UV fixed-point structure of the 3d Thirring model

Transcript

1 UV fixed-point structure of the 3d Thirring model arxiv: [hep-th] (PRD accepted) Lukas Janssen in collaboration with Holger Gies Friedrich-Schiller-Universität Jena Theoretisch-Physikalisches Institut Corfu Summer Institute, ERG 2010, 13/09/2010 Lukas Janssen (FSU Jena) UV fixed-point structure of 3d Thirring ERG 2010, 13/09/ / 11

2 Outline 1 Introduction 2 Chirality in 3d 3 Fermionic RG flow and phase transitions 4 Conclusions Lukas Janssen (FSU Jena) UV fixed-point structure of 3d Thirring ERG 2010, 13/09/ / 11

3 Introduction I 3d massless Thirring model: L = i ψ a γ µ µ ψ a + g ( ψ a γ µ ψ a ) 2, 2N f a = 1, 2,..., N f Effective model for high-t c cuprate superconductors [Herbut, PRL 94, (2005)] electronic structure of graphene [Herbut et al., PRB 79, (2009)] But also d Thirring model per se very interesting field theory! TEM image of graphene Lukas Janssen (FSU Jena) UV fixed-point structure of 3d Thirring ERG 2010, 13/09/ / 11

4 Introduction II Chiral symmetry and/or parity symmetry spontaneously broken? Gap equation (leading 1/N f ): No SSB % Dyson-Schwinger Eq.: [Kondo, NPB 450, 251 (1995)] [Sugiura, PTP 97, 311 (1997)] [Hong et al., PRD 49, 5507 (1994)] χsb (large coupling) N f N cr f 2, 3.24, 4.32,? MC simulations: [Christofi et al., PRD 75, (2007)] N cr f 6.6(1)? Questions for effective models (N f = 2): Cuprate in χsb phase? Graphene in vacuum an insulator? [Drut et al., PRL 102, (2009)] [Drut et al., arxiv: [cond-mat.str-el]]... Lukas Janssen (FSU Jena) UV fixed-point structure of 3d Thirring ERG 2010, 13/09/ / 11

5 Chirality in 3d: 4-component formalism L = i ψ a γ µ µ ψ a + g 2N f ( ψ a γ µ ψ a ) 2, a = 1, 2,..., N f ψ, ψ: 4-component spinors γ µ : 4 4 matrices 2 fifth-γ matrices γ 4 & γ 5 : {γ 4,5, γ µ } = 0, {γ 4, γ 5 } = 0 For each flavor a = 1,..., N f : U(2) chiral symmetry 2 possible U A (1) generated by γ 4, γ 5 : ψ e iαγ4 ψ, etc. 2 possible UV (1) generated by 1, γ 45 := iγ 4 γ 5 : ψ e iβγ45 ψ, etc. U(N f ) flavor rotations: ψ a U ab ψ b, etc. Symmetry of 3d massless Thirring model: U(2N f ) generated by λ i {1, γ 4, γ 5, γ 45 } 1, γ 4, γ 5, γ 45 : generators of U(2) λ 1, λ 2,..., λ N 2: f generators of U(N f ) (N f N f Gell-Mann matrices) Lukas Janssen (FSU Jena) UV fixed-point structure of 3d Thirring ERG 2010, 13/09/ / 11

6 Chirality in 3d: 4-component formalism L = i ψ a γ µ µ ψ a + g 2N f ( ψ a γ µ ψ a ) 2, a = 1, 2,..., N f ψ, ψ: 4-component spinors γ µ : 4 4 matrices 2 fifth-γ matrices γ 4 & γ 5 : {γ 4,5, γ µ } = 0, {γ 4, γ 5 } = 0 For each flavor a = 1,..., N f : U(2) chiral symmetry 2 possible U A (1) generated by γ 4, γ 5 : ψ e iαγ4 ψ, etc. 2 possible UV (1) generated by 1, γ 45 := iγ 4 γ 5 : ψ e iβγ45 ψ, etc. U(N f ) flavor rotations: ψ a U ab ψ b, etc. Symmetry of 3d massless Thirring model: U(2N f ) generated by λ i {1, γ 4, γ 5, γ 45 } 1, γ 4, γ 5, γ 45 : generators of U(2) λ 1, λ 2,..., λ N 2: f generators of U(N f ) (N f N f Gell-Mann matrices) Lukas Janssen (FSU Jena) UV fixed-point structure of 3d Thirring ERG 2010, 13/09/ / 11

