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R Shankar - One of the best experts on this subject based on the ideXlab platform.
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fermi surfaces in general codimension and a new controlled Nontrivial Fixed Point
Physical Review Letters, 2009Co-Authors: T Senthil, R ShankarAbstract:The energy of a $d$-dimensional Fermi system typically varies only along ${d}_{c}=1$ (``radial'') dimensions. We consider ${d}_{c}=1+\ensuremath{\epsilon}$ and study a transition to superconductivity in an $\ensuremath{\epsilon}$ expansion. The Nontrivial Fixed Point describes a scale invariant theory with an effective space-time dimension $D={d}_{c}+1$. Remarkably, the results can be reproduced by the Hertz-Millis action for the superconducting order parameter in higher effective space-time dimensions. We consider possible realizations of the transition at $\ensuremath{\epsilon}=1$, which corresponds to a linear Fermi surface in $d=3$.
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Fermi Surfaces in General Codimension and a New Controlled Nontrivial Fixed Point
2009Co-Authors: R Shankar, T SenthilAbstract:The energy of a d-dimensional Fermi system typically varies only along dc ¼ 1 (‘‘radial’’) dimensions. We consider dc ¼ 1 þ " and study a transition to superconductivity in an " expansion. The Nontrivial Fixed Point describes a scale invariant theory with an effective space-time dimension D ¼ dc þ 1. Remarkably, the results can be reproduced by the Hertz-Millis action for the superconducting order parameter in higher effective space-time dimensions. We consider possible realizations of the transition at " ¼ 1, which corresponds to a linear Fermi surface in d ¼ 3. DOI: 10.1103/PhysRevLett.102.046406 PACS numbers: 71.10.Hf The ground state of a fluid of noninteracting fermions in spatial dimension d � 2 is typically a Landau Fermi liquid with a sharp Fermi surface and quasiparticles. The Fermi liquid state can be understood within a fermionic renormalization group (RG) framework [1,2] in terms of a Fixed Point obtained as one focuses on modes within a bandwidth � of the Fermi surface and systematically reduces � . Unlike the BCS transition, transitions like the Stoner ferromagnetic transition that occur at finite coupling are not described by traditional flows within this approach. An alternate approach of Hertz [3], extended by Millis [4], performs a renormalization group analysis on the bosonic action for the order parameter obtained by integrating out the fermions and obtains an action nonanalytic in frequency and momentum and which yields results in agreement with those of Moriya [5] who used the self-consistent renormalization approach. However, there are reasons [6,7] to believe that integrating out the gapless fermionic modes can lead to singularity structure not evident in the Hertz-Millis analysis. When the noninteracting band structure consists of Fermi Points (as in graphene) the low energy description is a massless ‘‘relativistic’’ Dirac theory. In dimension d � 2 weak short-ranged interactions are irrelevant, though gap inducing phase transitions can occur at strong coupling. Without the complications of an extended Fermi surface these can be analyzed within conventional field theoretic framework for critical phenomena. We consider a general class of problems that fall in between these two cases. The low energy theory of a system of noninteracting fermions in d spatial dimension may be characterized by the codimension dc of the surface in momentum space where the energy gap vanishes. The ordinary Fermi surface has codimension dc ¼ 1 while the case of Fermi Points has codimension d. We study the general case of codimension d c with a eye toward gaining insight into dc ¼ 2 in d ¼ 3 which corresponds to fermions with line nodes in three dimensions which arise in unconventional three dimensional superconductors and the tight binding model on the diamond lattice at half filling [8]. Fermionic RG methods of Refs. [1,2] show that whenever dc > 1 all short-range interactions are irrelevant. With increasing interaction strength phase transitions which gap out the fermions are again possible. For dc
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fermi surfaces in general codimension and a new controlled Nontrivial Fixed Point
