1 , Daniel Rohe 2 , and funRG with discrete symmetry-breaking Roland Gersch, Carsten Honerkamp Walter Metzner Max Planck Institute for Solid State Research, Stuttgart 1 now Würzburg 2 now Paris funRG with discrete symmetry-breaking – p.1/20
� Problem statement T This region is inaccessible to symmetric-phase funRG techniques. T = 0 has been Until recently, funRG techniques were unable to 1 reproduce mean-field results for mean-field-exact models. T > 0 . BCS-model (U(1) symmetry) at 1 Salmhofer, Honerkamp, Metzner, Lauscher 2004 treated This talk: discrete-symmetry breaking, 1. Second-order phase transitions funRG with discrete symmetry-breaking – p.2/20
Q := ( � ; � ; : : : ) generates a d -dimensional charge-density wave. Hamiltonian P P P y y y U 0 H = " n � + � 0 0 0 k k ext k k ;k k N k k + Q k k + Q k + Q k At half-filling: a repulsive interaction restricted to momentum-transfers of � =U 0 : amplitude of the density wave. Appears as gap We assume the thermodynamic limit and a grand canonical ensemble at zero chemical potential U : effective interaction, effective coupling (Here, this implies half filling). t : Hopping integral, unit of energy. in the spectrum. 1. Second-order phase transitions funRG with discrete symmetry-breaking – p.3/20
Exact Diagrammatics � Resumming perturbation theory leads to the gap = + + + � � � = + = equation. � � � � � � � � = + + � � � = + RPA resummation for the effective interaction. � � � � � � � � � � 1. Second-order phase transitions funRG with discrete symmetry-breaking – p.4/20
Results Temperature-dependence of the gap, effective interaction 0.2 100 ∆ ext = 0.01t 250 0.06t Effective interaction [units of t] 0.40t 200 [units of t] 150 0.1 ∆ 100 100 ∆ ext = 0.00t 50 0.03t 0.16t 1.00t 0 0 0 0.1 0.2 0 0.1 0.2 Temperature [units of t] Temperature [units of t] The phase transition is “smeared out” by the external field and the singularity of the effective interaction is regularized. 1. Second-order phase transitions funRG with discrete symmetry-breaking – p.5/20
d � 1 S = G ( G ) G 0 d � funRG equations = Gap flow equation � � �� �� � � = �� �� Effective interaction flow equation ����� ����� �� �� V = U � = � i 0 , i ext �� �� ����� ����� ���� ���� �� �� �� �� ����� ����� ���� ���� � � � � Initial conditions: 1. Second-order phase transitions funRG with discrete symmetry-breaking – p.6/20
T = 0 The sub- T flows of the effective interaction and the funRG flows at gap resemble the temperature dependences. 0.4 400 100 ∆ ext = 100 ∆ ext = 0.01t 0.01t 0.03t 0.03t 0.3 300 0.06t 0.06t 0.16t 0.16t V [units of t] Σ [units of t] 0.40t 0.40t 1.00t 1.00t 0.2 200 0.1 100 0 0 0 0.1 0.2 0.3 0.4 0 0.1 0.2 0.3 0.4 Λ [units of t] Λ [units of t] Increasing the external field suppresses the effective interaction flow maximum and furthers the smearing of the transition. The self-energy’s final value changes by only 10% whi- le the initial gap varies over two orders of magnitude 1. Second-order phase transitions funRG with discrete symmetry-breaking – p.7/20
T > 0 funRG flows for 300 0.1 T= T= 0.090t 0.090t 0.094t 0.094t 0.098t 0.098t 250 0.102t 0.102t 0.106t 0.106t 0.110t 0.110t 200 V( Λ ) [units of t] Σ [units of t] 150 100 50 ) Graph of flow moves to 0 0 0 0.005 0.01 0.015 0.02 0.025 0.03 0 0.005 0.01 0.015 0.02 0.025 0.03 Λ [units of t] Λ [units of t] Increase in temperature lower scales 1. Second-order phase transitions funRG with discrete symmetry-breaking – p.8/20
Comparison to BCS flows BCS model: Nor- mal and anoma- lous effective interactions (all ar- rows in/out) exist. 0.4 400 Discrete-symmetry 100 ∆ ext = 100 ∆ ext = 0.01t 0.01t 0.3 300 breaking flow of V [units of t] Σ [units of t] the effective inter- 0.2 200 action resembles 0.1 100 the BCS flows’ sum. Interpreta- 0 0 0 0.1 0.2 0.3 0.4 0 0.1 0.2 0.3 0.4 tion: amplitude Λ [units of t] Λ [units of t] mode. 1. Second-order phase transitions funRG with discrete symmetry-breaking – p.9/20
