Fermi Arcs in the Superconducting Clustered State for Underdoped Cuprate Superconductors

The one-particle spectral function of a state formed by superconducting (SC) clusters is studied via Monte Carlo techniques. The clusters have similar SC amplitudes but randomly distributed phases. This state is stabilized by competition with the antiferromagnetism expected to be present in the cuprates and after quenched disorder is introduced. A Fermi surface composed of disconnected segments, i.e., Fermi arcs, is observed between the critical temperature $T_c$ and the cluster formation temperature scale $T^{*}$.

Figure 1

(a) Schematic $T$ vs doping ($x$) phase diagram [8] for the competition AF vs SC: $D=0$ ($D\ne 0$) is the clean (dirty) limit. The vertical green (gray) line is the $T$-range emphasized here: at $T_c$, long-range order develops, and at $T^{*}$ SC clusters (short-range order) are formed. In practice, the $\mathrm{AF}\to \mathrm{SC}$ transition is reached in Eq. (1) by decreasing ${\rho }_{\mathrm{SC}}$ from zero, favoring SC, and decreasing ${\rho }_{\mathrm{AF}}>0$, reducing AF, for example, by fixing the relation ${\rho }_{\mathrm{AF}}=1+{\rho }_{\mathrm{SC}}$ [8]. The disorder in the couplings is correlated with power $\alpha =0.8$ [17]. (b) SC correlation (SS) vs $T$ at short [vectorial distance (3,0)] and long [vectorial distance (16,16)] distance, in the clean and dirty limits. Results are from MC simulations on a $32\times 32$ cluster with periodic boundary conditions, using 2000 (3000) thermalization (measurement) steps, starting with a random initial configuration. ${\rho }_{\mathrm{SC}}$ was taken from a bimodal distribution with values $-1.1$ and $-0.1$ in the dirty limit, and uniformly $-1.1$ in the clean limit. SS is defined as $SS(\mathbf{i})=\frac{1}{N_{\mathrm{sites}}}\sum _{\mathbf{j}}|{\Delta }_{\mathbf{j}}||{\Delta }_{\mathbf{j}+\mathbf{i}}|\cos ({\varphi }_{\mathbf{j}}-{\varphi }_{\mathbf{j}+\mathbf{i}})$. Some results gathered on $64\times 64$ lattices show that size effects are not important. (c) Classical SC order parameter $\Delta$ for a typical MC-equilibrated configuration at several $T$’s (0.2, $1.0\sim T_c$, 1.5 and $2.0\sim T^{*}$). Color intensity (shades of gray) represents $⟨\Delta ⟩$ averaged over short MC times, and the actual colors (shades of gray) represent the SC angle $\varphi$ at each site (see wheel). The maximum value of $|\Delta |$ for the temperatures studied was $\sim 2.0$. The bimodal couplings configuration is also shown (d): blue (dark gray) is where SC dominates (${\rho }_{\mathrm{SC}}=-1.1$); red (light gray) where AF dominates (${\rho }_{\mathrm{SC}}=-0.1$).

Figure 2

(a) $A(\mathbf{k},\omega )$ for equally spaced $\mathbf{k}$’s along the direction from ($\pi /2$, $\pi /2$) (bottom) to (0, $\pi$) (top). The classical LG configurations used are obtained as described in Fig. 1 but for the case ${\rho }_{\mathrm{SC}}=-0.8$ (0) in the SC (AF) regions, using ${\rho }_{\mathrm{AF}}=1+{\rho }_{\mathrm{SC}}$, on a $32\times 32$ lattice. The $T$’s are, from left to right, $T=0.1$, 0.4, 1.0, and 2.0. In this case, $T_c\sim 0.4$, and $T^{*}\sim 1.2–1.5$. (b) $A(\mathbf{k},\omega =0)$ in the $k_x-k_y$ plane for the same parameters and temperatures as in (a). Results shown are those within a window $\Delta ϵ=0.1$ from the Fermi level. (c) SC gap (distance between peaks) vs momentum for the same parameters as in (a) but with ${\rho }_{\mathrm{SC}}=-1.1$ or $-0.1$ (bimodal distribution), at the $T$’s indicated. (d) $A(\mathbf{k},\omega =0)$ from (0, $\pi /2$) to ($\pi /2$, $\pi$), at $T=1.0$ and 0.1, and parameters as in (a).

Figure 3

(a) Length of the Fermi arc (as a % of the maximum length) vs $T$ for the case described in Fig. 2a. The Fermi arc was defined to exist at a certain momentum $\mathbf{k}$ when its intensity was within 35% of the maximum intensity. Other definitions lead to a similarly linear relation but with different $T\to 0$ limits. The energy cutoff $\Delta ϵ=0.1$ prevent the size of the nodes from being exactly zero in the low-$T$ state. (b) Schematic representation of the toy model configuration (see text). ${\varphi }_1$ and ${\varphi }_2$ refer to the SC order parameter phases in $4\times 4$ squares. (c) LDOS for the example shown in (b) with ${\varphi }_1=0$ and ${\varphi }_2=\pi$. The red (solid) line corresponds to a site at the center, and the blue (dashed) line to a site on the (border) of the $4\times 4$ square. The parameters used are the same at each site: $|\Delta |=1$, $V=0.25$, and $J=2$. (d) $A(\mathbf{k},\omega )$ for $\mathbf{k}$ along the direction ($\pi /2$, $\pi /2$) to (0, $\pi$) for the case shown in (b) with ${\varphi }_1={\varphi }_2=0$ (i.e., perfect $d$-wave superconductor). (e) Same as (d), but with ${\varphi }_1$ and ${\varphi }_2$ randomly chosen between 0 and $\pi$. (f) $A(\mathbf{k},\omega =0)$ in the $k_x-k_y$ plane for case (e) ($\Delta ϵ=0.1$).

Figure 4

(a) Schematic representation of a Josephson-junction-like structure. Top: two SC regions with order parameters ${\Delta }_1$ and ${\Delta }_2$, separated by a $\Delta =0$ area of width $w$ ( gray). Bottom: expected gap interpolation for ${\Delta }_1=-{\Delta }_2$. (b) LDOS on a $32\times 32$ lattice containing 4 equally-spaced $12\times 12$ SC clusters separated by $w=4$. The order parameters are staggered between clusters using ${\Delta }_1=1$ and ${\Delta }_2=-1$, with $V=0.25$ and without AF ($J=0$). Shown are results from the center of one SC region to a nearest neighbor. (c) $A(\mathbf{k},\omega =0)$ in the $k_x-k_y$ plane for ${\Delta }_1=-{\Delta }_2=1$ and $w=2$, with a setup as described in (b) but using $14\times 14$ SC clusters.