Fermionic functional renormalization group for first-order phase transitions A mean-field model Roland Gersch, Julius Reiss, Carsten Honerkamp Dept. Metzner, MPI for Solid State Research, Stuttgart Dresden, 27.02.2007 Fermionic functional renormalization group for first-order phase transitions – p.1/12
Outline The fRG can scan a system’s order parameter space for minima of the thermodynamic potential. 1. Some fRG 2. Bias as a challenge 3. Bias as a chance Roland Gersch: fRG for first-order phase transitions – p.2/12
Motivation X X i : a : � Q ( X )� ! � ( Q ( X ) � �( X ; Y ))� X X X Y X X Y Spontaneous symmetry breaking f R G ! physical observables Microscopic models Fermionic degrees of freedom kept Unbiased approach for multiple instabilities Zanchi and Schulz 1998, Halboth and Metzner 2000, Salmhofer, Honerkamp, Furukawa, Rice 2001, Metzner, Reiss, Rohe 2005, Dupuis 2005 . Rigorous error estimates available Salmhofer, Honerkamp 2001 1. Some fRG Roland Gersch: fRG for first-order phase transitions – p.3/12
Z Formalities D � (� ; Q =� (�)�) � V (�)+( J ; �) exp ( � W ( J )) = e det Q =� ( �) Generating functional of connected Green’s functions J : Grassmannian generating field, V : two-particle � (� ) makes W solvable, � (� ) � 1 . i f � 1 _ _ _ interaction � = T T r ( Q Q + G Q ) One-particle irreducible (1PI) flow Salmhofer, Honerkamp 2001 1. Some fRG Roland Gersch: fRG for first-order phase transitions – p.4/12
Z Formalities D � (� ; Q =� (�)�) � V (�)+( J ; �) exp ( � W ( J )) = e det Q =� ( �) Generating functional of connected Green’s functions J : Grassmannian generating field, V : two-particle � (� ) makes W solvable, � (� ) � 1 . i f � 1 _ _ _ interaction � = T T r ( Q Q + G Q ) One-particle irreducible (1PI) flow Salmhofer, Honerkamp 2001 Katanin 2004 1. Some fRG Roland Gersch: fRG for first-order phase transitions – p.5/12
X X V 0 y y y H = ( " � � ) � 0 0 k k k k k + Q k k + Q N The external field’s influence 0 k k ; k X y Charge-density-wave mean-field model Hamiltonian + (� � � ) ; Q = ( � ; � ; : : : ) i k + Q k k � : chemical potential, " : tight-binding dispersion, V 0 : � � � i : external field. T = � = 0 : nearest-neighbor repulsion, 200 100 ∆ ext = 100 ∆ ext = 0.2 0.01 0.01 0.03 0.03 0.06 0.06 150 Effective interaction 0.16 0.16 RG, Honer- 0.40 0.40 1.00 1.00 kamp, Rohe, 100 ∆ 0.1 Metzner 2005 50 Units: hopping integral 0 0 0 0.1 0.2 0 0.05 0.1 0.15 0.2 Λ Λ 2. Bias as a challenge Roland Gersch: fRG for first-order phase transitions – p.6/12
First-order phase transitions Grand canonical potential, µ =0.24t, V 0 =2t 0.0025 T 0.08 0.002 0.06 0.04 0.02 0.0015 0.001 Ω 0.0005 0 -0.0005 -0.001 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 ∆ 2. Bias as a challenge Roland Gersch: fRG for first-order phase transitions – p.7/12
X X V 0 y y y H = ( " � � ) � 0 0 k Counterterms: first attempt k k k k + Q k k + Q N 0 k k ; k Back to the CDW Hamiltonian. X y + (� � � ) i k + Q k k ! 1 i! � " + � � � � � � 1 G = To bare propagator � � � � � i! + " + � To initial self-energy � (�) 2f 0 ; 1 g _ ) G / � � ( � � � � ) ) � = / � � ( � � � � ) = 0 12 | {z } =: � � eff 2. Bias as a challenge Roland Gersch: fRG for first-order phase transitions – p.8/12
p � (�) � � , � = 0 , � = 1 equivalent to linearly turning on i f 0 to V 0 . Interaction flow the interaction from p Honerkamp, Rohe, Andergassen, Enss 2004 . � � j � Æ �( � � ) 6 = 0 e� �= Æ � _ � 6 = 0 . Advantages: � = � i : scanning the order parameter � Thus, i for mean-field models Chosing space for thermodynamic potential minima. Results independent of Large effective interactions restricted to the final region of the flow 3. Bias as a chance Roland Gersch: fRG for first-order phase transitions – p.9/12
First-order CDW phase transition T > T T < T t t 0.8 0.8 Order pa- 0.6 0.6 rameter ∆ eff ∆ eff 0.4 0.4 units: hopping 0.2 0.2 integral 0 0 0 0.2 0.4 0.6 0.8 0 0.2 0.4 0.6 0.8 Λ Λ T > T T < T t t 10 40 Effective interaction Effective interaction 9 35 Effective 8 30 interaction 7 25 6 20 5 15 units: hopping 4 10 3 5 integral 2 0 0 0.2 0.4 0.6 0.8 0 0.2 0.4 0.6 0.8 Λ Λ RG, Reiss, Honerkamp 2006 3. Bias as a chance Roland Gersch: fRG for first-order phase transitions – p.10/12
� � for T < T � for T > T t : t : Flow of -1.392 -1.394 -1.394 -1.396 -1.396 -1.398 Ω [units of t] Ω [units of t] -1.398 -1.4 -1.4 -1.402 -1.402 -1.404 -1.404 -1.406 -1.406 0.9 0.92 0.94 0.96 0.98 1 0.9 0.92 0.94 0.96 0.98 1 Λ Λ RG, Reiss, Honerkamp 2006 3. Bias as a chance Roland Gersch: fRG for first-order phase transitions – p.11/12
Conclusions The fRG is set up as a powerful tool for the study of symmetry breaking. A method to study stable and metastable states has been developed. Bias has turned from challenge to chance. Studies of models with extended momentum structure remain to be done (discretization: patching, expansion, ...?) Thank you very much! 3. Bias as a chance Roland Gersch: fRG for first-order phase transitions – p.12/12
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