APS, 2009Co-Authors: R Shankar, T SenthilAbstract:The energy of a d-dimensional Fermi system typically varies only along dc ¼ 1 (‘‘radial’’) dimensions. We consider dc ¼ 1 þ " and study a transition to superconductivity in an " expansion. The Nontrivial Fixed Point describes a scale invariant theory with an effective space-time dimension D ¼ dc þ 1. Remarkably, the results can be reproduced by the Hertz-Millis action for the superconducting order parameter in higher effective space-time dimensions. We consider possible realizations of the transition at " ¼ 1, which corresponds to a linear Fermi surface in d ¼ 3. DOI: 10.1103/PhysRevLett.102.046406 PACS numbers: 71.10.Hf The ground state of a fluid of noninteracting fermions in spatial dimension d � 2 is typically a Landau Fermi liquid with a sharp Fermi surface and quasiparticles. The Fermi liquid state can be understood within a fermionic renormalization group (RG) framework [1,2] in terms of a Fixed Point obtained as one focuses on modes within a bandwidth � of the Fermi surface and systematically reduces � . Unlike the BCS transition, transitions like the Stoner ferromagnetic transition that occur at finite coupling are not described by traditional flows within this approach. An alternate approach of Hertz [3], extended by Millis [4], performs a renormalization group analysis on the bosonic action for the order parameter obtained by integrating out the fermions and obtains an action nonanalytic in frequency and momentum and which yields results in agreement with those of Moriya [5] who used the self-consistent renormalization approach. However, there are reasons [6,7] to believe that integrating out the gapless fermionic modes can lead to singularity structure not evident in the Hertz-Millis analysis. When the noninteracting band structure consists of Fermi Points (as in graphene) the low energy description is a massless ‘‘relativistic’’ Dirac theory. In dimension d � 2 weak short-ranged interactions are irrelevant, though gap inducing phase transitions can occur at strong coupling. Without the complications of an extended Fermi surface these can be analyzed within conventional field theoretic framework for critical phenomena. We consider a general class of problems that fall in between these two cases. The low energy theory of a system of noninteracting fermions in d spatial dimension may be characterized by the codimension dc of the surface in momentum space where the energy gap vanishes. The ordinary Fermi surface has codimension dc ¼ 1 while the case of Fermi Points has codimension d. We study the general case of codimension d c with a eye toward gaining insight into dc ¼ 2 in d ¼ 3 which corresponds to fermions with line nodes in three dimensions which arise in unconventional three dimensional superconductors and the tight binding model on the diamond lattice at half filling [8]. Fermionic RG methods of Refs. [1,2] show that whenever dc > 1 all short-range interactions are irrelevant. With increasing interaction strength phase transitions which gap out the fermions are again possible. For dc
superconductivity. We show that this can be accessed within the fermionic RG through a controlled � expansion in the codimension dc1 þ ", yielding a Nontrivial scaling structure with effective space-time dimensionality dc þ 1. The transition can also be analyzed within the Moriya-Hertz-Millis approach which works with a bosonic order parameter that lives in the full d space dimensions. Despite the difference in dimensionalities both approaches give identical results. Comparison of the two approaches provides valuable insight into the nature of quantum criticality in fermionic systems with an extended gapless Fermi surface. Consider then a ‘‘generalized Fermi surface’’ of dimen�
T Senthil - One of the best experts on this subject based on the ideXlab platform.
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fermi surfaces in general codimension and a new controlled Nontrivial Fixed Point
Physical Review Letters, 2009Co-Authors: T Senthil, R ShankarAbstract:The energy of a $d$-dimensional Fermi system typically varies only along ${d}_{c}=1$ (``radial'') dimensions. We consider ${d}_{c}=1+\ensuremath{\epsilon}$ and study a transition to superconductivity in an $\ensuremath{\epsilon}$ expansion. The Nontrivial Fixed Point describes a scale invariant theory with an effective space-time dimension $D={d}_{c}+1$. Remarkably, the results can be reproduced by the Hertz-Millis action for the superconducting order parameter in higher effective space-time dimensions. We consider possible realizations of the transition at $\ensuremath{\epsilon}=1$, which corresponds to a linear Fermi surface in $d=3$.
