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arXiv:1805.05006 · Physical review. B./Physical review. B

Crossover of correlation functions near a quantum impurity in a Tomonaga-Luttinger liquid

Chung-Yu Lo, Yoshiki Fukusumi, Masaki Oshikawa, Ying-Jer Kao, Pochung Chen

Published Mar 7, 2019• 10 citations
arXivHTMLPDF

Abstract

An impurity in a Tomonaga-Luttinger liquid leads to a crossover between the short- and long-distance regime which describes many physical phenomena. However, a calculation of the entire crossover of correlation functions over different length scales has been difficult. We develop a powerful numerical method based on the infinite density matrix renormalization group. By utilizing infinite boundary conditions we can obtain correlation functions within a finite-size window that contains the impurity. For the $S=1/2$ chain, we demonstrate that a full crossover can be precisely obtained, and that their limiting behaviors show a good agreement with field-theory predictions.

DOI
10.1103/physrevb.99.121103

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An impurity in a Tomonaga-Luttinger liquid leads to a crossover between the short- and long-distance regime which describes many physical phenomena. However, a calculation of the entire crossover of correlation functions over different length scales has been difficult. We develop a powerful numerical method based on the infinite density matrix renormalization group. By utilizing infinite boundary conditions we can obtain correlation functions within a finite-size window that contains the impurity. For the S=1/2S=1/2S=1/2 chain, we demonstrate that a full crossover can be precisely obtained, and that their limiting behaviors show a good agreement with field-theory predictions.
We improve the recently developed functional renormalization group (fRG) for impurities and boundaries in Luttinger liquids by including renormalization of the two-particle interaction, in addition to renormalization of the impurity potential. Explicit flow equations are derived for spinless lattice fermions with nearest-neighbor interaction at zero temperature, and a fast algorithm for solving these equations for very large systems is presented. We compute spectral properties of single-particle excitations, and the oscillations in the density profile induced by impurities or boundaries for chains with up to 106{10}^{6}106 lattice sites. The expected asymptotic power laws at low energy or long distance are fully captured by the fRG. Results on the relevant energy scales and crossover phenomena at intermediate scales are also obtained. A comparison with numerical density matrix renormalization results for systems with up to 1000 sites shows that the fRG with the inclusion of vertex renormalization is remarkably accurate even for intermediate interaction strengths.
Using a fermionic renormalization group method we present a simple real space picture of the strong influence an impurity has on the electronic properties of a Luttinger liquid. We compute the flow of the renormalized impurity potential for a single impurity over the entire energy range - from the microscopic scale of a lattice-fermion model down to the low-energy limit. We confirm that low energy properties close to the impurity are as if the chain is cut in two pieces with open boundary conditions at the end points, but show that this universal behavior is only reached for extremely large systems. The accuracy of the renormalization group scheme is demonstrated by a direct comparison with data obtained from the density-matrix renormalization group method.
The Tomonaga-Luttinger liquid (TLL) concept is believed to generically describe the strongly correlated physics of one-dimensional systems at low temperatures.A hallmark signature in 1D conductors is the quantum phase transition between metallic and insulating states induced by a single impurity.However, this transition impedes experimental explorations of real-world TLLs.Furthermore, its theoretical treatment, explaining the universal energy rescaling of the conductance at low temperatures, has so far been achieved exactly only for specific interaction strengths.Quantum simulation can provide a powerful workaround.Here, a hybrid metal-semiconductor dissipative quantum circuit is shown to implement the analogue of a TLL of adjustable electronic interactions comprising a single, fully tunable scattering impurity.Measurements reveal the renormalization group "beta function" for the conductance that completely determines the TLL universal crossover to an insulating state upon cooling.Moreover, the characteristic scaling energy locating at a given temperature the position within this conductance renormalization flow is established over nine decades versus circuit parameters, and the out-of-equilibrium regime is explored.With the quantum-simulator quality demonstrated from the precise parameter-free validation of existing and novel TLL predictions, quantum simulation is achieved in a strong sense, by elucidating interaction regimes which resist theoretical solutions.
We examine the density-density correlation function in the Tomonaga-Luttinger liquid state for the one-dimensional extended Hubbard model with the on-site Coulomb repulsion UUU and the intersite repulsion VVV at quarter filling. By taking into account the effect of the marginally irrelevant umklapp scattering operator based on the bosonization and renormalization-group methods, we obtain the generalized analytical form of the correlation function. We show that, in the proximity to the gapped charge-ordered phase, the correlation function exhibits anomalous crossover between the pure power-law behavior and the power-law behavior with logarithmic corrections, depending on the length scale. Such a crossover is also confirmed by the highly accurate numerical density-matrix renormalization-group method.
The orthogonality catastrophe in a Tomonaga-Luttinger liquid with an impurity is reexamined for the case when the interaction parameter or the dimensionless conductance is g=1/2.g=1/2.g=1/2. By transforming bosons back to fermions, the Hamiltonian is reduced to a quadratic form, which allows for explicit calculation of the overlap integral and the local density of states at the defect site. The exponent of the orthogonality catastrophe due to a backward scattering center is found to be 1/8, in agreement with previous studies using different approaches. The time dependence of the core-hole Green's function is computed numerically, which shows a clear crossover from a nonuniversal short-time behavior to a universal long-time behavior. The local density of states vanishes linearly in the low-energy limit at g=1/2.g=1/2.g=1/2.
Transport through a spinless Tomonaga-Luttinger (TL) liquid is studied in the presence of a single impurity. A boundary bosonization technique is developed to treat an impurity of arbitrary strength. Our new bosonization formula (i) smoothly connects the ones for the strong and the weak impurity limits, and (ii) reproduces exact scaling equation for the transmission probability in the weak two-body interaction limit. With the use of our formula, TL parameter is determined from the long-distance correlation functions. The result indicates a possible renormalization of TL parameter due to the effect of the single impurity scattering.