Fermionic criticality is shaped by Fermi surface topology: a case study of the Tomonaga-Luttinger liquid

Abstract

We perform a unitary renormalization group (URG) study of the 1D fermionic Hubbard model. The formalism generates a family of effective Hamiltonians and many-body eigenstates arranged holographically across the tensor network from UV to IR. The URG is realized as a quantum circuit, leading to the entanglement holographic mapping (EHM) tensor network description. A topological $\Theta$-term of the projected Hilbert space of the degrees of freedom at the Fermi surface are shown to govern the nature of RG flow towards either the gapless Tomonaga-Luttinger liquid or gapped quantum liquid phases. This results in a nonperturbative version of the Berezenskii-Kosterlitz-Thouless (BKT) RG phase diagram, revealing a line of intermediate coupling stable fixed points, while the nature of RG flow around the critical point is identical to that obtained from the weak-coupling RG analysis. This coincides with a phase transition in the many-particle entanglement, as the entanglement entropy RG flow shows distinct features for the critical and gapped phases depending on the value of the topological $\Theta$-term. We demonstrate the Ryu-Takyanagi entropy bound for the many-body eigenstates comprising the EHM network, concretizing the relation to the holographic duality principle. The scaling of the entropy bound also distinguishes the gapped and gapless phases, implying the generation of very different holographic spacetimes across the critical point. Finally, we treat the Fermi surface as a quantum impurity coupled to the high energy electronic states. A thought-experiment is devised in order to study entanglement entropy generated by isolating the impurity, and propose ways by which to measure it by studying the quantum noise and higher order cumulants of the full counting statistics.