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abstract: 'We discuss the possibility that a class of non-Fermi liquid systems has a non-singular continuous transition. We show how such a transition in a 2D band with non-linear dispersion can be studied in terms of non-linear $\sigma$ models for the quantum critical point and the non-Fermi liquid fixed point, and discuss several cases. We also present the strong coupling theory in terms of a large N version of the sigma model for the non-Fermi liquid fixed point. The possibility of this kind of transition can be tested in several quantum critical points in 2D, including the integer and fractional quantum Hall transitions.'
author:
- 'Philip W. Phillips, Oriol Vendramin and Kwon Park'
title: 'Quantum critical and non-Fermi liquid fixed points in 2D'
---
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[ address=[Department of Physics, Massachusetts Institute of Technology, Cambridge, MA 02139]{} ]{}
Introduction
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The concept of quantum criticality has played an important role in the past decade in addressing the quantum phase transition problem in metals and other correlated electron systems, as shown by the wide acceptance of the Hertz-Mills-Millis theory. A central role in this approach has been played by the non-linear $\sigma$ model [@Hertz1976; @Millis1993]. In some cases, however, it has been noted that non-Fermi liquid behavior can occur at quantum critical points. It is this possibility that we wish to discuss in this paper, and the discussion will be focused on the 2D case.
The Landau theory of the normal phase is that the Fermi liquid theory fails at a quantum critical point near the onset of ferromagnetism [@Hertz1976; @Millis1993]. In the ferromagnetic phase, a Landau quasiparticle description of electrons is appropriate and the interaction should be short-ranged. This is the case in the metallic ferromagnetic phase of Chromium [@CrO2], which has an exchange interaction of 4 meV and is located at a pressure of 80 atm [@Morales2010]. Here the electronic mean-free path exceeds the size of the crystallites of Chromium oxide [@Morales2010]. The Landau theory then predicts that Fermi liquid behavior will be recovered as the system is tuned to the antiferromagnetic region near the ferromagnetic phase. However, in the antiferromagnetic region it has been noted by Chubukov and Pines [@Chubukov2003; @Chubukov2004] that in Chromium oxide the $f$ electrons have an entropy, obtained from the spin susceptibility, that does not go to zero even as the antiferromagnetic order is approached. This is surprising, since it was expected that the Kondo effect would lead to a well-defined quasiparticle resonance. This lack of quasiparticle resonance is taken as evidence for a non-Fermi liquid fixed point and for non-Fermi liquid behavior at the ferromagnetic critical point.
The possibility of non-Fermi liquid behavior has been examined for the 3D to 2D metal to insulator transition in Chromium [@Schroder1999] where there is good evidence for the existence of non-Fermi liquid behavior in the 2D metal.
Non-Fermi liquid behavior was also found in the magnetoresistance measurement in Chromium at its antiferromagnetic-ferromagnetic critical point [@Cooper1989] (although this has been questioned by others [@Jin1989]). At present it is not clear whether this experimental finding will turn out to be due to non-Fermi liquid fixed points or the presence of the quantum critical point in the antiferromagnetic phase and the associated quantum critical fluctuation.
More recently, at the metallic quantum critical point in $Sr_3CuPt_{x}Ir_{x}O_6$ ($x=0$, $0.1$, and $0.2$) [@Senthil2004], the Sommerfeld constant as a function of temperature shows a non-Fermi liquid behavior in the temperature region below the crossover to the Fermi liquid behavior and this is taken as evidence for non-Fermi liquid behavior [@Lobo2006]. It is noted that in the Fermi liquid region, the Sommerfeld constant goes as $1/3$ whereas in the non-Fermi liquid region, it goes as $\sim 1/\sqrt{T}$. At present there is no explanation for this observation.
There has also been some theoretical work on the fractional quantum Hall transition in 2D [@Girvin1990; @Girvin1994] (see also references therein) and on the possibility of non-Fermi liquid fixed points in that region [@Kivelson2002; @KunYang2003]. The existence of non-Fermi liquid fixed points near a fractional quantum Hall transition has been proposed in the presence of spin-orbit interaction and electron-electron interaction by using a $\sigma$ model in terms of the $S=\frac{1}{2}$ quantum Heisenberg model at the fractional quantum Hall critical point in the presence of interactions [@Sachdev1992] (see also references therein). It has also been argued that this could be examined in terms of a large N $\sigma$ model [@Furusaki1995; @Kwon2003] (see also reference [@Shankar1991])
In this paper we will consider 2D systems and explore non-Fermi liquid fixed points in certain cases. Since there are so many possibilities of such fixed points in 2D, we will mainly consider the possibility of non-Fermi liquid behavior associated with some of these fixed points.