T -dependence of � 0.4 t℄ 0.3 T emp erature [units of t℄ of [units 0.2 � 6/ � � 0.1 0 0 0.1 0.2 0.3 0.4 1. Second-order phase transitions funRG with discrete symmetry-breaking – p.10/20
X X X U 0 y y y H = ( " � � ) n � + � 0 0 k k ext k k + Q k k + Q k + Q k Hamiltonian N 0 k k ;k k 2. First-order phase transitions funRG with discrete symmetry-breaking – p.11/20
� = 0 Exact Diagrammatics = + = Same diagrams as for ����������� ����������� ������������ ������������ ����������� ����������� ������������ ������������ � � to the arguments of all Fermi � � � � � � ����������� ����������� ������������ ������������ � = 0 equations. � � � � � � ����������� ����������� ������������ ������������ � obtained by Simple rule: Add distributions (and derivatives) in 2 X � � 1 = � ln ( f ( � � � E ) f ( � � + E )) New: grand canonical potential V ol 2 U 2 0 k integrating the gap equation wrt the gap 2. First-order phase transitions funRG with discrete symmetry-breaking – p.12/20
Grand canonical potential and phases Grand canonical potential, ∆ ext =10 -10 t, 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. First-order phase transitions funRG with discrete symmetry-breaking – p.13/20
Phase diagram Phase transitions for V=2t, ∆ ext =10 -10 t 0.22 0.2 0.18 0.16 0.14 0.12 T 0.1 0.08 0.06 0.04 1st order 0.02 2nd order 0 0 0.05 0.1 0.15 0.2 0.25 µ 2. First-order phase transitions funRG with discrete symmetry-breaking – p.14/20
Challenge Grand canonical potential, ∆ ext =10 −10 t, µ =0.24t, V 0 =2t 0.0025 Flo w dire tion 0.002 0.0015 0.001 Starting p oin t � i Ω 0.0005 0 −0.0005 � without appreciably �( T ) . −0.001 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 ∆ Challenge: Starting at large changing 2. First-order phase transitions funRG with discrete symmetry-breaking – p.15/20
� . X X y H = ( " � � ) n � �� k k k + Q k Counter terms and interaction flow k k Back to the Hamiltonian. Now, add a counterterm X X U 0 y y y � + �� 0 0 ext k k + Q k k + Q k + Q k N 0 k ;k k � = � ext To naked propagator � � ext and cancel at all To initial self-energy � � Set ext and cancel only at the Momentum-shell cutoff: scales. Interaction “cutoff”: end of the flow. This implies: The initial self-energy can be chosen arbitrarily without changing the physics! 2. First-order phase transitions funRG with discrete symmetry-breaking – p.16/20
Interaction flow of the self-energy Interaction flows, 2d, V=2t, µ =0.245t, T=0.001t 0.2 0 -0.2 -0.4 -0.6 -0.8 ∆ -1 -1.2 -1.4 -1.6 -1.8 -2 0 0.2 0.4 0.6 0.8 1 Λ 2. First-order phase transitions funRG with discrete symmetry-breaking – p.17/20
Interaction flow of the coupling Interaction flows, 2d, V=2t, µ =0.245t, T=0.001t 10 8 6 V 4 2 0 0 0.2 0.4 0.6 0.8 1 Λ 2. First-order phase transitions funRG with discrete symmetry-breaking – p.18/20
~ � = � = V ol , flow equation (traditional 1PI): � � _ 1 d � 1 ~ � = T r ( G )( G � G ) 0 0 2 d � Grand canonical potential? _ ~ � = 2T r( G � � _ =� ) , same Mahan (slightly generalized): as above except for a constant factor. Numerics currently fail to reproduce exact grand canonical potential. 2. First-order phase transitions funRG with discrete symmetry-breaking – p.19/20
T . Exact Conclusions Flows in phases with a broken discrete symmetry possible and illustrated above and below results obtainable for mean-field models. 1st-order phase transitions treatable with funRG. 1. So far only using the interaction flow 2. Work in progress Outlook: Use the new methods study physically more relevant problems. Thanks to Sabine Andergassen, Tilman Enss, Andrey Katanin, Julius Reiss, and Manfred Salmhofer for useful discussions. 2. First-order phase transitions funRG with discrete symmetry-breaking – p.20/20
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