critical exponent experiment

Universality

When we started talking about critical opalescence, we spoke of water and carbon dioxide and the gas-liquid transition. The experiment described something different – two different liquids going through a “dissolved/not-dissolved” transition. But everything we understood about critical phenomena from the liquid-gas transition seemed to work in the experiment.

The connection is deeper than that. If you look at the way response functions like compressibility get big around the critical point (how fast as you bring the temperature close to the critical point), you find that you end up with the same numbers for lots of very diverse situations.

As scientists, we have to take one more important step: we have to be a little more concrete in the description of “sameness”. This requires a little math, but it’s totally worth the effort. So here it goes: things get big (diverge) around the critical temperature as

As T gets close to T c , the function diverges or “blows up” (gets really big).
There are three lines below corresponding to three different values of the critical exponent , x. Match the line to the critical exponent.
The critical exponent controls how fast the function blows up. The bigger the exponent, the faster the blow up. A gas-liquid critical point, like that for water or carbon dioxide, is associated with a bunch of different critical exponents – one for isothermal compressibility, one for heat capacity, and so on. We just recycle the same function with a different x .

The remarkable thing about critical phenomena is that a bunch of different kinds of quantities – response functions like compressibility and others – follow this rule close to the critical temperature. There might be a different critical exponent for each different kind of quantity, but the same kind of expression represents the behavior of all of these quantities.

The even more remarkable thing is that the critical exponents – the x 's – for totally different systems are the same .

Ask yourself…

What happens at the critical point that is always the same?
What all of these situations have in common are fluctuations about an average state (temperature and pressure). Right around the critical point, these fluctuations become magnified, and control the way the system behaves. Because the fluctuations grow in the same way – whether they are fluctuations of average density or magnetization or any other order parameter – the same rate of growth appears for all of these systems….WOW!

….or something like that – it’s a little more complicated. It turns out that the critical behavior is not always the same, but it doesn’t depend so much on the identity of the substance at play – more on its dimensionality and number of components.

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Experimental Critical-Exponent Values for Fluids

Published: 08 October 2009
Volume 137 , pages 857–877, ( 2009 )

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Jan V. Sengers 1 &
Joseph G. Shanks 2

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After a review of the history and an assessment of the current status of the subject, we present light-scattering data to determine the critical correlation-function exponents for an aqueous electrolyte solution yielding the experimental values γ =1.238±0.012, ν =0.629±0.003, η =0.032±0.013. We conclude the paper with some comments concerning the temperature dependence of non-asymptotic effective critical-exponent values of fluids.

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Critical Crossover Functions for Simple Fluids: Towards the Crossover Modelling Uniqueness

Fluctuation theory of critical phenomena in fluids.

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Sengers, J.V., Shanks, J.G. Experimental Critical-Exponent Values for Fluids. J Stat Phys 137 , 857–877 (2009). https://doi.org/10.1007/s10955-009-9840-z

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DOI : https://doi.org/10.1007/s10955-009-9840-z

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Universal window for two-dimensional critical exponents

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1 Department of Chemistry, University College London, 20 Gordon Street, London WC1H 0AJ, UK.
PMID: 21694394
DOI: 10.1088/0953-8984/20/27/275233

Two-dimensional condensed matter is realized in increasingly diverse forms that are accessible to experiment and of potential technological value. The properties of these systems are influenced by many length scales and reflect both generic physics and chemical detail. To unify their physical description is therefore a complex and important challenge. Here we investigate the distribution of experimentally estimated critical exponents, β, that characterize the evolution of the order parameter through the ordering transition. The distribution is found to be bimodal and bounded within a window ∼0.1≤β≤0.25, facts that are only in partial agreement with the established theory of critical phenomena. In particular, the bounded nature of the distribution is impossible to reconcile with the existing theory for one of the major universality classes of two-dimensional behaviour-the XY model with four-fold crystal field-which predicts a spectrum of non-universal exponents bounded only from below. Through a combination of numerical and renormalization group arguments we resolve the contradiction between theory and experiment and demonstrate how the 'universal window' for critical exponents observed in experiment arises from a competition between marginal operators.

