Fractionalization Waves in Two-Dimensional Dirac Fermions: Quantum Imprint from One Dimension

Seth M. Davis and Matthew S. Foster

Phys. Rev. Lett. 122, 065302 – Published 14 February 2019

Abstract

Particle fractionalization is believed to orchestrate the physics of many strongly correlated systems, yet its direct experimental detection remains a challenge. We propose a simple measurement for an ultracold matter system, in which correlations in initially decoupled 1D chains are imprinted via quantum quench upon two-dimensional Dirac fermions. Luttinger liquid correlations launch relativistic “fractionalization waves” along the chains, while coupling noninteracting chains induces perpendicular dispersion. These could be easily distinguished in an ultracold gas experiment.

Received 14 September 2018

DOI:https://doi.org/10.1103/PhysRevLett.122.065302

Physics Subject Headings (PhySH)

Cold atoms & matter waves Cold gases in optical lattices Quantum quench

Quantum wires

Bosonization Luttinger liquid model

Condensed Matter, Materials & Applied PhysicsAtomic, Molecular & OpticalStatistical Physics & Thermodynamics

Authors & Affiliations

Seth M. Davis¹ and Matthew S. Foster^1,2

¹Department of Physics and Astronomy, Rice University, Houston, Texas 77005, USA
²Rice Center for Quantum Materials, Rice University, Houston, Texas 77005, USA

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Issue

Vol. 122, Iss. 6 — 15 February 2019

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Images

Figure 1
Lattice setup for quench-induced fractionalization waves. We consider fermions hopping on a pi-flux square lattice (left), with horizontal bonds of strength $J$ and sign-staggered vertical bonds of strength $b J$ . The Hamiltonian for the noninteracting model is given by Eq. (1). We quench from the decoupled chain limit ( $b = 0$ ) to $b > 0$ . We assume that interactions can induce Luttinger liquid correlations (charge fractionalization) along the chains in the prequench state. The latter are imprinted by the quench upon the pi-flux band fermions, leading to the two-dimensional density wave dynamics depicted in Fig. 2 (fractionalized) and Fig. 3 (not fractionalized, initially noninteracting). The right panels in this figure show the pre- and postquench energy bands. The prequench state of decoupled chains is characterized by vertical nodal lines. The postquench band gaps these out, except for a pair of Dirac points at ${k_{x}, k_{y}} = {\pm π / 2 a, 0}$ . Energy bands are depicted over the reduced Brillouin zone of the pi-flux lattice; $a$ denotes the lattice constant.
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Figure 2
Quench from decoupled chains to the pi-flux lattice I: horizontal, “relativistic” fractionalization waves. A positive Gaussian density bump is superimposed by an external potential on top of decoupled Luttinger liquids in the initial state. Via instantaneous quench, the interchain coupling is turned on, while the bump is released by turning off the potential. In this figure we plot the time evolution of the post-quench density profile $ρ (t, x, y)$ , assuming charge fractionalization of the initial Luttinger liquid. Left (right) panels show the density in profile (contour) plots; negative density means a depletion of the filled Fermi sea. Total particle number is conserved [37]. The above plots give time-slice profiles at $t = 0$ , 3, 6, 9, 12 (where $t$ is in units of $v_{F} / a$ ). The degree of fractionalization is characterized by the fermion anomalous dimension $η = 0.7$ [Eq. (5)]. We also incorporate a small but positive initial Fermi momentum, $k_{F} = 0.1 / a$ . The other initial configuration parameters are ${Q = 0.1, Δ_{x} = 2 a, Δ_{y} = 3 a}$ [Eq. (6)], and we time evolve with the Hamiltonian parameters ${v_{F} = b = 1}$ . We choose a small bump to conserve computational resources, but in an experiment a larger bump would minimize lattice-scale detail neglected in Eq. (5) [37].
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Figure 3
Quench from decoupled chains to the pi-flux lattice II: dispersing vertical density waves from nonfractionalized (noninteracting) chains. The absence of relativistic propagation in the horizontal direction is due to Pauli blocking (see text and Fig. 4). We plot the same time evolution of the post-quench density profile as in Fig. 2, but for a vanishing prequench fermion anomalous dimension $η = 0$ . All other parameters are identical to Fig. 2. The plots show profiles at $t = 0$ , 3, 6, 9, 12.
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Figure 4
Wigner velocity distribution $δ n_{+} (v_{x}, v_{y}; R_{x}, R_{y})$ imprinted on the pi-flux band fermions at the time of the quench, induced by charge fractionalization and a Gaussian density bump in the initial state of decoupled chains. The distribution is evaluated at the center of the bump $R_{x} = R_{y} = 0$ . Unlike continuum 2D massless Dirac fermions, the allowed velocities span a disk due to the lattice regularization in the $k_{y}$ -direction, Eqs. (2a) and (8). The parameter $η$ is the fermion anomalous dimension [Eq. (5)]. For $η > 0$ , there is a divergence of the $x$ -velocities near the speed of light (band velocity $v_{F} = 1$ ). At $η = 0$ (noninteracting initial condition), large $x$ velocities are suppressed by Pauli blocking (see text).
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