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arXiv:1912.13040 · Physical review. B./Physical review. B

Tunneling density of states in a Y junction of Tomonaga-Luttinger liquid wires: A density matrix renormalization group study

Monalisa Singh Roy, Manoranjan Kumar, Sourin Das

Published Jul 17, 2020• 3 citations
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Abstract

It is well known that the pristine bulk of an interacting one-dimensional system in Tomonaga-Luttinger liquid (TLL) phase shows power law suppression of quasiparticle tunneling amplitude for all values of TLL parameter $g$, in the zero-energy limit. We perform a density matrix renormalization group (DMRG) study of a fully symmetric Y junction of TLL wires and observe an anomalous enhancement of the tunneling density of states (TDOS) in the vicinity of the junction for both (a) interacting bosons case and (b) interacting fermions case, when $g>1$. We also observe suppression of TDOS for $g<1$ for both bosonic and fermionic cases. We find that the TDOS enhancements follow different power laws for bosonic and fermionic cases which suggests that these represent distinct fixed points, owing to statistical correlations that play an important role at the Y junction. Analysis of static conductance for the junction indicates that the fixed point for $1<g<3$ resembles the mysterious $M$ fixed point of Y junction predicted by Oshikawa, Chamon, and Affleck [J. Stat. Mech. (2006) P02008]. We also show that the TDOS enhancement spans over a length scale of $\ensuremath{\propto}{\ensuremath{\omega}}^{\ensuremath{-}1}$ from the junction, for $1<g<3$.

DOI
10.1103/physrevb.102.035130

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It is well known that the pristine bulk of an interacting one-dimensional system in Tomonaga-Luttinger liquid (TLL) phase shows power law suppression of quasiparticle tunneling amplitude for all values of TLL parameter ggg, in the zero-energy limit. We perform a density matrix renormalization group (DMRG) study of a fully symmetric Y junction of TLL wires and observe an anomalous enhancement of the tunneling density of states (TDOS) in the vicinity of the junction for both (a) interacting bosons case and (b) interacting fermions case, when g>1g>1g>1. We also observe suppression of TDOS for g<1g<1g<1 for both bosonic and fermionic cases. We find that the TDOS enhancements follow different power laws for bosonic and fermionic cases which suggests that these represent distinct fixed points, owing to statistical correlations that play an important role at the Y junction. Analysis of static conductance for the junction indicates that the fixed point for 1<g<31<g<31<g<3 resembles the mysterious MMM fixed point of Y junction predicted by Oshikawa, Chamon, and Affleck [J. Stat. Mech. (2006) P02008]. We also show that the TDOS enhancement spans over a length scale of \ensuremath∝\ensuremathω\ensuremath−1\ensuremath{\propto}{\ensuremath{\omega}}^{\ensuremath{-}1}\ensuremath∝\ensuremathω\ensuremath−1 from the junction, for 1<g<31<g<31<g<3.
We calculate the tunneling density of states (TDOS) in a dissipationless three-wire junction of interacting spin-1/2 electrons, and find an anomalous enhancement of the TDOS in the zero-bias limit, even for repulsive interactions for several bosonic fixed points. This enhancement is physically related to the reflection of holes from the junction for incident electrons, and it occurs only in the vicinity of the junction ($x<{v}_{\mathrm{min}}/2\ensuremath{\omega}$, where vmin{v}_{\mathrm{min}}vmin​ is the minimum of the velocity of charge or spin excitations and $\ensuremath{\omega}$ is the bias frequency), crossing over to the bulk value, which is always suppressed at larger distances. The TDOS exponent can be directly probed in an STM experiment by measuring the differential tunneling conductance as a function of either the bias voltage or temperature as done in C. Blumenstein et al., Nat. Phys. 7, 776 (2011).
Power-law suppression of local electronic tunneling density of states (TDOS) in the zero-energy limit is a hallmark of the Luttinger liquid (LL) phase of the interacting one-dimensional electron system. We present a theoretical model which hosts the LL state with the surprising feature of enhancement rather than suppression in local TDOS originating from nonlocal and repulsive density-density interactions. Importantly, we find enhancement of TDOS in the manifold of parameter space where the system is stable in the renormalization group (RG) sense. We argue that enhancement of TDOS along with RG stability is possible only when the system has broken parity symmetry about the position of local TDOS enhancement. Such a model could be realized on the edge states of a bilayer quantum Hall system where both intralayer and interlayer density-density interactions are present mimicking the role of local and nonlocal interactions, respectively.
We study the tunneling density of states (TDOS) for a junction of three Tomonaga-Luttinger liquid wires. We show that there are fixed points which allow for the enhancement of the TDOS, which is unusual for Luttinger liquids. The distance from the junction over which this enhancement occurs is of the order of x=v/(2omega), where v is the plasmon velocity and omega is the bias frequency. Beyond this distance, the TDOS crosses over to the standard bulk value independent of the fixed point describing the junction. This finite range of distances opens up the possibility of experimentally probing the enhancement in each wire individually.
In a one-dimensional (1D) system of interacting electrons, excitations of spin and charge travel at different speeds, according to the theory of a Tomonaga-Luttinger liquid (TLL) at low energies. However, the clear observation of this spin-charge separation is an ongoing challenge experimentally. We have fabricated an electrostatically gated 1D system in which we observe spin-charge separation and also the predicted power-law suppression of tunneling into the 1D system. The spin-charge separation persists even beyond the low-energy regime where the TLL approximation should hold. TLL effects should therefore also be important in similar, but shorter, electrostatically gated wires, where interaction effects are being studied extensively worldwide.
We calculate the conductances of a three-terminal junction setup of spinless Luttinger liquid wires threaded by a magnetic flux, allowing for different interaction strength ${g}_{3}\ensuremath{\ne}g$ in the third wire. We employ the fermionic representation in the scattering state picture, allowing for a direct calculation of the linear response conductances, without the need of introducing contact resistances at the connection points to the outer ideal leads. The matrix of conductances is parametrized by three variables. For these we derive coupled renormalization group (RG) equations, by summing up infinite classes of contributions in perturbation theory. The resulting general structure of the RG equations may be employed to describe junctions with an arbitrary number of wires and arbitrary interaction strength in each wire. The fixed point structure of these equations (for the chiral Y junction) is analyzed in detail. For repulsive interaction (g,g3>0g,{g}_{3}>0g,g3​>0) there is only one stable fixed point, corresponding to the complete separation of the wires. For attractive interaction (g<0g<0g<0 and/or g3<0{g}_{3}<0g3​<0) four fixed points are found, the stability of which depends on the interaction strength. We confirm our previous weak-coupling result of lines of fixed points for special values of the interaction parameters reaching into the strong-coupling domain. We find new fixed points not discussed before, even at the symmetric line g=g3g={g}_{3}g=g3​, at variance with the results of M. Oshikawa et al. [J. Stat. Mech. (2006) P02008]. The pair-tunneling phenomenon conjectured by the latter authors is not found by us.
Junctions of multiple one-dimensional quantum wires of interacting electrons have received considerable theoretical attention as a basic constituent of quantum circuits. While results have been obtained on these models using bosonization and density-matrix renormalization-group (DMRG) methods, another powerful technique is based on direct perturbation theory in the bulk interactions combined with the renormalization group. This technique has so far only been applied to the case in which finite-length interacting wires are attached to noninteracting Fermi liquid leads. We extend this method to cover the case of infinite-length interacting leads, obtaining results on two- and three-lead junctions in good agreement with previous bosonization and DMRG results.