Colloquium: Theory of intertwined orders in high temperature superconductors

The electronic phase diagrams of many highly correlated systems, and, in particular, the cuprate high temperature superconductors, are complex, with many different phases appearing with similar (sometimes identical) ordering temperatures even as material properties, such as dopant concentration, are varied over wide ranges. This complexity is sometimes referred to as “competing orders.” However, since the relation is intimate, and can even lead to the existence of new phases of matter such as the putative “pair-density wave,” the general relation is better thought of in terms of “intertwined orders.” Some of the experiments in the cuprates which suggest that essential aspects of the physics are reflected in the intertwining of multiple orders, not just in the nature of each order by itself, are selectively analyzed. Several theoretical ideas concerning the origin and implications of this complexity are also summarized and critiqued.

Figure 1

Schematic phase diagram (temperature $T$ vs doping $x$) for an array of weakly coupled two-leg Hubbard ladders. The long-dashed curve is the spin gap, which vanishes at $x_c\approx 0.3$, and which also indicates a crossover temperature to a LE liquid regime. The short-dashed line indicates a crossover scale to a higher-dimensional Fermi liquid (FL) regime. $x_P\approx 0.1$ is the point at which $K_c=1$ and $P$ is the corresponding tetra-critical point discussed in the text. $C$ and $S$ are quantum critical points, between which SC and CDW orders coexist. We have shown $Q$ as a trictitical point, below which the transition (dark short-dashed line) between the CDW and the FL becomes first order, although there are other possibilities here. For details, see text.

Figure 2

Scales and phases of ${\mathrm{La}}_{2-x}{\mathrm{Ba}}_x{\mathrm{CuO}}_4$ near $x=1/8$: PDW fluctuations begin below the onset of charge order $T_{\mathrm{CDW}}$ and become pronounced below $T_{\mathrm{SDW}}$. The presumably PDW 2D SC phase occurs below $T_{\mathrm{KT}}$. The regime between $T_{3\mathrm{D}}$ and $T_c$ is expected to be an $XY$ glass, and the 3D $d$-wave SC lies below $T_c$. See text for details.

Figure 3

Schematic diagram summarizing doping dependence of gaps and characteristic temperatures. Above optimal doping, the experimental identification of the pseudogap can differ depending on whether one is looking at measurements in the normal or superconducting state; the behavior shown is consistent temperature-dependent tunneling measurements. From [73].

Figure 4

Schematic diagram summarizing ARPES measurements of gaps around the Fermi surface in underdoped cuprates. The combined gray regions indicate the $d$-wave gap, with magnitude ${\mathrm{\Delta }}_0$, observed at $T<T_{\mathrm{SC}}$. At $T_{\mathrm{SC}}$, the dark gray part of the gap closes, leaving a gapless arc of states in the near-nodal region and a pseudogap (light gray) in the antinodal region. The pseudogap loses definition at $T\sim T^{*}$. The energy scale for coherent pairing ${\mathrm{\Delta }}_c$ indicated by tunneling empirically corresponds to the magnitude of the $d$-wave gap at the wave vector corresponding to the end of the nodal arc.

Figure 5

Sketch of the phase diagram with SC and CDW orders in quasi-1D strongly correlated systems with a spin gap ${\mathrm{\Delta }}_s$. $P$ is a multicritical point.

Figure 6

Qualitative phase diagram for the melting of a unidirectional PDW state coexisting with $d$-wave SC order. See text for details. ${\mathrm{La}}_{2-x}{\mathrm{Ba}}_x{\mathrm{CuO}}_4$ is presumably along the dark broken vertical line.