Abstract An outstanding challenge involves understanding the many-particle entanglement of liquid states of quantum matter that arise in systems of interacting electrons. The Fermi liquid (FL) shows a violation of the area-law in real-space entanglement entropy of a subsystem, believed to be a signature of the ground state of a gapless quantum critical system of interacting fermions. Here, we apply a T = 0 renormalization group approach to the FL, unveiling the long-wavelength quantum fluctuations from which long-range entanglement arises. A similar analysis of non-Fermi liquids such as the 2D marginal Fermi liquid (MFL) and the 1D Tomonaga–Luttinger liquid reveals a universal logarithmic violation of the area-law in gapless electronic liquids, with a proportionality constant that depends on the nature of the underlying Fermi surface. We extend this analysis to classify the gapped quantum liquids emergent from the destabilisation of the Fermi surface by renormalisation group relevant quantum fluctuations arising from backscattering processes.
Abstract An outstanding challenge involves understanding the many-particle entanglement of liquid states of quantum matter that arise in systems of interacting electrons. The Fermi liquid (FL) shows a violation of the area-law in real-space entanglement entropy of a subsystem, believed to be a signature of the ground state of a gapless quantum critical system of interacting fermions. Here, we apply a T = 0 renormalization group approach to the FL, unveiling the long-wavelength quantum fluctuations from which long-range entanglement arises. A similar analysis of non-Fermi liquids such as the 2D marginal Fermi liquid (MFL) and the 1D Tomonaga–Luttinger liquid reveals a universal logarithmic violation of the area-law in gapless electronic liquids, with a proportionality constant that depends on the nature of the underlying Fermi surface. We extend this analysis to classify the gapped quantum liquids emergent from the destabilisation of the Fermi surface by renormalisation group relevant quantum fluctuations arising from backscattering processes.
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 Θ-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 Θ-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.
Motivated by recent development of the concept of the disorder operator and its relation with entanglement entropy in bosonic systems, here we show the disorder operator successfully probes many aspects of quantum entanglement in fermionic many-body systems. From both analytical and numerical computations in free and interacting fermion systems in 1D and 2D, we find the disorder operator and the entanglement entropy exhibit similar universal scaling behavior, as a function of the boundary length of the subsystem, but with subtle yet important differences. In 1D they both follow the \log{L} log L scaling behavior with the coefficient determined by the Luttinger parameter for disorder operator, and the conformal central charge for entanglement entropy. In 2D they both show the universal L\log L L log L scaling behavior in free and interacting Fermi liquid states, with the coefficients depending on the geometry of the Fermi surfaces. However at a 2D quantum critical point with non-Fermi-liquid state, extra symmetry information is needed in the design of the disorder operator, so as to reveal the critical fluctuations as does the entanglement entropy. Our results demonstrate the fermion disorder operator can be used to probe quantum many-body entanglement related to global symmetry, and provide new tools to explore the still largely unknown territory of highly entangled fermion quantum matter in 2 or higher dimensions.
Capturing the interplay between electronic correlations and many-particle entanglement requires a unified framework for Hamiltonian and eigenbasis renormalization. In this work, we apply the unitary renormalization group (URG) scheme developed in a companion work [1] to the study of two archetypal models of strongly correlated lattice electrons, one with translation invariance and one without. We obtain detailed insight into the emergence of various gapless and gapped phases of quantum electronic matter by computing effective Hamiltonians from numerical evaluation of the various RG equations, as well as their entanglement signatures through their respective tensor network descriptions. For the translationally invariant model of a single-band of interacting electrons, this includes results on gapless metallic phases such as the Fermi liquid and Marginal Fermi liquid, as well as gapped phases such as the reduced Bardeen-Cooper-Schrieffer, pair density-wave and Mott liquid phases. Additionally, a study of a generalised Sachdev-Ye model with disordered four-fermion interactions offers detailed results on many-body localised phases, as well as thermalised phase. We emphasise the distinctions between the various phases based on a combined analysis of their dynamical (obtained from the effective Hamiltonian) and entanglement properties. Importantly, the RG flow of the Hamiltonian vertex tensor network is shown to lead to emergent gauge theories for the gapped phases. Taken together with results on the holographic spacetime generated from the RG of the many-particle eigenstate (seen through, for instance, the holographic upper bound of the one-particle entanglement entropy), our analysis offer an ab-initio perspective of the gauge-gravity duality for quantum liquids that are emergent in systems of correlated electrons.
