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Physical review. B, Condensed matter

Renormalization-group study of a concentrated fluctuating-valence model

Werner R. Hanke, J. E. Hirsch

Published Jun 1, 1982• 28 citations
Publisher

Abstract

The ground state and gap properties of a spinless model of $f$ and $d$ electrons with hybridization $V$ and $f\ensuremath{-}d$ interaction ${U}_{\mathrm{fd}}$ are studied in one dimension using a real-space renormalizationgroup (RG) method. To understand the dependence of the RG results on the cell size chosen, we first study the problem for ${U}_{\mathrm{fd}}=0$ and arbitrary cell size $N$. It is found that even-site cells give qualitatively incorrect results for the gap, while odd-site cells (as small as $N=3$) reproduce very accurately the exact results for the gap and give a reasonable upper bound for the groundstate energy. On the basis of this result the phase diagram is studied as a function of arbitrary $V$ and ${U}_{\mathrm{fd}}$ using three site cells. We consider the half-filled band situation for which the ground state is found to be insulating for all values of ${U}_{\mathrm{fd}}(\ensuremath{\ne}0)$. The $f$-electron occupation number as a function of ${E}_{f}$ changes continuously, confirming recent mean-field results. This result indicates that the interacting $f\ensuremath{-}d$ system does not scale to a single impurity model as argued in previous studies. Using a Jordan-Wigner transformation from spin to fermion operators, the Kondo necklace model is mapped onto a fluctuating valence model and the transition from ferromagnetic to Kondo-state behavior is numerically studied using $N=3$.

