Tunable discrete scale invariance in transition-metal pentatelluride flakes

Log-periodic quantum oscillations discovered in transition-metal pentatelluride give a clear demonstration of discrete scale invariance (DSI) in solid-state materials. The peculiar phenomenon is convincingly interpreted as the presence of two-body quasi-bound states in a Coulomb potential. However, the modifications of the Coulomb interactions in many-body systems having a Dirac-like spectrum are not fully understood. Here, we report the observation of tunable log-periodic oscillations and DSI in ZrTe5 and HfTe5 flakes. By reducing the flakes thickness, the characteristic scale factor is tuned to a much smaller value due to the reduction of the vacuum polarization effect. The decreasing of the scale factor demonstrates the many-body effect on the DSI, which has rarely been discussed hitherto. Furthermore, the cut-offs of oscillations are quantitatively explained by considering the Thomas-Fermi screening effect. Our work clarifies the many-body effect on DSI and paves a way to tune the DSI in quantum materials.


INTRODUCTION
One of the most important concepts in the area of phase transitions is scale invariance. A scale invariant system reproduces itself on different temporal and spatial scales. This is described by the relation ( ) = ( ), where k is an arbitrary parameter, is the scaling dimension, and f is a physical field. Discrete scale invariance (DSI) is a weaker case of the scale invariance, where a system only obeys the scale invariance for specific choices of k 1,2 .With a fundamental scaling ratio λ and characteristic log-periodicity, DSI arises in various contexts, such as earthquakes, financial crashes, turbulence and so on 1 . After being introduced to bound-state problems of quantum systems by Vitaly Efimov in 1970 3 , the DSI had been observed only in cold atom systems for a long time [4][5][6][7] . It had not previously been observed in the solid state. The discovery of Dirac materials has changed that situation [8][9][10][11][12][13][14] . Especially in the topological transition-metal pentatelluride ZrTe 5 , the quantum oscillations with log-periodicity have revealed the existence of DSI in a solid state system 10 . The origin is attributed to the quasi-bound states induced by the supercritical Coulomb interaction between the massless Dirac fermions and charged impurities 10,12,13 . Further studies reported the log-periodic oscillations in HfTe 5 14 and elemental semiconductor tellurium 15 , confirming that the DSI feature can be a universal characteristic of Dirac materials with Coulomb impurities.
ZrTe 5 and HfTe 5 are predicted to be quantum spin Hall insulators in the two-dimensional (2D) limit and the 3D crystals are located near the phase boundary between weak and strong topological insulators 16 . Later studies indicated that the topological natures of both ZrTe 5 and HfTe 5 are very sensitive to the crystal lattice constant and detailed composition 17,18 . Taking ZrTe 5 as an example, some angle-resolved photoemission spectroscopy (ARPES) 19, 20 and magneto-infrared spectroscopy studies show that ZrTe 5 is a Dirac semimetal 21 . The observed negative magnetoresistance (MR) that is related to the chiral anomaly and the anomalous Hall effect support the hypothesis of a massless Dirac band structure [22][23][24] . However, other ARPES and scanning tunneling microscopy (STM) results suggest that ZrTe 5 is a topological insulator [25][26][27][28] . Thus, the transition-metal pentatelluride ZrTe 5 and HfTe 5 are ideal platforms to investigate different intriguing physical properties due to their high tunability [29][30][31][32][33][34] . In particular, the very small Fermi surface of the compounds has enabled some peculiar findings in the ultraquantum regime, such as log-periodic quantum oscillations and three-dimensional quantum Hall effect 10, 14, 35 . As described above, the log-periodic oscillations revealing the DSI feature can be convincingly explained by a two-body quasi-bound state model. However, the solids ZrTe 5 and HfTe 5 are in fact many-body systems, and the Coulomb interactions are modified by screening effects that are not fully understood. The screening of the Coulomb interaction is closely related to the carrier density. In the experiments we find that the carrier density in ZrTe 5 flakes changes with thickness 29 . The interaction between the layers of ZrTe 5 and HfTe 5 is comparable to graphene, making it easy to get flakes from the bulk samples by exfoliation 16 . Thus, it is interesting to study the log-periodic quantum oscillations in the transition-metal pentatelluride flakes with different carrier density by thickness control, which may provide insights into many-body effects on DSI.
In this work, we carried out systematic magnetotransport measurements on ZrTe 5

