Quantum quench spectroscopy of a Luttinger liquid: Ultrarelativistic density wave dynamics due to fractionalization in an $XXZ$ chain

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Quantum quench spectroscopy of a Luttinger liquid: Ultrarelativistic density wave dynamics due to fractionalization in an $X X Z$ chain

Matthew S. Foster, Timothy C. Berkelbach, David R. Reichman, and Emil A. Yuzbashyan

Phys. Rev. B 84, 085146 – Published 31 August 2011

Abstract

We compute the dynamics of localized excitations produced by a quantum quench in the spin-1/2 XXZ chain. Using numerics combining the density-matrix renormalization group and exact time evolution, as well as analytical arguments, we show that fractionalization due to interactions in the prequench state gives rise to “ultrarelativistic” density waves that travel at the maximum band velocity. The system is initially prepared in the ground state of the chain within the gapless $X Y$ phase, which admits a Luttinger liquid (LL) description at low energies and long wavelengths. The Hamiltonian is then suddenly quenched to a band insulator, after which the chain evolves unitarily. Through the gapped dispersion of the insulator spectrum, the postquench dynamics serve as a “velocity microscope,” revealing initial-state particle correlations via space-time density propagation. We show that the ultrarelativistic wave production is tied to the particular way in which fractionalization evades Pauli blocking in the zero-temperature initial LL state.

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Received 18 May 2011

DOI:https://doi.org/10.1103/PhysRevB.84.085146

Authors & Affiliations

Matthew S. Foster^1,*, Timothy C. Berkelbach^2,†, David R. Reichman², and Emil A. Yuzbashyan¹

¹Center for Materials Theory, Department of Physics and Astronomy, Rutgers University, Piscataway, New Jersey 08854, USA
²Department of Chemistry, Columbia University, New York, New York 10027, USA

^*psiborf@rci.rutgers.edu
^†tcb2112@columbia.edu

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Vol. 84, Iss. 8 — 15 August 2011