7 Chiral symmetry as part of flavor symmetry U(2N f ) in 2-component formalism: ( ) ψ a χ a L = i χ A σ µ µ χ A + ( χ A σ µ χ A ) 2 χ a+n f where a = 1,..., N f and A = 1,..., 2N f U(2N f ) symmetry manifest: χ A U AB χ B, etc. Chiral symmetry flavor symmetry Symmetry breaking mechanisms: χsb vs. PSB ψ a ψ a = χ a χ a χ a+n f χ a+n f 0 U(2N % f ) U(N f ) U(N f ), P " ψ a γ 45 ψ a = χ A χ A 0 U(2N f ) ", P% Lukas Janssen (FSU Jena) UV fixed-point structure of 3d Thirring ERG 2010, 13/09/ / 11

8 Chiral symmetry as part of flavor symmetry U(2N f ) in 2-component formalism: ( ) ψ a χ a L = i χ A σ µ µ χ A + ( χ A σ µ χ A ) 2 χ a+n f where a = 1,..., N f and A = 1,..., 2N f U(2N f ) symmetry manifest: χ A U AB χ B, etc. Chiral symmetry flavor symmetry Symmetry breaking mechanisms: χsb vs. PSB ψ a ψ a = χ a χ a χ a+n f χ a+n f 0 U(2N % f ) U(N f ) U(N f ), P " ψ a γ 45 ψ a = χ A χ A 0 U(2N f ) ", P% Lukas Janssen (FSU Jena) UV fixed-point structure of 3d Thirring ERG 2010, 13/09/ / 11

9 Classification of bilinears/4-fermi interactions Find all bilinears/4-fermi terms invariant under U(2N f )! Bilinears: No mass terms: % ψψ, ψγ % 45 ψ First order in derivative: i ψγ µ µ ψ " 4-fermi terms: ( ψ a γ µ ψ a ) 2 = ( χ A σ µ χ A ) 2 " ( ψ a γ 45 ψ a ) 2 = ( χ A χ A ) 2 " ( ψ a ψ b ) 2 ( ψ a γ 4 ψ b ) 2 ( ψ a γ 5 ψ b ) 2 + ( ψ a γ 45 ψ b ) 2 = ( χ A χ B ) 2 " ( ψ a γ µ ψ b ) 2 + ( ψ a σ µν ψ b ) 2 ( ψ a iγ µ γ 4 ψ b ) 2 ( ψ a iγ µ γ 5 ψ b ) 2 = ( χ A σ µ χ B ) 2 2 " where ( ψ a ψ b ) 2 ψ a ψ b ψ b ψ a, etc. Fierz: ( χ A χ B ) 2 = ( χ A σ µ χ A ) 2 ( χ A χ A ) 2, ( χ A σ µ χ B ) 2 = ( χ A σ µ χ A ) 2 3( χ A χ A ) 2 2 independent couplings Lukas Janssen (FSU Jena) UV fixed-point structure of 3d Thirring ERG 2010, 13/09/ / 11

10 Classification of bilinears/4-fermi interactions Find all bilinears/4-fermi terms invariant under U(2N f )! Bilinears: No mass terms: % ψψ, ψγ % 45 ψ First order in derivative: i ψγ µ µ ψ " 4-fermi terms: ( ψ a γ µ ψ a ) 2 = ( χ A σ µ χ A ) 2 " ( ψ a γ 45 ψ a ) 2 = ( χ A χ A ) 2 " ( ψ a ψ b ) 2 ( ψ a γ 4 ψ b ) 2 ( ψ a γ 5 ψ b ) 2 + ( ψ a γ 45 ψ b ) 2 = ( χ A χ B ) 2 " ( ψ a γ µ ψ b ) 2 + ( ψ a σ µν ψ b ) 2 ( ψ a iγ µ γ 4 ψ b ) 2 ( ψ a iγ µ γ 5 ψ b ) 2 = ( χ A σ µ χ B ) 2 2 " where ( ψ a ψ b ) 2 ψ a ψ b ψ b ψ a, etc. Fierz: ( χ A χ B ) 2 = ( χ A σ µ χ A ) 2 ( χ A χ A ) 2, ( χ A σ µ χ B ) 2 = ( χ A σ µ χ A ) 2 3( χ A χ A ) 2 2 independent couplings Lukas Janssen (FSU Jena) UV fixed-point structure of 3d Thirring ERG 2010, 13/09/ / 11

