American Institute of Mathematical Sciences

Advanced
March  2016, 21(2): 437-446. doi: 10.3934/dcdsb.2016.21.437

Classification of potential flows under renormalization group transformation

Sze-Bi Hsu 1, , Bernold Fiedler 2, and  Hsiu-Hau Lin 3,

1. 

Department of Mathematics, National Tsing Hua University, National Center of Theoretical Science, Hsinchu 300
2. 

Institute of Mathematics, Free University Berlin, Arnimallee 3, D-14195 Berlin, Germany
3. 

Department of Physics and The National Center for Theoretical Science, National Tsing Hua University, Hsinchu 30013, Taiwan

Received  December 2014 Revised  May 2015 Published  November 2015

Competitions between different interactions in strongly correlated electron systems often lead to exotic phases. Renormalization group is one of the powerful techniques to analyze the competing interactions without presumed bias. It was recently shown that the renormalization group transformations to the one-loop order in many correlated electron systems are described by potential flows. Here we prove several rigorous theorems in the presence of renormalization-group potential and find the complete classification for the potential flows. In addition, we show that the relevant interactions blow up at the maximal scaling exponent of unity, explaining the puzzling power-law Ansatz found in previous studies. The above findings are of great importance in building up the hierarchy for relevant couplings and the complete classification for correlated ground states in the presence of generic interactions.
Keywords: polar equations, blow-up in finite time, RG flow, gradient flow, Strongly correlated electron system, LaSalles' invariance principle..
Mathematics Subject Classification: 34, 3.
Citation: Sze-Bi Hsu, Bernold Fiedler, Hsiu-Hau Lin. Classification of potential flows under renormalization group transformation. Discrete & Continuous Dynamical Systems - B, 2016, 21 (2) : 437-446. doi: 10.3934/dcdsb.2016.21.437
References:
[1]

R. Shankar, Renormalization-group approach to interacting fermions,, Rev. Mod. Phys., 66 (1994), 129. doi: 10.1103/RevModPhys.66.129. Google Scholar

[2]

M. Salmhofer and C. Honerkamp, Fermionic renormalization group flows - technique and theory,, Progress of Theoretical Physics, 105 (2001), 1. doi: 10.1143/PTP.105.1. Google Scholar

[3]

W. Metzner, M. Salmhofer, C. Honerkamp, V. Meden and K. Schönhammer, Functional renormalization group approach to correlated fermion systems,, Rev. Mod. Phys., 84 (2012), 299. doi: 10.1103/RevModPhys.84.299. Google Scholar

[4]

M. Fabrizio, Role of transverse hopping in a two-coupled-chains model,, Phys. Rev. B, 48 (1993), 15838. doi: 10.1103/PhysRevB.48.15838. Google Scholar

[5]

L. Balents and M. P. A. Fisher, Weak-coupling phase diagram of the two-chain Hubbard model,, Phys. Rev. B, 53 (1996). doi: 10.1103/PhysRevB.53.12133. Google Scholar

[6]

H. J. Schulz, Phases of two coupled Luttinger liquids,, Phys. Rev. B, 53 (1996). doi: 10.1103/PhysRevB.53.R2959. Google Scholar

[7]

H.-H. Lin, L. Balents and M. P. A. Fisher, N-chain Hubbard model in weak coupling,, Phys. Rev. B, 56 (1997), 6569. doi: 10.1103/PhysRevB.56.6569. Google Scholar

[8]

H.-H. Lin, L. Balents and M. P. A. Fisher, Exact SO(8) symmetry in the weakly-interacting two-leg ladder,, Phys. Rev. B, 58 (1998), 1794. doi: 10.1103/PhysRevB.58.1794. Google Scholar

[9]

M.-H. Chang, W. Chen and H.-H. Lin, Renormalization group potential for quasi-one-dimensional correlated systems,, Prog. Theor. Phys. Suppl., 160 (2005), 79. doi: 10.1143/PTPS.160.79. Google Scholar

[10]