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Figure 1
Ultrarelativistic wave generated from an initial density bump, following an interacting quench. The Luttinger liquid ground state of an interacting $X X Z$ chain is time evolved according to a noninteracting, band insulator Hamiltonian. The density $δ ρ (t, x_{j})$ due to the inhomogeneity is plotted at time slices $t =$ 0, 12, 24, 36, and 48 after the quench; fainter (bolder) traces depict earlier (later) times. The evolution is symmetric about $x_{j} = 0$ . In this figure, red dashed lines were obtained from a combination of DMRG and exact time evolution for the $X X Z$ chain, while blue solid lines are the prediction of continuum sine-Gordon field theory. The curves marked “asymptotic” are the analytical result for the “regularized supersoliton” in Eq. (3.48). The initial coupling strength is $γ = - 0.872$ , corresponding to $σ = 0.7$ . The initial bump has width $Δ = 12$ and weight $Q = 0.10$ ; the mass gap is $M = 1 / 8$ . The continuum data is obtained from numerical integration of Eq. (3.47) with $α = 0.75$ and $ζ = 1$ .Reuse & Permissions
Figure 2
Dispersive decay of a density bump following a noninteracting ( $γ = σ = 0$ ), nonrelativistic ( $M Δ = 3 / 2$ ) quench. This is the same as Fig. 1, but for a quench from a noninteracting Fermi gas to a band insulator. Parameters $Δ$ , $Q$ , and $M$ are as in Fig. 1. Red dashed lines are the results of exact diagonalization of the lattice Hamiltonian, while blue solid lines are the continuum predictions.Reuse & Permissions
Figure 3
Postquench evolution of a density bump: dependence on initial-state interparticle interaction strength. Each subpanel exhibits a three-dimensional view of a lattice quench (obtained by DMRG + exact time evolution) into the gapped band insulator; $Q$ , $Δ$ , and $M$ are the same as in Figs. 1 and 2. The four frames depict quenches with increasing initial-state interactions, $σ = 0$ (noninteracting), 0.4, 0.7, and 1.0 (top to bottom). The cyan line demarks the maximum band propagation velocity (“speed of light”), $v_{\max} (M = 1 / 8) \approx 1.77$ .Reuse & Permissions
Figure 4
$T = 0$ ground-state phase diagram for the $X X Z$ chain (Ref. 50). Red dashed lines correspond to constant fermion density contours.Reuse & Permissions
Figure 5
Examples of single-particle, massive Dirac equation wave-packet propagation, obtained via numerical integration of Eqs. (3.6) and (3.7). Case (a) corresponds to a nonrelativistic initial condition, the Gaussian in Eq. (3.5) with $Δ = 5 / M$ . The relativistic case is illustrated in (b), with an initial $Δ = 0.5 / M$ . Here $M \equiv 1$ , and data are shown at times $t = 0$ , 15, 35, and 55; fainter (bolder) traces depict earlier (later) times.Reuse & Permissions
Figure 6
The right-moving supersoliton obtained in the interacting sine-Gordon quench. The number density evolution after a Luttinger liquid to insulator quench is depicted for a Gaussian initial density profile (heavy black line), with $σ = 0.7$ , $Δ = 3$ , and $M = 15 / 16$ , obtained via numerical integration of the exact bosonization result [Eqs. (3.13) and (3.20), using Eq. (3.22)]. Time series for two different $Q$ are plotted; the densities are normalized relative to these. The evolution is reflection symmetric about $x = 0$ .Reuse & Permissions
Figure 7
The number density evolution as in Fig. 6, but for the noninteracting quench $K = 1$ ( $σ = 0$ ). The initial bump (heavy black line) has area $Q = 0.1$ in the main figure and $Q = 1$ in the inset; in both cases $Δ = 3$ and $M = 15 / 16$ . The evolution is reflection symmetric about $x = 0$ . Now there is no fractionalization of the initial LL quasiparticles with respect to the insulator and, consequently, the dynamics are simply dispersive with no supersolitons or inhomogeneity growth.Reuse & Permissions
Figure 8
Occupancy of momentum states in the upper (conduction) band after the lattice $X X Z$ quench (obtained by DMRG), for the four values of the dynamical exponent $σ$ given in the legend. The occupancies are plotted for four different choices of the band-gap parameter, $M =$ 3/40 (a), 1/8 (b), 1/4 (c), and 3/8 (d). In each subplot, the dashed vertical line marks the wave number $k_{\max} (M)$ at which the final-state Hamiltonian band group velocity is maximized, $v (k_{\max}; M) = max [d E_{k} / d k] \equiv v_{\max} (M)$ ; see Sec. 4b2, Eqs. (4.3) and (4.4).Reuse & Permissions
Figure 9
The $ζ$ -regularized supersoliton obtained by numerically integrating Eq. (3.47). Here we have set $ζ = 1$ , $Δ = 6$ , and plotted data for four values of $α$ . The interaction exponent $σ = 0.7$ . We have assigned $M = 3 / 2 Δ$ , so that the dynamics reside within the “nonrelativistic” transport regime, as discussed in Sec. 3a. For all but the smallest value $α$ depicted, the characteristic “s” shape of the supersoliton is identified at sufficiently long times. In comparison to the pure sine-Gordon model result shown in Fig. 6, the amplitude of the regularized supersoliton saturates at times $t ≳ t_{ζ}$ . Two inequivalent definitions for $t_{ζ}$ are provided by Eqs. (3.49) and (3.51).Reuse & Permissions
Figure 10
Dynamical regimes of the lattice quench between $H^{(i)}$ and $H^{(f)}$ in Eq. (2.3), based on considerations of the regularized sine-Gordon model, as discussed in the text. The quench occurs at $t = 0$ . The cutoff time $t_{ζ}$ can be defined by either Eqs. (3.49) or (3.51).Reuse & Permissions
Figure 11
