# American Institute of Mathematical Sciences

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

## Classification of potential flows under renormalization group transformation

 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.
Citation: Sze-Bi Hsu, Bernold Fiedler, Hsiu-Hau Lin. Classification of potential flows under renormalization group transformation. Discrete and 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), 129-192. doi: 10.1103/RevModPhys.66.129. [2] M. Salmhofer and C. Honerkamp, Fermionic renormalization group flows - technique and theory, Progress of Theoretical Physics, 105 (2001), 1-35. doi: 10.1143/PTP.105.1. [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), 299-352. doi: 10.1103/RevModPhys.84.299. [4] M. Fabrizio, Role of transverse hopping in a two-coupled-chains model, Phys. Rev. B, 48 (1993), 15838-15860. doi: 10.1103/PhysRevB.48.15838. [5] L. Balents and M. P. A. Fisher, Weak-coupling phase diagram of the two-chain Hubbard model, Phys. Rev. B, 53 (1996), p12133. doi: 10.1103/PhysRevB.53.12133. [6] H. J. Schulz, Phases of two coupled Luttinger liquids, Phys. Rev. B, 53 (1996), R2959-R2962. doi: 10.1103/PhysRevB.53.R2959. [7] H.-H. Lin, L. Balents and M. P. A. Fisher, N-chain Hubbard model in weak coupling, Phys. Rev. B, 56 (1997), 6569-6593. doi: 10.1103/PhysRevB.56.6569. [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), 1794-1825. doi: 10.1103/PhysRevB.58.1794. [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), 79-113. doi: 10.1143/PTPS.160.79. [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), p155110. [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), 121107(R). doi: 10.1103/PhysRevB.81.121107. [12] A. Goriely and C. Hyde, Finite time blow-up in dynamical systems, Phys. Lett. A, 250 (1998), 311-318. doi: 10.1016/S0375-9601(98)00822-6. [13] A. Goriely and C. Hyde, Necessary and sufficient conditions for finite time singularity in ordinary differential equations, J. of diff. eq., 161 (2000), 422-448. doi: 10.1006/jdeq.1999.3688. [14] S. B. Hsu, Ordinary Differential Equations (second edition), World Scientific Press, 2013. doi: 10.1142/8744. [15] J. K. Hale, Ordinary Differential Equations, Wiley-Interscience, 1969. [16] A. V. Chubukov, D. V. Efremov and I. Eremin, Magnetism, superconductivity, and pairing symmetry in iron-based superconductors, Phys. Rev. B, 78 (2008), 134512. doi: 10.1103/PhysRevB.78.134512. [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), 047005. doi: 10.1103/PhysRevLett.102.047005. [18] F. Wang and D.-H. Lee, The electron-pairing mechanism of iron-based superconductors, Science, 332 (2011), 200-204. doi: 10.1126/science.1200182.

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##### References:
 [1] R. Shankar, Renormalization-group approach to interacting fermions, Rev. Mod. Phys., 66 (1994), 129-192. doi: 10.1103/RevModPhys.66.129. [2] M. Salmhofer and C. Honerkamp, Fermionic renormalization group flows - technique and theory, Progress of Theoretical Physics, 105 (2001), 1-35. doi: 10.1143/PTP.105.1. [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), 299-352. doi: 10.1103/RevModPhys.84.299. [4] M. Fabrizio, Role of transverse hopping in a two-coupled-chains model, Phys. Rev. B, 48 (1993), 15838-15860. doi: 10.1103/PhysRevB.48.15838. [5] L. Balents and M. P. A. Fisher, Weak-coupling phase diagram of the two-chain Hubbard model, Phys. Rev. B, 53 (1996), p12133. doi: 10.1103/PhysRevB.53.12133. [6] H. J. Schulz, Phases of two coupled Luttinger liquids, Phys. Rev. B, 53 (1996), R2959-R2962. doi: 10.1103/PhysRevB.53.R2959. [7] H.-H. Lin, L. Balents and M. P. A. Fisher, N-chain Hubbard model in weak coupling, Phys. Rev. B, 56 (1997), 6569-6593. doi: 10.1103/PhysRevB.56.6569. [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), 1794-1825. doi: 10.1103/PhysRevB.58.1794. [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), 79-113. doi: 10.1143/PTPS.160.79. [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), p155110. [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), 121107(R). doi: 10.1103/PhysRevB.81.121107. [12] A. Goriely and C. Hyde, Finite time blow-up in dynamical systems, Phys. Lett. A, 250 (1998), 311-318. doi: 10.1016/S0375-9601(98)00822-6. [13] A. Goriely and C. Hyde, Necessary and sufficient conditions for finite time singularity in ordinary differential equations, J. of diff. eq., 161 (2000), 422-448. doi: 10.1006/jdeq.1999.3688. [14] S. B. Hsu, Ordinary Differential Equations (second edition), World Scientific Press, 2013. doi: 10.1142/8744. [15] J. K. Hale, Ordinary Differential Equations, Wiley-Interscience, 1969. [16] A. V. Chubukov, D. V. Efremov and I. Eremin, Magnetism, superconductivity, and pairing symmetry in iron-based superconductors, Phys. Rev. B, 78 (2008), 134512. doi: 10.1103/PhysRevB.78.134512. [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), 047005. doi: 10.1103/PhysRevLett.102.047005. [18] F. Wang and D.-H. Lee, The electron-pairing mechanism of iron-based superconductors, Science, 332 (2011), 200-204. doi: 10.1126/science.1200182.
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