Frequency dependence of the vertex function for the fRG and beyond - - PowerPoint PPT Presentation

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frequency dependence of the vertex function for the frg

Frequency dependence of the vertex function for the fRG and beyond - - PowerPoint PPT Presentation

Nov 05, 2023 366 likes •782 views

Frequency dependence of the vertex function for the fRG and beyond Ciro Taranto Max-Planck-Institute for Solid State Research Main collaborators Stuttgart: Walter Metzner, Demetrio Vilardi ( next talk ) Vienna: Nils Wentzell ERG 2016,

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Frequency dependence of the vertex function for the fRG and beyond

Ciro Taranto

ERG 2016, Trieste

Max-Planck-Institute for Solid State Research Main collaborators Stuttgart: Walter Metzner, Demetrio Vilardi (next talk) Vienna: Nils Wentzell

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Outline

Vertex frequency dependence (Part I)

– Definitions, 1PI and 2PI vertexes – Diagrammatic understanding of the vertex structures – Vertex decomposition (and reconstruction)

Towards strong coupling (Part II)

– Apllication: combining DMFT and fRG (DMF2RG)

continued in next talk

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1PI vertex in (fermionic) fRG

2-particle picture:

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1PI vertex in (fermionic) fRG

2-particle picture:

Review: Metzner et al.,RMP '12

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1PI vertex in (fermionic) fRG

2-particle picture: fRG: Momentum dependence→ leading instabilities; calculation of susceptibilities This talk: Systematic analysis of frequency dependence

Review: Metzner et al.,RMP '12

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Notation conventions

“fermionic” 4-vector “bosonic” 4-vector

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Notation conventions

“fermionic” 4-vector “bosonic” 4-vector

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Notation conventions

“fermionic” 4-vector “bosonic” 4-vector Computed in fRG Diagonal & horizontal frequency structure → Large frequency behavior Plot a fixed transfer frequency

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Decomposing the vertex

Parquet equation

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Decomposing the vertex

Parquet equation 2-particle irreducible

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Decomposing the vertex

Parquet equation

Rohringer,Valli and Toschi,PRB'12

2-particle irreducible

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Decomposing the vertex

Parquet equation

Rohringer,Valli and Toschi,PRB'12

2-particle irreducible ED result No two-particle irreducible terms in fRG at one-loop truncation level

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Decomposing the vertex

Parquet equation 2-particle reducible fRG: integrate each channel separately

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Decomposing the vertex

Parquet equation 2-particle reducible fRG: integrate each channel separately

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Decomposing the vertex

Parquet equation 2-particle reducible fRG: integrate each channel separately

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Decomposing the vertex

Parquet equation 2-particle reducible fRG: integrate each channel separately

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Decomposing the vertex

Parquet equation 2-particle reducible fRG: integrate each channel separately

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Decomposing the vertex

Parquet equation 2-particle reducible Can one further understand the structures? fRG: integrate each channel separately

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Diagrammatic classification

Assumption: Bare interaction Local and frequency independent Lowest order: Dependence on transfer arguments only,

ften used in fRG:

Generalization this argument for higher order diagrams? Karrasch, et al.,JPCM 2008 (frequencies); Husemann and Salmhofer, PRB 2009 (momenta) Bauer, Heyder and von Delft, PRB 2014 (inhomogeneous systems)

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Diagrammatic classification

Direct connection with susceptibilities

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Diagrammatic classification

Direct connection with susceptibilities

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Diagrammatic classification

Direct connection with susceptibilities

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Diagrammatic classification

scan

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Diagrammatic classification

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Diagrammatic classification

=

 Subleading at weak coupling  Full argument dependence: numerically expensive  Possibly relevant for d-wave scattering

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Vertex: decomposition and reconstruction

Computed in a finite box

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Vertex: decomposition and reconstruction

fRG: separate channel integration From the full vertex: Bethe- Salpeter equations Computed in a finite box

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Vertex: decomposition and reconstruction

fRG: separate channel integration From the full vertex: Bethe- Salpeter equations Computed in a finite box Extract Extend

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Vertex: decomposition and reconstruction

fRG: separate channel integration From the full vertex: Bethe- Salpeter equations Computed in a finite box Extract Extend Extract

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Vertex: decomposition and reconstruction

fRG: separate channel integration From the full vertex: Bethe- Salpeter equations Computed in a finite box Extract Extend Extract Extract Check that it decays inside the box

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Part I: conclusions

1.The interaction vertex shows a nontrivial frequency structure 2.The vertex structure can be understood diagrammatically 3.The knowledge of the vertex asymptotic can be used to reduce computational effort

Part II: DMF2RG and strong coupling

Dynamical mean field theory in a nutshell
Starting fRG from a correlated starting point

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Flowing from infinite to d-dimensions

Goal: combine non-perturbative local physics from DMFT

with nonlocal fluctuations from fRG Taranto, et al.,PRL 2014;

Mapping on an Anderson Impurity model embedded in a self-consistent

frequency-dependent bath (MF in space) Georges et al., RMP 1996 Georges and Kotliar, PRB 1992 Georges et al., RMP 1996

In the ∞-dimensional limit local approximation for the self-energy becomes exact

Metzner and Vollhardt, PRB 1989; (1) Approximate a lattice model with an ∞-dimensional lattice (with the same DOS) (2) Exactly solve the problem in infinite dimensions (3) Flow from the infinite dimensional lattice to the original one using fRG Conceptual steps:

The Anderson Impurity model can be exactly solved (QMC, ED, …) good starting point

for the flow equations

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Flowing from infinite to d-dimensions

Taranto, et al.,PRL 2014;

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Flowing from infinite to d-dimensions

Taranto, et al.,PRL 2014;

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Flowing from infinite to d-dimensions

Taranto, et al.,PRL 2014; DMFT self-consistency condition for the Weiss field Georges,cond-mat/0403123 (2004)

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Flowing from infinite to d-dimensions

Taranto, et al.,PRL 2014; DMFT self-consistency condition for the Weiss field Georges,cond-mat/0403123 (2004)

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Flowing from infinite to d-dimensions

Taranto, et al.,PRL 2014; DMFT self-consistency condition for the Weiss field Georges,cond-mat/0403123 (2004) More freedom in the regulator

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Flowing from infinite to d-dimensions

Taranto, et al.,PRL 2014;

Effect of the frequency dependence in the next talk

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Part I: conclusions

1.The interaction vertex shows a nontrivial frequency structure 2.The vertex structure can be understood diagrammatically 3.The knowledge of the vertex asymptotic can be used to reduce computational effort

Part II: DMF2RG and strong coupling

Dynamical mean field theory in a nutshell
Starting fRG from a correlated starting point