E. Szirmai and J. Solyom, Possible phases of two coupled n-component fermionic chains determined using an analytic renormalization group method,, Phys. Rev. B, 74 (2006). Google Scholar

[11]

H.-Y. Shih, W.-M. Huang, S.-B. Hsu and H.-H. Lin, Hierarchy of relevant couplings in perturbative renormalization group transformations,, Phys. Rev. B, 81 (2010). doi: 10.1103/PhysRevB.81.121107. Google Scholar

[12]

A. Goriely and C. Hyde, Finite time blow-up in dynamical systems,, Phys. Lett. A, 250 (1998), 311. doi: 10.1016/S0375-9601(98)00822-6. Google Scholar

[13]

A. Goriely and C. Hyde, Necessary and sufficient conditions for finite time singularity in ordinary differential equations,, J. of diff. eq., 161 (2000), 422. doi: 10.1006/jdeq.1999.3688. Google Scholar

[14]

S. B. Hsu, Ordinary Differential Equations (second edition),, World Scientific Press, (2013). doi: 10.1142/8744. Google Scholar

[15]

J. K. Hale, Ordinary Differential Equations,, Wiley-Interscience, (1969). Google Scholar

[16]

A. V. Chubukov, D. V. Efremov and I. Eremin, Magnetism, superconductivity, and pairing symmetry in iron-based superconductors,, Phys. Rev. B, 78 (2008). doi: 10.1103/PhysRevB.78.134512. Google Scholar

[17]

F. Wang, H. Zhai, Y. Ran, A. Vishwanath and D.-H. Lee, Functional renormalization-group study of the pairing symmetry and pairing mechanism of the FeAs-based high-temperature superconductor,, Phys. Rev. Lett., 102 (2009). doi: 10.1103/PhysRevLett.102.047005. Google Scholar

[18]

F. Wang and D.-H. Lee, The electron-pairing mechanism of iron-based superconductors,, Science, 332 (2011), 200. doi: 10.1126/science.1200182. Google Scholar

show all references

References:
[1]

R. Shankar, Renormalization-group approach to interacting fermions,, Rev. Mod. Phys., 66 (1994), 129. doi: 10.1103/RevModPhys.66.129. Google Scholar

[2]

M. Salmhofer and C. Honerkamp, Fermionic renormalization group flows - technique and theory,, Progress of Theoretical Physics, 105 (2001), 1. doi: 10.1143/PTP.105.1. Google Scholar

[3]

W. Metzner, M. Salmhofer, C. Honerkamp, V. Meden and K. Schönhammer, Functional renormalization group approach to correlated fermion systems,, Rev. Mod. Phys., 84 (2012), 299. doi: 10.1103/RevModPhys.84.299. Google Scholar

[4]

M. Fabrizio, Role of transverse hopping in a two-coupled-chains model,, Phys. Rev. B, 48 (1993), 15838. doi: 10.1103/PhysRevB.48.15838. Google Scholar

[5]

L. Balents and M. P. A. Fisher, Weak-coupling phase diagram of the two-chain Hubbard model,, Phys. Rev. B, 53 (1996). doi: 10.1103/PhysRevB.53.12133. Google Scholar

[6]

H. J. Schulz, Phases of two coupled Luttinger liquids,, Phys. Rev. B, 53 (1996). doi: 10.1103/PhysRevB.53.R2959. Google Scholar

[7]

H.-H. Lin, L. Balents and M. P. A. Fisher, N-chain Hubbard model in weak coupling,, Phys. Rev. B, 56 (1997), 6569. doi: 10.1103/PhysRevB.56.6569. Google Scholar

[8]

H.-H. Lin, L. Balents and M. P. A. Fisher, Exact SO(8) symmetry in the weakly-interacting two-leg ladder,, Phys. Rev. B, 58 (1998), 1794. doi: 10.1103/PhysRevB.58.1794. Google Scholar

[9]

M.-H. Chang, W. Chen and H.-H. Lin, Renormalization group potential for quasi-one-dimensional correlated systems,, Prog. Theor. Phys. Suppl., 160 (2005), 79. doi: 10.1143/PTPS.160.79. Google Scholar