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Ising critical exponents in real fluids: An experiment

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Experimental Critical-Exponent Values for Fluids

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Scaling behavior of the quantum phase transition from a quantum-anomalous-Hall insulator to an axion insulator

Xinyu Wu ORCID: orcid.org/0000-0001-9654-3285 1 ,
Di Xiao ORCID: orcid.org/0000-0002-2894-034X 2 ,
Chui-Zhen Chen 3 ,
Jian Sun ORCID: orcid.org/0000-0002-4814-1676 1 ,
Ling Zhang 2 ,
Moses H. W. Chan 2 ,
Nitin Samarth 2 ,
X. C. Xie 1 , 4 , 5 ,
Xi Lin ORCID: orcid.org/0000-0003-3383-2017 1 , 4 , 5 &
Cui-Zu Chang ORCID: orcid.org/0000-0003-3515-2955 2

Nature Communications volume 11 , Article number: 4532 ( 2020 ) Cite this article

Phase transitions and critical phenomena
Quantum Hall
Topological insulators

The phase transitions from one plateau to the next plateau or to an insulator in quantum Hall and quantum anomalous Hall (QAH) systems have revealed universal scaling behaviors. A magnetic-field-driven quantum phase transition from a QAH insulator to an axion insulator was recently demonstrated in magnetic topological insulator sandwich samples. Here, we show that the temperature dependence of the derivative of the longitudinal resistance on magnetic field at the transition point follows a characteristic power-law that indicates a universal scaling behavior for the QAH to axion insulator phase transition. Similar to the quantum Hall plateau to plateau transition, the QAH to axion insulator transition can also be understood by the Chalker–Coddington network model. We extract a critical exponent κ ~ 0.38 ± 0.02 in agreement with recent high-precision numerical results on the correlation length exponent of the Chalker–Coddington model at ν ~ 2.6, rather than the generally-accepted value of 2.33.

Quantized resistance revealed at the criticality of the quantum anomalous Hall phase transitions

Probing the mesoscopic size limit of quantum anomalous Hall insulators

High spin axion insulator

Introduction.

The study of quantum phase transitions is a rich topic of research in condensed matter physics. Quantum phase transitions, such as the plateau-to-plateau transition in the quantum Hall (QH) effect at high magnetic field, can be accessed by varying only one physical parameter (e.g., magnetic field) near absolute zero temperature 1 . The fundamental physics of quantum phase transition can be revealed by investigating its scaling behaviors at the crossing point of magnetoresistance at different temperatures 2 , 3 , 4 , 5 . The universal scaling behavior associated with such a phase transition is usually studied by characterizing a temperature-dependent resistivity/conductivity that is governed by a single exponent κ 2 , 3 , 4 , 5 . A similar feature of crossing magnetoresistance at different temperatures has also been observed in the QH to Anderson or Hall insulator transition 6 , 7 , 8 , 9 , 10 .

The quantum anomalous Hall (QAH) state, a zero magnetic field manifestation of the integer QH state, is also a topological phase of quantum matter 11 , 12 , 13 . Similar to the QH effect 14 , the QAH effect harbors dissipationless chiral edge states with quantized Hall resistance and vanishing longitudinal resistance 15 . These edge states are spin-polarized and their chirality is determined by the internal magnetization of the sample. The QAH effect was envisioned by Duncan Haldane 11 and was first realized in magnetically doped topological insulator (TI) thin films 13 , 16 . To date, the QAH effect at zero magnetic field has been realized in the Cr- and/or V-doped TI systems 13 , 16 , 17 . An axion insulator state was recently realized in V-doped TI/TI/Cr-doped TI sandwich structures 18 , 19 . When the external magnetic field is swept between the coercive fields of the two magnetic TI films, i.e., the magnetizations of the two surfaces are antiparallel, the axion insulator state appears. The axion insulator is characterized by zero Hall resistance R xy and a very large longitudinal resistance R xx 18 , 19 . When the magnetizations of the two magnetic TI layers are driven by an external magnetic field from the antiparallel to parallel alignment, a quantum phase transition from an axion insulator to a QAH insulator has been demonstrated 18 , 19 .

