Extended quantum criticality of low-dimensional superconductors near a spin-density-wave instability

We use the renormalization group method to study normal-state properties of quasi-one-dimensional superconductors nearby a spin-density-wave instability. On the basis of one-loop scattering amplitudes for the quasi-one-dimensional electron gas, the integration of the renormalization group equations for the two-loop single-particle Matsubara self-energy leads to a non-Fermi-liquid temperature downturn of the momentum-resolved quasiparticle weight over most part of the Fermi surface. The amplitude of the downturn correlates with the entire instability line for superconductivity, defining an extended quantum critical region of the phase diagram as a function of nesting deviations of the Fermi surface. One also extracts the downward renormalization of interchain hopping amplitudes at arbitrary low temperature in the normal phase. By means of analytical continuation of the Matsubara self-energy, one-particle spectral functions are obtained with respect to both energy and temperature and their anomalous features analyzed in connection with the sequence of instability lines of the phase diagram. The quasiparticle scattering rate is found to develop an unusual temperature dependence, which is best described by the superimposition of a linear and quadratic $T$ dependencies. The non-Fermi-liquid linear-$T$ component correlates with the temperature scale $T_c$ of the superconducting instability over an extended range of nesting deviations, whereas its anisotropy along the Fermi surface is predicted to parallel the momentum profile of a $d$-wave pairing gap on the Fermi surface. We examine the implications of our results for low-dimensional unconventional superconductors, in particular, the Bechgaard salts series of quasi-one-dimensional organic conductors, but also the pnictide and cuprate superconductors where several common features are observed.

Figure 1

The Fermi surface of the quasi-one-dimensional electron gas model. Each dashed line ends to a patch $p_{i=1\cdots 32}$ on each Fermi sheet.

Figure 2

Flow equations for (a) the $g_{1,2}$ (open square) and umklapp $g_3$ (full square) scattering amplitudes for right (continuous line) and left (dashed line) moving electrons, and (b) the one-particle Matsubara self-energy ${\Sigma }_{+}$. The crossed and slashed lines refer to the high-energy intervals ${\Delta }_{\pm }$ and the last outer energy shell, respectively (permutations between crossed and slashed lines are not shown).

Figure 3

Momentum dependence of the projected coupling constants $g_1(k_{\perp }^{\prime },-k_{\perp }^{\prime };-k_{\perp },k_{\perp })+g_2(k_{\perp }^{\prime },-k_{\perp }^{\prime };k_{\perp },-k_{\perp })$ and $g_3(k_{\perp }^{\prime },-k_{\perp }^{\prime };k_{\perp },-k_{\perp })$ in the $(k_{\perp },k_{\perp }^{\prime })$ plane for the normal phase ($T=25$ K) of the SDW instability ($t_{\perp }^{\prime }=25$ K).

Figure 4

(a) Calculated phase diagram of the quasi-one-dimensional electron gas model from the renormalization group method at the one-loop level; $T_{\mathit{CW}}$ and $\Theta$ define the temperature region. The CW behavior of the inverse normalized SDW response function shown in (b) for different $t_{\perp }^{\prime }$.

Figure 5

Low-temperature momentum dependence of the coupling constants $g_1(k_{\perp }^{\prime },-k_{\perp }^{\prime };-k_{\perp },k_{\perp })+g_2(k_{\perp }^{\prime },-k_{\perp }^{\prime };k_{\perp },-k_{\perp })$ (left) and $g_3(k_{\perp }^{\prime },-k_{\perp }^{\prime };k_{\perp },-k_{\perp })$ (right) in the $(k_{\perp },k_{\perp }^{\prime })$ plane for the normal phase of the SCd instability ($t_{\perp }^{\prime }=29$ K); $T=15$ (top), 8 (middle), and 4 K (bottom).

Figure 6

Variation of the quasiparticle weight on the Fermi surface as a function of $k_{\perp }$ at different temperatures. (a) SDW ($t_{\perp }^{\prime }=25\text{K}<t_{\perp }^{\prime *},T_{\mathrm{SDW}}\simeq 12\text{K}$), and (b) SCd ($t_{\perp }^{\prime }=26.8\text{K}>t_{\perp }^{\prime *},T_c=0.8\text{K}$).