11 Classification of bilinears/4-fermi interactions Find all bilinears/4-fermi terms invariant under U(2N f )! Bilinears: No mass terms: % ψψ, ψγ % 45 ψ First order in derivative: i ψγ µ µ ψ " 4-fermi terms: ( ψ a γ µ ψ a ) 2 = ( χ A σ µ χ A ) 2 " ( ψ a γ 45 ψ a ) 2 = ( χ A χ A ) 2 " ( ψ a ψ b ) 2 ( ψ a γ 4 ψ b ) 2 ( ψ a γ 5 ψ b ) 2 + ( ψ a γ 45 ψ b ) 2 = ( χ A χ B ) 2 " ( ψ a γ µ ψ b ) 2 + ( ψ a σ µν ψ b ) 2 ( ψ a iγ µ γ 4 ψ b ) 2 ( ψ a iγ µ γ 5 ψ b ) 2 = ( χ A σ µ χ B ) 2 2 " where ( ψ a ψ b ) 2 ψ a ψ b ψ b ψ a, etc. Fierz: ( χ A χ B ) 2 = ( χ A σ µ χ A ) 2 ( χ A χ A ) 2, ( χ A σ µ χ B ) 2 = ( χ A σ µ χ A ) 2 3( χ A χ A ) 2 2 independent couplings Lukas Janssen (FSU Jena) UV fixed-point structure of 3d Thirring ERG 2010, 13/09/ / 11

12 Fermionic RG flow Truncation: Full basis of fermionic 4-point functions ( Γ k = d 3 x iz k ψ a γ µ µ ψ a + ḡk ( ψ a γ µ ψ a ) 2 + ḡ ) k ( ψ a γ 45 ψ a ) 2 2N f 2N f Wetterich Eq. β functions: β i = g i + g j A (i) jk g k Explicitly: k k g = g 4l(F) 1 π 2 k k g = g + 4l(F) 1 π 2 η k := k k ln Z k = 0 g, g : dim less couplings l (F) 1 : threshold function [ 2Nf 1 g 2 3 gg 1 ] g 2 2N f 2N f N f [ 1 gg + 2N ] f + 1 g 2 2N f 6N f l (F) 1 = { 2/3 linear cut-off 1 sharp cut-off Lukas Janssen (FSU Jena) UV fixed-point structure of 3d Thirring ERG 2010, 13/09/ / 11

13 RG flow for N f = 1 Analytic expressions for FPs & critical exponents Generically for NGFP: Bg = g Θ = 1 Separatrices regions I IV Lukas Janssen (FSU Jena) UV fixed-point structure of 3d Thirring ERG 2010, 13/09/ / 11

14 RG flow for N f = 1 Analytic expressions for FPs & critical exponents Generically for NGFP: Bg = g Θ = 1 Separatrices regions I IV Interpretation of FP A: Tune along g = 0: Interaction g( ψγ 45 ψ) 2 Large g > 0: ψγ45 ψ 0 A governs PSB Interpretation of Thirring FP C: Fierz Interaction (2g g)( ψγ µ ψ) 2 g[( ψψ) 2 ( ψγ 4 ψ) 2 ( ψγ 5 ψ) 2 ] If g / 2g g 1 : ψψ = 0 χsb If g / 2g g 1 : vector condensate ψγ µψ χsb % C : g / 2g g 4.3 expect: C governs χsb Lukas Janssen (FSU Jena) UV fixed-point structure of 3d Thirring ERG 2010, 13/09/ / 11

15 RG flow for N f 1 (a) FPs for N f = 1, 2, 4, 10, 100: (b)... and separatrices: g N f = 1 N f = 100 N f = O A C 1 g 2 3 N f = 100 N f = B 5 N f = 1 Large N f : C (0, 3) g / 2g g 1: χsb % Transition region g / 2g g 1 N f 7/4: Nf cr O(7/4) Quantum phase transition at Nf cr occurs because of competing large-n f d.o.f. not because of change in UV critical structure! Lukas Janssen (FSU Jena) UV fixed-point structure of 3d Thirring ERG 2010, 13/09/ / 11

16 Conclusions Summary: Fermionic RG flow for full basis of fermionic 4-point functions 2 possible phase transitions: parity breaking or χsb If N f Nf cr : χsb % even for large coupling Justification for MC/DSE studies building on microscopic definition fixed only with Thirring coupling g In attractive domain of Thirring FP But: No universality of FP values Outlook: Include composite bosonic degrees of freedom Competing order parameters: V µ χ A σ µ χ A vs. ϕ AB χ A χ B Direct calculation of Nf cr Implications for graphene/cuprates (N f = 2)? Lukas Janssen (FSU Jena) UV fixed-point structure of 3d Thirring ERG 2010, 13/09/ / 11

17 Conclusions Summary: Fermionic RG flow for full basis of fermionic 4-point functions 2 possible phase transitions: parity breaking or χsb If N f Nf cr : χsb % even for large coupling Justification for MC/DSE studies building on microscopic definition fixed only with Thirring coupling g In attractive domain of Thirring FP But: No universality of FP values Outlook: Include composite bosonic degrees of freedom Competing order parameters: V µ χ A σ µ χ A vs. ϕ AB χ A χ B Direct calculation of Nf cr Implications for graphene/cuprates (N f = 2)? Lukas Janssen (FSU Jena) UV fixed-point structure of 3d Thirring ERG 2010, 13/09/ / 11