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arXiv:1302.2667

Quantum Bowling: Particle-hole transmutation in one-dimensional strongly interacting lattice models

Martin Ganahl, Masudul Haque, Hans Gerd Evertz

Published Feb 11, 2013• 4 citations
arXivHTMLPDF

Abstract

We study the scattering of a soliton-like propagating particle with a wall of bound particles, in several strongly interacting one-dimensional lattice models with discrete degrees of freedom. We consider spin-polarized fermions (anisotropic Heisenberg spin chain), the fermionic Hubbard model, and the Bose Hubbard model, using precise numerical time dependent Density Matrix Renormalization Group techniques. We show that in all integrable models studied, there is no reflection. Instead, an incoming particle experiences particle-hole transmutation upon entry and exit of the wall, and travels inside the wall as a hole, analoguous to Klein tunneling, even though the dispersion is highly nonlinear and there is no external potential. {\em Two} particles are added to the wall on the incoming side and removed on the opposite side. For spin-polarized fermions a single transmitted particle thus shifts the wall by two lattice sites, in complete contrast to classical physics. For both Hubbard models, the wall shifts by one doubly occupied single site. In the nonintegrable models studied, the same process occurs in linear superposition with backscattering events. We demonstrate a corresponding fermionic quantum Newton's cradle and a metamaterial with "tachyonic" modes travelling faster than in an empty system. We present a possible atomic scale signal counter for spintronics. Our scenario should be realizable in future cold atom experiments.