Temperature dependence of resistance
ZrTe 5 and HfTe 5 belong to the orthorhombic space group Cmcm ( 2ℎ 17 ) 36 . Figure 1(a) shows the crystal structure of ZrTe 5 and HfTe 5 . Within the a-c plane, the trigonal prismatic chains of "ZrTe 3 " or "HfTe 3 " run along the a axis and are linked by parallel zigzag chains of "Te 2 " along the c axis. The layers of ZrTe 5 and HfTe 5 are stacked along the b axis. The false-color scanning electron microscopy (SEM) image of a fabricated ZrTe 5 flake (grey color) with six electrodes (gold color) is displayed in Fig. 1(b). A schematic measurement structure of standard six-electrode-method is illustrated in the inset of Fig. 1(c). The current is applied along the a axis and the magnetic field is along the b axis for all measurements. and ZrTe 5 (~190 nm thick) flakes. Resistance peaks at T p ~ 65 K and 74 K can be detected for HfTe 5 and ZrTe 5 flakes, respectively. It is noted that the T p of the flakes are definitely higher than that of the bulk 10,14 . The origin of the resistance peak in ZrTe 5 and HfTe 5 has been discussed for decades, and recently the view of the Lifshitz transition during the changing of temperatures explains some findings in the compounds 28 . However, in other reports, the Hall resistances of ZrTe 5 and HfTe 5 remain p type with increasing temperatures, which is not consistent with the picture of a Lifshitz transition 29,37 . An alternative two-band model has been presented to be responsible for the transport anomaly and the various T p in different samples 14,29 . In this picture, at low temperatures, the metallic R-T curve mainly relies on a semi-metallic Dirac band, while a semiconducting band dominates at higher temperatures.
The combination and competition between the two bands lead to the resistance peak at T p . In showing that the quality of the device is still reserved during the process of the thin flake fabrication.
The thinner flakes are further studied for comparison. Figure 3 shows

DISCUSSION
As discussed in previous works, the log-periodic quantum oscillations can be attributed to the two-body quasi-bound states composed of a massless Dirac particle and a stationary attractive Coulomb impurity 10,14 . For DSI to occur, we need the supercritical condition α > 1 to occur in the system, where = 2 / ℏ is the effective fine-structure constant of the impurity.
Here is the ionicity, is the background dielectric constant and is the Fermi velocity.
When the supercritical condition holds, the radius of the quasi-bound states of the impurity is and has little influence on . The vacuum polarization is a specific feature of Dirac and Weyl semimetals and comes from the fact that the joint density of states for particle-hole excitations is large at low energies 38,39 . The introduction of an impurity charge excites many virtual particle-hole pairs in Dirac and Weyl semimetals, much more than in a typical insulator with a hard gap. In Ref. 38 it was found in two dimensions that this effect renormalizes Z downwards: where is proportional to the Fermi wavevector and thus to the square root of the carrier concentration n e . a is a short-distance cutoff at the atomic scale and Q is a numerical factor of order 1. A review of interaction effects in the Coulomb impurity problem in graphene concluded that this is the main effect 40 . There is no such complete theory in three dimensions to our knowledge, but the physics is expected to be similar. The renormalization of Z is substantial due to its large starting value and indeed if n e a 2 is of order 10 -4 , as in our thin flakes, then changes in λ by a factor of 2 are easily obtained.
Essentially, in the thinner flakes the vacuum polarization effect decreases and the effective charge becomes larger, which gives an increase in s 0 and a smaller . Therefore, we propose that vacuum polarization plays a crucial role in the decrease of in flakes. Further theoretical efforts are still needed to make this connection fully quantitative.
Although at 4.2 K there are more than three oscillating cycles detected in the thin flakes due to the smaller scale factor, the oscillation signal becomes clear only above 3-5 T for the thin flakes, much higher than that for thick flakes or bulk 10 . We explain that the critical field factor is observed to decrease with decreasing thickness, which is attributed to the weaker vacuum polarization effect resulting from higher carrier density. This work offers a way to tune the DSI and log-periodic oscillations in quantum materials. It also provides deep insights into many-body effects with a Dirac-like spectrum.

