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arXiv:1701.07557 · Physical review. B./Physical review. B

Role of finite-size effects in the critical behavior of itinerant fermions

Avraham Klein, Andrey V. Chubukov

Published Jul 27, 2017• 3 citations
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Abstract

We study the role of finite-size effects on critical behavior near a metallic $q=0$ critical point and compare the results with a recent extensive quantum Monte Carlo (QMC) study [Y. Schattner et al., Phys. Rev. X 6, 031028 (2016)]. This study found several features in both bosonic and fermionic responses, in disagreement with the expected critical behavior with a dynamical exponent $z=3$. We show that finite-size effects are particularly strong for $z=3$ criticality and give rise to a behavior different from that of an infinite system, over a wide range of momenta and frequencies. We argue that by taking finite-size effects into account, the QMC results can be explained within $z=3$ theory.

DOI
10.1103/physrevb.96.041125

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We study the role of finite-size effects on critical behavior near a metallic q=0q=0q=0 critical point and compare the results with a recent extensive quantum Monte Carlo (QMC) study [Y. Schattner et al., Phys. Rev. X 6, 031028 (2016)]. This study found several features in both bosonic and fermionic responses, in disagreement with the expected critical behavior with a dynamical exponent z=3z=3z=3. We show that finite-size effects are particularly strong for z=3z=3z=3 criticality and give rise to a behavior different from that of an infinite system, over a wide range of momenta and frequencies. We argue that by taking finite-size effects into account, the QMC results can be explained within z=3z=3z=3 theory.
We study the effects of finite temperature on normal state properties of a metal near a quantum critical point to an antiferromagnetic or Ising-nematic state.At T ¼ 0, bosonic and fermionic self-energies are traditionally computed within Eliashberg theory, and they obey scaling relations with characteristic power laws.Corrections to Eliashberg theory break these power laws but only at very small frequencies.Quantum Monte Carlo (QMC) simulations have shown that, already at much larger frequencies, there are strong systematic deviations from these predictions, casting doubt on the validity of the theoretical analysis.We extend Eliashberg theory to finite T and argue that in the T range accessible in the QMC simulations above the superconducting transition, the scaling forms for both fermionic and bosonic self-energies are quite different from those at T ¼ 0. We compare finite T results with QMC data and find good agreement for both systems.We argue that this agreement resolves the key apparent contradiction between the theory and the QMC simulations.
We consider the effect of the coupling between two-dimensional (2D) quantum rotors near an XY ferromagnetic quantum critical point and spins of itinerant fermions. We analyze how this coupling affects the dynamics of rotors and the self-energy of fermions. A common belief is that near a q=0q=0q=0 ferromagnetic transition, fermions induce an $\mathrm{\ensuremath{\Omega}}/q$ Landau damping of rotors (i.e., the dynamical critical exponent is z=3z=3z=3) and Landau overdamped rotors give rise to non-Fermi liquid fermionic self-energy $\mathrm{\ensuremath{\Sigma}}\ensuremath{\propto}{\ensuremath{\omega}}^{2/3}$. This behavior has been confirmed in previous quantum Monte Carlo (QMC) studies. Here we show that for the XY case the behavior is different. We report the results of large-scale quantum Monte Carlo simulations, which show that at small frequencies z=2z=2z=2 and $\mathrm{\ensuremath{\Sigma}}\ensuremath{\propto}{\ensuremath{\omega}}^{1/2}$. We argue that the new behavior is associated with the fact that a fermionic spin is by itself not a conserved quantity due to spin-spin coupling to rotors, and a combination of self-energy and vertex corrections replaces 1/q1/q1/q in the Landau damping by a constant. We discuss the implication of these results to experiments.
Metallic quantum critical phenomena are believed to play a key role in many strongly correlated materials, including high-temperature superconductors. Theoretically, the problem of quantum criticality in the presence of a Fermi surface has proven to be highly challenging. However, it has recently been realized that many models used to describe such systems are amenable to numerically exact solution by quantum Monte Carlo (QMC) techniques, without suffering from the fermion sign problem. In this review, we examine the status of the understanding of metallic quantum criticality and the recent progress made by QMC simulations. We focus on the cases of spin-density wave and Ising nematic criticality. We describe the results obtained so far and their implications for superconductivity, non-Fermi liquid behavior, and transport near metallic quantum critical points. Some of the outstanding puzzles and future directions are highlighted.
Metallic quantum criticality is among the central themes in the understanding of correlated electronic systems, and converging results between analytical and numerical approaches are still under review. In this work, we develop a state-of-the-art large-scale quantum Monte Carlo simulation technique and systematically investigate the itinerant quantum critical point on a 2D square lattice with antiferromagnetic spin fluctuations at wavevector [Formula: see text]-a problem that resembles the Fermi surface setup and low-energy antiferromagnetic fluctuations in high-Tc cuprates and other critical metals, which might be relevant to their non-Fermi-liquid behaviors. System sizes of [Formula: see text] ([Formula: see text]) are comfortably accessed, and the quantum critical scaling behaviors are revealed with unprecedented high precision. We found that the antiferromagnetic spin fluctuations introduce effective interactions among fermions and the fermions in return render the bare bosonic critical point into a different universality, different from both the bare Ising universality class and the Hertz-Mills-Moriya RPA prediction. At the quantum critical point, a finite anomalous dimension [Formula: see text] is observed in the bosonic propagator, and fermions at hotspots evolve into a non-Fermi liquid. In the antiferromagnetically ordered metallic phase, fermion pockets are observed as the energy gap opens up at the hotspots. These results bridge the recent theoretical and numerical developments in metallic quantum criticality and can serve as the stepping stone toward final understanding of the 2D correlated fermions interacting with gapless critical excitations.
We study finite-temperature properties of metals close to an Ising-nematic quantum critical point in two spatial dimensions. In particular we show that at any finite temperature there is a regime where order parameter fluctuations are characterized by a dynamical critical exponent z=2z=2z=2, in contrast to z=3z=3z=3 found at zero temperature. Our results are based on a simple Eliashberg-type approach, which gives rise to a boson self-energy proportional to $\mathrm{\ensuremath{\Omega}}/\ensuremath{\gamma}(T)$ at small momenta, where $\ensuremath{\gamma}(T)$ is the temperature dependent fermion scattering rate. These findings might shed some light on recent Monte Carlo simulations at finite temperature, where results consistent with z=2z=2z=2 were found.
The validity of modified finite-size scaling above the upper critical dimension is demonstrated for the quantum phase transition whose dynamical critical exponent is z=2. We consider the N -component Bose-Hubbard model, which is exactly solvable and exhibits mean-field type critical phenomena in the large- N limit. The modified finite-size scaling holds exactly in that limit. However, the usual procedure, taking the large system-size limit with fixed temperature, does not lead to the expected (and correct) mean-field critical behavior because of the limited range of applicability of the finite-size scaling form. By quantum Monte Carlo simulation, it is shown that the modified finite-size scaling holds in the case of N=1.