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arXiv:1808.01314 · Physical Review A

Impenetrable SU(N) fermions in one-dimensional lattices

Yicheng Zhang, Lev Vidmar, Marcos Rigol

Published Oct 25, 2018• 15 citations
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Abstract

We study $\mathrm{SU}(N)$ fermions in the limit of infinite on-site repulsion between all species. We focus on states in which every pair of consecutive fermions carries a different spin flavor. Since the particle order cannot be changed (because of the infinite on-site repulsion) and contiguous fermions have a different spin flavor, we refer to the corresponding constrained model as the model of distinguishable quantum particles. We introduce an exact numerical method to calculate equilibrium one-body correlations of distinguishable quantum particles based on a mapping onto noninteracting spinless fermions. In contrast to most many-body systems in one dimension, which usually exhibit either power-law or exponential decay of off-diagonal one-body correlations with distance, distinguishable quantum particles exhibit a Gaussian decay of one-body correlations in the ground state, while finite-temperature correlations are well described by stretched exponential decay.

DOI
10.1103/physreva.98.042129

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We study SU(N)\mathrm{SU}(N)SU(N) fermions in the limit of infinite on-site repulsion between all species. We focus on states in which every pair of consecutive fermions carries a different spin flavor. Since the particle order cannot be changed (because of the infinite on-site repulsion) and contiguous fermions have a different spin flavor, we refer to the corresponding constrained model as the model of distinguishable quantum particles. We introduce an exact numerical method to calculate equilibrium one-body correlations of distinguishable quantum particles based on a mapping onto noninteracting spinless fermions. In contrast to most many-body systems in one dimension, which usually exhibit either power-law or exponential decay of off-diagonal one-body correlations with distance, distinguishable quantum particles exhibit a Gaussian decay of one-body correlations in the ground state, while finite-temperature correlations are well described by stretched exponential decay.
We study quantum quench dynamics in the Fermi-Hubbard model, and its SU(N)\mathrm{SU}(N)SU(N) generalizations, in one-dimensional lattices in the limit of infinite onsite repulsion between all flavors. We consider families of initial states with generalized N\'eel order, namely, initial states in which there is a periodic NNN-spin pattern with consecutive fermions carrying distinct spin flavors. We introduce an exact approach to describe the quantum evolution of those systems, and study two unique transient phenomena that occur during expansion dynamics in finite lattices. The first one is the dynamical emergence of Gaussian one-body correlations during the melting of sharp (generalized) N\'eel domain walls. Those correlations resemble the ones in the ground state of the SU(N)\mathrm{SU}(N)SU(N) model constrained to the same spin configurations. This is explained using an emergent eigenstate solution to the quantum dynamics. The second phenomenon is the transformation of the quasimomentum distribution of the expanding strongly interacting SU(N)\mathrm{SU}(N)SU(N) gas into the rapidity distribution after long times. Finally, we study equilibration in SU(N)\mathrm{SU}(N)SU(N) gasses and show that observables after equilibration are described by a generalized Gibbs ensemble. Our approach can be used to benchmark analytical and numerical calculations of dynamics of strongly correlated SU(N)\mathrm{SU}(N)SU(N) fermions at large UUU.
We develop a self-consistent variant of the constrained path quantum Monte Carlo approach which ensures its independence of the trial wave function, and apply the method to compute ground-state correlations in the two-dimensional SU(3) Fermi-Hubbard Hamiltonian at 13\frac{1}{3}31​ filling, modeling fermions with three possible spin flavors moving on a square lattice with an average of one particle per site. We provide clear evidence of a quantum critical point separating a nonmagnetic uniform metallic phase from a regime where long-range ``spin'' order is present. This discovery of multiple successive transitions to magnetic states with regular, long-range alternation of the different flavors, whose symmetry changes as the interaction strength increases, significantly extends previous work in the Heisenberg limit to itinerant fermions. In addition to the rich quantum magnetism, this important physical system allows one to study integer filling and the associated Mott transition disentangled from nesting, in contrast to the usual SU(2) model, while preserving the square-lattice geometry. Our results also provide a significant step towards the interpretation of present and future experiments on fermionic alkaline-earth atoms, and other realizations of SU(N)\mathrm{SU}(N)SU(N) physics.
