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TU Wien, April 2014

`. M. Isachenkov , I.K., V. Schomerus , arXiv : 1403.6857. Chiral Primaries in Strange Metals Ingo Kirsch DESY Hamburg, Germany. Based on work with V . Schomerus , M . Isachenkov. TU Wien, April 2014. Chiral Primaries in Strange Metals .

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TU Wien, April 2014

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  1. ` M. Isachenkov, I.K., V. Schomerus, arXiv: 1403.6857 ChiralPrimaries in Strange Metals Ingo Kirsch DESY Hamburg, Germany Based on work with V. Schomerus, M. Isachenkov TU Wien, April 2014

  2. Chiral Primaries in Strange Metals Motivation: Compressible quantum matter at T=0 • 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 … • Strangemetals: Fermi surface is hidden (since Green’s function not gauge invariant), and characterized by singular, non-quasi-particle low-energy excitations • Dispersion relation: Technische Universität Wien -- Ingo Kirsch

  3. Chiral Primaries in Strange Metals An exampleof a strangemetalatT=0 • EtMe3Sb[Pd(dmit)2]2Yamashita 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) Technische Universität Wien -- Ingo Kirsch

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

  5. Chiral Primaries in Strange Metals Strange metal model in d=1 spatial dimensions • 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 Technische Universität Wien -- Ingo Kirsch

  6. Chiral Primaries in Strange Metals Strange metal model in d=1 spatial dimensions (cont.) • 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 SUSYin 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 • ferminonsseparately!) • central charge: Technische Universität Wien -- Ingo Kirsch

  7. Chiral Primaries in Strange Metals Strange metal model in d=1 spatial dimensions (cont.) • 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 cosetfor higher N! Technische Universität Wien -- Ingo Kirsch

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

  9. Chiral Primaries in Strange Metals NumeratorpartitionfunctionZN 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 Technische Universität Wien -- Ingo Kirsch

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

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

  12. Chiral Primaries in Strange Metals Modular invariant partitionfunction Total partition function: Substituting the matrices and , we find Ex: N=2 Problem: The modular invariant possessesnon-integer coefficients. This canbefixedby a procedureknownasfixed-point resolution (Schellekens, Yankielowicz). Technische Universität Wien -- Ingo Kirsch

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

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

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

  16. Chiral Primaries in Strange Metals N=2: Characters (cont.) 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) Fixed-point resolution e.g. for x=1: Technische Universität Wien -- Ingo Kirsch

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

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

  19. Chiral Primaries in Strange Metals Chiral primaries 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 . Technische Universität Wien -- Ingo Kirsch

  20. Chiral Primaries in Strange Metals Regular chiral primaries • 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 ) Technische Universität Wien -- Ingo Kirsch

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

  22. Chiral Primaries in Strange Metals Regular and exceptional chiral primaries at N=4 Necklaces for N=4: (h, Q) of the ground states in the NS sectors (id, a)and (v, a) exceptionalchiralprimary Technische Universität Wien -- Ingo Kirsch

  23. Chiral Primaries in Strange Metals Regular and exceptional chiral primaries at N=5 Threeexceptional CPs Technische Universität Wien -- Ingo Kirsch

  24. Chiral Primaries in Strange Metals Chiral Ring 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. Technische Universität Wien -- Ingo Kirsch

  25. Chiral Primaries in Strange Metals Chiral Ring for N=2, 3 Visualization of the chiral ring by tree diagrams: An arrow represents the action of x on a field, e.g. OPE (N=3) Technische Universität Wien -- Ingo Kirsch

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

  27. Chiral Primaries in Strange Metals Chiral Ring for N=5 N=5: In the large N limit, the number of chiral primaries is governed by the partition function p(6h). Technische Universität Wien -- Ingo Kirsch

  28. Chiral Primaries in Strange Metals Conclusions • I discussed cosettheories of the type • Gopakumar et al. studied this space for N=2, 3, for which the coset can be related to supersymmetricminimal models. • I developed new techniques to study the coset for higher N: • N=4, 5: - I explicitly derivedthe 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 + AdSdual description Technische Universität Wien -- Ingo Kirsch

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

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