Many-body energy invariant for $T$-linear resistivity

The description of dynamics of strongly correlated quantum matter is a challenge, particularly in physical situations where a quasiparticle description is absent. In such situations, however, the many-body Kubo formula from linear response theory, involving matrix elements of the current operator computed with many-body wavefunctions, remains valid. Working directly in the many-body Hilbert space and not making any reference to quasiparticles (or lack thereof), we address the puzzle of linear in temperature ($T$-linear) resistivity seen in non-Fermi liquid phases that occur in several strongly correlated condensed matter systems. We derive a simple criterion for the occurrence of $T$-linear resistivity based on an analysis of the contributions to the many-body Kubo formula, determined by an energy invariant "$f$-function" involving current matrix elements and energy eigenvalues that describes the DC conductivity of the system in the microcanonical ensemble. Using full diagonalization, we test this criterion for the $f$-function in the spinless nearest neighbor Hubbard model and in a system of Sachdev-Ye-Kitaev dots coupled by weak single particle hopping. We also study the $f$-function for the spin conductivity in the 2D Heisenberg model and arrive at similar conclusions. Our work suggests that a general principle, formulated in terms of many-body Hilbert space concepts, is at the core of the occurrence of $T$-linear resistivity in a wide range of systems, and precisely translates $T$-linear resistivity into a notion of energy scale invariance far beyond what is typically associated with quantum critical points.