In this study, we studied the scaling behavior of the quantum phase transition from the QAH insulator to the axion insulator in magnetic TI sandwich samples in the temperature range between 45 and 100 mK. We found the temperature dependence of the derivative of the longitudinal resistance R xx on the magnetic field B evaluated at the critical field B c follows a characteristic power-law behavior, i.e. $(\frac{{dR_{xx}}}{{dB}})_{B = B_{\mathrm{{c}}}} \propto T^{ - \kappa }$ . The exponent κ is found to be 0.38 ± 0.02, smaller than the previously believed value κ ∼ 0.43 for the QH plateau-to-plateau transition 4 , 20 , 21 , but in good agreement with the recent high-precision numerical results on the Chalker–Coddington network model, which yields a correlation length exponent ν ~ 2.6 if the dynamic exponent p = 2 22 , 23 , 24 , 25 . The experimental results are supported by theoretical arguments that the QAH to axion insulator phase transition, just like the QH plateau-to-plateau transition, can also be described using the Chalker–Coddington network model. In addition, by using a quasi-DC measurement circuit, we measured the two-terminal resistance R 12 in the axion insulator regime and found it to be as high as 5.0 × 10 4 h/e 2 (~1.3 GΩ) (Supplementary Note 1 ). This value suggests the highly insulating property of the topology-induced axion insulator state. By analyzing the temperature dependence of R 12 , we probed two different insulating behaviors in two adjacent temperature ranges, validating the reliability of our scaling study for the QAH to axion insulator transition.

Sample structures and sample characterizations

The heterostructure samples used in this work are sandwiches with an undoped TI layer (5QL (Bi, Sb) 2 Te 3 layer) inserted between two magnetic TI layers with a 3QL Cr-doped (Bi, Sb) 2 Te 3 layer in the bottom and a 3QL V-doped (Bi, Sb) 2 Te 3 layer on top (Fig. 1a ). Specifically, the samples are 3QL (Bi, Sb) 1.89 V 0.11 Te 3 /5QL (Bi, Sb) 2 Te 3 /3QL (Bi, Sb) 1.85 Cr 0.15 Te 3 sandwich heterostructures. These samples were grown on 0.5 mm-thick heat-treated SrTiO 3 (111) substrate in a molecular beam epitaxy (MBE) chamber with a base vacuum of 2 × 10 −10 mbar. The Bi/Sb ratio in each layer was optimized to tune the chemical potential near the charge neutral point. The transport studies were performed in two dilution refrigerators (Leiden 6 mK/14 T and 8 mK/9 T) with the magnetic field B applied perpendicular to the film plane. We used an excitation current of no more than 0.3 nA with a lowest trusted electron temperature ~45 mK (Supplementary Note 2 ). Six-terminal Hall bars with a bottom-gate electrode (Fig. 1b ) were used for electrical transport studies.

a Schematic of the magnetic/nonmagnetic/magnetic TI sandwich heterostructure. b Schematic of the Hall bar device used in electrical transport measurements. c Magnetic field B dependence of R xy (blue) at V g = V g 0 = +21.5 V. d B dependence of R xx (red) and R 12 (green) at V g = V g 0 = +21.5 V. The four-terminal longitudinal resistance R xx data are shown in red, whereas the two-terminal resistance R 12 data are shown in green. When the R 12 value is comparable with or higher than the impedance of parasitic capacitance of the measurement circuit, the R 12 values measured with AC two-terminal method are not reliable and are represented by the green dashed lines. R xy values in the highly insulating regime due to the inaccuracy caused by mutual mixing from huge R xx are not shown. Measurements in c and d were taken when the environmental temperature of the dilution fridge is 15 mK.

