funRG with discrete symmetry-breaking Roland Gersch, Carsten - - PowerPoint PPT Presentation

funRG with discrete symmetry-breaking

Roland Gersch, Carsten Honerkamp

Walter Metzner Max Planck Institute for Solid State Research, Stuttgart

funRG with discrete symmetry-breaking – p.1/20

Problem statement

• 1. Second-order phase transitions
• T

This region is inaccessible to symmetric-phase funRG techniques. Until recently, funRG techniques were unable to reproduce mean-field results for mean-field-exact models. BCS-model (U(1) symmetry) at

treated

This talk: discrete-symmetry breaking,

funRG with discrete symmetry-breaking – p.2/20

Hamiltonian

• 1. Second-order phase transitions

At half-filling: a repulsive interaction restricted to momentum-transfers of

• ;

• U

• ext

We assume the thermodynamic limit and a grand canonical ensemble at zero chemical potential (Here, this implies half filling).

in the spectrum.

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Exact Diagrammatics

• 1. Second-order phase transitions

Resumming perturbation theory leads to the gap equation.

• +

• =

• RPA resummation for the effective interaction.
• +

• =

funRG with discrete symmetry-breaking – p.4/20

Results

• 1. Second-order phase transitions

Temperature-dependence of the gap, effective interaction

0.1 0.2 0.1 0.2 [units of t] Temperature [units of t] 100 ∆ext= 0.00t 0.03t 0.16t 1.00t

∆

50 100 150 200 250 0.1 0.2 Effective interaction [units of t] Temperature [units of t] 100 ∆ext= 0.01t 0.06t 0.40t

The phase transition is “smeared out” by the external field and the singularity of the effective interaction is regularized.

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funRG equations

• 1. Second-order phase transitions

Gap flow equation

• =

Effective interaction flow equation

• =

Initial conditions:

• i

• ext

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funRG flows at

• 1. Second-order phase transitions

The sub-T

gap resemble the temperature dependences.

0.1 0.2 0.3 0.4 0.1 0.2 0.3 0.4 Σ[units of t] Λ [units of t] 100∆ext= 0.01t 0.03t 0.06t 0.16t 0.40t 1.00t 100 200 300 400 0.1 0.2 0.3 0.4 V [units of t] Λ [units of t] 100∆ext= 0.01t 0.03t 0.06t 0.16t 0.40t 1.00t

Increasing the external field suppresses the effective interaction flow maximum and furthers the smearing

• f the transition.

The self-energy’s final value changes by only 10% whi- le the initial gap varies over two orders of magnitude

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funRG flows for

• 1. Second-order phase transitions

50 100 150 200 250 300 0.005 0.01 0.015 0.02 0.025 0.03 V(Λ) [units of t] Λ [units of t] T= 0.090t 0.094t 0.098t 0.102t 0.106t 0.110t 0.1 0.005 0.01 0.015 0.02 0.025 0.03 Σ[units of t] Λ [units of t] T= 0.090t 0.094t 0.098t 0.102t 0.106t 0.110t

Increase in temperature

lower scales

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Comparison to BCS flows

• 1. Second-order phase transitions

0.1 0.2 0.3 0.4 0.1 0.2 0.3 0.4 Σ[units of t] Λ [units of t] 100∆ext= 0.01t 100 200 300 400 0.1 0.2 0.3 0.4 V [units of t] Λ [units of t] 100∆ext= 0.01t

BCS model: Nor- mal and anoma- lous effective interactions (all ar- rows in/out) exist. Discrete-symmetry breaking flow

• f

the effective inter- action resembles the BCS flows’ sum. Interpreta- tion: amplitude mode.

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• 1. Second-order phase transitions

0.1 0.2 0.3 0.4 0.1 0.2 0.3 0.4

• [units
• f

• f

• 6/
• funRG with discrete symmetry-breaking – p.10/20

Hamiltonian

• 2. First-order phase transitions

• )n

• U

• ext

funRG with discrete symmetry-breaking – p.11/20

Exact Diagrammatics

• 2. First-order phase transitions

Same diagrams as for

• =
• =
• =

Simple rule: Add

distributions (and derivatives) in

• =

New: grand canonical potential

• btained by

integrating the gap equation wrt the gap

• V
• l

• 2

• 1

• E

funRG with discrete symmetry-breaking – p.12/20

Grand canonical potential and phases

• 2. First-order phase transitions
• 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, ∆ext=10-10t, µ=0.24t, V0=2t T 0.08 0.06 0.04 0.02

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Phase diagram

• 2. First-order phase transitions

0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 0.22 0.05 0.1 0.15 0.2 0.25 T µ Phase transitions for V=2t, ∆ext=10-10t 1st order 2nd order

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Challenge

• 2. First-order phase transitions

−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, ∆ext=10−10 t, µ=0.24t, V

0=2t

• in

• i

Challenge: Starting at large

changing

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Counter terms and interaction flow

• 2. First-order phase transitions

Back to the Hamiltonian. Now, add a counterterm

• .

• )n

• X

• y

• U

• ext

To naked propagator To initial self-energy Set

• =
• ext

Momentum-shell cutoff:

• ext and
• cancel at all

scales. Interaction “cutoff”:

• ext and
• cancel only at the

end of the flow. This implies: The initial self-energy can be chosen arbitrarily without changing the physics!

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Interaction flow of the self-energy

• 2. First-order phase transitions
• 2
• 1.8
• 1.6
• 1.4
• 1.2
• 1
• 0.8
• 0.6
• 0.4
• 0.2

0.2 0.2 0.4 0.6 0.8 1 ∆ Λ Interaction flows, 2d, V=2t, µ=0.245t, T=0.001t

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Interaction flow of the coupling

• 2. First-order phase transitions

2 4 6 8 10 0.2 0.4 0.6 0.8 1 V Λ Interaction flows, 2d, V=2t, µ=0.245t, T=0.001t

funRG with discrete symmetry-breaking – p.18/20

Grand canonical potential?

• 2. First-order phase transitions

• =

• l, flow equation (traditional 1PI):

• =

• d

• G

• Mahan (slightly generalized):

• =

• =), same

as above except for a constant factor. Numerics currently fail to reproduce exact grand canonical potential.

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Conclusions

• 2. First-order phase transitions

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.

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