Gabriele Bellomia

Faithful entanglement description of strongly correlated electrons

• Published: spatial entanglement properties across the Mott-Hubbard transition on the square lattice [Phys. Rev. B 109, 115104](arXiv preprint){open code for reproducibility}
  
  > Among the most debated topics in condensed matter physics, the fundamental nature of the Mott-Hubbard transition still escapes a full theoretical account.  Lately, a fresh bloom of ideas has followed the advent of the two new central paradigms for the classification of phase transition, beyond symmetry breaking arguments. On one side the recent proposal of a local mapping to an amazingly simple SSH-type transition in the infinite coordination limit [1], relating Mottness to topological order in "hidden" degrees of freedom. On the other side, the increasingly relevant role of entanglement measures as a probe for the complexity of quantum states and the related developments in cold-atom experiments, has inspired a full characterization of the Mott-Hubbard phase diagram through the single-site von Neumann entropy and mutual information [2,3,4], which capture the entanglement between a single electronic orbital and the rest of the lattice, at all lenght scales. 
  The two concepts share a common root in the seminal proposal by Anderson of describing a Mott insulator as a resonating valence bond state, characterized by strong entanglement between spatially separated atomic orbitals, acting as a signature of nonlocal topological order. To develop a deeper understanding of this connection, we investigate the spatial entanglement properties of cluster dynamical mean-field solutions of the Hubbard model on the square lattice, at zero temperature.
  Analyzing the behavior of reduced density matrices, ranging from a single site up to 2x4 clusters, we first establish the size-scaling of the corresponding von Neumann entropies: our results hold the comparison with recent work [5], devising a non-equilibrium extension of inhomogeneous dynamical mean-field theory to target the second Renyi entropy and mutual information for subsystems up to 128 sites in the square lattice. Overall, von Neumann and generalized Renyi entropies, from single sites up to mesoscopic chunks of lattice, all describe the Mott metal-insulator transition as a gradual decrease of highly nonlocal quantum correlations. Yet, we find that the quantum entanglement and classical correlations between two orbitals, without any contribution from the rest of the lattice, recover the inherent character of strongly correlated insulators: as demonstrated through suitable upper and lower bound to their magnitude, they both increase abruptly at the Mott transition, and are quickly damped with inter-site distance at all interaction values. This suggests that Mott localization might be fundamentally linked to a concept of strong quasilocal entanglement, supporting the relevance of the local topological picture introduced in [1], and enhancing the analysis of nonlocal entanglement properties introduced in [2,3,4] towards a space-resolved characterization of the RVB structure in the Mott state. 
   
• In preparation:  Two-orbital entanglement description of the doping-driven insulator-to-metal transition and entanglement properties of the intermediate bad metal phase.
  
  > When a Mott insulator is doped with holes, charge transport is immediately recovered, but not by a direct transition to a Fermi liquid state. Intermediate exotic conducting phase, with no concept of quasiparticle and distinct thermodynamic and spectral properties arise in between the Mott insulator and the conventional metal. Such bad metallic phases have attracted a lot of attention for their relevance in determining the normal state of high-temperature superconductors and intensely investigated within a multitude of computational methods. 
  We join the effort, focusing on the so-called pseudogap phase, whose crucial aspects at the quasilocal length scale can be effectively simulated within cluster dynamical mean-field theory at zero temperature. In this framework, we have already established a pathway to access the spatial entanglement properties of both Fermi liquids and Mott insulators at half-filling (see previous work). Hence, we naturally extend the analysis to the hole-doped regime, finding that the nearest-neighbor entanglement of the pseugodap metal resembles much closely the Mott insulator than the weakly correlated metal, despite the strikingly different transport properties.
  This is especially true if we apply the charge superselection rule, that kills all local charge fluctuations, thus effectively killing the trivial real-space entanglement of conventional metals.
   

