Free fermion systems enjoy a privileged place in physics. With their simple structure they can explain a variety of effects, ranging from insulating and metallic behaviours to superconductivity and the integer quantum Hall effect. Interactions, e.g. in the form of Coulomb repulsion, can dramatically alter this picture by giving rise to emerging physics that may not resemble free fermions. Examples of such phenomena include high-temperature superconductivity, fractional quantum Hall effect, Kondo effect and quantum spin liquids. The non-perturbative behaviour of such systems remains a major obstacle to their theoretical understanding that could unlock further technological applications. Here, we present a pedagogical review of “interaction distance" [Nat. Commun. 8, 14926 (2017)] – a systematic method that quantifies the effect interactions can have on the energy spectrum and on the quantum correlations of generic many-body systems. In particular, the interaction distance is a diagnostic tool that identifies the emergent physics of interacting systems. We illustrate this method on the simple example of a two-site Fermi-Hubbard model.
Free fermion systems enjoy a privileged place in physics. With their simple structure they can explain a variety of effects, ranging from insulating and metallic behaviours to superconductivity and the integer quantum Hall effect. Interactions, e.g. in the form of Coulomb repulsion, can dramatically alter this picture by giving rise to emerging physics that may not resemble free fermions. Examples of such phenomena include high-temperature superconductivity, fractional quantum Hall effect, Kondo effect and quantum spin liquids. The non-perturbative behaviour of such systems remains a major obstacle to their theoretical understanding that could unlock further technological applications. Here, we present a pedagogical review of “interaction distance" [Nat. Commun. 8, 14926 (2017)] – a systematic method that quantifies the effect interactions can have on the energy spectrum and on the quantum correlations of generic many-body systems. In particular, the interaction distance is a diagnostic tool that identifies the emergent physics of interacting systems. We illustrate this method on the simple example of a two-site Fermi-Hubbard model.
Interacting bosons or fermions give rise to some of the most fascinating phases of matter, including high-temperature superconductivity, the fractional quantum Hall effect, quantum spin liquids and Mott insulators. Although these systems are promising for technological applications, they also present conceptual challenges, as they require approaches beyond mean-field and perturbation theory. Here we develop a general framework for identifying the free theory that is closest to a given interacting model in terms of their ground-state correlations. Moreover, we quantify the distance between them using the entanglement spectrum. When this interaction distance is small, the optimal free theory provides an effective description of the low-energy physics of the interacting model. Our construction of the optimal free model is non-perturbative in nature; thus, it offers a theoretical framework for investigating strongly correlated systems.
Quantum many-body systems realize many different phases of matter characterized by their exotic emergent phenomena. While some simple versions of these properties can occur in systems of free fermions, their occurrence generally implies that the physics is dictated by an interacting Hamiltonian. The interaction distance has been successfully used to quantify the effect of interactions in a variety of states of matter via the entanglement spectrum [C. J. Turner, K. Meichanetzidis, Z. Papic and J. K. Pachos, Nat. Commun. 8 (2017) 14926, Phys. Rev. B 97 (2018) 125104]. The computation of the interaction distance reduces to a global optimization problem whose goal is to search for the free-fermion entanglement spectrum closest to the given entanglement spectrum. In this work, we employ techniques from machine learning in order to perform this same task. In a supervised learning setting, we use labeled data obtained by computing the interaction distance and predict its value via linear regression. Moving to a semi-supervised setting, we train an autoencoder to estimate an alternative measure to the interaction distance, and we show that it behaves in a similar manner.
Strongly correlated electron systems (SCES) represent a class of quantum materials where electron-electron interactions play a dominant role in determining physical properties, leading to a wide array of exotic phenomena such as high-temperature superconductivity, quantum spin liquids, Mott transitions, and heavy fermion behavior. Unlike weakly interacting systems described adequately by Fermi-liquid theory, SCES require more advanced theoretical frameworks due to the breakdown of single-particle descriptions. This article reviews the fundamental interactions governing SCES, emphasizing Coulomb repulsion, spin-charge separation, and entanglement-driven effects. The role of lattice geometry, orbital degeneracy, and dimensionality is explored to understand emergent behaviors. Through theoretical insights and recent experimental findings, this work outlines how these fundamental interactions shape the rich phase diagrams of SCES and their implications in quantum technologies.
