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We show that the magnetic Raman response of a spin liquid is independent of the polarizations of the light for triangular symmetries. In contrast, a ground state that has a broken-symmetry shows characteristic oscillations when the polarizations are rotated. This would allow one to detect weak broken-symmetries and emergent order parameters. We focus on the kagome antiferromagnet wherein no conventional long-range order has been found so far and present the Raman cross section of a spin liquid and a valence bond crystal by using a random phase approximation.

We revisit the relationship between three classical measures of particle number, namely, the chemical doping $x$, the Hall number ${x}_{\mathit{\text{Hall}}}$, and the particle number inferred from the optical sum rule ${x}_{\mathit{opt}}$. We study the $t\text{\ensuremath{-}}{t}^{\ensuremath{'}}\text{\ensuremath{-}}J$ model of correlations on a square lattice, as a minimal model for high ${T}_{c}$ systems, using numerical methods to evaluate the low temperature Kubo conductivities. These measures disagree significantly in this type of system, owing to Mott Hubbard correlations. The Hall constant has a complex behavior with several changes of sign as a function of filling $x$, depending on the model parameters. Thus, ${x}_{\mathit{\text{Hall}}}$ depends sensitively on ${t}^{\ensuremath{'}}$ and $J$ due to a kind of quantum interference.

Thermal response functions of strongly correlated electron systems are of appreciable interest to the larger scientific community both theoretically and technologically. Here, we focus on the infinitely correlated $t\text{\ensuremath{-}}J$ model on a geometrically frustrated two-dimensional triangular lattice. Using exact diagonalization on a finite sized system, we calculate the dynamical thermal response functions in order to determine the thermopower, Lorenz number, and dimensionless figure of merit. The dynamical thermal response functions are compared to the infinite frequency limit and shown to be very weak functions of frequency, hence, establishing the validity of the high frequency formalism recently proposed by Shastry [Phys. Rev. B 73, 085117 (2006)] for the thermopower, Lorenz number, and the dimensionless figure of merit. Further, the thermopower is demonstrated to have a low to intermediate temperature enhancement when the sign of the hopping parameter $t$ is switched from positive to negative for the geometrically frustrated lattice [A. P. Ramirez, in More Is Different, edited by N. P. Org and R. N. Bhatt (Princeton University Press, New Jersey, 2001), p. 255] considered.

In this paper, we study the thermal response functions for two one-dimensional models, namely, the Hubbard and spinless fermion $t\text{\ensuremath{-}}V$ model, respectively. By exactly diagonalizing finite sized systems, we calculate dynamical, electrical, thermoelectrical, and thermal conductivities via the Kubo formalism [J. Phys. Soc. Jpn. 12, 570 (1957)]. The thermopower (Seebeck coefficient), Lorenz number, and dimensionless figure of merit are then constructed, which are quantities of great interest to the physics community both theoretically and experimentally. We also geometrically frustrate these systems and destroy integrability by the inclusion of a second-neighbor hop in the kinetic energy operator. These frustrated systems are shown to have enhanced thermopower and Lorenz number at intermediate and low temperatures.

We present a finite temperature $(T)$ study of the $t\text{\ensuremath{-}}J$ model on the two-dimensional triangular lattice for the negative hopping $t$, as relevant for the electron-doped ${\mathrm{Na}}_{x}\mathrm{Co}{\mathrm{O}}_{2}$ (NCO). We study several thermodynamic and transport properties in this study: the $T$-dependent chemical potential, specific heat, magnetic susceptibility, and the dynamic Hall coefficient across the entire doping range. We show systematically how this simplest model for strongly correlated electrons describes a crossover as function of doping $(x)$ from a Pauli-like weakly spin-correlated metal close to the band limit (density $n=2$) to the Curie-Weiss metallic phase $(1.5lnl1.75)$ with pronounced antiferromagnetic (AFM) correlations at low temperatures and Curie-Weiss-type behavior in the high-temperature regime. Upon further reduction of the doping, a different energy scale, dominated by spin-interactions $(J)$ emerges. It is apparent both in specific heat and susceptibility, and we identify an effective interaction ${J}_{\mathit{eff}}(x)$, valid across the entire doping range. This is in contrast to the formula by Anderson et al. [J. Phys.: Condens. Matter 16, R755 (2004)] for the square lattice. NCO has $tl0$, hence the opposite sign of the Nagaoka-ferromagnetic situation, this expression includes the subtle effect of weak kinetic AFM [Haerter and Shastry, Phys. Rev. Lett. 95, 087202 (2005)], as encountered in the infinitely correlated situation $(U=\ensuremath{\infty})$ for electronic frustration. By explicit computation of the Kubo formulas, we address the question of practical relevance of the high-frequency expression for the Hall coefficient ${R}_{H}^{*}$ [Shastry et al., Phys. Rev. Lett. 70, 2004 (1993)]. We hope to clarify some open questions concerning the applicability of the $t\text{\ensuremath{-}}J$ model to real experimental situations through this study.