TU Wien, April 2014 Chiral Primaries in Strange Metals Ingo Kirsch DESY Hamburg, Germany ` M. Isachenkov, I.K., V. Schomerus, arXiv: 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-  Q is compressible if changes smoothly as a function of the chemical potential  (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: 2 Technische Universität Wien -- Ingo Kirsch Chiral Primaries in Strange Metals

EtMe 3 Sb[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!  thermal transport of fermions near a Fermi : surface  ground state: spinons (carry spin half but no charge) An example of a strange metal at T=0 3 Technische Universität Wien -- Ingo Kirsch Chiral Primaries in Strange Metals

Outline: I.Motivation: Strange metals at T=0 II.Strange metal model in d=1 spatial dimensions: Coset CFT III.Partition function Z N (for higher N) IV.The characters of the coset theory V.Chiral ring of chiral primaries Conclusions Overview ETH Zurich, 30 June 2010 Chiral Primaries in Strange Metals 4 Technische Universität Wien -- Ingo Kirsch

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(2N 2 -2) 1 affine algebra Strange metal model in d=1 spatial dimensions 5 Chiral Primaries in Strange Metals 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.) 6 Chiral Primaries in Strange Metals Technische Universität Wien -- Ingo Kirsch

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.) 7 Chiral Primaries in Strange Metals Technische Universität Wien -- Ingo Kirsch

Part III: Partition function Z N 4 University of Chicago, 23 April 2007Technische Universität Wien -- Ingo Kirsch Chiral Primaries in Strange Metals

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 Z N 9 Chiral Primaries in Strange Metals Technische Universität Wien -- Ingo Kirsch

group: representations: conformal weights: identification current: monodromy charge: denominator partition function (D-type): Denominator partition function Z D 10 Chiral Primaries in Strange Metals Technische Universität Wien -- Ingo Kirsch

Example: N=2 So, and Denominator partition function Z D (cont.) 11 Chiral Primaries in Strange Metals Technische Universität Wien -- Ingo Kirsch

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 12 Chiral Primaries in Strange Metals Technische Universität Wien -- Ingo Kirsch

Part IV: The characters 4 University of Chicago, 23 April 2007Technische Universität Wien -- Ingo Kirsch Chiral Primaries in Strange Metals

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 14 Chiral Primaries in Strange Metals Technische Universität Wien -- Ingo Kirsch

The partition function is given by and the branching functions are N=2: Characters 15 Chiral Primaries in Strange Metals Technische Universität Wien -- Ingo Kirsch

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.) 16 Chiral Primaries in Strange Metals Technische Universität Wien -- Ingo Kirsch Fixed-point resolution e.g. for x=1:

The partition function is given by and the branching functions are N=3: Characters 17 Chiral Primaries in Strange Metals Technische Universität Wien -- Ingo Kirsch

Part V: Chiral Primaries and Chiral Ring 4 University of Chicago, 23 April 2007Technische Universität Wien -- Ingo Kirsch Chiral Primaries in Strange Metals

Chiral primaries O are superconformal primaries ([S , O] ~ 0) that are also annihilated by some of the supercharges: [Q , O] ~ 0 chiral primaries: bound on chiral primaries: They can be read off from the characters… find terms with. Chiral primaries 19 Chiral Primaries in Strange Metals Technische Universität Wien -- Ingo Kirsch

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 Y as 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 20 Chiral Primaries in Strange Metals Technische Universität Wien -- Ingo Kirsch

(2x) N=4: Characters 21 Chiral Primaries in Strange Metals Technische Universität Wien -- Ingo Kirsch

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 22 Chiral Primaries in Strange Metals Technische Universität Wien -- Ingo Kirsch exceptional chiral primary

Regular and exceptional chiral primaries at N=5 23 Chiral Primaries in Strange Metals Technische Universität Wien -- Ingo Kirsch 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 24 Chiral Primaries in Strange Metals Technische Universität Wien -- 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 for N=2, 3 25 Chiral Primaries in Strange Metals Technische Universität Wien -- Ingo Kirsch

N=4: Chiral Ring for N=4 26 Chiral Primaries in Strange Metals Technische Universität Wien -- Ingo Kirsch

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 27 Chiral Primaries in Strange Metals Technische Universität Wien -- Ingo Kirsch

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 Z N (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 Chiral Primaries in Strange Metals 28 Technische Universität Wien -- Ingo Kirsch

Spectrum: coset elements and their conformal weights Parallel computing on DESY’s theory and HPC clusters N=2 N=3 N=4 29 Chiral Primaries in Strange Metals we also have N=5... Technische Universität Wien -- Ingo Kirsch