Sample information
The high-quality ZrTe 5 and HfTe 5 single crystals were grown by the Te-flux method as described in previous reports 10,14,37 . The crystals were characterized by powder X-ray diffraction, scanning electron microscopy with energy dispersive X-ray spectroscopy, and transmission electron microscopy.

Device fabrication
The ZrTe 5 and HfTe 5 flakes were exfoliated by using the Scotch tape method onto 300 nm-thick SiO 2 /Si substrates. After spin coating of poly (methyl methacrylate) (PMMA), the standard electron beam lithography in a FEI Helios NanoLab 600i Dual Beam System was carried out to define electrodes. Metal electrodes (Pd/Au, 6.5/300 nm) were deposited in a LJUHV E-400L E-Beam Evaporator after Ar plasma cleaning.

Transport measurements
Electronic transport measurements in this work were mainly conducted in the pulsed high magnetic field facility (53 T) at Wuhan National High Magnetic Field Center. Standard six-electrode-method was used for the measurements. The current is applied along the a axis and the magnetic field is along the b axis for all measurements. Gold wires are attached to the electrodes by the silver epoxy and fixed on the substrate by GE varnish to avoid the vibration under pulsed field (see Supplementary Figure 9).

DATA AVAILABILITY
The data that support the findings of this study are available from the corresponding author on reasonable request.

Analysis of log-periodic quantum oscillations
The log-periodic oscillations are superimposed in a large smooth magnetoresistance background.
The oscillating signals are relatively weak, and thus we need to subtract the background as generally do for quantum oscillations 1 . We used two different methods to extract the oscillating term. As shown in Supplementary Figure 4, the oscillations can be obtained by subtracting a smooth background 2,3 . Alternatively, we computed the second derivative for the measured raw data to get the oscillations, which are shown in Supplementary Figures 1-3

Number of Oscillations
It is noticed that only a finite number of oscillations are observed in the experiments, even though the log-periodicity is accurately preserved. In this section, we show how to compute the field range over which the oscillations should be observed, and N, the total number of oscillations.
The basic idea is that there is a single oscillation for each bound state (labeled by n) when the magnetic length ℓ n and the quasi-bound state radius R n coincide up to a factor: with 0 = √ 2 − 1. Hence the number of oscillations is in principle the same as the number of bound states that satisfy DSI. This assumes that there are no limits on the field strength, which is of course not actually the case. We deal with this issue below.
There are natural cutoffs for R n at both ends. At the short-distance end there is r 0 , the cutoff length close to the atomic scale. This is unfortunately not very accurately known. At the long-distance end we have the Thomas-Fermi screening length ξ. When the distance from the impurity is greater than ξ, the potential becomes exponential and DSI is lost. So the oscillations occur when the inequality r 0 < R n < ξ is satisfied. Inverting the equation above, we find that the oscillations occur over a field range However, if r 0 = 0.2 nm, then the upper limit on B is of order 10 3 T , so it's always true in practice that the upper limit is set by the maximum experimental field B m . In the current experiments, this is B m = 53 T. To get the lower limit we need the screening length ξ, which is determined from the equations −2 = 4 2 , = ℏ , and = 3 /6 2 . Here n is the carrier density of holes from Dirac band, μ is the chemical potential and = 4 is the number of Weyl nodes. Inserting the measured electron densities we find = 11.1 nm for the thick (bulk-like) films and = 2.20 nm for the thin films. Using the above formula, we find that the oscillations will be present for > = 0.79 T for the thick film, while in thin films they only appear once > = 3.5 T. Both of these values are in agreement with experiment. 29 The total number of oscillations is given for the thick films by