Figures 1c , d show the magnetic field B dependence of the Hall resistance R xy (blue) and the longitudinal resistance R xx (red) of the magnetic TI sandwich sample at T = 15 mK and V g = V g 0 = +21.5 V. Here, V g 0 is the charge neutral point, determined by achieving the largest two-terminal resistance in the axion insulator regime. When the magnetization of the bottom Cr-doped TI layer is aligned in parallel with that of the top V-doped TI layer, the sample displays a perfect QAH state: at B = 0 T, R xy (0) = ± h/e 2 (within 0.07% of quantization) and R xx (0) < 17 Ω. When the magnetization of the two magnetic TI layers is antiparallel, i.e., B c1 < B < B c2 , where B c1 and B c2 are the coercive fields of Cr-doped and V-doped TI layers, the sample shows an axion insulator state 19 . As the axion insulator is a topology-induced insulating state, its R xx value diverges. The traditional four-terminal measurement method is not suitable to measure such a large resistance. The two-terminal method with an AC signal source could deal with larger resistance but the largest resistance is still limited due to the unavoidable parasitic capacitance in the measurement circuit. Neither the traditional four-terminal measurement method nor the two-terminal method with an AC signal source provides reliable resistance values for such an insulating state. Therefore, we show four-terminal data only between 0 and ~10 h/e 2 , and AC two-terminal data from ~10 h/e 2 to ~1000 h/e 2 in Fig. 1c . At higher resistance range, we used a quasi-DC measurement scheme. This quasi-DC measurement method can yield reliable two-terminal resistance up to ~2 GΩ (~7.7 × 10 4 h/e 2 ) (Supplementary Note 1 ). This method, however, is inconvenient for measurement when sweeping the magnetic field, because it requires very long measurement time and does not provide a “continuous” field-dependent resistance value as the AC lock-in technique.

QAH insulator to axion insulator quantum phase transition

When B > B c2 > B c1 , the sample shows the perfect QAH state (Fig. 1c ), so there must be a magnetic-field-driven quantum phase transition from an axion insulator to a QAH insulator near B c2 . To study the scaling behavior of this quantum phase transition, we measured the magnetic field B dependence of R xx around B c2 at different temperatures, as shown in Fig. 2a . All R xx − B curves in the temperature range 45 mK ≤ T ≤ 80 mK cross each other at one point, specifically at the critical magnetic field B = B c ~ 0.979 T. B c separates the R xx − B curves into two regions. For B < B c , R xx increases with lowering temperature, showing an “insulating” behavior. This “insulating” state corresponds to the axion insulator state with a large R xx and zero R xy near-zero temperature 18 , 19 . For B > B c , R xx decreases with lowering temperature, exhibiting a “metallic” behavior. This “metallic” state corresponds to the chiral edge channel of the QAH insulator state with zero R xx and quantized R xy at zero temperature 13 , 16 . This “metal” to insulator quantum phase transition is further demonstrated from the temperature dependence of R xx measured at various magnetic fields (Fig. 2b ). Figure 2c shows the value of R xx at B = B c for different temperatures. The R xx value at T = 100 mK shows a clear deviation from the values measured at lower temperatures, which implies that we should confine our investigation of the scaling behavior of the QAH to axion insulator transition in the temperature range of 45 ≤ T ≤ 80 mK. It is noteworthy that one more transition from the QAH insulator to the axion insulator occurs at B c1 ~ 0.20 T. We experienced instability problem of our magnet at low magnetic field region, so the data acquired near B c1 cannot be used for scaling analysis.

a Magnetic field dependence of R xx at various temperatures. All curves cross at one critical point at B c ~ 0.979 T. b Temperature dependence of R xx at different magnetic fields. The sample shows “insulating” behavior for B < B c and “metallic” behavior for B > B c . At B = 0.980 T, R xx is nearly independent of temperature. c R xx value at the crossing point as a function of temperature.

