Conductivity of hard core bosons: A paradigm of a bad metal

Two-dimensional hard core bosons suffer strong scattering in the high-temperature resistive state at half filling. The dynamical conductivity $\sigma \left(\omega \right)$ is calculated using nonperturbative tools such as continued fractions, series expansions, and exact diagonalization. We find a large temperature range with linearly increasing resistivity and broad dynamical conductivity, signaling a breakdown of Boltzmann-Drude quasiparticle transport theory. At zero temperature, a high-frequency peak in $\sigma \left(\omega \right)$ appears above a “Higgs mass” gap and corresponds to order-parameter magnitude fluctuations. We discuss the apparent similarity between conductivity of hard core bosons and phenomenological characteristics of cuprates, including the universal scaling of Homes et al. [Nature (London) 430, 539 (2004)].

Figure 1

(Color online) Temperature-dependent resistivity of two-dimensional hard core bosons at half filling. High-temperature line (blue online) is calculated up to order $1/T$, with error margins depicted by dashed lines. The critical region above the BKT transition is given by the vortex plasma theory of HN. For illustration, a layered system with weak interlayer coupling shows a rapid rise in resistivity above the three-dimensional transition temperature $T_c^{3\text{D}}$, as depicted by a solid black line.

Figure 2

(Color online) Infinite temperature current fluctuations function $G_{\infty }^{″}\left(\omega \right)$. Solid lines (blue online, indicated by vho) depict the results of the variational harmonic-oscillator expansion and (green online, indicated by extr) depicts the result of the recurrents extrpolation, shown in Fig. 3. Circles (red online, indicated by ed) depict the exact diagonalization result computed on a $4\times 4$ lattice. Inset: convergence of the variational harmonic-oscillator expansion. $\left|\Delta {\tilde{G}}^{″}\right|$ are the distances between consecutive approximants and $n_{max}$ is the number of computed moments.

Figure 3

(Color online) The recurrents of $G_{\infty }^{″}\left(\omega \right)$ at infinite temperature and their extrapolation to large index $n$.

Figure 4

(Color online) Dynamical conductivity in the high-temperature resistive phase. The function can be fit by Eq. (73).

Figure 5

(Color online) The temperature-dependent kinetic energy $⟨-K_x⟩$ (solid, red online), calculated by Refs. 5, 6, 43 and the sum rule for the high-temperature expansion of the dynamical conductivity of Fig. 4 (solid, blue online). The vertical dashed line marks the superconducting phase below $T_{\text{BKT}}$.

Figure 6

Low-order current fluctuations function in the relativistic Gross-Pitaevskii field theory, as calculated by Eq. (84). Dashed line describes gapped magnitude fluctuations. Solid line describes gapless phase fluctuations.

Figure 7

(Color online) The recurrents of $G_{T=0}^{″}\left(\omega \right)$ at zero temperature. Dashed line emphasizes the different behavior of even and odd recurrents, which is related to the gap structure in Fig. 8.

Figure 8

(Color online) Zero-temperature dynamical conductivity of HCB. The Higgs mass gap $m$ is depicted. Solid line (blue online) is determined by variationally fitting the recurrents to those in Fig. 7. Dashed line (red online) is computed by exact diagonalization of a 20-site cluster, with $\delta$-function broadening of $0.5J$.

Figure 9

(Color online) Comparison of Homes law of HCB and cuprate superconductors. HCB calculation is depicted by the dashed line (blue online). Cuprate data, compiled from Refs. 24, 63, describe the two-dimensional critical conductivity ${\sigma }_c$, as measured by optical conductivity exprapolated to zero frequency, at the transition temperature $T_c$. ${\rho }_s$ values are compiled from measured plasma frequencies by Eq. (100). $R_Q=6453\Omega$ is the boson quantum of resistance. Different data points of the same symbol correspond to different doping concentrations of the same compound.