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Quantum dynamics in one-dimensional and two-leg ladder systems

Naushad Ahmad Kamar

Published Jan 1, 2019• 2 citations
Publisher

Abstract

In this thesis, we work on three problems. In the first problem, we work on a local time-dependent potential in a one-dimensional interacting bath. This work is motivated by cold atomic experiments, where one can shake the ultracold systems by using a time-dependent potential. We use linear response theory, bosonization, and renormalization group to compute the energy deposition in the one-dimensional bath by the time-dependent potential. We find that the energy deposition both for weak and strong coupling regimes shows a power-law behavior on the frequency of oscillation, and shows a different power-law exponent in strong and weak coupling limit. In the second problem, we work on the dynamics of a mobile impurity in a two-leg bosonic ladder. The theoretical and experimental studies of a mobile impurity in a one-dimensional bath in context of ultracold systems motivate this work. We confine the impurity to move in one of the legs of the ladder. We combine bosonization, renormalization group, and density matrix renormalization group to understand the dynamics of the impurity. We compute the Green's function of the impurity both theoretically and numerically. For a small interaction between the impurity and the bath, we find theoretically that the Green's function of the impurity decays as a power-law, and our numerical results show a good agreement with analytical result. Furthermore, we also give an analytical result of the Green's function for an infinite interaction, we find again that the Green's function decays as a power-law, and shows a good agreement with numerical results. Moreover, finally we give a semi-analytical expression for the power-law exponent at zero momentum as a function of interaction, again in good agreement with the numerical results. In the third problem, we are again investigating the dynamics of a mobile impurity in a two-leg bosonic ladder, but in this case, the impurity also tunnels in both longitudinal and transverse directions. We use similar methods than for the second problem. We compute the Green's function in the bonding and the anti-bonding modes of the impurity. The Green's function in the anti-bonding mode decays as power-law at small momentum and small interaction, while the Green's function in the bonding decays exponentially, and our numerical results show a good agreement with analytical ones.

DOI
10.13097/archive-ouverte/unige:128219

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•Jan 1, 2011•0 citations
In this thesis, we work on three problems. In the first problem, we work on a local time-dependent potential in a one-dimensional interacting bath. This work is motivated by cold atomic experiments, where one can shake the ultracold systems by using a time-dependent potential. We use linear response theory, bosonization, and renormalization group to compute the energy deposition in the one-dimensional bath by the time-dependent potential. We find that the energy deposition both for weak and strong coupling regimes shows a power-law behavior on the frequency of oscillation, and shows a different power-law exponent in strong and weak coupling limit. In the second problem, we work on the dynamics of a mobile impurity in a two-leg bosonic ladder. The theoretical and experimental studies of a mobile impurity in a one-dimensional bath in context of ultracold systems motivate this work. We confine the impurity to move in one of the legs of the ladder. We combine bosonization, renormalization group, and density matrix renormalization group to understand the dynamics of the impurity. We compute the Green's function of the impurity both theoretically and numerically. For a small interaction between the impurity and the bath, we find theoretically that the Green's function of the impurity decays as a power-law, and our numerical results show a good agreement with analytical result. Furthermore, we also give an analytical result of the Green's function for an infinite interaction, we find again that the Green's function decays as a power-law, and shows a good agreement with numerical results. Moreover, finally we give a semi-analytical expression for the power-law exponent at zero momentum as a function of interaction, again in good agreement with the numerical results. In the third problem, we are again investigating the dynamics of a mobile impurity in a two-leg bosonic ladder, but in this case, the impurity also tunnels in both longitudinal and transverse directions. We use similar methods than for the second problem. We compute the Green's function in the bonding and the anti-bonding modes of the impurity. The Green's function in the anti-bonding mode decays as power-law at small momentum and small interaction, while the Green's function in the bonding decays exponentially, and our numerical results show a good agreement with analytical ones.
We study the dynamics of a mobile impurity in a two-leg bosonic ladder. The impurity moves both along and across the legs and interacts with a bath of interacting bosonic particles present in the ladder. We use both analytical (Tomonaga-Luttinger liquid) and numerical [Density Matrix Renormalization Group (DMRG)] methods to compute the Green's function of the impurity. We find that for a small impurity-bath interaction, the symmetric mode of the impurity effectively couples only to the gapless mode of the bath while the antisymmetric mode of the impurity couples to both gapped and gapless modes of the bath. We compute the time dependence of the Green's function of the impurity, for impurity created in either the antisymmetric or symmetric mode with a given momentum. The latter case leads to a decay as a power law below a critical momentum and exponential above, while the former case exhibits both power-law and exponential decay depending on the transverse tunneling of the impurity. We compare the DMRG results with analytical results using the linked cluster expansion and find good agreement. In addition, we use DMRG to extract the lifetime of the quasiparticle, when the Green's function decays exponentially. We also treat the case of an infinite bath-impurity coupling for which both the symmetric and antisymmetric modes are systematically affected. For this case, the impurity Green's function in the symmetric mode decays as a power law at zero momentum. The corresponding exponent increases with increasing transverse tunneling of the impurity. We compare our results with other impurity problems for which the motion of either the impurity or the bath is limited to a single chain. Finally, we comment on the consequences of our findings for experiments with the ultracold gases.
We analyze the behavior of a mobile quantum impurity, restricted to a one-dimensional motion, in a bath formed by a two-leg bosonic ladder through a combination of field theory [Tomonaga-Luttinger liquid (TLL)] and numerical (density-matrix renormalization group) techniques. We compute the Green's function of the impurity as a function of time at different momenta. We find a power-law decay at zero momentum, which signals the breakdown of a quasiparticle description of the impurity motion. We compute the exponent both for the limits of weak and strong impurity-bath interactions. At small impurity-bath interaction, the impurity experiences the ladder as a single channel one-dimensional bath, but with an effective coupling reduced by a factor of 2\sqrt{2}2​. We compare the numerical results for the exponent at zero momentum with a semianalytical expression, initially established for the chain, and find excellent agreement without adjustable parameters. We find an increase of the exponent with increasing transverse tunneling in the bath. At small tunneling, the exponent is compatible with the TLL prediction, while larger tunneling shows strong deviations. Finally, we show that, as a function of the momentum of the impurity, two different regimes of decay of the Green's function exist, similar to the single chain case. The power-law regime occurs for small momentum, while at large momentum the Green's function shows a faster decay, corresponding to the one expected in a polaronic regime. In this last regime, we compute the lifetime of the polaron numerically. We compute the critical momentum marking the transition between these two regimes. We compare with analytical predictions based on the structure factor of the bath and find good agreement with the numerical results. Finally, we discuss the consequences of our results for cold atomic experiments.
We study the superfluid response and localization dynamics from static and mobile impurities. The superfluidity is formed in the rung-Mott phase of a bosonic ladder model producing spin-Meissner currents induced by a U(1)\mathbb{U}(1)U(1) gauge field or a uniform magnetic field. Impurities are described through two-state systems, which act as a two-peak random potential. An impurity sits either at the top or at the bottom of the ladder on each rung equally, producing a telegraph signal. The impurities-matter coupling gives rise to a classical Ising symmetry for static and mobile impurities associated to the inversion symmetry of the two legs of the ladder. From the decoupled rungs limit, we also identify a local Z2{\mathbb{Z}}_{2}Z2​ gauge theory for mobile impurities. The properties of the system are studied from an effective quantum spin model including the possibility of four-body coupling in the limit of a strong interaction between bosons and impurities. Through analytical approaches and numerical exact diagonalization, we study the superfluid currents both in the weakly-coupled and strongly-coupled rungs limits for the bosons. In the weakly-coupled rungs situation, we find a smooth power-law localization whereas the strongly-coupled rungs limit produces a steep localization or insulating phase for various configurations of the two-peak random potential. In the strongly disordered situation, through entanglement and bipartite fluctuation measures, we also identify a many-body localization regime in time after a quench of the system when prepared in a N\'eel state.
We study the motion of an impurity in a two-leg ladder interacting with two fermionic baths along each leg, a simple model bridging cold atom quantum simulators with an idealized description of the basic transport processes in a layered heterostructure. Using the linked-cluster expansion, we obtain exact analytical results for the single-particle Green's function and find that the long-time behavior is dominated by an intrinsic orthogonality catastrophe associated to the motion of the impurity in each one-dimensional chain. We explore both the case of two identical legs as well as the case where the legs are characterized by different interaction strengths: In the latter case, we observe a subleading correction which can be relevant for intermediate-time transport at an interface between different materials. In all the cases, we do not find significant differences between the intra- and interleg Green's functions in the long-time limit.
We investigate two-leg Hubbard ladders in the Luttinger liquid regime by the density-matrix renormalization group method. Applicability of the bond concept in this region is discussed in a quantitative comparison of power-law decay exponents, energy components and anti-bond occupations between ladders and chains. The interaction between bond and anti-bond bands induced by finite Hubbard repulsion U is emphasized by our numerical results. Our quantitative results are useful for the justification of the validity of various analytical approaches.
The physics of ladder systems and the Kondo problem have an essentially one-dimensional nature. There are powerful tools, such as bosonization and conformal field theory, applicable to these systems. In the first part of this dissertation, the stripe phase in 4-leg ladders is investigated by bosonization and weak-coupling renormalization group equations. A new type of charge-four correlation, bipairing, is proposed and gives a simple explanation of the stripe phase. In the second part, we discuss boundary effects on the entanglement entropy in quantum spin chains and calculate the entanglement entropy in the Kondo-like quantum impurity model. The result on impurity entanglement entropy is generalized to the system with finite size or finite temperature by employing conformal field theory methods. The theoretical calculations are also compared with numerical results obtained by density matrix renormalization group methods.