Spin-Charge Subordination in the Infinite-UUU SU(N)SU(N)SU(N) Hubbard Chain
We obtain the exact full counting statistics of charge and flavor transport in the one-dimensional infinite-UUU SU(N)SU(N)SU(N) Hubbard model. The no-passing constraint freezes the ordered flavor sequence while the particle coordinates evolve as free spinless fermions. Consequently, charge transfer is governed by a free-fermion determinant, whereas flavor transfer is exactly subordinated to the number of particles crossing the observation cut. For an arbitrary traceless Cartan generator this yields an exact relation between the charge and flavor cumulants. The complete asymptotic flavor distribution is a universal non-Gaussian M-Wright law. Together, these results establish spin--charge subordination as a kinematic mechanism for anomalously slow internal-state transport in one-dimensional impenetrable quantum gases.
Fermionization is what happens when repulsive interacting bosons in one spatial dimension feel infinite interaction. Due to their strong repulsion the bosons try to minimize their spatial overlap like the fermions. Fermionized bosons and noninteracting fermions resemble each other and we trace these similarities of strongly correlated one-dimensional bosonic systems and fermionic system by exploring their ground state properties using the one- and two-body density matrix in coordinate space. We solve the N-body Schrödinger equation accurately and from first principles using the multiconfigurational time-dependent Hartree for bosons (MCTDHB) method. Using the one- and two-body density in momentum space, fermionized bosons can be distinguished from noninteracting fermions.
In the framework of exactly solvable quantum many-body systems we study models of interacting spin one-half Fermions in one dimension. The first part deals with systems of spin one-half Fermions which interact via repulsive contact interaction. A reformulation of the Bethe-Ansatz solvable many-body wave function is presented. This simplifies considerably the calculations for the highly imbalanced case, where very few particles of one species (minority Fermions) are present. For the other particle species (majority Fermions) the thermodynamic limit is taken. We assume the majority Fermions to be in the ground state such that their non-interacting momentum distribution is a Fermi-sea. Upon this we consider excitations where the particles of the minority species may occupy an arbitrary state within the Fermi-sea. In the case of only a single minority Fermion, the many-body wave function can be expressed as a determinant. This allows us to derive exact thermodynamic expressions for several expectation values as well as for the density-density correlation function. Moreover it is possible to find closed expressions for the single particle Green's function. All of the above mentioned quantities show a non-trivial dependence on the minority particle's momentum. In particular the Green's function in the Tonks-Girardeau regime of hardcore interaction is shown to undergo a transition from the one of impenetrable Bosons to that of free Fermions as the extra particle's momentum varies from the core to the edge of the Fermi-sea. This transition becomes manifest in an algebraic asymptotic decay of the Green's function. If two minority Fermions are present, the many-body wave function turns out to be more complicated. Nevertheless it is possible to derive exact expressions for the two and the three particle density-density correlation functions. Furthermore we calculate the system's total energy and based on that, identify terms which have a natural interpretation as effective interaction energy for the two minority Fermions in the presence of the Fermi-sea. The second part is devoted to the study of one-dimensional systems consisting of two fermionic particle species with different masses. We show that for specific kinds of interaction potentials and for certain relations between the masses and the coupling constants, the particle creation and annihilation operators of such a system can be constructed exactly.
Motivated by the possibility to load multicolor fermionic atoms in optical lattices, we study the entropy dependence of the properties of the one-dimensional antiferromagnetic SU(N) Heisenberg model, the effective model of the SU(N) Hubbard model with one particle per site (filling 1/N) in the large U/t limit. Using continuous-time world-line Monte Carlo simulations for N=2-5, we show that characteristic short-range correlations develop at low temperature as a precursor of the ground state algebraic correlations. We also calculate the entropy as a function of temperature, and we show that the first sign of short-range order appears at an entropy per particle that increases with N and already reaches 0.8k(B) at N=4, in the range of experimentally accessible values.