral primaries O are superconformal primaries ([Sa , O] ~ 0) that are also annihilated by some of the supercharges:

[Qa, O] ~ 0

chiral primaries:

bound on chiral primaries:

They can be read off from the characters… find terms with .

Chiral primaries

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There is a large set of chiral primaries which Y’ Y N=4 can be constructed for any N:

Consider all Young diagrams Y’ with

Then we can construct a Young diagram Yas follows (graphical construction):

• complete to matrix• rotate complement and attach from left • remove those which are in the same orbit , appear

only once

(e.g. N=4 )

Regular chiral primaries

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(2x)

N=4: Characters

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Necklaces for N=4:

(h, Q) of the ground states in the NS sectors (id, a) and (v, a)

Regular and exceptional chiral primaries at N=4

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exceptional chiral primary

Regular and exceptional chiral primaries at N=5

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Three exceptional CPs

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 us check that the previously found chiral primaries indeed form a closed algebra under fusion…

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

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

regular NS chiral primary fields.

Chiral Ring

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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 for N=2, 3

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N=4:

Chiral Ring for N=4

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N=5:

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

Chiral Ring for N=5

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I discussed coset theories of the type

Gopakumar et al. studied this space for N=2, 3, for which the coset can be related to supersymmetric minimal models.

I developed new techniques to study the coset for higher N:

• N=4, 5: - I explicitly derived the q-expansion of ZN (up to some order) - identified the chiral primary fields - established a classification scheme for CPs (regulars vs. exceptionals) - found a representation of CPs (and orbits) in terms of necklaces

- argued that they form a chiral ring under fusion

Outlook (work in progress):

• Large N limit + AdS dual description

Conclusions



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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we also have N=5 ...


TU Wien, April 2014

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