Competition and duality correspondence between chiral and - PowerPoint PPT Presentation

Competition and duality correspondence between chiral and superconducting channels in (2+1)-dimensional four-fermion models D. Ebert 1 T. Khunjua 2 K. Klimenko 3 V. Zhukovsky 2 1 Institute of Physics, Humboldt-University Berlin, Berlin, Germany 2 Faculty of Physics, Moscow State University, Moscow, Russia 3 Institute for High Energy Physics, NRC ”Kurchatov Institute”, Protvino, Moscow Region, Russia April 2016 JINR, Dubna 2016 () 1 / 29

List of content 1 Introduction 2 The model and its thermodynamical potential Lagrangian of the model Thermodynamical potential (TDP) 3 Numerical calculations Vacuum case: µ = 0 , µ 5 = 0 Selfdual case: g 1 = g 2 General case: g 1 � = g 2 4 Discussions/Summary Alternative model symmetric under U γ 3 (1) - group 4F theory in (1+1) dimensions Conclusions Bibliography JINR, Dubna 2016 () 2 / 29

Introduction Introduction JINR, Dubna 2016 () 3 / 29

Introduction Introduction Models with four-fermion interactions It is well known that relativistic quantum field models with four-fermion interactions serve as effective theories for low energy considerations of different real phenomena in a variety of physical branches: Meson spectroscopy, neutron star and heavy-ion collision physics are often investigated in the framework of (3+1)-dimensional 4F theories. Physics of (quasi)one-dimensional organic Peierls insulators (polyacetylene) is well described in terms of the (1+1)-dimensional 4F Gross-Neveu (GN) model. The quasirelativistic treatment of electrons in planar systems like high-temperature superconductors or in graphene is also possible in terms of (2+1)-dimensional GN models. It is important to note that the low-dimensional versions of the 4F theories provide just a method to describe solid state matter and to check the theoretical mechanism experimentally. JINR, Dubna 2016 () 4 / 29

Introduction Introduction Chiral symmetry breaking vs. superconductivity competition and duality correspondence In this talk we demonstrate that there exists a dual correspondence between chiral symmetry breaking phenomenon and superconductivity in the framework of some (2+1)-dimensional 4F theories. Before now, such a duality correspondence was a well-known feature of only some (1+1)-dimensional 4F theories: In 1977 Ojima and Fukuda mentioned that as a result of Pauli–G¨ ursey symmetry the chiral phase in (1+1)–dimensional 4F model could be interpreted as a difermion supercondicting phase. [Prog. Theor. Phys. 57 , 1720 (1977)] In 2003 Thies showed that in addition to the duality between condensates there is also duality between fermion number- µ and chiral charge- µ 5 chemical potentials. [Phys. Rev. D 68 , 047703 (2003)] In 2014 Ebert et al. investigated chiral symmetry breaking vs. superconductivity competition taking into account µ, µ 5 - chemical potentials and inhomogeneous patterns for the condensates. The duality correspondence was also investigated in details. [Phys. Rev. D 90 , 045021 (2014)] It is worth to note that in recent years properties of media with nonzero chiral chemical potential µ 5 , i.e. chiral media, attracted considerable interest. In nature, chiral media might be realized in heavy-ion collisions, compact stars, condensed matter systems, etc. JINR, Dubna 2016 () 5 / 29

The model and its thermodynamical potential The model and its thermodynamical potential JINR, Dubna 2016 () 6 / 29

The model and its thermodynamical potential Lagrangian of the model Lagrangian of the model � γ ν i∂ ν + µγ 0 + µ 5 γ 0 γ 5 � ψ k + G 1 N (4 F ) ch + G 2 L = ¯ ψ k N (4 F ) sc , where � � � � � ¯ � 2 + � ¯ � 2 , ψ k iγ 5 ψ k ψ j C ¯ ¯ ψ T ψ T (4 F ) ch = ψ k ψ k (4 F ) sc = k Cψ k . j Definitions ψ k ( k = 1 , ..., N ) – fundamental multiplet of the O ( N ) ψ k – four-component (reducible) Dirac spinor γ ν ( ν = 0 , 1 , 2) and γ 5 – gamma-matrices C ≡ γ 2 – charge conjugation matrix JINR, Dubna 2016 () 7 / 29

The model and its thermodynamical potential Lagrangian of the model Lagrangian of the model � γ ν i∂ ν + µγ 0 + µ 5 γ 0 γ 5 � ψ k + G 1 N (4 F ) ch + G 2 L = ¯ ψ k N (4 F ) sc , where � � � � � ¯ � 2 + � ¯ � 2 , ψ k iγ 5 ψ k ψ j C ¯ ¯ ψ T ψ T (4 F ) ch = ψ k ψ k (4 F ) sc = k Cψ k . j Notations µ – fermion number chemical potential µ 5 – chiral (axial) chemical potential G 1 , G 2 – coupling constants JINR, Dubna 2016 () 7 / 29

The model and its thermodynamical potential Lagrangian of the model Lagrangian of the model � γ ν i∂ ν + µγ 0 + µ 5 γ 0 γ 5 � ψ k + G 1 N (4 F ) ch + G 2 L = ¯ ψ k N (4 F ) sc , where � � � � � ¯ � 2 + � ¯ � 2 , ψ k iγ 5 ψ k ψ j C ¯ ¯ ψ T ψ T (4 F ) ch = ψ k ψ k (4 F ) sc = k Cψ k . j Symmetries Lagrangian is invariant under transformations from the U V (1) × U γ 5 (1) group Fermion number conservation group U V (1) : ψ k → exp( iα ) ψ k Continuous chiral transformations U γ 5 (1) : ψ k → exp( iαγ 5 ) ψ k Lagrangian is also invariant under transformations from the internal auxiliary O ( N ) group JINR, Dubna 2016 () 7 / 29