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Fermi Surfaces in General Codimension and a New Controlled Nontrivial Fixed Point
2009Co-Authors: R Shankar, T SenthilAbstract:The energy of a d-dimensional Fermi system typically varies only along dc ¼ 1 (‘‘radial’’) dimensions. We consider dc ¼ 1 þ " and study a transition to superconductivity in an " expansion. The Nontrivial Fixed Point describes a scale invariant theory with an effective space-time dimension D ¼ dc þ 1. Remarkably, the results can be reproduced by the Hertz-Millis action for the superconducting order parameter in higher effective space-time dimensions. We consider possible realizations of the transition at " ¼ 1, which corresponds to a linear Fermi surface in d ¼ 3. DOI: 10.1103/PhysRevLett.102.046406 PACS numbers: 71.10.Hf The ground state of a fluid of noninteracting fermions in spatial dimension d � 2 is typically a Landau Fermi liquid with a sharp Fermi surface and quasiparticles. The Fermi liquid state can be understood within a fermionic renormalization group (RG) framework [1,2] in terms of a Fixed Point obtained as one focuses on modes within a bandwidth � of the Fermi surface and systematically reduces � . Unlike the BCS transition, transitions like the Stoner ferromagnetic transition that occur at finite coupling are not described by traditional flows within this approach. An alternate approach of Hertz [3], extended by Millis [4], performs a renormalization group analysis on the bosonic action for the order parameter obtained by integrating out the fermions and obtains an action nonanalytic in frequency and momentum and which yields results in agreement with those of Moriya [5] who used the self-consistent renormalization approach. However, there are reasons [6,7] to believe that integrating out the gapless fermionic modes can lead to singularity structure not evident in the Hertz-Millis analysis. When the noninteracting band structure consists of Fermi Points (as in graphene) the low energy description is a massless ‘‘relativistic’’ Dirac theory. In dimension d � 2 weak short-ranged interactions are irrelevant, though gap inducing phase transitions can occur at strong coupling. Without the complications of an extended Fermi surface these can be analyzed within conventional field theoretic framework for critical phenomena. We consider a general class of problems that fall in between these two cases. The low energy theory of a system of noninteracting fermions in d spatial dimension may be characterized by the codimension dc of the surface in momentum space where the energy gap vanishes. The ordinary Fermi surface has codimension dc ¼ 1 while the case of Fermi Points has codimension d. We study the general case of codimension d c with a eye toward gaining insight into dc ¼ 2 in d ¼ 3 which corresponds to fermions with line nodes in three dimensions which arise in unconventional three dimensional superconductors and the tight binding model on the diamond lattice at half filling [8]. Fermionic RG methods of Refs. [1,2] show that whenever dc > 1 all short-range interactions are irrelevant. With increasing interaction strength phase transitions which gap out the fermions are again possible. For dc
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fermi surfaces in general codimension and a new controlled Nontrivial Fixed Point
APS, 2009Co-Authors: R Shankar, T SenthilAbstract:The energy of a d-dimensional Fermi system typically varies only along dc ¼ 1 (‘‘radial’’) dimensions. We consider dc ¼ 1 þ " and study a transition to superconductivity in an " expansion. The Nontrivial Fixed Point describes a scale invariant theory with an effective space-time dimension D ¼ dc þ 1. Remarkably, the results can be reproduced by the Hertz-Millis action for the superconducting order parameter in higher effective space-time dimensions. We consider possible realizations of the transition at " ¼ 1, which corresponds to a linear Fermi surface in d ¼ 3. DOI: 10.1103/PhysRevLett.102.046406 PACS numbers: 71.10.Hf The ground state of a fluid of noninteracting fermions in spatial dimension d � 2 is typically a Landau Fermi liquid with a sharp Fermi surface and quasiparticles. The Fermi liquid state can be understood within a fermionic renormalization group (RG) framework [1,2] in terms of a Fixed Point obtained as one focuses on modes within a bandwidth � of the Fermi surface and systematically reduces � . Unlike the BCS transition, transitions like the Stoner ferromagnetic transition that occur at finite coupling are not described by traditional flows within this approach. An alternate approach of Hertz [3], extended by Millis [4], performs a renormalization group analysis on the bosonic action for the order parameter obtained by integrating out the fermions and obtains an action nonanalytic in