[10]

E. Szirmai and J. Solyom, Possible phases of two coupled n-component fermionic chains determined using an analytic renormalization group method,, Phys. Rev. B, 74 (2006). Google Scholar

[11]

H.-Y. Shih, W.-M. Huang, S.-B. Hsu and H.-H. Lin, Hierarchy of relevant couplings in perturbative renormalization group transformations,, Phys. Rev. B, 81 (2010). doi: 10.1103/PhysRevB.81.121107. Google Scholar

[12]

A. Goriely and C. Hyde, Finite time blow-up in dynamical systems,, Phys. Lett. A, 250 (1998), 311. doi: 10.1016/S0375-9601(98)00822-6. Google Scholar

[13]

A. Goriely and C. Hyde, Necessary and sufficient conditions for finite time singularity in ordinary differential equations,, J. of diff. eq., 161 (2000), 422. doi: 10.1006/jdeq.1999.3688. Google Scholar

[14]

S. B. Hsu, Ordinary Differential Equations (second edition),, World Scientific Press, (2013). doi: 10.1142/8744. Google Scholar

[15]

J. K. Hale, Ordinary Differential Equations,, Wiley-Interscience, (1969). Google Scholar

[16]

A. V. Chubukov, D. V. Efremov and I. Eremin, Magnetism, superconductivity, and pairing symmetry in iron-based superconductors,, Phys. Rev. B, 78 (2008). doi: 10.1103/PhysRevB.78.134512. Google Scholar

[17]

F. Wang, H. Zhai, Y. Ran, A. Vishwanath and D.-H. Lee, Functional renormalization-group study of the pairing symmetry and pairing mechanism of the FeAs-based high-temperature superconductor,, Phys. Rev. Lett., 102 (2009). doi: 10.1103/PhysRevLett.102.047005. Google Scholar

[18]

F. Wang and D.-H. Lee, The electron-pairing mechanism of iron-based superconductors,, Science, 332 (2011), 200. doi: 10.1126/science.1200182. Google Scholar

[1]

Van Tien Nguyen. On the blow-up results for a class of strongly perturbed semilinear heat equations. Discrete & Continuous Dynamical Systems - A, 2015, 35 (8) : 3585-3626. doi: 10.3934/dcds.2015.35.3585
[2]

Marius Ghergu, Vicenţiu Rădulescu. Nonradial blow-up solutions of sublinear elliptic equations with gradient term. Communications on Pure & Applied Analysis, 2004, 3 (3) : 465-474. doi: 10.3934/cpaa.2004.3.465
[3]

Miaoqing Tian, Sining Zheng. Global boundedness versus finite-time blow-up of solutions to a quasilinear fully parabolic Keller-Segel system of two species. Communications on Pure & Applied Analysis, 2016, 15 (1) : 243-260. doi: 10.3934/cpaa.2016.15.243
[4]

Yanbing Yang, Runzhang Xu. Nonlinear wave equation with both strongly and weakly damped terms: Supercritical initial energy finite time blow up. Communications on Pure & Applied Analysis, 2019, 18 (3) : 1351-1358. doi: 10.3934/cpaa.2019065
[5]

Timothy Blass, Rafael De La Llave, Enrico Valdinoci. A comparison principle for a Sobolev gradient semi-flow. Communications on Pure & Applied Analysis, 2011, 10 (1) : 69-91. doi: 10.3934/cpaa.2011.10.69
[6]

Monica Marras, Stella Vernier-Piro, Giuseppe Viglialoro. Blow-up phenomena for nonlinear pseudo-parabolic equations with gradient term. Discrete & Continuous Dynamical Systems - B, 2017, 22 (6) : 2291-2300. doi: 10.3934/dcdsb.2017096
[7]

Sachiko Ishida, Tomomi Yokota. Blow-up in finite or infinite time for quasilinear degenerate Keller-Segel systems of parabolic-parabolic type. Discrete & Continuous Dynamical Systems - B, 2013, 18 (10) : 2569-2596. doi: 10.3934/dcdsb.2013.18.2569
[8]