Numerous correlated electron systems exhibit a strongly scale-dependent behavior. Upon lowering the energy scale, collective phenomena, bound states, and new effective degrees of freedom emerge. Typical examples include (i) competing magnetic, charge, and pairing instabilities in two-dimensional electron systems; (ii) the interplay of electronic excitations and order parameter fluctuations near thermal and quantum phase transitions in metals; and (iii) correlation effects such as Luttinger liquid behavior and the Kondo effect showing up in linear and nonequilibrium transport through quantum wires and quantum dots. The functional renormalization group is a flexible and unbiased tool for dealing with such scale-dependent behavior. Its starting point is an exact functional flow equation, which yields the gradual evolution from a microscopic model action to the final effective action as a function of a continuously decreasing energy scale. Expanding in powers of the fields one obtains an exact hierarchy of flow equations for vertex functions. Truncations of this hierarchy have led to powerful new approximation schemes. This review is a comprehensive introduction to the functional renormalization group method for interacting Fermi systems. A self-contained derivation of the exact flow equations is presented and frequently used truncation schemes are described. Reviewing selected applications it is shown how approximations based on the functional renormalization group can be fruitfully used to improve our understanding of correlated fermion systems.
A state-of-the-art review is given of the investigations of the Hubbard model, which is the main model used in the theory of strongly correlated electron systems. It is shown how the main correlation effects — the metal–insulator phase transition, the appearance of localised magnetic moments, and the breakdown of the Fermi-liquid behaviour — appear in the case of strong electron correlations when U ~ W, where U is the Coulomb repulsion parameter and W is the band width. Different approaches to the problem, not based on perturbation theory in terms of the parameter U/W or W/U, are presented. A new method in the theory of strongly correlated systems, in which the limit of infinite number of dimensions of space, d = ∞, is the main feature of the review. The physical results obtained by this approach are compared with those deduced by traditional methods. An overview of the behaviour of strongly correlated systems is given for wide ranges of the model parameters U and W, and of the electron density n.
The study of the interplay between interaction-induced correlations and nonequilibrium initial conditions in many-body systems has recently attracted a lot of attention. New experimental techniques provide high control over many-body systems and systematic access to their nonequilibrium regime. In many systems the nonequilibrium properties of a Fermi liquid can be relevant. A first approach to their understanding is the main content of this thesis. At the beginning, nonequilibrium phenomena, Landau's theory of a Fermi liquid, the Hubbard model, time evolution in quantum mechanics, related experiments, and the flow equation method are reviewed. The key observation of this thesis, namely a characteristic mismatch of expectation values in equilibrium and nonequilibrium, is first illustrated for the squeezed oscillator. For this one-particle model Hamiltonian the perturbative approach can be compared with the exact solution and it is shown that the mismatch holds even beyond perturbation theory. Afterwards these observations are generalized to a larger class of one-particle models. Then the nonequilibrium behavior of a Fermi liquid is examined by analyzing the Fermi liquid phase of the Hubbard model in more than one dimension. After a sudden switch-on of a weak two-particle interaction to the noninteracting Fermi gas the relaxation of the many-body system is observed. For this purpose, the flow equation transformation is implemented for the Hubbard Hamiltonian. Then the discussion of the momentum distribution function and of the kinetic energy displays a three-step relaxation behavior of the Fermi liquid from the initial perturbation until thermalization is reached. Firstly, the sudden switch inserts excitation energy into the system which drives the following dynamics. Then a rapid initial build-up of correlations is caused by initial dephasing and leads to the establishment of a quasiparticle picture. By that the system enters into a quasi-stationary state which can be long-lasting for weak interaction. This state shows prethermalization of the kinetic energy which already has relaxed to its final values. However, the momentum distribution still resembles a zero temperature Fermi liquid. Its later relaxation on a second time scale is caused by a residual two-particle interaction which allows for scattering processes and can be described by a quantum Boltzmann equation. The physical origin of this delayed relaxation can be traced back to an interplay of translational invariance of the Hubbard model and the Pauli principle for fermions; together they restrict the phase space for scattering events. Comparing with similar work I conjecture on the generic nature of the findings made for the quenched Fermi liquid in other many-body systems. Finally I point out to the potential relevance of the delayed relaxation for the observation of further nonequilibrium phenomena, for instance in BCS systems. In order to extend the study of sudden switching to arbitrary switching processes the calculation is repeated using the Keldysh perturbation theory. First evaluations for a linear increase of the interaction strength are given. Extensions of this work are suggested and may motivate further research.