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arXiv:cond-mat/9407045 · Physical Review Letters

New Criticality of 1D Fermions

Michael Lässig

Published Jul 25, 1994• 17 citations
arXivHTMLPDF

Abstract

One-dimensional massive quantum particles [or (1 + 1)-dimensional random walks] with short-ranged multiparticle interactions are studied by exact renormalization group methods. With repulsive pair forces, such particles are known to scale as free fermions. With finite $m$-body forces ($m=3,4,\dots{}$), a critical instability is found, indicating the transition to a fermionic bound state. These unbinding transitions represent new universality classes of interacting fermions relevant to polymer and membrane systems. Implications for massless fermions, e.g., in the Hubbard model, are also noted.

DOI
10.1103/physrevlett.73.561

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New Criticality of 1D Fermions [Phys. Rev. Lett. 73, 561 (1994)]

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One-dimensional massive quantum particles [or (1 + 1)-dimensional random walks] with short-ranged multiparticle interactions are studied by exact renormalization group methods. With repulsive pair forces, such particles are known to scale as free fermions. With finite mmm-body forces (m=3,4,…m=3,4,\dots{}m=3,4,…), a critical instability is found, indicating the transition to a fermionic bound state. These unbinding transitions represent new universality classes of interacting fermions relevant to polymer and membrane systems. Implications for massless fermions, e.g., in the Hubbard model, are also noted.
The authors study the phase transition between thermal and many-body localized phases in one-dimensional systems with quenched randomness and short-range interactions. The approach uses a renormalization group model, very similar to those of previous works, that puts the avalanche instability of the many-body localized phase at center stage. Within that model, the universal critical behavior is determined analytically, and the phase transition is found to belong to a new universality class that differs from the previously found Kosterlitz-Thouless result in important ways.
Quantum critical phenomena may be qualitatively different when massless Dirac fermions are present at criticality. Using our recently-discovered fermion-sign-free Majorana quantum Monte Carlo method introduced by us in (Li et al 2015 Phys. Rev. B 91 241117 ), we investigate the quantum critical phenomena of spinless Dirac fermions at their charge-density-wave phase transitions on the honeycomb lattice having sites with largest L = 24. By finite-size scaling, we accurately obtain critical exponents of this so-called Gross–Neveu chiral-Ising universality class of two (two-component) Dirac fermions in 2+1D: , , and , which are qualitatively different from the mean-field results but are reasonably close to the ones obtained from renormalization group calculations.
We investigate the ground-state properties of trapped fermion systems described by the Hubbard model with an external confining potential. We discuss the universal behaviors of systems in different regimes: from few particles, i.e., in dilute regimes, to the trap thermodynamic limit. The asymptotic trap-size (TS) dependence in the dilute regime (increasing the trap size $\ensuremath{\ell}$ keeping the particle number NNN fixed) is described by a universal TS scaling controlled by the dilute fixed point associated with the metal-to-vacuum quantum transition. This scaling behavior is numerically checked by DMRG simulations of the one-dimensional (1D) Hubbard model. In particular, the particle density and its correlations show crossovers among different regimes: for strongly repulsive interactions they approach those of a spinless Fermi gas, for weak interactions those of a free Fermi gas, and for strongly attractive interactions they match those of a gas of hard-core bosonic molecules. The large-NNN limit keeping the ratio $N/\ensuremath{\ell}$ fixed corresponds to a 1D trap thermodynamic limit. We address issues related to the accuracy of the local density approximation (LDA). We show that the particle density approaches its LDA in the large-$\ensuremath{\ell}$ limit. When the trapped system is in the metallic phase, corrections at finite $\ensuremath{\ell}$ are $O({\ensuremath{\ell}}^{\ensuremath{-}1})$ and oscillating around the center of the trap. They become significantly larger at the boundary of the fermion cloud, where they get suppressed as $O({\ensuremath{\ell}}^{\ensuremath{-}1/3})$ only. This anomalous behavior arises from the nontrivial scaling at the metal-to-vacuum transition occurring at the boundaries of the fermion cloud.
We use the two-step density-matrix renormalization group method to elucidate the long-standing issue of the universality class of the Mott transition in the Hubbard model in two dimensions. We studied a spatially anisotropic two-dimensional Hubbard model with a nonperfectly nested Fermi surface at half-filling. We find that unlike the pure one-dimensional case where there is no metallic phase, the quasi-one-dimensional model displays a genuine metal-insulator transition at a finite value of the interaction. The critical exponent of the correlation length is found to be $\ensuremath{\nu}\ensuremath{\approx}1.0$. This implies that the fermionic Mott transition belongs to the universality class of the 2D Ising model.
The behavior of low-dimensional manifolds such as interfaces, membranes, or polymers in an external potential is studied in d=${\mathit{d}}_{\mathrm{\ensuremath{\parallel}}}$+${\mathit{d}}_{\mathrm{\ensuremath{\perp}}}$ dimensions. If the potential has several minima, the manifolds can undergo nontrivial transitions from a localized to a delocalized state at a characteristic temperature T=T∗{\mathit{T}}_{\mathrm{*}}T∗​. For interfaces in d=1+1, several universality classes must be distinguished which can be determined exactly by transfer-matrix methods. The same classification also applies to (self-intersecting) polymers in d=1+${\mathit{d}}_{\mathrm{\ensuremath{\perp}}}$. Within a functional renormalization-group approach, the critical behavior is governed by a line of fixed points in close analogy with unbinding transitions. However, in contrast to unbinding, this fixed point line now contains the trivial fixed point at which one can determine the spectrum of eigenperturbations exactly. The number of relevant eigenperturbations increases monotonically with the decay exponent \ensuremath{\tau}=${\mathit{d}}_{\mathrm{\ensuremath{\parallel}}}$/\ensuremath{\zeta} where the roughness exponent \ensuremath{\zeta} characterizes the delocalized manifold.