We investigate two exotic states in condensed matter: mesoscopic magnetism of integrable systems and Cooper pairing mediated by multiple-spin exchange.To study mesoscopic magnetism of integrable systems, we investigate, analytically and numerically, the orbital magnetism of free electron gas in a rectangular box, which is, classically, a model integrable system.We find that both the average orbital susceptibility and its fluctuations are determined by the two-level van Vleck susceptibility that involves the last occupied (Fermi) level and the first unoccupied level.This is in line with previous results for disordered (classically chaotic) systems.The mesoscopic fluctuations, however, are much larger in integrable systems and there is a pronounced non-self-averaging effect for phyical quantities in integrable systems.We find that, in units of µ 2 B /∆, where ∆ is the mean-level-spacing, the average orbital susceptibility isis the orbital angular momentum, N is the number of electrons in the box, and ε f -ε i is the level spacing (in units of ∆) between the last occupied level and the first unoccupied level.(ε f -ε i ) -1 is principally divergent and hence determined by the relevant cut-off.i
We investigate two exotic states in condensed matter: mesoscopic magnetism of integrable systems and Cooper pairing mediated by multiple-spin exchange.To study mesoscopic magnetism of integrable systems, we investigate, analytically and numerically, the orbital magnetism of free electron gas in a rectangular box, which is, classically, a model integrable system.We find that both the average orbital susceptibility and its fluctuations are determined by the two-level van Vleck susceptibility that involves the last occupied (Fermi) level and the first unoccupied level.This is in line with previous results for disordered (classically chaotic) systems.The mesoscopic fluctuations, however, are much larger in integrable systems and there is a pronounced non-self-averaging effect for phyical quantities in integrable systems.We find that, in units of µ 2 B /∆, where ∆ is the mean-level-spacing, the average orbital susceptibility isis the orbital angular momentum, N is the number of electrons in the box, and ε f -ε i is the level spacing (in units of ∆) between the last occupied level and the first unoccupied level.(ε f -ε i ) -1 is principally divergent and hence determined by the relevant cut-off.i
The progress of ultracold atoms renders numerous possibilities to investigate exotic magnetism and superfluidity, which are rarely observed in solid state systems. In this thesis, we will introduce two novel physical descriptions: "frustrated Cooper pairing" and "large-hyperfine spin physics". Geometric frustration in quantum magnetism refers to which magnetic interactions on different bonds cannot be simultaneously minimized, and usually Cooper pairing favors uniform phases among different lattice sites. Here, we introduce "frustration" in Cooper pairing in a fermionic p-orbital model. By mean- field calculations, we show that the system exhibits behavior analogous to frustrated magnetism, and an unconventional supersolid state with the f-wave symmetry. Next, we introduce large-spin physics. In usual condensed matter systems, large spin is not intriguing because large values of spin suppress quantum fluctuations. In contrast, in ultracold fermion systems, large-hyperfine spin enhances quantum fluctuations and brings exotic quantum magnetism. Here the simplest large-spin fermionic system, a spin-3/2 exchange system is proposed, which can be characterized by an Sp(4)/SO(5) symmetry. In one dimension, the ground states exhibit either a dimerized state with a finite spin gap or a gapless spin liquid state by means of the density matrix renormalization group method. In the latter case, the spin-spin correlation functions are identified to have 4-site periodicities, which behaves similarly to the SU(4) chain. In two dimension, we infer that there exist three competing phases: Neel ordering, columnar dimerization and 2 x 2 plaquette formation, in the thermodynamic limit by exact diagonalization calculation on small sizes. Finally we perform the projector Quantum Monte Carlo method to study another large-spin system: the half-filled SU(N) Hubbard model. We show that at half-filling there is no sign problem such that our simulations are accurate. By finite size scaling, it is clearly found that the magnetic Neel ordering can exist not only for N = 2 but also in the N = 4 case at strong interactions. For N ̲> 6 or N = 4 at small U, the numerical results do not have any prominent signal that the long-range ordering exists in the thermodynamic limit. Due to strong finite size effects and finite numerical accuracy, however, we are unable to make any conclusion to identify the physics in the regimes
The nature of superfluid, superconducting, and magnetic ordering is elucidated for mesoscopic systems in which the single-particle level spacing is much larger than both the temperature and the critical temperature of ordering. Ordering is defined as a spontaneous violation of symmetry, the gauge invariance and time reversal being by definition symmetries violated in superfluidity (superconductivity) and magnetism contexts, respectively. Superfluidity and superconductivity are realized in thermodynamic equilibrium states with a non-integral average number of particles and are accompanied by the spontaneous violation of time homogeneity. In Fermi systems two types of superfluidity and superconductivity are possible which are characterized by the presence of pair or single-particle 'condensates'. The latter is remarkable in that spontaneous violation of fundamental symmetries such as spatial 2? rotation and double time reversal takes place. Possible experiments on metallic nanoparticles and ultracold atomic gases in magnetic traps are discussed.
