Renormalization Group Theory SineGordon Model Mariana Malard Renormalization

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Renormalization Group Theory & Sine-Gordon Model Mariana Malard

Renormalization Group Theory & Sine-Gordon Model Mariana Malard

Renormalization Group Theory & Sine-Gordon Model SUMMARY OF THE LECTURES Lecture 1 January 16

Renormalization Group Theory & Sine-Gordon Model SUMMARY OF THE LECTURES Lecture 1 January 16 th Renormalization Group Theory Ø Ø Conceptual overview. General procedure 0: Define a field theory Action. General procedure I: Decompose in slow and fast modes. General procedure II: Expressing S in terms of Green’s functions.

Renormalization Group Theory & Sine-Gordon Model SUMMARY OF THE LECTURES Lecture 2 January 21

Renormalization Group Theory & Sine-Gordon Model SUMMARY OF THE LECTURES Lecture 2 January 21 st Renormalization Group Theory Ø Finish general procedure II: Expressing S in terms of Green’s functions. Ø General procedure III: Averaging in the fast modes’ ground state. Sine-Gordon Model Ø Conceptual overview. Ø From 1 D interacting electrons to the sine-Gordon model: Schematics of bosonization. Ø The model.

Renormalization Group Theory & Sine-Gordon Model SUMMARY OF THE LECTURES Lecture 3 January 23

Renormalization Group Theory & Sine-Gordon Model SUMMARY OF THE LECTURES Lecture 3 January 23 rd Sine-Gordon Model Ø Re-scaled Action for the sine-Gordon model. Ø Renormalization group flows equations of the sine-Gordon model. Kosterlitz-Thouless Phase Diagram Ø Gap

Sine-Gordon Model Re-scaled Action for the sine-Gordon model Effective residual Action for the slow

Sine-Gordon Model Re-scaled Action for the sine-Gordon model Effective residual Action for the slow modes: Substitute as and bs and compute the integrals. Loooong calculation. See review paper.

Sine-Gordon Model Re-scaled Action for the sine-Gordon model Effective Action for the slow modes:

Sine-Gordon Model Re-scaled Action for the sine-Gordon model Effective Action for the slow modes:

Sine-Gordon Model Re-scaled Action for the sine-Gordon model Re-scaled Action: Back to the original

Sine-Gordon Model Re-scaled Action for the sine-Gordon model Re-scaled Action: Back to the original scale

Sine-Gordon Model Re-scaled Action for the sine-Gordon model Re-scaled Action:

Sine-Gordon Model Re-scaled Action for the sine-Gordon model Re-scaled Action:

Sine-Gordon Model Renormalization group flow equations of the sine-Gordon model The phylosophy is: Full

Sine-Gordon Model Renormalization group flow equations of the sine-Gordon model The phylosophy is: Full theory Average with respect to the fast modes’ ground state Effective theory for the slow modes Re-scale theory back to the full cutoff Flow equations re a mp o C Renormalized theory

Sine-Gordon Model Renormalization group flow equations of the sine-Gordon model Renormalized parameters:

Sine-Gordon Model Renormalization group flow equations of the sine-Gordon model Renormalized parameters:

Sine-Gordon Model Renormalization group flow equations of the sine-Gordon model Flow equations:

Sine-Gordon Model Renormalization group flow equations of the sine-Gordon model Flow equations:

Sine-Gordon Model Renormalization group flow equations of the sine-Gordon model Flow equations: u =

Sine-Gordon Model Renormalization group flow equations of the sine-Gordon model Flow equations: u = 0: du dl = d. K dl = 0 No flow. Line of fixed points. Flow of u : u u > 0, K < 1: du dl > 0 u > 0, K > 1: du dl < 0 upward flow downward flow u < 0, K < 1: du dl < 0 u < 0, K > 1: du dl > 0 downward flow upward flow Any u ≠ 0, K = 1: du dl = 0 (but d. K dl ≠ 0 ) horizontal flow 1 K

Sine-Gordon Model Renormalization group flow equations of the sine-Gordon model Flow equations: Flow of

Sine-Gordon Model Renormalization group flow equations of the sine-Gordon model Flow equations: Flow of k : Any u ≠ 0, K > 0: d. K dl < 0 backward flow u Any u ≠ 0, K < 0: d. K dl > 0 forward flow Any u ≠ 0, K = 0: d. K dl = 0 (but du dl ≠ 0 ) vertical flow 1 K

Sine-Gordon Model Renormalization group flow equations of the sine-Gordon model Flow equations: u u

Sine-Gordon Model Renormalization group flow equations of the sine-Gordon model Flow equations: u u 1 K 1 u 1 K K

Sine-Gordon Model Renormalization group flow equations of the sine-Gordon model Kosterlitz-Thouless phase-diagram u .

Sine-Gordon Model Renormalization group flow equations of the sine-Gordon model Kosterlitz-Thouless phase-diagram u . . . 0. . . . 1. . . . K

Kosterlitz-Thouless Phase Diagram For K ~ 1. See review paper. The expression inside parenthesis

Kosterlitz-Thouless Phase Diagram For K ~ 1. See review paper. The expression inside parenthesis is an invariant.

Kosterlitz-Thouless Phase Diagram Family of hiperbolae characterized by the sign of c.

Kosterlitz-Thouless Phase Diagram Family of hiperbolae characterized by the sign of c.

Kosterlitz-Thouless Phase Diagram u c > 0 : Flows outside the separatrices c =

Kosterlitz-Thouless Phase Diagram u c > 0 : Flows outside the separatrices c = 0 : Separatrices c < 0 : Flows enclosed by the separatrices . . . 0. . . . 1. . . . c > 0 : Flows outside the separatrices K

Kosterlitz-Thouless Phase Diagram opens up a gap (mass) u marginally relevant interaction strongly relevant

Kosterlitz-Thouless Phase Diagram opens up a gap (mass) u marginally relevant interaction strongly relevant interaction . . . 0. . . . 1. . . . unstable fixed points bosons remain gapless (massless) irrelevant interaction K stable fixed points

Kosterlitz-Thouless Phase Diagram • • • u 1 x lc • At this critical

Kosterlitz-Thouless Phase Diagram • • • u 1 x lc • At this critical scale, interactions become too strong, i. e. strong enough to win over the kinetic term in H. Past this scale, R. G. is no longer valid. System undergoes a phase transition from gapless to gapped bosons. Bosons become trapped at the cosine minima. . . . 0. . . . 1. . . . K

Kosterlitz-Thouless Phase Diagram Gap long (short) critical scale small (large) gap What is the

Kosterlitz-Thouless Phase Diagram Gap long (short) critical scale small (large) gap What is the value of lc ? Flow equations up to first order in u: