Breakdown of the periodic potential ansatz in correlated electron systems

Abstract

Our electronic structure theory for crystalline solids is commonly built on the periodic potential assumption $V(\mathbf r)=V(\mathbf r+\mathbf R)$ for every lattice translation $\mathbf R$, enabling Bloch eigenstates, crystal momentum as a good quantum number, and the standard quasiparticle-based description of the behavior of metals. Because the zero-point motion of the ions, however, in correlated electron systems the electronic environment experienced by an itinerant electron is neither static nor self-averaging at the single-particle level, even in perfectly stoichiometric crystals, leading to a distribution of local Kondo scales that spans two orders of magnitude in temperature. We discuss, through a comparison between uniform scenarios and one that breaks with perfect lattice translational symmetry, how incorporating this distribution yields a unified description for all heavy-fermion systems at the quantum critical point.

Figures (5)

• Figure 1: Figure adapted from Ref. Bretana2023. The instantaneous distribution of Kondo energy scales for stoichiometric URu$_2$Si$_2$ (solid line, left axis) and corresponding fraction of unshielded moments (dotted curve, right axis) calculated from eqn \ref{['TKspread']}. The three parameters used were an average $T_K=70$ K estimated from the resistivity coherence temperature, a bandwidth $D=0.4$ eV from band-structure estimates Elgazzar2009, and a zero-point displacement of $0.006$ nm obtained from EXAFS experiments Dudschus2019.
• Figure 2: dM/dT data (filled symbols) for YbRh$_2$Si$_2$ digitized from Tokiwa et al.Tokiwa2010 measured for B= 0.06T applied in the easy plane, corresponding to the critical field $B_c$. The solid curve is the prediction for a collection of superspins at the percolation threshold for a 2-level system with $E= \pm \mu_{\text{eff}}\mu_B B$ for $\mu_{\text{eff}}$= 1.74 with clusters of size 2 having their AF simulated superspin replaced by that corresponding to an ordering wave vector $\bold{Q}$= (0.14,0.14,0) Stock2012, the dotted curve is for clusters upto size 5 having their AF moments corrected. Note that no vertical or horizontal adjustment factors were applied in the comparison between simulation and experiment. Had the simulated clusters not acquired a superspin, then the simulated curves would have been raised by a factor $\sim$20.
• Figure 3: Figure collated and adapted from Ref. Bretana2023. Panel (A) displays the measured susceptibility for Ce(Ru$_{0.24}$Fe$_{0.76}$)$_2$Ge$_2$Montfrooij2019, panel (B) for UCu$_4$Pd at $B$= 0.5 T (open symbols)Bernal1995 and at 0.01 T (filled symbols) Vollmer2000, panel (C) for YbRh$_2$Si$_2$Gegenwart2006 at $B$= 0.025 T. The solid blue curves are the prediction of eqn. \ref{['main']} for the contribution of the infinite cluster using the measured entropy $S$ with $\mu_{\text{eff}}$= 1.43, 0.80, and 1.74 $\mu_B$, respectively. The red dashed-dotted curves in panels (A), (B), and (C) are the response of the superspin collective (evaluated at the percolation threshold) added to eqn. \ref{['main']}, using the same effective moments. The range in panels (A) and (C) arises from the fact that the simulations were done for an AF-arrangement (see Ref. Bretana2023 for further details). We opted for a slightly different plotting in panel (B) to highlight the perfect agreement for $T >$ 3 K between the measured entropy and the measured susceptibility (scaled using the susceptibility of a free moment). Panel (D): The field dependence of the maxima in $c/T$ (closed circles) and ac-susceptibility (open circles) Custers2010. The lines denote the maxima for a simulated distribution of clusters at $p_c$ for a ground state doublet ($\mu_{\text{eff}}$ = 1.74$\mu_B$), with the hatched areas indicative of the sensitivity Bretana2023 of the predicted maxima to the ordering wave vector (dotted curve: AF ordering; solid curves: replacing the AF-zero moment of clusters of size 2 with the average moment calculated for Q = (0.14, 0.14, 0); dashed-dotted curves: replacing clusters up to size five.
• Figure 4: The temperature dependence of the Hall coefficient $R_H$ calculated for a system where a fraction $f$ of 28% of the local moments remain unshielded at the percolation threshold. The lines have been calculated for T=[29 mK, 75 mK, 0.2 K, 0.3 K, 0.5K] (from left to right), using the same $\mu_{\text{eff}}$ as in Fig. \ref{['ybrhsi']}. For a 2-level system, $R_H$ is given by $R_H=R_0(1-f)B+a_0 f\mu_{\text{eff}}$tanh$(\mu_{\text{eff}} \mu_B B/k_BT)$, where we have used the $R_0$ value from Ref. Paschen2004 and taken $a_0=R_0$ for lack of a robust estimate.
• Figure 5: An updated sketch of the Doniach phasediagram. The vertical axis is the temperature of the system, the horizontal axis is a measure of both the average coupling $<J>$ between the conduction electrons and the magnetic moments as well as the average fraction $p$ of magnetic moments still present at any moment in time. The filled circle on the $T$= 0 axis corresponds to the state of the system where the isolated clusters can no longer line up with each other at any temperature. The shapes in the cube indicate the isolated, ordered clusters (in blue), as well as the lattice spanning cluster (in red). The open circle on the $T$= 0 axis indicates the percolation threshold where the infinite cluster fragments into smaller clusters. For higher occupancies, the infinite cluster survives and its moments will line up below a $p$-dependent transition temperature $T_N$, producing long range order. Above $T_N$ the moments on the infinite cluster remain disordered. For lower occupancies, the infinite cluster breaks up into smaller clusters that will acquire a superspin. Under the dome marked 'cluster glass', the long-range part of the RKKY interaction aligns their superspins (see text). This dome can be destroyed with the application of a magnetic field or through chemical pressure. The dotted arrow indicates where YbRh$_2$Si$_2$ is (likely) located. Note that there exists a range of coupling strength, for $p > p_c$, where we can expect to see two magnetic transitions on cooling, corresponding to ordering on the infinite cluster and to a superspin alignment transition.