frequency and momentum and which yields results in agreement with those of Moriya [5] who used the self-consistent renormalization approach. However, there are reasons [6,7] to believe that integrating out the gapless fermionic modes can lead to singularity structure not evident in the Hertz-Millis analysis. When the noninteracting band structure consists of Fermi Points (as in graphene) the low energy description is a massless ‘‘relativistic’’ Dirac theory. In dimension d � 2 weak short-ranged interactions are irrelevant, though gap inducing phase transitions can occur at strong coupling. Without the complications of an extended Fermi surface these can be analyzed within conventional field theoretic framework for critical phenomena. We consider a general class of problems that fall in between these two cases. The low energy theory of a system of noninteracting fermions in d spatial dimension may be characterized by the codimension dc of the surface in momentum space where the energy gap vanishes. The ordinary Fermi surface has codimension dc ¼ 1 while the case of Fermi Points has codimension d. We study the general case of codimension d c with a eye toward gaining insight into dc ¼ 2 in d ¼ 3 which corresponds to fermions with line nodes in three dimensions which arise in unconventional three dimensional superconductors and the tight binding model on the diamond lattice at half filling [8]. Fermionic RG methods of Refs. [1,2] show that whenever dc > 1 all short-range interactions are irrelevant. With increasing interaction strength phase transitions which gap out the fermions are again possible. For dc
superconductivity. We show that this can be accessed within the fermionic RG through a controlled � expansion in the codimension dc1 þ ", yielding a Nontrivial scaling structure with effective space-time dimensionality dc þ 1. The transition can also be analyzed within the Moriya-Hertz-Millis approach which works with a bosonic order parameter that lives in the full d space dimensions. Despite the difference in dimensionalities both approaches give identical results. Comparison of the two approaches provides valuable insight into the nature of quantum criticality in fermionic systems with an extended gapless Fermi surface. Consider then a ‘‘generalized Fermi surface’’ of dimen�
Abdelmalek Abdesselam - One of the best experts on this subject based on the ideXlab platform.
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A Complete Renormalization Group Trajectory Between Two Fixed Points
Communications in Mathematical Physics, 2007Co-Authors: Abdelmalek AbdesselamAbstract:We give a rigorous nonperturbative construction of a massless discrete trajectory for Wilson’s exact renormalization group. The model is a three dimensional Euclidean field theory with a modified free propagator. The trajectory realizes the mean field to critical crossover from the ultraviolet Gaussian Fixed Point to an analog recently constructed by Brydges, Mitter and Scoppola of the Wilson-Fisher Nontrivial Fixed Point.
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Towards a complete renormalization group trajectory between two Fixed Points
Communications in Mathematical Physics, 2007Co-Authors: Abdelmalek AbdesselamAbstract:We give a rigorous nonperturbative construction of a massless discrete trajectory for Wilson’s exact renormalization group. The model is a three dimensional Euclidean field theory with a modified free propagator. The trajectory realizes the mean field to critical crossover from the ultraviolet Gaussian Fixed Point to an analog recently constructed by Brydges, Mitter and Scoppola of the Wilson-Fisher Nontrivial Fixed Point.
Dine Ousmane Samary - One of the best experts on this subject based on the ideXlab platform.
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Ward-constrained melonic renormalization group flow for the rank-four $\phi^6$ tensorial group field theory.
arXiv: High Energy Physics - Theory, 2020Co-Authors: Vincent Lahoche, Dine Ousmane SamaryAbstract:The Nontrivial Fixed Point discovered for $\phi^4$-marginal couplings in tensorial group field theories have been showed to be incompatible with Ward-Takahashi identities. In this previous analysis we have stated that the case of models with interactions of order greater than four could probably lead to a Fixed Point compatible with local Ward's identities. In this paper we focus on a rank-4 Abelian $\phi^6$-just renormalizable tensorial group field theory and describe the renormalization group flow over the sub-theory space where Ward constraint is satisfied along the flow, by using an improved version of the effective vertex expansion. We show that this model exhibit a Nontrivial Fixed Points in this constrained subspace. Finally, the well-known asymptotically freedom of this model is highlighted.