Figure 7

quasiparticle weight $z\left(k_{\perp }\right)$ as a function of temperature (logarithmic scale) and antinesting parameter $t_{\perp }^{\prime }$ for $k_{\perp }=0$, $\pi /4$, and $\pi /2$. The dashed line corresponds to the power law decay $z_{1\mathrm{D}}$ of the one-dimensional limit.

Figure 8

Low-temperature dependence of the quasiparticle weight $z\left(k_{\perp }\right)$ for different antinesting parameters $t_{\perp }^{\prime }$ for $k_{\perp }=0$ (full circles), and nodal point $\pi /2$ (open circles). The continuous lines correspond to a fit to Eq. (18) of the marginal liquid theory at $k_{\perp }=0$.

Figure 9

Renormalized interchain hopping ${\overline{t}}_{\perp }$ and antinesting term ${\overline{t}}_{\perp }^{\prime }$ by self-energy corrections as a function of temperature and for different bare values of $t_{\perp }^{\prime }$. The dashed dot line corresponds to the result of extended scaling hypothesis for the renormalization of $t_{\perp }$.

Figure 10

Typical frequency dependence of the real (${\Sigma }^{\prime }$) and imaginary (${\Sigma }^{\prime \prime }$) parts of the one-particle self-energy at different temperatures in the SDW ($t_{\perp }^{\prime }=25$ K, $T=20,100,300$ K, left) and SC ($t_{\perp }^{\prime }=26.8$ K, $T=5,100,300$ K, right) parts of the phase diagram in Fig. 4.

Figure 11

Low-frequency dependence of the real (${\Sigma }_{+}^{\prime }$) and imaginary (${\Sigma }_{+}^{\prime \prime }$) parts of the one-particle self-energy at different $t_{\perp }^{\prime }$ and regions of the Fermi surface.

Figure 12

Spectral weight at $k_{\perp }=0$ on the Fermi surface vs frequency (log-log scale) in the metallic domain of the SDW ($t_{\perp }^{\prime }=25$ K) and SCd ($t_{\perp }^{\prime }=26.8$ K) sectors of the phase diagram.

Figure 13

Low-frequency dependence of the single-particle spectral weight at different $t_{\perp }^{\prime }$ and regions of the Fermi surface.

Figure 14

(a) Temperature dependence of the electron-electron scattering rate at $k_{\perp }=0$ for different $t_{\perp }^{\prime }$ in the SDW and SCd regions of the phase diagram and (b) Curie-Weiss temperature region in the SCd domain where the continuous lines correspond to a fit to Eq. (25).

Figure 15

Momentum dependence of the quasiparticle scattering rate on the Fermi surface in the SDW($t_{\perp }^{\prime }=25$ K) and SCd ($t_{\perp }^{\prime }=26.8$ and 27.7 K).

Figure 16

(a) Temperature dependence of the electron-electron scattering rate averaged on the Fermi surface for different $t_{\perp }^{\prime }$ in the superconducting sector of the phase diagram and (b) polynomial fit (continuous lines) of the temperature dependence of the mean scattering rate (26).

Figure 17

Evolution of the linear (a) and quadratic (b) coefficients of the scattering rate as a function of the ratio $T_c/T_c^{*}$ for $k_{\perp }=0$ (open circles) and $k_{\perp }=5\pi /8$ (crosses). The full circles correspond to the averages $a$ and $b$ over the Fermi surface (26).

Figure 18

Not to scale variation of the $d$-wave superconducting gap modulus $\left|\Delta \right(k_{\perp }\left)\right|$ along the $+k_F$ Fermi surface sheet (top) and $T$-linear coefficient anisotropy of the quasiparticle scattering rate along the Fermi surface and at different $t_{\perp }^{\prime }$ (bottom).

Figure 19

Outer shell contribution to the one-particle self-energy of the $p=+$ branch for normal ($\delta {\Sigma }_{+}^{(1,2)}$) and umklapp ($\delta {\Sigma }_{+}^{\left(3\right)}$) scattering processes. $n$ and $m$ refer to the energy indexes in the cascades of contraction and $N$ to the index of the running energy shell at $E_N=E_0\left(\ell \right)$.