Fermionic Lattice Systems Research Articles

A Two-Dimensional Analogue of the Luttinger Model

We present a fermion model that is, as we suggest, a natural 2D analogue of the Luttinger model. We derive this model as a partial continuum limit of a 2D spinless lattice fermion system with local interactions and away from half filling. In this derivation, we use certain approximations that we motivate by physical arguments. We also present mathematical results that allow an exact treatment of parts of the degrees of freedom of this model by bosonization, and we propose to treat the remaining degrees of freedom by mean field theory.

Fermionic multiscale entanglement renormalization ansatz

In a recent contribution [P. Corboz, G. Evenbly, F. Verstraete, and G. Vidal, arXiv:0904.4151 (unpublished)] entanglement renormalization was generalized to fermionic lattice systems in two spatial dimensions. Entanglement renormalization is a real-space coarse-graining transformation for lattice systems that produces a variational ansatz, the multiscale entanglement renormalization ansatz (MERA), for the ground states of local Hamiltonians. In this paper we describe in detail the fermionic version of the MERA formalism and algorithm. Starting from the bosonic MERA, which can be regarded both as a quantum circuit or in relation to a coarse-graining transformation, we indicate how the scheme needs to be modified to simulate fermions. To confirm the validity of the approach, we present benchmark results for free and interacting fermions on a square lattice with sizes between $6\ifmmode\times\else\texttimes\fi{}6$ and $162\ifmmode\times\else\texttimes\fi{}162$ and with periodic boundary conditions. The present formulation of the approach applies to generic tensor network algorithms.

Superfluidity and magnetism in two-dimensional fermionic optical lattice systems

We study the ground-state properties of the two-dimensional Hubbard model with a confining potential. To discuss how the attractive (repulsive) interactions induce a spatially modulated superfluid (magnetically ordered) state at zero temperature, we use the variational Monte Carlo method, which can incorporate not only onsite but also intersite correlations. We then elucidate how the intersite correlations affect the ground-state properties of the system.

Competition between Anderson Localization and Antiferromagnetism in Correlated Lattice Fermion Systems with Disorder

The magnetic ground state phase diagram of the disordered Hubbard model at half-filling is computed in dynamical mean-field theory supplemented with the spin resolved, typical local density of states. The competition between many-body correlations and disorder is found to stabilize paramagnetic and antiferromagnetic metallic phases at weak interactions. Strong disorder leads to Anderson localization of the electrons and suppresses the antiferromagnetic long-range order. Slater and Heisenberg antiferromagnets respond characteristically differently to disorder. The results can be tested with cold fermionic atoms loaded into optical lattices.

Ferromagnetism and Metal-Insulator Transitions in Correlated Electron Systems with Alloy Disorder

Ferromagnetism and Mott–Hubbard metal–insulator transitions (MIT) in charged lattice fermion systems originate from strong repulsive interactions. Both phenomena occur at intermediate coupling strengths and are notoriously difficult to investigate since they require the application of non–perturbative theoretical methods. Further complications arise in the presence of disorder. Indeed, even in the absence of any interactions the disorder–induced delocalization–localization transition, e.g., caused by a random distribution of two different atoms in an alloy (“alloy disorder”), occurs at a disorder strength comparable to the band width. For this reason disorder also requires a non–perturbative treatment. The simultaneous presence of interactions and disorder therefore leads to a highly non–trivial many–body problem [1, 2, 3, 4] which is still far from understood. In this paper we review our recent investigations of interacting, alloy– disordered lattice fermions. By solving the Hubbard model and the periodic Anderson model (PAM) within the dynamical mean–field theory (DMFT), we show that alloy disorder can lead to (i) non–monotonic changes of the Curie temperature Tc as a function of some control parameter, and even to an enhancement of Tc compared to the non–disordered

Supersolid State of Ultracold Fermions in Optical Lattice

We study ultracold fermionic atoms trapped in an optical lattice with harmonic confinement by dynamical mean-field approximation. It is demonstrated that a supersolid state, where an s-wave superfluid coexists with a density-wave state with a checkerboard pattern, is stabilized by attractive onsite interactions on a square lattice. Our new finding here is that a confining potential plays an invaluable role in stabilizing the supersolid state. We establish a rich phase diagram at low temperatures, which clearly shows how an insulator, a density wave and a superfluid compete with each other to produce an interesting domain structure. Our results shed light on the possibility of the supersolid state in fermionic optical lattice systems.

Asymptotic distinguishability measures for shift-invariant quasifree states of fermionic lattice systems

We apply the recent results of Hiai et al. [J. Math. Phys. 49, 032112 (2008)] to the asymptotic hypothesis testing problem of locally faithful shift-invariant quasifree states on a CAR algebra. We use a multivariate extension of Szegő’s theorem to show the existence of the mean Chernoff and Hoeffding bounds and the mean relative entropy and show that these quantities arise as the optimal error exponents in suitable settings.

