Tendencies of enhanced electronic nematicity in the Hubbard model and a comparison with Raman scattering on high-temperature superconductors

The pseudogap regime of the cuprate high-temperature superconductors is characterized by a variety of competing orders, the nature of which are still widely debated. Recent experiments have provided evidence for electron nematic order, in which the electron fluid breaks rotational symmetry while preserving translational invariance. Raman spectroscopy, with its ability to symmetry resolve low energy excitations, is a unique tool that can be used to assess nematic fluctuations and nematic ordering tendencies. Here we compare results from determinant quantum Monte Carlo simulations of the Hubbard model to experimental results from Raman spectroscopy in ${\mathrm{La}}_{2-x}{\mathrm{Sr}}_x{\mathrm{CuO}}_4$, which show a prominent increase in the $B_{1g}$ response around 10% hole doping as the temperature decreases, indicative of a rise in nematic fluctuations at low energy. Our results support a picture of nematic fluctuations with $B_{1g}$ symmetry occurring in underdoped cuprates, which may arise from melted stripes at elevated temperatures.

Figure 1

Real part ${\chi }_{B_{1g}}^{\prime }\left(T\right)$ of the static Raman susceptibility in $B_{1g}$ symmetry as a function of temperature for various doping levels $x$. The doping ranges below and above $x=0.1$ are shown separately for clarity in (a) and (b), respectively. A clear Curie-like increase of ${\chi }_{B_{1g}}^{\prime }\left(T\right)$ at low temperatures can only be resolved for $x=0.08$, 0.1, and 0.12.

Figure 2

Continuous color plots of the susceptibility ${\chi }^{\prime }(x,T)$. (a) In $B_{1g}$ symmetry ${\chi }^{\prime }\left(T\right)$ increases in the region around $x=0.1$. (b) In comparison, the $B_{2g}$ susceptibility is weakly temperature dependent at the onset of superconductivity and remains flat otherwise.

Figure 3

Nematic/Raman susceptibility calculated using DQMC as a function of temperatures for various doping levels for (a) $B_{1g}$ symmetry and (b) $B_{2g}$ symmetry. Both susceptibility increases with decreasing temperature. For $B_{1g}$, the rate of increase for different doping becomes different as temperature is lowered through the point indicated by the arrow, which is near $T=2t/3=J$ (the data point to the right of the arrow), the magnetic exchange. The $B_{2g}$ susceptibility does not show similar behavior, and is gapped at half-filling.

Figure 4

Curie-Weiss temperature $T_0$ obtained from Curie-Weiss fit of the $B_{1g}$ susceptibility as a function of doping level for different values of $t^{\prime }$, all showing a peak at $\langle x\rangle =0.1$. The peak moves slightly to higher doping level with increasing $-t^{\prime }/t$, which may be due to the van-Hove point moving to higher doping level with increasing $-t^{\prime }/t$.

Figure 5

Raman spectra obtained via maximum entropy analytic continuation for (a) $B_{1g}$ and (b) $B_{2g}$ symmetries for different doping levels, at $\beta =4.5/t$ and $t^{\prime }=-0.25t$. The broad peak at around $\omega /t=6$ arises from interband transitions across the Mott gap.

Figure 6

Real part ${\chi }_{B_{2g}}^{\prime }\left(T\right)$ of the static Raman susceptibility in $B_{2g}$ symmetry as a function of temperature for various doping levels $x$. The doping ranges below and above $x=0.1$ are shown separately for clarity in (a) and (b), respectively.

Figure 7

Curie-Weiss temperature $T_0$ for the $B_{1g}$ susceptibility as a function of doping, for $L=8$ and $L=10$ square clusters, both showing a peak around $\langle x\rangle =0.1$. The next-nearest-neighbor hopping is $t^{\prime }=-0.2t$.