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Ward-constrained melonic renormalization group flow for the $\phi^6_4$ tensorial group field theory
arXiv: High Energy Physics - Theory, 2019Co-Authors: Vincent Lahoche, Dine Ousmane SamaryAbstract:The Nontrivial Fixed Point discovered for $\phi^4$-marginal couplings in tensorial group field theories have been showed to be incompatible with Ward-Takahashi identities. In this previous analysis we have stated that the case of models with interactions of order greater than four could probably lead to a Fixed Point compatible with local Ward's identities. In this paper we focus on a rank-4 Abelian $\phi^6$-just renormalizable tensorial group field theory and describe the renormalization group flow over the sub-theory space where Ward constraint is satisfied along the flow, by using an improved version of the effective vertex expansion. We show that this model exhibit a Nontrivial Fixed Points in this constrained subspace. Finally, the well-known asymptotically freedom of this model is highlighted.
Hans-rudolf Jauslin - One of the best experts on this subject based on the ideXlab platform.
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Universality for the breakup of invariant tori in Hamiltonian flows
Physical Review E, 1998Co-Authors: Cristel Chandre, M. Govin, Hans-rudolf Jauslin, Hans KochAbstract:In this article, we describe a new renormalization-group scheme for analyzing the breakup of invariant tori for Hamiltonian systems with two degrees of freedom. The transformation, which acts on Hamiltonians that are quadratic in the action variables, combines a rescaling of phase space and a partial elimination of irrelevant (non-resonant) frequencies. It is implemented numerically for the case applying to golden invariant tori. We find a Nontrivial Fixed Point and compute the corresponding scaling and critical indices. If one compares flows to maps in the canonical way, our results are consistent with existing data on the breakup of golden invariant circles for area-preserving maps.
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KOLMOGOROV-ARNOLD-MOSER RENORMALIZATION-GROUP APPROACH TO THE BREAKUP OF INVARIANT TORI IN HAMILTONIAN SYSTEMS
Physical Review E, 1998Co-Authors: Cristel Chandre, M. Govin, Hans-rudolf JauslinAbstract:We analyze the breakup of invariant tori in Hamiltonian systems with two degrees of freedom using a combination of Kolmogorov-Arnold-Moser (KAM) theory and renormalization-group techniques. We consider a class of Hamiltonians quadratic in the action variables that is invariant under the chosen KAM transformations, following the approach of Thirring. The numerical implementation of the transformation shows that the KAM iteration converges up to the critical coupling at which the torus breaks up. By combining this iteration with a renormalization, consisting of a shift of resonances and rescalings of momentum and energy, we obtain a more efficient method that allows one to determine the critical coupling with high accuracy. This transformation is based on the physical mechanism of the breakup of invariant tori. We show that the critical surface of the transformation is the stable manifold of codimension one of a Nontrivial Fixed Point, and we discuss its universality properties.
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Kolmogorov-Arnold-Moser–Renormalization-Group Analysis of Stability in Hamiltonian Flows
Physical Review Letters, 1997Co-Authors: M. Govin, Cristel Chandre, Hans-rudolf JauslinAbstract:We study the stability and breakup of invariant tori in Hamiltonian flows using a combination of Kolmogorov-Arnold-Moser (KAM) theory and renormalization-group techniques. We implement the scheme numerically for a family of Hamiltonians quadratic in the actions to analyze the strong coupling regime. We show that the KAM iteration converges up to the critical coupling at which the torus breaks up. Adding a renormalization consisting of a rescaling of phase space and a shift of resonances allows us to determine the critical coupling with higher accuracy. We determine a Nontrivial Fixed Point and its universality properties.
