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

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.

Figure 1

Coupling of bosonic fluctuations and fermions in a finite system. The figure depicts the reciprocal space of a typical square lattice. The solid black line is the Fermi surface that would exist in an infinite system. Black (gray) dots are the filled (empty) states in $\mathrm{k}$ space. A bosonic fluctuation of wave vector $\mathrm{q}$ (blue arrows) can couple to fermions by exciting an electron-hole pair. In an infinite system, this coupling can occur at any point on the Fermi surface. However, in a finite system, the filled (empty) states actually appear as a series of terraces (dashed lines). As a result, for small enough $q$, there is a region of the Fermi surface where excitations cannot occur (shaded region). The right panel depicts an approximation to the left panel, replacing the lattice with the electron/hole continuum with a gap in momentum space with the width $q_1=\pi /L$, as described in Eqs. (7) and (8).

Figure 2

Dynamic polarization bubble $\mathrm{\Pi }(q,{\mathrm{\Omega }}_m)$. In an infinite system, $\mathrm{\Pi }(q,{\mathrm{\Omega }}_m)=\gamma [1-|{\mathrm{\Omega }}_m|/\sqrt{{\left(v_{F}q\right)}^2+{\mathrm{\Omega }}_m^2}]$, i.e., the slope of $1-\mathrm{\Pi }(q,{\mathrm{\Omega }}_m)/\gamma$ scales as $1/q$. In a finite system, this behavior is modified because $\mathrm{\Pi }(q,{\mathrm{\Omega }}_m)$ vanishes at $q_1=\pi /L$. (a) A depiction of $1-\mathrm{\Pi }(q,{\mathrm{\Omega }}_m)/\gamma$ vs ${\mathrm{\Omega }}_m$ at different $q$ in a finite square lattice of size $L=20a$ (solid lines) and an infinite system (dashed lines). For the finite system there is a large intermediate region (dotted ellipse), where the slope of $\mathrm{\Pi }$ appears roughly independent of $q$. (b) Frequency variation of $\delta \mathrm{\Pi }=\mathrm{\Pi }(q,{\mathrm{\Omega }}_m)-\mathrm{\Pi }(q,0)$ in our model vs QMC data from Ref. [17] (up to a constant shift [27]). (c) Phenomenological form of $\delta \mathrm{\Pi }(q,{\mathrm{\Omega }}_m)$, Eq. (2), vs QMC data [27].

Figure 3

Self-energy of a finite system. In an infinite system, the self-energy has a nonanalytic behavior at low temperatures, $\mathrm{\Sigma }\left(\omega \right)\sim {\omega }^{2/3}$. In a finite system, the nonanalytic behavior is cut off at a scale of $\omega \sim v_F{q}_1$. The left panel depicts $\mathrm{\Sigma }\left(\omega \right)$ for a system of size $L=20a$. The dashed gray lines are guides to the eye of $\omega ,{\omega }^{2/3}$. Note how the FL-type behavior extends over a numerically large region near $v_F{q}_1$. The right panel depicts the quasiparticle residue as obtained from Eq. (16) [27].