DOI
10.1103/physrevb.25.6748

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The ground state and gap properties of a spinless model of fff and ddd electrons with hybridization VVV and f\ensuremath−df\ensuremath{-}df\ensuremath−d interaction Ufd{U}_{\mathrm{fd}}Ufd​ are studied in one dimension using a real-space renormalizationgroup (RG) method. To understand the dependence of the RG results on the cell size chosen, we first study the problem for Ufd=0{U}_{\mathrm{fd}}=0Ufd​=0 and arbitrary cell size NNN. It is found that even-site cells give qualitatively incorrect results for the gap, while odd-site cells (as small as N=3N=3N=3) reproduce very accurately the exact results for the gap and give a reasonable upper bound for the groundstate energy. On the basis of this result the phase diagram is studied as a function of arbitrary VVV and Ufd{U}_{\mathrm{fd}}Ufd​ using three site cells. We consider the half-filled band situation for which the ground state is found to be insulating for all values of Ufd(\ensuremath≠0){U}_{\mathrm{fd}}(\ensuremath{\ne}0)Ufd​(\ensuremath=0). The fff-electron occupation number as a function of Ef{E}_{f}Ef​ changes continuously, confirming recent mean-field results. This result indicates that the interacting f\ensuremath−df\ensuremath{-}df\ensuremath−d system does not scale to a single impurity model as argued in previous studies. Using a Jordan-Wigner transformation from spin to fermion operators, the Kondo necklace model is mapped onto a fluctuating valence model and the transition from ferromagnetic to Kondo-state behavior is numerically studied using N=3N=3N=3.
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Two aspects of strongly correlated electron systems are studied in this thesis. Hubbard-type models are thought to be at the basis of high-critical-temperature superconductivity. One interaction which has not had much study is the nearest-neighbour hopping of on-site singlet pairs. We refine earlier renormalization group arguments and, using the density-matrix renormalization group method, numerically confirm their prediction that a spin gap opens at infinitesimal pair-hopping amplitude V > 0 in the one-dimensional tight-binding model. We also find a phase separation transition at a finite V > 0 as well as a spin-gap transition at a finite V < 0. The exotic magnetic behaviour of heavy-fermion materials involves an interplay between the screening of local moments from each other and the formation of a magnetic state of long range order. While the single-impurity Kondo Hamiltonian is thought to model some aspects of this behaviour, the properties of the Kondo lattice model away from half-filling are largely unknown. We determine the presence of a spin-gap region in the phase diagram of the one-dimensional Heisenberg-Kondo lattice model and make predictions about certain concealed dimerization order parameters.
Abstract We present a T = 0 K renormalization group (RG) phase diagram for the electronic Hubbard model in two dimensions on the square lattice at half filling. The RG procedure treats quantum fluctuations in the single particle occupation number nonperturbatively via the unitarily decoupling of one electronic state at every RG step. The resulting phase diagram thus possesses the quantum fluctuation energy scale (ω) as one of its axes. A relation is derived between ω and the effective temperature scale upto which gapless, as well as emergent gapped phases can be obtained. We find that the normal and insulating phases of the half-filled Hubbard model correspond, for any on-site repulsion, to a marginal Fermi liquid normal phase and a topologically-ordered gapped Mott insulating liquid respectively. The marginal Fermi liquid is found to arise from singular forward scattering in directions normal to the nested Fermi surface, while singular backscattering events lead to Mott liquid state. The transition between these two phases involves passage through a pseudogapped phase bookended by Fermi surface topology-changing Lifshitz transitions. The pseudogap phase is observed to arise from the electronic differentiation encoded within the nested Fermi surface, and involves the gradual gapping of the Fermi surface from antinodes to nodes via charge and spin excitations that are mutually entangled. We obtain effective Hamiltonians for various phases, as well as wavefunctions for the low-energy many-body eigenstates of the Mott liquid. Benchmarking of the ground-state energy per particle and the double-occupancy fraction for the Mott liquid against existing numerical results yields excellent agreement. Presence of a Néel ordering symmetry-breaking perturbation in the RG leads to an antiferromagnetic spin-ordered charge insulating Mott state. Our results thus offer novel insights on a variety of aspects of the Mott–Hubbard problem, and can be extended to the doped system.
We use the numerical renormalization group method to investigate the spectral properties of a single-impurity Anderson model with a gap $\ensuremath{\delta}$ across the Fermi level in the conduction-electron spectrum. For any finite $\ensuremath{\delta}>0$, at half filling the ground state of the system is always a doublet. Away from half filling a quantum phase transition (QPT) occurs as function of the gap value $\ensuremath{\delta}$, and the system evolves from the strong-coupling (SC) Kondo-type state, corresponding to $\ensuremath{\delta}<{\ensuremath{\delta}}_{C}$ toward a localized moment (LM) regime for $\ensuremath{\delta}>{\ensuremath{\delta}}_{C}$. The opening of the gap leads to the formation of one (two) bound states when the system is in the SC (LM) regime. The evolution across the QPT of their positions and the corresponding weights together with the dynamic properties of the model are investigated.
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We analyze the antiferromagnetic spin-1/2 XXZ model on the kagome lattice at finite external magnetic field with the help of a non-perturbative zero-temperature renormalization group (RG) technique. The exact nature of the ground and excited state properties (e.g. gapped or gapless spectrum etc) of this system are still debated. Approximate methods have typically been adopted towards understanding the low-energy spectrum. Following the work of Kumar et al (2014 Phys. Rev. B 90 174409), we use a Jordan–Wigner transformation to map the spin problem into one of spinless fermions (spinons) in the presence of a statistical gauge field, and with nearest-neighbor interactions. While the work of Kumar et al was confined mostly to the plateau at 1/3-filling (magnetization per site) in the XY regime, we analyze the role of inter-spinon interactions in shaping the phases around this plateau in the entire XXZ model. The RG phase diagram obtained contains three spin liquid phases whose position is determined as a function of the exchange anisotropy and the energy scale for fluctuations arising from spinon scattering. Two of these spins liquids are topologically ordered states of matter with gapped, degenerate states on the torus. The gap for one of these phases corresponds to the one-spinon band gap of the Azbel–Hofstadter spectrum for the XY part of the Hamiltonian, while the other arises from two-spinon interactions. The Heisenberg point of this problem is found to lie within the interaction gapped spin liquid phase, in broad agreement with a recent experimental finding. The third phase is an algebraic spin liquid with a gapless Dirac spectrum for spinon excitations, and possess properties that show departures from the Fermi liquid paradigm. The three phase boundaries correspond to critical theories, and meet at a SU (2)-symmetric multicritical point. This special critical point agrees well with the gap-closing transition point predicted by Kumar et al . We discuss the relevance of our findings to various recent experiments, as well as results obtained from other theoretical analyses.