law is enhanced by a logarithmic factor. In Ref. [1], we use multi-dimensional bosonization to provide a simple derivation of this result, and show that the logarithimic factor has a 1D origin. More importantly the bosonization technique allows us to take into account the Fermi liquid interactions, and obtain the leading scaling behavior of the entanglement entropy of Fermi liquids. The central result of our work is that Fermi liquid interactions do not alter the leading scaling behavior of the entanglement entropy, and the logarithmic enhancement of area law is a robust property of the Fermi liquid phase. In sharp contrast to the fermioic systems with Fermi surfaces, quantum critical (or gapless) bosonic systems do not violate the area law above 1D (except for the case discussed below). The fundamental difference lies in the fact that gapless excitations live near a single point (usually origin of momentum space) in such bosonic systems, while they live around an (extended) Fermi surface in Fermi liquids. In Ref. [2], we studied entanglement properties of some specific examples of the so called Bose metal states, in which bosons neither condense (and become a superfluid) nor localize (and insulate) at T=0. The system supports gapless excitations around “Bose surfaces,” instead of isolated points in momentum space. We showed that similar to free Fermi gas and Fermi liquids, these states violate the entanglement area law in a logarithmic fashion. Compared to ground states, much less is known concretely about entanglement in (highly) excited states. Going back to free fermion systems, in Ref. [3] we show that there exists a duality relation between ground and excited states, and the area law obeyed by ground state turns into a volume law for excited states, something widely expected but hard to prove. Most importantly, we find in appropriate limits the reduced density matrix of a subsystem takes the form of thermal density matrix, providing an explicit example of the eigenstate thermalization hypothesis. Our work [3] explicitly demonstrates how statistical physics emerges from entanglement in a single eigenstate.
We analyze in detail the effect of nontrivial band topology on the area-law behavior of the entanglement entropy in Kitaev's honeycomb model. By mapping the translationally invariant 2D spin model onto 1D fermionic subsystems, we identify those subsystems responsible for universal entanglement contributions in the gapped phases and those responsible for critical entanglement scaling in the gapless phases. For the gapped phases, we analytically show how the topological edge states contribute to the entanglement entropy and provide a universal lower bound for it. For the gapless semimetallic phases and topological phase transitions, the identification of the critical subsystems shows that they fall always into the Ising or the XY universality classes. As our study concerns the fermionic degrees of freedom in the honeycomb model, qualitatively similar results are expected to apply also to generic topological insulators and superconductors.
States with gapless degrees of freedom are typically more complicated and less well- understood than systems possessing a gap. In this thesis, we study strongly-correlated systems described by gapless fermions. In most of the systems we consider, the fermionic excitations are emergent, i.e. they are not adiabatically connected to the electrons which constitute the fundamental building blocks of the system in question.\nWe start by proposing several novel states of matter which we claim are relevant to strongly-correlated systems. The rst state we present is a fractionalized Fermi liquid on the surface of a topological Kondo insulator. A Kondo insulator is a material which becomes insulating at low temperatures as a result of strong electron-spin interactions. When spin- orbit coupling is present, this insulator might be topological and, consequently, host robust gapless surface modes. Given the strong interactions and the decreased dimensionality of the surface, we propose that the spins and electrons there may decouple, resulting in the formation of a fractionalized Fermi liquid.\nWe next argue that quantum electrodynamics in 2+1 dimensions (QED3) with Nf = 4 fermion avours may describe a continuous, decon ned phase transition connecting the 120◦ coplanar Néel phase and the √12 × √12 valence bond solid phase of the triangular lattice antiferromagnet (AF). In addition to being a critical point, QED3 is also believed to describe a critical phase of matter called the Dirac spin liquid. Regardless of whether QED3 is manifest as a phase or a critical point, impurities and imperfections are always present in the real world, and it is therefore important to understand what e ects this may have. We show that when QED3 is perturbed by weak disorder, under certain circumstances, it ows to a new critical point in which both interactions and disorder are present.\nConclusively identifying a system described by QED3 has proven to be a di cult task, not only experimentally, but in numerical studies as well. Since QED3 is a critical theory, its spectrum on a torus is a universal quantity dependent only on low-energy degrees of freedom, such as the torus area. It follows that this data is accessible via exact diagonalization and can serve as an identifying signature of the theory. To this end, we calculate the QED3 torus spectrum using path integral methods.\nWe conclude this thesis with a comprehensive study of gapped Z2 spin liquids of the square lattice antiferrogmagnet, using a critical theory of Dirac fermions coupled to an SU(2) gauge eld as a starting point. There are many di erent Z2 spin liquid groundstates which preserve the same symmetry group. Even if one knows that a system has a Z2 spin liquid groundstate, it is by no means obvious which Z2 spin liquid is being realized. By starting from a gapless theory, the set of possibilities can be reduce considerably. Further, using certain recently proposed dualities, comparisons with theories formulated using bosons can be made.