Scaling analysis of the QAH insulator to axion insulator transition

Next, we analyze the scaling behavior of the QAH to axion insulator phase transition. We follow the scaling analysis of the QH plateau-to-plateau transition 2 , 3 and scale the B dependence of R xx at different temperatures to a single parameter relation: $(\frac{{dR_{xx}}}{{dB}})_{B = B_{\mathrm{{c}}}} \propto T^{ - \kappa }$ . Figure 3a shows the $(\frac{{dR_{xx}}}{{dB}})_{B = B_{\mathrm{{c}}}}$ at different temperatures plotted on log–log scale. The perfect linear fit in the temperature range 45 mK ≤ T ≤ 80 mK demonstrates the QAH to axion insulator quantum phase transition does show the universal scaling behavior. For T < 45 mK, $(\frac{{dR_{xx}}}{{dB}})_{B = B_{\mathrm{{c}}}}$ starts to deviate from the linear dependence, leading us to suspect that the real electron temperature of the sample is higher than the value registered by the thermometer on the mixing chamber in this temperature range (Supplementary Note 2 ). We note that κ can also be acquired in the R xy analysis over the same temperature range, but it has a lower accuracy due to mutual mixing from huge R xx (Supplementary Note 3 ).

a Temperature dependence of $(\frac{{dR_{xx}}}{{dB}})_{B = B_{\mathrm{{c}}}}$ on log–log scale. The black dashed line is the linear fit. The inset shows R xx as a function of the s c aled magnetic field | B − B c |· T −κ at different temperatures. b The two-terminal DC resistance R 12 as a function of 1/ T (bottom axis) or T (top axis) when the sample shows the most insulating state (i.e., B = 0.40 T). The vertical axis is on a log scale. The inset shows the measurement circuit we used to accurately measure the huge R 12 of the axion insulator state.

From the linear fit in Fig. 3a , we extracted a critical exponent κ ~ 0.38 ± 0.02. To further demonstrate the universal scaling behavior of the QAH to axion insulator transition, we used the one-parameter scaling relation $R_{xx} = f[\left| {B - B_{\mathrm{c}}} \right| \cdot T^{ - \kappa }]$ to scale the B dependence of R xx at different temperatures as a function of scaled magnetic field $\left| {B - B_{\mathrm{c}}} \right| \cdot T^{ - \kappa }$ 26 , 27 . All curves except the curve at T = 100 mK are well-scaled to a single curve around the critical point. Figure 3a shows that including the 100 mK data for the scaling fit does not change κ , but if more high-temperature data points are included, a larger κ is found but not reliable (Supplementary Note 4 ). Below, we propose a theoretical picture to explain why the QAH to the axion insulator transition shares the same universality class of QH plateau-to-plateau transition and also demonstrate that κ ~ 0.38 ± 0.02 is in agreement with the latest high-precision numerical results of the correlation length exponent ν ~ 2.6 22 , 23 , 24 , 25 .

Theoretical analysis of the QAH insulator to axion insulator transition

The top and bottom surface states in V-doped TI/TI/Cr-doped TI sandwich heterostructures can be described by the effective Hamiltonian:

Here, s x,y,z are Pauli matrices acting on spin space, k x,y are wave vectors in x and y directions, and v F is the Fermi velocity. The Dirac mass term M t,b and Δ t,b (r) represent spatial-averaged and random parts of the exchange field in the z -direction due to the magnetization on top and bottom surfaces, respectively, where “ t ” and “ b ” denote the top and bottom surfaces. The random scalar potential V t,b ( r ) denotes an inhomogeneous onsite potential inside the sandwich sample. In the clean limit, V t,b ( r ) = 0 and Δ t,b ( r ) = 0, so the Hall conductance of the top and bottom surfaces ${\upsigma}_{xy}^{{\mathrm{t}},{\mathrm{b}}} = {\mathrm{sign}}(M_{{\mathrm{t}},{\mathrm{b}}})e^2/2{\mathrm{h}}$ is determined by the sign of magnetization sign( M t,b ). The QAH phase with ${\upsigma}_{xy}^{\mathrm{t}} = {\upsigma}_{xy}^{\mathrm{b}}$ and the axion insulator phase with ${\upsigma}_{xy}^{\mathrm{t}} = - {\upsigma}_{xy}^{\mathrm{b}}$ require parallel and antiparallel magnetizations on the top and bottom surfaces, respectively. During the magnetization reversal process, the magnetization of one surface switches direction and a magnetic-field-driven quantum phase transition from the QAH insulator to the axion insulator occurs. Therefore, we consider only one surface in the following and denote it as top surface for concreteness. At the critical point of the QAH insulator to the axion insulator transition, the spatial-averaged magnetization of the top surface M t = 0, whereas ${\Delta}_{\mathrm{t}}\left( {{r}} \right) \ne 0$ with an equal population of upward and downward magnetic domains (Fig. 4a ). Now, the top surface state can be described by the random Dirac Hamiltonian with randomness in the mass Δ t ( r ) and the scalar potential V t ( r ):