In section II we will consider 2D fermions with Coulomb repulsion and dispersion nonlinearity, so that the quantum critical point is described by the Hamiltonian $$H=\sum_{\textbf{k},\sigma}\epsilon_{\textbf{k}}(\textbf{k})n_{\textbf{k},\sigma}+\sum_{\textbf{k},\textbf{q}}\frac{U}{|\textbf{k}-\textbf{q}|}n_{\textbf{k},\uparrow}n_{\textbf{q},\downarrow}$$ where $\epsilon_{\textbf{k}}(\textbf{k})=-\sum_{i=1}^{d}2g_{i}k_{i}^{2}-\mu$ and $U$ is the strength of the on-site Coulomb repulsion between $f$ electrons. We will also discuss 2D fermions with on-site Coulomb repulsion and spin orbit interaction, where the non-Fermi liquid critical point is described by the Hamiltonian $$\begin{aligned}
H & = & \sum_{\textbf{k},\sigma}\epsilon_{\textbf{k}}(\textbf{k})n_{\textbf{k},\sigma}+ \frac{J}{2}\sum_{\textbf{k},\textbf{q},\sigma} \left(
c_{\textbf{k},\sigma}^{\dag}c_{\textbf{q},\sigma}^{\dag}c_{\textbf{q}-\textbf{k},\sigma}c_{\textbf{k},\sigma} + c_{\textbf{k},\sigma}^{\dag}c_{\textbf{-q}-\textbf{k},\sigma}
c_{\textbf{-q},\sigma}^{\dag}c_{\textbf{k},\sigma}\right) \\ \nonumber & + & \alpha\sum_{\textbf{k},\sigma,\sigma'}\gamma_{\textbf{k}} \vec{\sigma}_{\sigma\sigma'}.\left(c_{\textbf{k},\sigma}^{\dag}\vec{\tau}_{\textbf{k}}
c_{\textbf{k},\sigma'}-c_{\textbf{k},\sigma'}^{\dag}\vec{\tau}_{\textbf{-k}}c_{\textbf{k},\sigma}\right)\end{aligned}$$ Here the dispersion has the form $\epsilon_{\textbf{k}}(\textbf{k})=-\sum_{i=1}^{2}2g_{i}k_{i}^{2}-\mu$, $\alpha$ is the strength of spin-orbit coupling, $\gamma_{\textbf{k}}=\gamma_{\textbf{k}}^{x}+\gamma_{\textbf{k}}^{y}$, $\gamma_{\textbf{k}}^{x}=\cos k_{x}$ and $\gamma_{\textbf{k}}^{y}=\sin k_{y}$. $J$ is the superexchange coupling between the $d$ electrons and the strength of spin orbit coupling is given by $\alpha$. The spin $\vec{\sigma}$ has two components, while $\vec{\tau}$ has $3$ components. For $J>0$, the non-Fermi liquid critical point is described by the Hamiltonian $$H = \sum_{\textbf{k},\sigma} \epsilon_{\textbf{k}}(\textbf{k})n_{\textbf{k},\sigma}+\sum_{\textbf{q},\textbf{k},\sigma} (c_{\textbf{q},\sigma}^{\dag} c_{\textbf{k},\sigma}
+c_{\textbf{k},\sigma}^{\dag} c_{\textbf{-q},\sigma})
\left(\frac{J}{2}\left(\gamma_{\textbf{q}} \gamma_{\textbf{-q}}+\alpha \gamma_{\textbf{k}} \gamma_{\textbf{q}}\right)+\frac{1}{J}\right)$$ where $\textbf{q}=-\textbf{k}-\textbf{k}'$ and $\textbf{k}\neq \textbf{q}$.
This Hamiltonian reduces to the quantum critical Hamiltonian of the $\sigma$ model, as explained in Appendix A. The latter will be used to understand the critical behavior near the non-Fermi liquid critical point.
In section III we will study a system of spinless fermions with Coulomb repulsion
