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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

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SLIDE 1

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

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SLIDE 2

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

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SLIDE 3

Motivation

  • 1. Some fRG

Spontaneous symmetry breaking

X X
  • X
Q ( X ) X i:a: ! X X Y
  • X
(Q (X )
  • (X
; Y )) Y

Microscopic models

f R G ! physical observables

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

Roland Gersch: fRG for first-order phase transitions – p.3/12

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SLIDE 4

Formalities

  • 1. Some fRG

Generating functional of connected Green’s functions

exp (W (J )) = Z D
  • det
Q =( ) e (;Q =())V ()+(J ;) J: Grassmannian generating field, V : two-particle

interaction

( i ) makes W solvable, ( f )
  • 1.

One-particle irreducible (1PI) flow Salmhofer, Honerkamp 2001

_
  • =
T T r ( _ Q Q 1 + G _ Q )

Roland Gersch: fRG for first-order phase transitions – p.4/12

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SLIDE 5

Formalities

  • 1. Some fRG

Generating functional of connected Green’s functions

exp (W (J )) = Z D
  • det
Q =( ) e (;Q =())V ()+(J ;) J: Grassmannian generating field, V : two-particle

interaction

( i ) makes W solvable, ( f )
  • 1.

One-particle irreducible (1PI) flow Salmhofer, Honerkamp 2001

_
  • =
T T r ( _ Q Q 1 + G _ Q )

Katanin 2004

Roland Gersch: fRG for first-order phase transitions – p.5/12

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SLIDE 6

The external field’s influence

  • 2. Bias as a challenge

Charge-density-wave mean-field model Hamiltonian

H = X k (" k
  • )
y k k
  • V
N X k;k y k k+Q y k k +Q + X k (
  • i
) y k+Q k ; Q = ( ;
  • ;
: : : ) : chemical potential, ": tight-binding dispersion, V 0:

nearest-neighbor repulsion,

  • i: external field.
T =
  • =
:

RG, Honer- kamp, Rohe, Metzner 2005 Units: hopping integral

0.1 0.2 0.1 0.2 ∆ Λ 100∆ext= 0.01 0.03 0.06 0.16 0.40 1.00 50 100 150 200 0.05 0.1 0.15 0.2 Effective interaction Λ 100∆ext= 0.01 0.03 0.06 0.16 0.40 1.00

Roland Gersch: fRG for first-order phase transitions – p.6/12

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SLIDE 7

First-order phase transitions

  • 2. Bias as a challenge
  • 0.001
  • 0.0005

0.0005 0.001 0.0015 0.002 0.0025 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 Ω ∆ Grand canonical potential, µ=0.24t, V0=2t T 0.08 0.06 0.04 0.02

Roland Gersch: fRG for first-order phase transitions – p.7/12

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SLIDE 8

Counterterms: first attempt

  • 2. Bias as a challenge

Back to the CDW Hamiltonian.

H = X k (" k
  • )
y k k
  • V
N X k;k y k k+Q y k k +Q + X k (
  • i
) y k+Q k

To bare propagator To initial self-energy

G 1 = 1
  • i!
  • "
+
  • i!
+ " +
  • !
) G 12 /
  • (
  • |
{z } =: eff ) ()2f0;1g ) _
  • =
/
  • (
  • )
=

Roland Gersch: fRG for first-order phase transitions – p.8/12

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SLIDE 9

Interaction flow

  • 3. Bias as a chance
()
  • p
,
  • i
= 0,
  • f
= 1 equivalent to linearly turning on

the interaction from

0 to V 0.

Honerkamp, Rohe, Andergassen, Enss 2004.

Advantages:

  • e
j =Æ
  • p
Æ ( ) 6=

Thus,

_
  • 6=
0.

Chosing

  • i
=
  • : scanning the order parameter

space for thermodynamic potential minima. Results independent of

  • i for mean-field models

Large effective interactions restricted to the final region of the flow

Roland Gersch: fRG for first-order phase transitions – p.9/12

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SLIDE 10

First-order CDW phase transition

  • 3. Bias as a chance

Order pa- rameter

T < T t

units: hopping integral

0.2 0.4 0.6 0.8 0.2 0.4 0.6 0.8 ∆eff Λ

T > T t

0.2 0.4 0.6 0.8 0.2 0.4 0.6 0.8 ∆eff Λ

Effective interaction

T < T t

units: hopping integral

2 3 4 5 6 7 8 9 10 0.2 0.4 0.6 0.8 Effective interaction Λ

T > T t

5 10 15 20 25 30 35 40 0.2 0.4 0.6 0.8 Effective interaction Λ

RG, Reiss, Honerkamp 2006

Roland Gersch: fRG for first-order phase transitions – p.10/12

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SLIDE 11

Flow of

  • 3. Bias as a chance
for T < T t:
  • 1.406
  • 1.404
  • 1.402
  • 1.4
  • 1.398
  • 1.396
  • 1.394
  • 1.392

0.9 0.92 0.94 0.96 0.98 1 Ω [units of t] Λ

for T > T t:
  • 1.406
  • 1.404
  • 1.402
  • 1.4
  • 1.398
  • 1.396
  • 1.394

0.9 0.92 0.94 0.96 0.98 1 Ω [units of t] Λ

RG, Reiss, Honerkamp 2006

Roland Gersch: fRG for first-order phase transitions – p.11/12

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SLIDE 12

Conclusions

  • 3. Bias as a chance

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!

Roland Gersch: fRG for first-order phase transitions – p.12/12