The model and its thermodynamical potential Lagrangian of the model Gamma matrices in the four-dimensional spinor space Irreducible representation of the SO (2 , 1) group � 1 � � 0 � � � 0 i 0 1 γ 0 = σ 3 = γ 1 = iσ 1 = γ 2 = iσ 2 = ˜ , ˜ , ˜ . 0 − 1 i 0 − 1 0 Note that the definition of chiral symmetry is slightly unusual in (2+1)-dimensions. The formal reason is simply that there exists no other 2 × 2 matrix anticommuting with the γ ν which would allow the introduction of a γ 5 -matrix. The important Dirac matrices ˜ concept of chiral symmetries and their breakdown by mass terms can nevertheless be realized by considering a four-component reducible representation for Dirac fields: Reducible representation of the SO (2 , 1) group � ˜ � � ˜ � γ µ 0 ψ 1 ( x ) γ µ = ; ψ ( x ) = . ˜ γ µ 0 − ˜ ψ 2 ( x ) There exist two matrices, γ 3 and γ 5 , which anticommute with all γ µ and with themselves: � � � 0 , � 0 , I I γ 3 = i γ 5 = − γ 0 γ 1 γ 2 γ 3 = , . − I , 0 I , 0 JINR, Dubna 2016 () 8 / 29

The model and its thermodynamical potential Lagrangian of the model Duality correspondence and Pauli–G¨ ursey transformation Pauli–G¨ ursey transformation of the fields → 1 2(1 − γ 5 ) ψ k ( x ) + 1 2(1 + γ 5 ) C ¯ ψ T PG : ψ k ( x ) − k ( x ) . Taking into account that all spinor fields anticommute with each other, it is easy to see that under the action of the PG-transformation the 4F structures of the Lagrangian are converted into themselves: P G (4 F ) ch ← → (4 F ) sc , and, moreover, each Lagrangian L ( G 1 , G 2 ; µ, µ 5 ) is transformed into another one according to the following rule: P G L ( G 1 , G 2 ; µ, µ 5 ) ← → L ( G 2 , G 1 ; − µ 5 , − µ ) . JINR, Dubna 2016 () 9 / 29

The model and its thermodynamical potential Lagrangian of the model Semi-bosonized version of the Lagrangian Let us introduce the semi-bosonized version of the Lagrangian that contains only quadratic powers of fermionic fields as well as auxiliary bosonic fields σ ( x ), π ( x ), ∆( x ) and ∆ ∗ ( x ): � � γ ν i∂ ν + µγ 0 + µ 5 γ 0 γ 5 − σ − iγ 5 π L = ¯ � ψ k ψ k − − N ( σ 2 + π 2 ) − N ∆ ∗ ∆ − ∆ ∗ k Cψ k ] − ∆ 2 [ ψ T 2 [ ¯ ψ k C ¯ ψ T k ] , where 4 G 1 4 G 2 Bosonic fields σ = − 2 G 1 π = − 2 G 1 N ( ¯ N ( ¯ ψ k iγ 5 ψ k ); ψ k ψ k ) , ∆ = − 2 G 2 ∆ ∗ = − 2 G 2 N ( ¯ ψ k C ¯ N ( ψ T ψ T k Cψ k ) , k ); σ and π – are real fields ∆ and ∆ ∗ – are Hermitian conjugated complex fields JINR, Dubna 2016 () 10 / 29

The model and its thermodynamical potential Lagrangian of the model Properties of the bosonic fields Under the chiral U γ 5 (1) group the fields ∆ , ∆ ∗ are singlets, but the fields σ, π are transformed in the following way: U γ 5 (1) : σ → cos(2 α ) σ + sin(2 α ) π, π → − sin(2 α ) σ + cos(2 α ) π Clearly, all the fields are also singlets with respect to the auxiliary O ( N ) group, since the representations of this group are real. Moreover, with respect to the parity transformation P : ψ k ( t, x, y ) → iγ 5 γ 1 ψ k ( t, − x, y ) , P : k = 1 , ..., N, the fields σ ( x ), ∆( x ) and ∆ ∗ ( x ) are even quantities, i.e. scalars, but π ( x ) is a pseudoscalar. If � ∆ � � = 0, then the Abelian fermion number conservation U V (1) symmetry of the model and parity invariance is spontaneously broken down and the superconducting phase is realized in the model. If � σ � � = 0 then the continuous U γ 5 (1) chiral symmetry of the model is spontaneously broken. JINR, Dubna 2016 () 11 / 29

The model and its thermodynamical potential Thermodynamical potential (TDP) Effective action The effective action S eff ( σ, π, ∆ , ∆ ∗ ) of the considered model is expressed by means of the path integral over fermion fields: � � � � � N exp( i S eff ( σ, π, ∆ , ∆ ∗ )) = [ d ¯ L d 3 x � ψ l ][ dψ l ] exp i , l =1 where � N � � 4 G 1 ( σ 2 + π 2 ) + N S eff ( σ, π, ∆ , ∆ ∗ ) = − d 3 x 4 G 2 ∆∆ ∗ + � S eff , and � � � � � � ψ ( γ ν i∂ ν + µγ 0 + µ 5 γ 0 γ 5 − σ − iγ 5 π ) ψ − ∆ ∗ ψ T ) ¯ 2 ( ψ T Cψ ) − ∆ 2 ( ¯ ψC ¯ d 3 x S eff ) = i e ( i � [ d ¯ ψ l ][ dψ l ] e Henceforth we omit the index k from quark fields. JINR, Dubna 2016 () 12 / 29