Time slices of a noninteracting quench with a gapless final Hamiltonian [ $M = 0$ in Eq. (2.3b), a “Fermi gas to Fermi gas” quench] at times $t =$ 0, 15, 30, 45, and 60; fainter (bolder) traces depict earlier (later) times. Blue solid lines are the continuum prediction [the first term of Eq. (3.48)], and red dashed lines are the result of exact diagonalization of the lattice Hamiltonian. The evolution is symmetric about $x_{j} = 0$ . The relevant quench parameters are $Q = 0.10$ , $Δ = 4$ .Reuse & Permissions
Figure 12
The same as in Fig. 11, but with $Δ = 12$ .Reuse & Permissions
Figure 13
Time slices of a noninteracting quench into a gapped final Hamiltonian (a noninteracting Fermi gas to band insulator quench) at times $t =$ 0, 15, 30, 45, and 60; fainter (bolder) traces depict earlier (later) times. Blue solid lines are the continuum predictions and red dashed lines are the result of exact diagonalization of the lattice Hamiltonian. The continuum data results from a numerical integration of Eq. (3.47) with $σ = 0$ . The evolution is symmetric about $x_{j} = 0$ . The relevant quench parameters are $Q = 0.10$ , $Δ = 4$ , $M = 3 / 8$ .Reuse & Permissions
Figure 14
Comparison of the initial-state correlation function $C (x_{j}, 0)$ defined in Eq. (2.13), as predicted by the regularized ( $α = 0.75$ , $ζ = 1$ ) LL theory [Eq. (3.46), blue dots connected by lines] and as calculated with DMRG (red open circles). The inset is a closeup of the same data. The relevant parameters are $σ =$ 1.0, $Q =$ 0.0.Reuse & Permissions
Figure 15
Envelopes of the correlation function $C (x_{j}, 0)$ calculated by DMRG for the dynamical exponents $σ = 1.0$ (black circles), 0.7 (red squares), and 0.4 (green diamonds)—all for $Q$ = 0.0. The dashed lines are the prediction of Luttinger liquid theory for a finite-size system with periodic boundary conditions (Ref. 79).Reuse & Permissions
Figure 16
Time slices of an interacting quench with $σ = 1.0$ into a gapped final Hamiltonian (an interacting Luttinger liquid to band insulator quench) at times $t =$ 0, 15, 30, 45, and 60; fainter (bolder) traces depict earlier (later) times. Blue solid lines are the continuum predictions and red dashed lines are the results of DMRG calculations combined with exact time evolution of the lattice Hamiltonian. The continuum data obtains from a numerical integration of Eq. (3.47), with $α = 0.75$ and $ζ = 1$ . The evolution is symmetric about $x_{j} = 0$ . The relevant quench parameters are $Q = 0.10$ , $Δ = 4$ , $M = 3 / 8$ .Reuse & Permissions
Figure 17
Interacting quench with $σ = 1.0$ as in Fig. 16, but with $Δ = 6$ , $M = 1 / 4$ . The continuum result [numerical integration of Eq. (3.47)] can be obtained for much larger times and system sizes than is currently practical with the interacting numerics (DMRG + dynamics). The top panel shows the continuum evolution over a window of length 600.Reuse & Permissions
Figure 18
Interacting quench with $σ = 1.0$ as in Fig. 16, but at times $t =$ 0, 12, 24, 36, and 48, with $Δ = 12$ , $M = 1 / 8$ .Reuse & Permissions
Figure 19
Interacting quench with $σ = 1.0$ as in Fig. 16, but at times $t =$ 0, 10, 20, 30, and 40, with $Δ = 20$ , $M = 3 / 40$ .Reuse & Permissions
Figure 20
Time slices of an interacting quench with $σ = 0.7$ at times $t =$ 0, 15, 30, 45, and 60. The continuum data are obtained from a numerical integration of Eq. (3.47), with $α = 0.64$ and $ζ = 1$ . The relevant quench parameters are $Q = 0.10$ , $Δ = 4$ , $M = 3 / 8$ .Reuse & Permissions
Figure 21
Interacting quench with $σ = 0.7$ as in Fig. 20, but with $Δ = 6$ , $M = 1 / 4$ . The curve marked “asymptotic” is the analytical result for the “regularized supersoliton” in Eq. (3.48). The top panel shows the numerical continuum evolution over a window of length 600.Reuse & Permissions
Figure 22
Time slices of an interacting quench with $σ = 0.4$ at times $t =$ 0, 15, 30, 45, and 60. The continuum data obtains from a numerical integration of Eq. (3.47), with $α = 0.50$ and $ζ = 1$ . The relevant quench parameters are $Q = 0.10$ , $Δ = 4$ , $M = 3 / 8$ .Reuse & Permissions
Figure 23
Interacting quench with $σ = 0.4$ as in Fig. 22, but with $Δ = 6$ , $M = 1 / 4$ . The curve marked “asymptotic” is the analytical result for the “regularized supersoliton” in Eq. (3.48). The top panel shows the numerical continuum evolution over a window of length 600.Reuse & Permissions
Figure 24
Interacting quench with $σ = 0.4$ as in Fig. 22, but at times $t =$ 0, 12, 24, 36, and 48, with $Δ = 12$ , $M = 1 / 8$ . The curve marked “asymptotic” is the analytical result for the “regularized supersoliton” in Eq. (3.48).Reuse & Permissions
Figure 25
The maximum band velocity versus the band-gap parameter $M$ as calculated by Eqs. (4.3) and (4.4) (black dashed line) and the propagation velocity of the relativistically moving wave packet in the interacting lattice quench for the values $σ =$ 0.4 (green diamonds), 0.7 (red squares), and 1.0 (black circles).Reuse & Permissions
Figure 26
Three-dimensional view of the Fermi gas to Fermi gas lattice quench ( $σ = 0$ , $M = 0$ ) with $Q = 0.1$ , $Δ = 6$ . The cyan line demarks the maximal propagation speed, $v_{\max} = v_{F} = 2$ .Reuse & Permissions
Figure 27
Three-dimensional view of lattice quenches with $Q = 0.1$ and $Δ = 6$ , into a gapped final Hamiltonian with $M = 1 / 4$ for four different interaction strengths, $σ = 0$ , 0.4, 0.7, and 1.0 (top to bottom). The cyan line demarks the maximal propagation speed, $v_{\max} (M = 1 / 4) \approx 1.56$ .Reuse & Permissions

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