Cazalilla and Marston Reply:

Abstract

Cazalilla and Marston Reply:The authors of the Comment [1] propose an improvement to our timedependent density-matrix renormalization group (TdDMRG) algorithm as described in our Letter [2], providing few details about the proposed extension. Luo, Xiang, and Wang (LXW) [1] then point out that the oscillations at late times (t > 18) shown in two of the lines plotted in Fig. 2 of our manuscript are an artifact of the truncation of the Hilbert space.LXW are correct that the oscillations we reported in the one case of V=w 1:1 are in fact spurious. We had systematically examined the errors induced by truncations of the Hilbert space in our Letter. However, we failed to notice that, for the case of V=w 1:1 when the leads are insulating, the eigenvalues of the reduced density matrices become so small so quickly (due to the gap to excitations) that our code discarded the same portion of the Hilbert space for truncations M 256 and 512. Thus we were left with the illusion that we had achieved convergence in this one case, when in fact we had not.Unfortunately the Comment [1] does not provide enough details regarding the construction of the (reduced) density matrix. As Eq. (1) of the Comment [1] shows, LXW's density matrix requires knowledge of the wave function at different times; but to obtain the timeevolved wave function, one needs the density matrix in the first place. Thus it seems that LXW use an iterative procedure: LXW apparently carry out repeated integrations forward in time for each chain size starting with the shortest chain, of length four sites. Once the density matrix is constructed for a chain of a given length, it is then lengthened by the addition of two sites via the usual (infinite-size) density-matrix renormalization group (DMRG) algorithm, and time integration is repeated again from the beginning. The process is then repeated until a sufficiently long chain is built up, tailored to the particular choice of applied bias. Thus the proposed modification of our TdDMRG algorithm is computationally intensive and time-consuming. We further point out

Multimer formation in one-dimensional two-component gases and trimer phase in the asymmetric attractive Hubbard model

“…This picture has been, for instance, proposed at large interaction and low-density limit [50]. Bose-Fermi mixtures of 1D models have extensively studied in recent years [17,20,21,23,24,26,27,[51][52][53][54] and a similar picture emerges in certain regimes of three-component Fermi-gases [55]. When |U | is large, ↑ and ↓ fermions naturally form onsite pairs which are effectively hard-core bosons which we label b.…”

“…The quadratic part H 0 (φ ↑ ) + H 0 (φ ↓ ) + H 1 can be diagonalized by a Bogoliubov transformation [37], which could give a starting point for a perturbative RG calculation [8,17,19,20,23]. Due to the velocity asymmetry, additional couplings are generated and velocities are renormalized along the flow.…”

“…The role of mass asymmetry for a two-component Luttinger liquid has been investigated in the renormalization group (RG) framework originally in the context of solid-state physics [8,15,16] and recently revisited mostly in the context of cold atoms [17][18][19][20][21][22][23][24][25] and supplemented by numerical investigations [26][27][28][29]. Overall, the consensus was that the only new instability arising due to asymmetry is the collapse (demixing) instability for large asymmetry and/or strong interspecies attraction (repulsion).…”

Time-step targeting methods for real-time dynamics using the density matrix renormalization group

“…where the ground state energy E 0 is introduced to reduce the amplitude of the oscillations by making the diagonal elements of H smaller [6]. We use a time dependent Hamiltonian to include the case where a timedependent perturbation V (t) is added to the time independent Hamiltonian H 0 .…”

“…In exchange for the small loss of efficiency, we gain the ability to treat longer range interactions, ladder systems, and narrow two-dimensional strips. In addition, the accuracy is much improved over the lowest order Trotter method.We want to find the solution to the time-dependent Schrödinger equation:where the ground state energy E 0 is introduced to reduce the amplitude of the oscillations by making the diagonal elements of H smaller [6]. We use a time dependent Hamiltonian to include the case where a timedependent perturbation V (t) is added to the time independent Hamiltonian H 0 .…”