A robust and tunable Luttinger liquid in correlated edge of transition -metal second -order topological insulator Ta2Pd3Te5 Anqi Wang12 Yupeng Li1 Guang Yang1 Dayu Yan1 Yuan Huang3 Zhaopeng Guo1

2025-04-24 0 0 3.86MB 41 页 10玖币

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A robust and tunable Luttinger liquid in correlated edge of transition-metal

second-order topological insulator Ta2Pd3Te5

Anqi Wang1,2,†, Yupeng Li1,†, Guang Yang1,†, Dayu Yan1,†, Yuan Huang3, Zhaopeng Guo1,

Jiacheng Gao1,2, Jierui Huang1,2, Qiaochu Zeng1, Degui Qian1, Hao Wang1, Xingchen Guo1,2,

Fanqi Meng1, Qinghua Zhang1,4, Lin Gu1,2,5, Xingjiang Zhou1,2,5, Guangtong Liu1,5, Fanming

Qu1,2,5, Tian Qian1,5, Youguo Shi1,2,5*, Zhijun Wang1,2*, Li Lu1,2,5*, Jie Shen1,5*

1Beijing National Laboratory for Condensed Matter Physics, Institute of Physics,

Chinese Academy of Sciences, Beijing 100190, China

2School of Physical Sciences, University of Chinese Academy of Sciences, Beijing

100049, China

3Advanced Research Institute of Multidisciplinary Science, Beijing Institute of

Technology, Beijing 100081, China

4Yangtze River Delta Physics Research Center Co. Ltd, Liyang 213300, China

5Songshan Lake Materials Laboratory, Dongguan 523808, China

†These authors contributed equally to this work

*Corresponding author. Email: ygshi@iphy.ac.cn (Y.-G.S.), wzj@iphy.ac.cn (Z.-J.W.),

lilu@iphy.ac.cn (L.L.), shenjie@iphy.ac.cn (J.S.)

Abstract

The interplay between topology and interaction always plays an important role in

condensed matter physics and induces many exotic quantum phases, while rare transition

metal layered material (TMLM) has been proved to possess both. Here we report a

TMLM Ta2Pd3Te5 has the two-dimensional second-order topology (also a quadrupole

topological insulator) with correlated edge states - Luttinger liquid. It is ascribed to the

unconventional nature of the mismatch between charge- and atomic- centers induced by

a remarkable double-band inversion. This one-dimensional protected edge state

preserves the Luttinger liquid behavior with robustness and universality in scale from

micro- to macro- size, leading to a significant anisotropic electrical transport through

two-dimensional sides of bulk materials. Moreover, the bulk gap can be modulated by the

thickness, resulting in an extensive-range phase diagram for Luttinger liquid. These

provide an attractive model to study the interaction and quantum phases in correlated

topological systems.

Main Text:

Introduction

Whereas topological systems without consideration of correlation have been studied widely

in the last decades, the exotic quantum phases derived from the interaction between electrons

and topological state have recently attracted great attention1, such as quantum spin liquids2 and

topological Mott insulators3 which possess one-dimensional (1D) bosonic mode with spin-

charge separation, strongly correlated Chern insulators in flat bands of magic-angle twisted

bilayer graphene4,5 and topological superconductors for topological quantum computation6,7.

The aforementioned correlated 1D bosonic mode is described as Tomonoga-Luttinger liquid

(simplified as Luttinger liquid, LL), and characterized by a power-law vanishing to anomaly

zero density-of-state at the Fermi level. This power-law behavior in electrical transport

measurements has been harnessed to experimentally test LL in the edge states of quantum spin

Hall insulator candidates8-10 and quantum Hall systems11-16. Thus, LL quantum phase provides

a powerful tool to analyze the correlated 1D edge state emerging from topological order.

However, in these systems, the LL relies on the microscopic details and is lack of experimental

evidence for its universality and robustness, let alone extending to macro-scale, which is

supposed to be a signature for topological order. Moreover, this vulnerability leads to the

absence of the quantitative test of LL, e.g., its exponent factor as the Coulomb interaction

strength, in the edge state of topological system, though a large amount of theories have

discussed the possibility of taking into account different interactions17-21.