DOI
10.48550/arxiv.1302.2667

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We study the scattering of a soliton-like propagating particle with a wall of bound particles, in several strongly interacting one-dimensional lattice models with discrete degrees of freedom. We consider spin-polarized fermions (anisotropic Heisenberg spin chain), the fermionic Hubbard model, and the Bose Hubbard model, using precise numerical time dependent Density Matrix Renormalization Group techniques. We show that in all integrable models studied, there is no reflection. Instead, an incoming particle experiences particle-hole transmutation upon entry and exit of the wall, and travels inside the wall as a hole, analoguous to Klein tunneling, even though the dispersion is highly nonlinear and there is no external potential. {\em Two} particles are added to the wall on the incoming side and removed on the opposite side. For spin-polarized fermions a single transmitted particle thus shifts the wall by two lattice sites, in complete contrast to classical physics. For both Hubbard models, the wall shifts by one doubly occupied single site. In the nonintegrable models studied, the same process occurs in linear superposition with backscattering events. We demonstrate a corresponding fermionic quantum Newton's cradle and a metamaterial with "tachyonic" modes travelling faster than in an empty system. We present a possible atomic scale signal counter for spintronics. Our scenario should be realizable in future cold atom experiments.
The fate of a local two-hole doublon excitation in the one-dimensional Fermi-Hubbard model is systematically studied for strong Hubbard interaction UUU in the entire filling range using the density-matrix renormalization group (DMRG) and the Bethe ansatz. For strong UUU, two holes at the same site form a compound object whose decay is impeded by the lack of phase space. Still, a partial decay is possible on an extremely short time scale where phase-space arguments do not yet apply. We argue that the initial decay and the resulting intermediate state are relevant for experiments performed with ultracold atoms loaded into an optical lattice as well as for (time-resolved) CVV Auger-electron spectroscopy. The detailed discussion comprises the mixed ballistic-diffusive real-time propagation of the doublon through the lattice, its partial decay on the short time scale as a function of filling and interaction strength, as well as the analysis of the decay products, which are metastable on the intermediate time scale that is numerically accessible and which show up in the two-hole excitation (Auger) spectrum. The ambivalent role of singly occupied sites is key to understanding the doublon physics; for high fillings, ground-state configurations with single occupancies are recognized to strongly relax the kinematic constraints and to open up decay channels. For fillings close to half-filling, however, their presence actually blocks the doublon decay. Finally, the analysis of the continua in the two-hole spectrum excludes a picture where the doublon decays into unbound electron holes for generic fillings, different from the limiting case of the completely filled band. We demonstrate that the decay products as well as the doublon propagation should rather be understood in terms of Bethe ansatz eigenstates.
Motivated by the recent experiments that reported signatures of many-body localization of ultracold atoms in optical lattices [M. Schreiber et al., Science 349, 842 (2015)], we study dynamics of highly excited states in the strongly disordered Hubbard model in one dimension. Owing to the SU(2)SU(2)SU(2) spin symmetry, spin degrees of freedom form a delocalized thermal bath with a narrow bandwidth. The spin bath mediates slow particle transport, eventually leading to delocalization of particles. The particle hopping rate is exponentially small in t/W(t,Wt/W\phantom{\rule{4pt}{0ex}}(t,Wt/W(t,W being hopping and disorder scales) owing to the narrow bandwidth of the spin bath. We find the optimal length scale for particle hopping and show that the particle transport rate depends strongly on the density of singly occupied sites in the initial state. The delocalization rate is zero for initial states with only doubly occupied or empty sites, suggesting that such states are truly many-body localized, and therefore the Hubbard model may host both localized and delocalized states. Full many-body localization can be induced by breaking spin rotational symmetry.
A technique to perform real-time quantum propagation of many-particle systems on finite lattices is presented and applied to the motion of a single hole in the Heisenberg and Ising magnets (t-J and t-Jz{\mathit{J}}_{\mathit{z}}Jz​ models) for finite two-dimensional lattices of sizes 3\ifmmode\times\else\texttimes\fi{}3 and 4\ifmmode\times\else\texttimes\fi{}4. Use of Trotter's formula together with a checkerboard decomposition allows a direct real-time calculation of the many-fermion dynamics to be made for asymptotically long times. Starting from localized initial states, the hole-density distribution in the single-hole system is evaluated as a function of time. We find statistical behavior already appears in the 4\ifmmode\times\else\texttimes\fi{}4 lattice due to the large Hilbert space of the fermion background (\ensuremath{\approxeq}105{10}^{5}105 states). For J\ensuremath{\ll}t, the hole appears to move almost incoherently in both models, with small quantum fluctuations. For larger values of the spin coupling J, the t-Jz{\mathit{J}}_{\mathit{z}}Jz​ model shows a strong tendency to localize, while incorporation of the Heisenberg spin-flip term removes this feature and introduces an increasing component of low-frequency coherent motion for larger J.
We consider a Hubbard-like model of strongly interacting spinless fermions and hardcore bosons on a square lattice, such that nearest neighbor occupation is forbidden. Stripes (lines of holes across the lattice forming antiphase walls between ordered domains) are a favorable way to dope this system below half filling. The problem of a single stripe can be mapped to fermions in one dimension, which allows understanding of its elementary excitations and calculation of the stripe's effective mass for transverse vibrations. Using Lanczos exact diagonalization, we investigate the excitation gap and dispersion of a hole on a stripe, and the interaction of two holes. We also study the interaction of a few stripes, finding that they repel and the interaction energy decays with stripe separation as if they are hardcore particles moving in one (transverse) direction. To determine the stability of an array of stripes against phase separation into a particle-rich phase and a hole-rich liquid, we evaluate the liquid's equation of state, finding that the stripe-array is not stable for bosons but is possibly stable for fermions.
We study nonequilibrium dynamics in the one-dimensional Bose-Hubbard model starting from an initial product state with one boson per site and a single-hole defect. We find that for parameters close to the quantum critical point, the hole splits into a core showing a very slow diffusive dynamics, and a fast mode which propagates ballistically. Using an effective fermionic model at large Hubbard interactions UUU, we show that the ballistic mode is a consequence of an interference between slow-holon and fast-doublon dynamics, which occurs once the hole defect starts propagating into the bosonic background at unit filling. Within this model, the signal velocity of the fast ballistic mode is given by the maximum slope of the dispersion of the doublon quasiparticle in good agreement with the numerical data. Furthermore, we contrast this global quench with the dynamics of a single-hole defect in the ground state of the Bose-Hubbard model and show that the dynamics in the latter case is very different even for large values of UUU. This can also be seen in the entanglement entropy of the time-evolved states which grows much more rapidly in time in the global quench case.
We show that pronounced modulations in spin and charge densities can be induced by the insertion of a single hole in an otherwise half-filled two-leg Hubbard ladder. Accompanied with these modulations is a loosely bound structure of the doped charge with a spin-1/2, in contrast to the tightly bound case where such modulations are absent. These behaviors are caused by the interference of the Berry phases associated with a string of flipped spins (or ``phase strings'') left behind as a hole travels through a spin bath with a short-range antiferromagnetic order. The key role of the phase strings is also reflected in how the system responds to increasing spin polarization and the on-site repulsion, addition of a second hole, and increasing asymmetry between intra- and interchain hopping. Remarkably, all these properties persist down to ladders as short as $\ensuremath{\sim}10$ sites, as the smoking gun of the phase-string effect. They can therefore be studied in cold-atom experiments using the recently developed fermion microscope.