Lattice potentials and fermions in holographic non Fermi-liquids: hybridizing local quantum criticality

TL;DR

The paper investigates how a periodic lattice modulates strongly coupled fermions at finite density in a holographic RN-AdS setup. Using a weak-potential perturbative expansion, it reveals two intertwined effects: standard band-gap formation for background domain-wall fermions and, crucially, a novel hybridization of local quantum critical CFT$_1$ propagators in the AdS$_2$ IR that yields momentum-dependent scaling exponents displaced by Umklapp vectors. This leads to crossovers between high-energy AdS$_2$ scaling and low-energy Brillouin-zone-specific exponents, and implies that quasi-Fermi-surface metals cannot be localized into band insulators while AdS$_2$-metal regimes exhibit a lattice-tuned algebraic pseudo-gap. The results offer a distinctive, experimentally testable fingerprint for AdS$_2$-like pseudogap metals and connect holographic non-Fermi liquid behavior to lattice phenomena observed in strongly correlated materials, with clear avenues for extending the analysis to more general lattices and backreacted geometries.

Abstract

We study lattice effects in strongly coupled systems of fermions at a finite density described by a holographic dual consisting of fermions in Anti-de-Sitter space in the presence of a Reissner-Nordstrom black hole. The lattice effect is encoded by a periodic modulation of the chemical potential with a wavelength of order of the intrinsic length scales of the system. This corresponds with a highly complicated "band structure" problem in AdS, which we only manage to solve in the weak potential limit. The "domain wall" fermions in AdS encoding for the Fermi surfaces in the boundary field theory diffract as usually against the periodic lattice, giving rise to band gaps. However, the deep infrared of the field theory as encoded by the near horizon AdS2 geometry in the bulk reacts in a surprising way to the weak potential. The hybridization of the fermions bulk dualizes into a linear combination of CFT1 "local quantum critical" propagators in the bulk, characterized by momentum dependent exponents displaced by lattice Umklapp vectors. This has the consequence that the metals showing quasi-Fermi surfaces cannot be localized in band insulators. In the AdS2 metal regime, where the conformal dimension of the fermionic operator is large and no Fermi surfaces are present at low T/μ, the lattice gives rise to a characteristic dependence of the energy scaling as a function of momentum. We predict crossovers from a high energy standard momentum AdS2 scaling to a low energy regime where exponents found associated with momenta "backscattered" to a lower Brillioun zone in the extended zone scheme. We comment on how these findings can be used as a unique fingerprint for the detection of AdS2 like "pseudogap metals" in the laboratory.

Figures (5)

• Figure 1: A cartoon of our results of the band structure for different $k_F$. The system under consideration has a lattice structure only in $x$ direction. The red line curve is the Fermi surface ($\omega=0$) and the black dashed line is the first BZ boundary. The left plot is for $k_F>\frac{K}{2}$ and the right plot is for $k_F=\frac{K}{2}$. We have a band gap at the first Brillouin Zone boundary $k_x=\pm\frac{K}{2}$ (the black dashed line) for generic $k_F>\frac{K}{2}$ (i.e. $k_y\neq 0$) which will close when $k_F=\frac{K}{2}$ (i.e. $k_y= 0$) . At generic $k_x$, the self-energy receives a second order correction related to the lattice effect, i.e. $\Sigma=\alpha_{\vec{k}}\, \omega^{2 \nu_{\vec{k}}} + \beta^{(-)}_{\vec{k}}\omega^{2 \nu_{\vec{k}-\vec{K}}}+\beta^{(0)}_{\vec{k}}\omega^{2 \nu_{\vec{k}}}\ln\omega+\beta^{(+)}_{\vec{k}}\omega^{2 \nu_{\vec{k} +\vec{K}}} +\ldots$. (See Eq. \ref{['CFT1generalSelf']}). Note that this picture is a result in the periodic Brillouin zone scheme.
• Figure 2: The behavior of the different powers of $\omega^{2\nu_{k-K}},\omega^{2\nu_k},\omega^{2\nu_{k+K}}$ in the lattice AdS${}_2$-metal spectral function as a function of $k$. The IR of the Green's function is controlled by the lowest branch: $\omega^{2\nu_{k+K}}$ in the $\ell=-1$ Brillioun zone, $\omega^{2\nu_{k}}$ in the $\ell=0$ Brillioun zone, and $\omega^{2\nu_{k-K}}$ in the $\ell=1$ Brillioun zone.
• Figure 3: This sequence of AdS$_2$ metal spectral functions for a fixed generic $k$ shows how the Umklapp contribution takes over at low frequencies. For $\ell=0$, inside the 1st BZ, we show the full corrected spectral function $A_{\text{full}}(\omega,\vec{k})\sim \text{Im} G$ (red), the original "bare" holographic spectral function $A_{\text{pure AdS}}(\omega,\vec{k})\sim \text{Im} {G}_0$ (brown), and the Umklapp correction due to the periodic chemical potential modulation $\delta A_{\text{lattice}}(\omega,\vec{k})\sim \text{Im} \delta G$ (black).
• Figure 4: The behavior of the AdS$_2$ metal. The spectral function in three distinct regimes: the first Brillion zone, the Brillioun edge and higher Brillioun zones. In each case the full spectral function $\text{Im}{G}(\omega, \vec{k}) =\text{Im} {G}_0+\text{Im} \delta{G}$ is plotted in red, the "bare" component $G_0$ in brown and the Umklapp contribution, $\text{Im} \delta{G}$ in black. The frequency scale is chosen such that the contributions are comparable by zooming as in Fig. \ref{['cartoon5']}. Below the ratio of the full spectral function in units of the "bare" spectral function is plotted for the full range of frequencies. This shows the excess states at low frequency that appear due to the lattice.
• Figure 5: A cartoon of the dispersion driven pseudogap: standard band gap (black)vs a pseudogap driven by self-energy corrections that persist to the lowest frequencies (red), i.e. the spectral function is only singular at one point right at the chemical potential.