One of the central goals of condensed matter theory is to understand the behavior of electrons under various circumstances. This dissertation focuses on the some relatively unexplored effects brought by lattices, in a general setting.One ingredient that lattices introduce is the orbital degrees of freedom, resulting from the interaction between electrons and background ions. We have non-perturbatively studied a strongly coupled multi-orbital electron liquid, which has a fully polarized ground state. Both Hund’s coupling and electron itinerancy are essential to building up the global spin coherence. At finite temperature, the system exhibits both Curie-Weiss type spin susceptibility and finite compressibility, demonstrating the coexistence of spin coherence and charge incoherence. At low temperature, the single particle excitation extends over the entire Brillouin zone rather than the vicinity of the Fermi surface due to the spin fluctuation.Another dramatic effect of lattices on electrons is the formation of Mott insulators. We consider a special type of Mott insulators occurring at non-integer fillings, which has a finite single-particle gap but strong local charge fluctuation. As an example, we studied 1D Hubbard model with SU(N) symmetry at half-filling. The systems are insulators for all values of N, but the filling is half-integer when N is odd. We demonstrate that the systems with N even and N odd exhibit distinct energy scales of charge and spin gaps. In the case of odd N, the local charge fluctuation leads to spin gap much greater than the super exchange energy scale. The difference between the systems with opposite parity of N is gradually smeared out as N increases.The symmetry of lattices can also enforce band touchings and band crossing, leading to nodal-line and nodal-point semi-metals. We explore the symmetry constraints on electrons imposed by generalized lattices with space-time mixing periodicities. which include the Floquet lattice systems as a special case. Compared to space and magnetic groups, the symmetry group of such systems is augmented by “time-screw” rotations and “time-glide” reflections involving fractional time translations. A complete classification of the 13 space-time groups in 1+1D and 275 space-time groups in 2+1D are performed. Kramers-type degeneracy can arise from space-time symmetries without the half-integer spinor structure, which constrains the winding number patterns of spectral dispersions. In 2+1D, non-symmorphic space-time symmetries enforce spectral degeneracies, leading to protected Floquet semi-metal states.
We address the question of real space pairing induced by exchange interaction, customarily associated with the presence of the broken-symmetry magnetism. For such mechanism of pairing, the appearance of superconductivity (or superfluidity) without magnetism presents itself still as a highly nontrivial problem and it seems, it can be practically realized only when the kinetic energy of individual-particle motion is comparable or even smaller than the magnitude of exchange interaction that takes place among strongly correlated fermions , where the kinetic exchange and/or the Hund's rule coupling can become relatively strong in the vicinity of the onset of the Mott insulating state. In that situation, the pairing arises from a coherent correlated pair motion rather than from a free motion of a tightly bound pair. We elaborate on this question in general terms, as well as provide selected results in the mean-field approximation in concrete situations.
The spectrum of collective (pair) excitations in the ground state of a dilute twodimensional (2D) attractive Fermi-gas is studied within the functional integral formalism. The linearized equations for the fluctuations about the non-trivial saddle point are analyzed for all coupling regimes, which are characterized by the ratio ε 0 /ε F , where ε 0 is the two-fermion binding energy and ε F is the Fermi energy. The approximation takes into account propagation of the fluctuations and their interaction with the condensate. In the strong-coupling, or ‘;Bosegas’, regime (ε 0 /ε F > 1) the spectrum is continuous and has the Bogolubov form, but in the weak-coupling limit (ε 0 /ε F «1) there are two types of excitations (different from the two-fermion scattering states): (i) long-wavelength sound-like excitations with the cut-off at momentum q c ≃ l/ξ 0 (where ξ 0 is the Cooper pair size), and (ii) pair excitations with q ≃ (8 m μ) ½ , where μ is the chemical potential and m is the fermion mass. The crossover between weakand strong-coupling behavior of the excitation spectrum is found to occur at the value of the coupling parameter ε 0 /ε F ≃¼. Introduction Since the discovery of high- T c superconductivity there has been growing interest in studying 2D Fermi gases with attractive interaction, especially in the crossover regime, when the pair size ξ 0 is of the order of the interparticle distance. The importance of such a model, which can be regarded as a semiphenomenological model of a 2D superconductor, is highlighted by experimental evidence that the high- T c superconductors, most of which have a layered structure, have a short coherence length, so that k F ξ 0 ≃1, where k F is the Fermi momentum.
Bond-operator analytical approach for the t-J model
We present a bond-operator theory (BOT) for analytical consideration of the t-J model and its extensions with longer-rage hopping terms. This technique is based on previously suggested representation of electron operators via localized spins 1/2 and spinless Fermi-operators of mobile holons which requires no constraint between them. We introduce a representation of operators of spins, holons, and electrons in (extended) unit cell containing several lattice sites via a zoo of Bose- and Fermi-operators acting in the Hilbert space of all quantum states of the whole unit cell. BOT provides a regular expansion of physical quantities in powers of 1/n using conventional diagrammatic technique, where n≥1 is the maximum number of introduced quasiparticles (bosons and fermions) which can occupy a unit cell. The suggested representation reproduces commutation algebra of all operators at any n>0 and allows to consider both magnetically ordered and disordered phases. Some elementary excitations described in the BOT by separate bosons or fermions appear in common approaches as bound states of conventional quasiparticles. In particular, there are two-hole bound states (Cooper pairs of two holes) which are described within the BOT by separate bosons. We discuss in detail properties of the t-J model on the square lattice with no more than two holes (polarons). Although the expansion parameter 1/n is not small in the physically meaningful case of n=1, we obtain a good quantitative agreement with previous numerical findings at n=1 even in the first order in 1/n after taking into account a few simple diagrams. Self-consistent calculations in the first order in 1/n bring our results to a very good quantitative agreement with previous numerical findings of the ground state energy, staggered magnetization, and spectra of magnons, polarons, and lowest-energy two-hole bound states.