On the other hand, unlike a two-dimensional (2D) topological insulator with gapless edge

states, a 2D second-order topological insulator (SOTI, or a quadrupole topological insulator)

is supposed to have gapped edge states and in-gap corner states (in the presence of chiral

symmetry)22. Although many 2D SOTI candidates have been proposed in literature23-27, rare

transition-metal compound has been predicted to be a 2D SOTI. After considering many-body

interactions, superconductivity, excitonic condensation and Luttinger liquid can be expected in

bulk and edge states of the transition-metal SOTI.

Here, we have observed the LL inhabiting the edge states, which are as the remnants of the

2D second-order topology, of a TMLM Ta2Pd3Te5. Through the first-principles calculations,

we demonstrate that this topological state originates from a remarkable double-band inversion

and results in an unconventional nature with an essential band representation A”@4c28. Then

we have performed the electrical transport measurements of tens of Ta2Pd3Te5 bulk and thin-

film devices, and detected the coexistence of the bulk insulating gap, the in-gap edge

conducting state and the edge gap as the signature of the 2D SOTI. Interestingly, the edge state

displays a universal scaling of power-law relation with a zero-energy anomaly as a function of

temperature (T) and energy (signed as the voltage bias here), which is considered as the critical

evidence of LL. The weak van der Waals (vdW) interaction between layers allows this LL to

be reproducible in the edge channels of all the devices from atomic- to macro- scale, with a

conductance proportional to the edge geometry. That is the 2D side facets of macroscopic bulk

also show 1D LL anisotropic electrical transport. This reveals for the first time the evidence of

a scale connection from atomic- to macro- LL, and supports this LL quantum state with

universality and robustness irrespective of the microscopic details. In addition, the coexistence

of the three states induces a tunability of the LL power-law exponent to represent the strength

of Coulomb interaction, as well as a reversible LL-Fermi liquid (FL) transition and a full phase

diagram. All these demonstrate a robust and universal LL with the many precise-controlled

parameters, as an effective tool to characterize the interaction and the quantum phase transition

in the correlated topological system.

Crystal structure

Ta2Pd3Te5 is a vdW material with quasi-one-dimensional (Q1D) Ta2Te5 chains and

crystallizes in needle-like single crystals along the b-direction (see the crystal structure and the

optical image of the typical samples in Fig. 1a and ref. 29). The single crystal is synthesized

with an orthorhombic structure Pnma (space group No.62)29. The Q1D characteristic and

layered structure are confirmed by high-resolution scanning transmission electron microscope

(STEM) images of Ta2Pd3Te5 single crystal along [010] and [001] projections, revealing a

thickness of ~0.7 nm for monolayer (Fig. 1b and Supplementary Fig. 1, b and c). Because of

the strong anisotropic bonding energy29, this material is easy to mechanically exfoliate into

large and thin films with flat and uniform edges along the Q1D chain (Supplementary Fig. 1a).

Importantly, the Fermi level naturally locates in the bottom of conduction bands (Fig. 1c and

Supplementary Fig. 1d), enabling a high-efficient modulation by the electrostatic gate. The

coupling between vdW layers is weak, proved by the angle resolved photoemission

spectroscopy (APRES) measurement (Fig. 1d and Supplementary Fig. 1, e and f).

Theoretical calculation

Recently, the Perdew-Burke-Ernzerhof (PBE) calculations with spin-orbit coupling (SOC)

predict that its monolayer is a quantum spin Hall state with a tiny SOC gap and the bulk

topological crystalline insulator without a global band gap (ref. 29). However, both of the

scanning tunneling microscopy (STM) (ref. 30) and APRES (Fig. 1c) measurements detect a

bulk gap of ~50 meV. As reported in ref. 29, the nearly zero-energy band gap is sensitive to the

lattice constant. On the other hand, the experimental energy gap can be induced by many-body

interactions in the Q1D structure, like electron-phonon interaction or electron-hole attractive

interaction, as the excitonic state is thermodynamically most stable in a zero-gap

semiconductor. Although the exact reason of the bulk gap opening needs further investigations,

the double-band inversion happens clearly between the {Y2;GM1-,GM3-} and

{Y1;GM1+,GM3+} bands, which are 0.3 eV (or higher) away from the Fermi level in the

monolayer structure of SG59 (imposing inversion; Supplementary Table. 1 and 2)28. The

schematic of the double band inversion in the system is given in Fig. 1e. Hence, in this

manuscript, we focus on the modified band structure of the monolayer Ta2Pd3Te5 (by adding

small onsite energy to simulate the many-body gap opening) in Fig. 1f31. The band

representation (BR) decomposition of the occupied bands shows that there is an essential BR

at vacancies, suggesting the unconventional nature/obstructed atomic limit with second-order

topology31, 32. A 2D quadrupole topological insulator with second-order topology possesses 1D

gapped edge states and isolated corner states (Detailed information can be found in Ref. 28).