Evgeny Galakhov, Olga Salieva. Blow-up for nonlinear inequalities with gradient terms and singularities on unbounded sets. Conference Publications, 2015, 2015 (special) : 489-494. doi: 10.3934/proc.2015.0489
[9]

Johannes Lankeit. Infinite time blow-up of many solutions to a general quasilinear parabolic-elliptic Keller-Segel system. Discrete & Continuous Dynamical Systems - S, 2018, 0 (0) : 233-255. doi: 10.3934/dcdss.2020013
[10]

Juan Dávila, Manuel Del Pino, Catalina Pesce, Juncheng Wei. Blow-up for the 3-dimensional axially symmetric harmonic map flow into $ S^2 $. Discrete & Continuous Dynamical Systems - A, 2019, 0 (0) : 1-31. doi: 10.3934/dcds.2019237
[11]

Victor A. Galaktionov, Juan-Luis Vázquez. The problem Of blow-up in nonlinear parabolic equations. Discrete & Continuous Dynamical Systems - A, 2002, 8 (2) : 399-433. doi: 10.3934/dcds.2002.8.399
[12]

Wenqing Hu, Chris Junchi Li. A convergence analysis of the perturbed compositional gradient flow: Averaging principle and normal deviations. Discrete & Continuous Dynamical Systems - A, 2018, 38 (10) : 4951-4977. doi: 10.3934/dcds.2018216
[13]

Cristophe Besse, Rémi Carles, Norbert J. Mauser, Hans Peter Stimming. Monotonicity properties of the blow-up time for nonlinear Schrödinger equations: Numerical evidence. Discrete & Continuous Dynamical Systems - B, 2008, 9 (1) : 11-36. doi: 10.3934/dcdsb.2008.9.11
[14]

José M. Arrieta, Raúl Ferreira, Arturo de Pablo, Julio D. Rossi. Stability of the blow-up time and the blow-up set under perturbations. Discrete & Continuous Dynamical Systems - A, 2010, 26 (1) : 43-61. doi: 10.3934/dcds.2010.26.43
[15]

Ying Fu, Changzheng Qu, Yichen Ma. Well-posedness and blow-up phenomena for the interacting system of the Camassa-Holm and Degasperis-Procesi equations. Discrete & Continuous Dynamical Systems - A, 2010, 27 (3) : 1025-1035. doi: 10.3934/dcds.2010.27.1025
[16]

Nguyen Thanh Long, Hoang Hai Ha, Le Thi Phuong Ngoc, Nguyen Anh Triet. Existence, blow-up and exponential decay estimates for a system of nonlinear viscoelastic wave equations with nonlinear boundary conditions. Communications on Pure & Applied Analysis, 2020, 19 (1) : 455-492. doi: 10.3934/cpaa.2020023
[17]

Seyedeh Marzieh Ghavidel, Wolfgang M. Ruess. Flow invariance for nonautonomous nonlinear partial differential delay equations. Communications on Pure & Applied Analysis, 2012, 11 (6) : 2351-2369. doi: 10.3934/cpaa.2012.11.2351
[18]

Monica Marras, Stella Vernier Piro. Bounds for blow-up time in nonlinear parabolic systems. Conference Publications, 2011, 2011 (Special) : 1025-1031. doi: 10.3934/proc.2011.2011.1025
[19]

Matthias Erbar, Jan Maas. Gradient flow structures for discrete porous medium equations. Discrete & Continuous Dynamical Systems - A, 2014, 34 (4) : 1355-1374. doi: 10.3934/dcds.2014.34.1355
[20]

Bertram Düring, Daniel Matthes, Josipa Pina Milišić. A gradient flow scheme for nonlinear fourth order equations. Discrete & Continuous Dynamical Systems - B, 2010, 14 (3) : 935-959. doi: 10.3934/dcdsb.2010.14.935

2018 Impact Factor: 1.008

Tools

Metrics

  • PDF downloads (7)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences