A mobile impurity in a higher-dimensional quantum bath shows a quasi-particle (QP) nature, and the impurity behaves like a free particle but with a renormalized mass and a lifetime. However, the recent studies [1, 2] of a mobile impurity in a one-dimensional (1D) bath show a very different class of dynamics and contradict the QP description of the impurity. The impurity generates an infinite number of excitations in the bath, and the Green's function of the impurity decays as a power-law below a critical momentum of the impurity.
In this talk, I will discuss matrix product states (MPS) as a variational ansatz to compute the ground state of a local-gapped Hamiltonian and the time evolution of MPS.
I use matrix product states to describe my joint work  with my Ph.D. advisor Thierry Giamarchi and co-advisor Adrian Kantian on the dynamics of a mobile impurity in a two-leg bosonic ladder bath. Both legs are coupled by inter-leg tunneling, and the impurity moves in one of the legs of the ladder.
We compute the Green's function of the impurity using MPS and field theory (Bosonization) . We find that the Green's function of the impurity decays as a power-law, but the power-law exponent is reduced by a factor of $1/sqrt(2)$ than one in the 1D bath . We also compute the Green's function of the impurity in the limit of an infinite impurity-bath interaction and give a semi-analytical result for the power-law exponent at zero momentum of the impurity. Our numerical results show a good agreement with analytical findings. I will discuss the extension of our results for the case of an impurity in an $N$-leg ladder case.
I will also discuss some of the results of the ongoing work of a mobile impurity in the two-leg ladder bath, but in this case, the impurity moves in both horizontal and transverse directions.