• Currently working on:  Entanglement description of SU(4) and SU(2) Mott insulators and their crossover, as realized in cold-atom experiments.

  > Inspired by the recent experimental observation of flavour-selective localization in SU(3) systems [1], we extend the picture to an SU(4) system, where a tunable hopping amplitude is able to tune a crossover to an effective flavor-selective SU(2) description. We characterize both the extreme regimes and the crossover by means of local inter-flavor entanglement measures, amenable to experimental verification.

Mottness, magnetism and topology of strongly-correlated Dirac fermions.

• Currently working on: Two-orbital entanglement description of correlated topological phase transitions
  
  > Recent research pursued by collaborators from the CNR-IOM group [1] has shown how symmetry protected topological phase transitions can drastically change character when strong correlations come into play. The role of local (inter-orbital) versus nonlocal (inter-atomic) correlation has been so far studied indirectly, by looking at the structure of the self-energy matrix, within a minimal cluster dynamical mean-field description [2]. We aim to target the issue more directly, from a quantum information perspective, by means of our recently introduced two-orbital entanglement description of strongly correlated materials [see above].
   
• In preparation: Engineering of strongly correlated magnetic states in a quantum spin-Hall
  insulator {open code for reproducibility}
  
  > We consider the effect of a local Hubbard repulsion on the Kane-Mele model, arguably the simplest
  description of interacting topological Dirac fermions. We use dynamical mean-field theory at zero
  temperature to explore the phase-diagram of the model in the infinite lattice size limit. At finite
  spin-orbit coupling, we assess the presence of a discontinuous interaction-driven phase transition
  between the ℤ₂ topological phase of the noninteracting model and two anisotropic antiferromagnetic
  solutions, which are separated by a small but finite energy gap. Approaching the discontinuous
  jump in the magnetization, from the weak-coupling side, we find evidence of an intermediate
  regime of coexisting topological and magnetic order. This defines a candidate antiferromagnetic
  topological insulator (AFMTI) regime, at intermediate coupling. An information-theory inspired
  quantification of quantum and classical correlations contained in a single electronic orbital
  demonstrates that the AFMTI regime displays much larger local correlations than both the quantum
  spin-Hall insulator, at weak-coupling, and the topologically trivial antiferromagnets, at strong
  coupling. Furthermore, it reveals that the easy-plane ground state is significantly less correlated
  than the frustrated off-plane state positioned at higher energy. We can invert the energy difference
  between the two states moving from the infinite lattice to suitable quantum dots, thus stabilizing a
  strongly correlated topological and magnetic state, at the nanoscale. Finally, we present a direct
  comparison with the local correlations developed across the transition from the ℤ₂ insulator to a
  paramagnetic Mott insulator, complementing the picture presented in our previous work on the
  quasilocal entanglement across the standard metal-to-insulator paramagnetic Mott transition.
  This allows us to discuss, on rigorous grounds, the close connection between magnetic frustration
  and strong local and quasilocal orbital correlations.

Quantum geometry and its relationship to measurable quantities.

• Published: solution to a long-standing paradox about how to properly evaluate the many-electron inverse adiabatic inertia —aka the Drude weight— within bounded crystalline systems. [Phys. Rev. B 102, 205123] (arXiv preprint) {open code for reproducibility}
  
  > The state-of-the-art theory of adiabatic transport, founded by Kohn in the 60s, predicts a vanishing Drude weight at any finite size within open boundary conditions, reflecting the obvious rule out of steady dc currents in a bounded system. Nevertheless at the thermodynamic limit the boundary condition choice must become irrelevant and so the finite value of the Drude weight in a metal has to be recovered also in the OBC case. Inspecting the details of this nontrivial limit has lead to a new effective definition of the OBC Drude-like current response, allowing for highly accurate estimations of the Drude weight even at finite size. 
   
• Long term goal: a local formulation of the Drude weight and its possible relationship with the modern local theory of orbital magnetization.