Entanglement of fermions at quantum criticality

The study of entanglement properties of quantum critical many-particle systems has become a subject of intense interest. While the basic features of entanglement scaling for critical spin-1/2 systems (coupled qubits) are by now fairly well understood, entanglement properties of critical fermions (or bosons) are less well studied. In an effort to contribute to this problem, we have analyzed the single-site entanglement of a generic spin-1/2 lattice fermion system and found that (under certain provisos) this measure can be used as a reliable marker to identify and partly characterize a quantum critical point. We illustrate our findings by exact analytical results for the single-site entanglement at the magnetic and Mott-Hubbard transitions of the 1D Hubbard model and at the Mott-Hubbard transitions of the 1D Hubbard model with long-range hopping.

Statistics Dependence of the Entanglement Entropy

The entanglement entropy of a distinguished region of a quantum many-body system reflects the entanglement in its pure ground state. Here we establish scaling laws for this entanglement in critical quasifree fermionic and bosonic lattice systems, without resorting to numerical means. We consider the setting of D-dimensional half-spaces which allows us to exploit a connection to the one-dimensional case. Intriguingly, we find a difference in the scaling properties depending on whether the system is bosonic-where an area law is proven to hold-or fermionic where we determine a logarithmic correction to the area law, which depends on the topology of the Fermi surface. We find Lifshitz quantum phase transitions accompanied with a nonanalyticity in the prefactor of the leading order term.

Strong pairing and microscopic inhomogeneity of lattice fermion systems

In order to study an interplay between electronic inhomogeneity and superconductivity as seen in high-Tc cuprates, we numerically examine attractive Hubbard model in the absence of the translation symmetry. We systematically calculate the particle density profile for both 1-D and 2-D attractive Hubbard models with harmonic potential wells using the exact diagonalization and the density-matrix renormalization group methods. The numerical results reveal that fine inhomogeneous zig-zag patterns universally emerge in the 1-D model case and the zig-zag structure becomes checkerboard type in the 2-D one. Moreover, it is numerically and theoretically found that such inhomogeneities are caused by double occupation component, i.e., pairs tightly bound on a site.

Entanglement scaling in lattice systems

We review some recent rigorous results on scaling laws of entanglement properties in quantum many body systems. More specifically, we study the entanglement of a region with its surrounding and determine its scaling behaviour with its size for systems in the ground and thermal states of bosonic and fermionic lattice systems. A theorem connecting entanglement between a region and the rest of the lattice with the surface area of the boundary between the two regions is presented for non-critical systems in arbitrary spatial dimensions. The entanglement scaling in the field limit exhibits a peculiar difference between fermionic and bosonic systems. In one-spatial dimension a logarithmic divergence is recovered for both bosonic and fermionic systems. In two spatial dimensions in the setting of half-spaces however we observe strict area scaling for bosonic systems and a multiplicative logarithmic correction to such an area scaling in fermionic systems. Similar questions may be posed and answered in classical systems.

Ramping fermions in optical lattices across a Feshbach resonance

We study the properties of ultracold Fermi gases in a three-dimensional optical lattice when crossing a Feshbach resonance. By using a zero-temperature formalism, we show that three-body processes are enhanced in a lattice system in comparison to the continuum case. This poses one possible explanation for the short molecule lifetimes found when decreasing the magnetic field across a Feshbach resonance. Effects of finite temperatures on the molecule formation rates are also discussed by computing the fraction of double-occupied sites. Our results show that current experiments are performed at temperatures considerably higher than expected: lower temperatures are required for fermionic systems to be used to simulate quantum Hamiltonians. In addition, by relating the double occupancy of the lattice to the temperature, we provide a means for thermometry in fermionic lattice systems, previously not accessible experimentally. The effects of ramping a filled lowest band across a Feshbach resonance when increasing the magnetic field are also discussed: fermions are lifted into higher bands due to entanglement of Bloch states, in good agreement with recent experiments.

Entanglement entropy and the Berry phase in the solid state

The entanglement entropy (von Neumann entropy) has been used to characterize the complexity of many-body ground states in strongly correlated systems. In this paper, we try to establish a connection between the lower bound of the von Neumann entropy and the Berry phase defined for quantum ground states. As an example, a family of translational invariant lattice free fermion systems with two bands separated by a finite gap is investigated. We argue that, for one dimensional (1D) cases, when the Berry phase (Zak's phase) of the occupied band is equal to $\pi \times ({odd integer})$ and when the ground state respects a discrete unitary particle-hole symmetry (chiral symmetry), the entanglement entropy in the thermodynamic limit is at least larger than $\ln 2$ (per boundary), i.e., the entanglement entropy that corresponds to a maximally entangled pair of two qubits. We also discuss this lower bound is related to vanishing of the expectation value of a certain non-local operator which creates a kink in 1D systems.

