# American Institute of Mathematical Sciences

June  2002, 1(2): 135-159. doi: 10.3934/cpaa.2002.1.135

## Peierls instability with electron-electron interaction: the commensurate case

 1 Dipartimento di Matematica, Università di Roma "Tor Vergata", Roma, I-00133, Italy

Revised  August 2001 Published  March 2002

We consider a quantum many-body model describing a system of electrons interacting with themselves and hopping from one ion to another of a one dimensional lattice. We show that the ground state energy of such system, as a functional of the ionic configurations, has local minima in correspondence of configurations described by smooth $\frac{\pi}{pF}$ periodic functions, if the interaction is repulsive and large enough and pF is the Fermi momentum of the electrons. This means physically that a $d=1$ metal develop a periodic distortion of its reticular structure (Peierls instability). The minima are found solving the Eulero-Lagrange equations of the energy by a contraction method.
Citation: V. Mastropietro. Peierls instability with electron-electron interaction: the commensurate case. Communications on Pure and Applied Analysis, 2002, 1 (2) : 135-159. doi: 10.3934/cpaa.2002.1.135
 [1] Jan J. Sławianowski, Vasyl Kovalchuk, Agnieszka Martens, Barbara Gołubowska, Ewa E. Rożko. Essential nonlinearity implied by symmetry group. Problems of affine invariance in mechanics and physics. Discrete and Continuous Dynamical Systems - B, 2012, 17 (2) : 699-733. doi: 10.3934/dcdsb.2012.17.699 [2] 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 [3] Ning Sun, Shaoyun Shi, Wenlei Li. Singular renormalization group approach to SIS problems. Discrete and Continuous Dynamical Systems - B, 2020, 25 (9) : 3577-3596. doi: 10.3934/dcdsb.2020073 [4] Florian Méhats, Olivier Pinaud. A problem of moment realizability in quantum statistical physics. Kinetic and Related Models, 2011, 4 (4) : 1143-1158. doi: 10.3934/krm.2011.4.1143 [5] I. Moise, Roger Temam. Renormalization group method: Application to Navier-Stokes equation. Discrete and Continuous Dynamical Systems, 2000, 6 (1) : 191-210. doi: 10.3934/dcds.2000.6.191 [6] G. A. Braga, Frederico Furtado, Vincenzo Isaia. Renormalization group calculation of asymptotically self-similar dynamics. Conference Publications, 2005, 2005 (Special) : 131-141. doi: 10.3934/proc.2005.2005.131 [7] Wenlei Li, Shaoyun Shi. Singular perturbed renormalization group theory and its application to highly oscillatory problems. Discrete and Continuous Dynamical Systems - B, 2018, 23 (4) : 1819-1833. doi: 10.3934/dcdsb.2018089 [8] Chiu-Ya Lan, Chi-Kun Lin. Asymptotic behavior of the compressible viscous potential fluid: Renormalization group approach. Discrete and Continuous Dynamical Systems, 2004, 11 (1) : 161-188. doi: 10.3934/dcds.2004.11.161 [9] Hans Koch. A renormalization group fixed point associated with the breakup of golden invariant tori. Discrete and Continuous Dynamical Systems, 2004, 11 (4) : 881-909. doi: 10.3934/dcds.2004.11.881 [10] Nathan Glatt-Holtz, Mohammed Ziane. Singular perturbation systems with stochastic forcing and the renormalization group method. Discrete and Continuous Dynamical Systems, 2010, 26 (4) : 1241-1268. doi: 10.3934/dcds.2010.26.1241 [11] Michael Field, Ian Melbourne, Matthew Nicol, Andrei Török. Statistical properties of compact group extensions of hyperbolic flows and their time one maps. Discrete and Continuous Dynamical Systems, 2005, 12 (1) : 79-96. doi: 10.3934/dcds.2005.12.79 [12] Vincenzo Michael Isaia. Numerical simulation of universal finite time behavior for parabolic IVP via geometric renormalization group. Discrete and Continuous Dynamical Systems - B, 2017, 22 (9) : 3459-3481. doi: 10.3934/dcdsb.2017175 [13] G. A. Braga, Frederico Furtado, Jussara M. Moreira, Leonardo T. Rolla. Renormalization group analysis of nonlinear diffusion equations with time dependent coefficients: Analytical results. Discrete and Continuous Dynamical Systems - B, 2007, 7 (4) : 699-715. doi: 10.3934/dcdsb.2007.7.699 [14] Laura Cremaschi, Carlo Mantegazza. Short-time existence of the second order renormalization group flow in dimension three. Discrete and Continuous Dynamical Systems, 2015, 35 (12) : 5787-5798. doi: 10.3934/dcds.2015.35.5787 [15] João Lopes Dias. Brjuno condition and renormalization for Poincaré flows. Discrete and Continuous Dynamical Systems, 2006, 15 (2) : 641-656. doi: 10.3934/dcds.2006.15.641 [16] Helmut Kröger. From quantum action to quantum chaos. Conference Publications, 2003, 2003 (Special) : 492-500. doi: 10.3934/proc.2003.2003.492 [17] Alberto Ibort, Alberto López-Yela. Quantum tomography and the quantum Radon transform. Inverse Problems and Imaging, 2021, 15 (5) : 893-928. doi: 10.3934/ipi.2021021 [18] Vadim S. Anishchenko, Tatjana E. Vadivasova, Galina I. Strelkova, George A. Okrokvertskhov. Statistical properties of dynamical chaos. Mathematical Biosciences & Engineering, 2004, 1 (1) : 161-184. doi: 10.3934/mbe.2004.1.161 [19] David Lubicz. On a classification of finite statistical tests. Advances in Mathematics of Communications, 2007, 1 (4) : 509-524. doi: 10.3934/amc.2007.1.509 [20] Denis Gaidashev, Tomas Johnson. Spectral properties of renormalization for area-preserving maps. Discrete and Continuous Dynamical Systems, 2016, 36 (7) : 3651-3675. doi: 10.3934/dcds.2016.36.3651

2021 Impact Factor: 1.273

## Metrics

• HTML views (0)
• Cited by (0)

• on AIMS