**Universal entanglement signatures of quantum liquids as a guide to fermionic criticality**
metals gapped liquids entanglement MERG Hubbard-model

Feb 12, 2023 New J. Phys (2023)
arXiv:2205.11123

## Some background

Our work is devoted towards understanding whether there exist any signatures of universality in the nature of many-particle entanglement encoded within the ground state and lowest-lying excited state wavefunctions of quantum liquids that are emergent within systems of interacting electrons. The novelty of such an investigation can be gauged by recalling that 20th century condensed matter physics was governed primarily by studies of systems that fall within the Ginzburg-Landau-Wilson (GLW) paradigm for the description of ordered states of matter. Such ground states are emergent from the spontaneous breaking of symmetries, and whose order is characterised by real space order parameters that correspond to the ground state expectation values of local operators. Importantly, the short-range entangled nature of these ground states is responsible for their satisfying the cluster decomposition property of all correlation functions. Starting with the discovery of the fractional quantised Hall effect in two-dimensional electron gases and high-temperature superconductivity in the hole-doped Mott insulating cuprates, the quantum condensed matter community has increasingly become interested in understanding the physics that can lead to long-range entangled ground states, and whether this can enable the classification of quantum matter. A variety of quantum liquids have emerged as candidate systems in which to meet these challenges.

Researches theorise that electrons in a strange metal are in a highly entangled quantum state. Their resistance depends on Planck’s constant, suggesting that the electrons are dissipating energy as fast as physics allows (Planckian dissipation). [Source]

Many-body quantum systems with locally interacting constituents are expected to show the area-law in the entanglement entropy of a subsystem, reflecting the minimal entanglement with the rest of the system across common boundaries. In our work, we focus on quantum liquid phases of interacting electrons that are noteworthy exceptions to this rule. Of these, gapped topologically ordered liquids (e.g., the Laughlin states of the FQHE) show long-range entanglement in the form of topological entanglement entropy, while gapless liquids (e.g., the Fermi liquid in two spatial dimensions) display scaling according to a log-modified area law in real-space calculations of entanglement entropy or even volume law scaling in some cases of quantum criticality. The long-range entanglement in both types of liquids arises from the prevalence of quantum fluctuations that defy the GLW paradigm, and we begin by identifying them in several gapless quantum liquids (e.g., the 2D Fermi liquid (FL), 2D marginal Fermi liquid (MFL), and the 1D Tomonaga-Luttinger liquid).

## Main results: unification of the entanglement signatures of gapless liquids

Gapless quantum liquids are described by effective theories that contain only the physics of the low-energy long-wavelength degrees of freedom proximate to the Fermi surface. It is then important to investigate theoretically the momentum-space quantum fluctuations related to forward and tangential scattering in a system of interacting electrons that are resolved under the renormalisation group (RG) flow from UV to IR. Thus, by employing a state-of-the-art T=0 numerical RG method based purely on many-particle unitary transformations, we confirm that a log-modified area law for the scaling of the entanglement entropy of the 2D FL for a subsystem (defined within a momentum-space window proximate to the Fermi surface) with its size arises from precisely such quantum fluctuations. A similar analysis of non-Fermi liquids such as the 2D MFL and the 1D TLL reveals striking evidence for the universal nature of this logarithmic violation of the area-law, with a proportionality constant that depends on the nature of the underlying Fermi surface. These results suggest a unification of the Fermi liquid paradigm with non-Fermi liquid metals (that are qualitatively different in terms of their low-lying excitations) based on the signatures of many-particle entanglement they display.

## Main results: entanglement as an indicator of the presence of a gap

We extend this analysis in order to classify the gapped quantum liquids emergent from the destabilisation of the Fermi surface by RG relevant quantum fluctuations arising from backscattering processes. Here, we find that the momentum-space entanglement signatures of gapped quantum liquids appear to be governed by the nature of the Fermi surface from which they emerge, as well as the nature of their parent gapless metallic liquid. The importance of the nature of the Fermi surface is further confirmed by our finding an enhanced entanglement entropy for the nodal MFL present at the quantum critical point recently discovered in the 2D Hubbard model at optimal hole-doping, and lying proximate to a point-like singular Fermi surface. Finally, in a supplementary materials note, we report on various quantum information theoretic measures that display the holographic entanglement of various gapless as well as gapped quantum liquids.

Our analysis, thus, enables an entanglement-based classification of various quantum liquids emergent from the criticality of interacting fermionic quantum matter. To the best of our knowledge, our work is the first of its kind, and paves the way for further explorations of quantum matter based on their entanglement properties.