Critical Temperature and Thermodynamics of Attractive Fermions at Unitarity

The unitarity regime of the BCS-BEC crossover can be realized by diluting a system of two-component lattice fermions with an on-site attractive interaction. We perform a systematic-error-free finite-temperature simulation of this system by diagrammatic determinant Monte Carlo method. The critical temperature in units of Fermi energy is found to be T(C)/epsilonF=0.152(7). We also report the behavior of the thermodynamic functions, and discuss the issues of thermometry of ultracold Fermi gases.

Single-site entanglement of fermions at a quantum phase transition

We show that the single-site entanglement of a generic spin-1/2 fermionic lattice system can be used as a reliable marker of a finite-order quantum phase transition, given certain provisos. We discuss the information contained in the single-site entanglement measure, and provide illustrations from the Mott--Hubbard metal-insulator transitions of the one-dimensional (1D) Hubbard model, and the (1D) Hubbard model with long-range hopping.

Equilibrium States and Their Macroscopic Uniformity in Fermion Lattice Systems

We study equilibrium states and their macroscopic uniformity in the fermion lattice system satisfying the assumptions presented by Araki and Moriya [5]. We show that every translation-invariant KMS state of the even part of the fermion algebra extends uniquely to a translation-invariant KMS state of the whole fermion algebra. The macroscopic uniformity is established for an extremal translation-invariant equilibrium state of the fermion algebra.

The Stability of the Non-Equilibrium Steady States

We show that the non-equilibrium steady state (NESS) of the free lattice Fermion model far from equilibrium is macroscopically unstable. The problem is translated to that of the spectral analysis of {\it Liouville Operator}. We use the method of positive commutators to investigate it. We construct a positive commutator on the lattice Fermion system, whose dispersion relation is $\omega(k)=\cos k-\gamma$.

EQUILIBRIUM STATISTICAL MECHANICS OF FERMION LATTICE SYSTEMS

We study equilibrium statistical mechanics of Fermion lattice systems which require a different treatment compared with spin lattice systems due to the non-commutativity of local algebras for disjoint regions.Our major result is the equivalence of the KMS condition and the variational principle with a minimal assumption for the dynamics and without any explicit assumption on the potential. Its proof applies to spin lattice systems as well, yielding a vast improvement over known results.All formulations are in terms of a C*-dynamical systems for the Fermion (CAR) algebra [Formula: see text] with all or a part of the following assumptions:(I) The interaction is even, namely, the dynamics αtcommutes with the even-oddness automorphism Θ. (Automatically satisfied when (IV) is assumed.)(II) The domain of the generator δαof αtcontains the set [Formula: see text] of all strictly local elements of [Formula: see text].(III) The set [Formula: see text] is the core of δα.(IV) The dynamics αtcommutes with lattice translation automorphism group τ of [Formula: see text].A major technical tool is the conditional expectation from [Formula: see text] onto its C*-subalgebras [Formula: see text] for any subset I of the lattice, which induces a system of commuting squares. This technique overcomes the lack of tensor product structures for Fermion systems and even simplifies many known arguments for spin lattice systems.In particular, this tool is used for obtaining the isomorphism between the real vector space of all *-derivations with their domain [Formula: see text], commuting with Θ, and that of all Θ-even standard potentials which satisfy a specific norm convergence condition for the one point interaction energy. This makes it possible to associate a unique standard potential to every dynamics satisfying (I) and (II). The convergence condition for the potential is a consequence of its definition in terms of the *-derivation and not an additional assumption.If translation invariance is imposed on *-derivations and potentials, then the isomorphism is kept and the space of translation covariant standard potentials becomes a separable Banach space with respect to the norm of the one point interaction energy. This is a crucial basis for an application of convex analysis to the equivalence proof in the major result.Everything goes in parallel for spin lattice systems without the evenness assumption (I).

Calculation of reduced density matrices from correlation functions

It is shown that for solvable fermionic and bosonic lattice systems, the reduced density matrices can be determined from the properties of the correlation functions. This provides the simplest way to these quantities which are used in the density-matrix renormalization group method.

Local Thermodynamical Stability of Fermion Lattice Systems

Within the framework for equilibrium statistical mechanics of Fermion lattice systems formulated in our preceding work, we study the local thermodynamical stability (LTS) as an alternative characterization of equilibrium states, which works without the translation invariance assumption for the states.