In the computed (01)-edge spectrum of Fig 1g, the gapped edge states are clearly shown in the

gap of the bulk continuum. The observed in-gap edge states as the remnants of the quadrupole

topological insulator can exhibit 1D feature well, which is responsible for the observed LL

behavior (will be analyzed later). In addition, since the layers of the bulk are vdW coupled (Fig.

1d and Supplementary Fig. 1, e and f), the 1D edge states can still survive on the side surfaces

of the bulk.

Three features in dI/dV curves

To investigate the electronic properties, we show the typical result for thin-film devices

#A1,2 with thickness ~10 nm in Fig 2, with a back-gate voltage Vbg tuning the chemical

potential as illustrated in Fig. 2a. Figure 2b and c show the four-terminal measurement

configurations and gate tunable dI/dV for devices #A1,2. Once applying the voltage bias (VDC),

the differential conductance (dI/dV(VDC)) curve of #A1 at charge neutral point (CNP, Vbg =-

28V) shows distinguishable three regions (Fig. 2d, indicated by the purple-green-orange

background). The first region at small VDC indicates a gap structure (highlighted by the purple

background, also see the conductance valley plotted by the symmetric VDC in the inset of Fig.

2d), which closes around T = 3.75 K; The second region at medium VDC reveals a power-law

shape with an exponent of ~0.33 (highlighted by the green background); The third region at

high VDC shows a new shape of enhanced conductance with increasing VDC. Those regions are

also repeated in other devices (Fig. 2e for device #A2 and Supplementary Fig. 12, l and m).

Notably, for device #A2, the T-dependent dI/dV(T) curve at CNP (purple curve in Fig. 2f)

shows similar three regions, which are semiconductor behavior with an energy gap of ~27 meV

from 300 to ~50 K, a power-law dependence with the exponential factor α of ~0.79 from ~30

K to 6 K and another semiconductor behavior with a smaller energy gap of ~ 0.6 meV at lower

T. This is clearer in the log(dI/dV) v.s. 1/T plot in the inset where the yellow dot dash lines are

fitting curves of Arrhenius equation dI/dV(T)  exp(−Δ/2kBT). Similar behavior is repeatable,

e.g., device #A4 with a ~ 33 meV gap at high T and a ~ 0.9 meV gap at low T in Supplementary

Fig. 2. Apparently, this three-region feature fits the theoretical result of band structure with the

bulk gap Eb, edge conducting state and edge gap Ee very well.

It should be noted that these devices are measured with four-terminal configuration with

standard lock-in measurement and the phases θ of the lock-in amplifier are also small enough.

So, the very-low-T behaviors do not come from the contact resistance which is not included in

the measurement. In addition, the gap size we extracted from the Arrhenius equation is

consistent with the thermal excitation energy of the temperature at which the gap behavior

appeared. We also measured dI/dV(VDC) and dI/dV(T) from CNP to positive back-gate voltage

(Vbg) (Fig. 2, e and f). As the Fermi level moves up via applying positive Vbg sketched in Fig.

2a, the three-region feature gradually transitions into that of two regions without Ee in both

dI/dV(VDC) and dI/dV(T) curves, further consistent with the calculated band structure.

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ArobustandtunableLuttingerliquidincorrelatededgeoftransition-metalsecond-ordertopologicalinsulatorTa2Pd3Te5AnqiWang1,2,†,YupengLi1,†,GuangYang1,†,DayuYan1,†,YuanHuang3,ZhaopengGuo1,JiachengGao1,2,JieruiHuang1,2,QiaochuZeng1,DeguiQian1,HaoWang1,XingchenGuo1,2,FanqiMeng1,QinghuaZhang1,4,LinGu1,2,5,Xin...

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A robust and tunable Luttinger liquid in correlated edge of transition -metal second -order topological insulator Ta2Pd3Te5 Anqi Wang12 Yupeng Li1 Guang Yang1 Dayu Yan1 Yuan Huang3 Zhaopeng Guo1

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