There exists a chiral edge mode between two regions with opposite masses for the Dirac Hamiltonian 28 . Due to a spatially varying random Dirac mass Δ t ( r ), the random Dirac Hamiltonian H t exhibits chiral edge modes confined to the zero-mass contours Δ t ( r ) = 0 (Fig. 4a ) 28 .

a The chiral zero-energy edge states along contours of zero-mass domain walls of Dirac fermions. “ + ” and “−” denote domains of positive and negative masses of Dirac fermions, respectively. b The Chalker–Coddington network model on a square lattice. c At each node (red dashed square in b ), a scattering matrix S describes the scattering from incoming (labeled by Z i = 1,3 ) to outgoing modes (labeled by Z i = 2,4 ). The chiral edge modes can be created by Dirac Hamiltonian with four domains of opposite Dirac masses labeled by ±Δ .

Moreover, it is known that the Chalker–Coddington network model can be mapped to the random Dirac Hamiltonian 28 , 29 , 30 . Figure 4b shows the Chalker–Coddington network model on a square lattice structure. At each node (red dashed square in Fig. 4b ), a scattering matrix S describes the scattering from incoming (labeled by Z i = 1,3 ) to outgoing channels (labeled by Z i = 2,4 ) (Fig. 4c ). Here, the incoming and outgoing channels in the network model correspond to the chiral edge modes at zero-mass domain walls of the random Dirac Hamiltonian (Fig. 4a ). Figure 4c shows the chiral edge channels at the four domains of opposite masses (labeled by ±Δ t ). All chiral edge channels in the network model can be created similarly and the Chalker–Coddington model is thus equivalent to the random Dirac Hamiltonian 28 , 30 . As the QH plateau-to-plateau transition (without electron–electron interactions) can also be described by the Chalker–Coddington model 29 , the quantum phase transition from the QAH insulator to the axion insulator must share the same universality class of QH plateau-to-plateau transition. Indeed, the critical exponent κ ~ 0.38 ± 0.02 revealed in our experiment is close to the generally accepted value κ ∼ 0.43 in the QH plateau-to-plateau transition 4 , 20 , 21 . The slight discrepancy of κ = p /2 ν may result from a decreased dynamic exponent p or an increased correlation exponent ν . However, the theoretical problem of the QH plateau-to-plateau transition with strong electron–electron interactions remains open and a reliable method for calculating these exponents is yet to be established 31 , 32 , 33 , 34 , 35 , 36 . Recently, the high-precision numerical calculations show that the correlation length exponent ν of the Chalker–Coddington model is ~2.6 22 , 23 , 24 instead of ~2.33. It is noteworthy that the larger ν -value is also supported by the recent field theory analysis 25 . As a consequence, the critical exponent κ = p /2 ν ∼ 0.38, if we assume p = 2 is in good agreement with the κ- value obtained in our experiment. Here, the exponent p = 2, exceeding the non-interacting value p = 1, may be induced by the appearance of the Coulomb interaction, similar to the discussion for the QH plateau-to-plateau transition 31 , 32 , 33 , 34 , 35 , 36 . In our samples, as the density of states of Dirac fermions is low, the long-range Coulomb interaction is unscreened and the triplet interaction channel is eliminated due to the strong spin-orbital coupling of TI materials 32 . We note that the QAH to axion insulator transition in magnetic TI sandwiches belongs to the QH-type instead of the Berezinskii–Kosterlitz–Thouless-type transition as we predicted in an individual magnetic TI thin film with random domains 37 . In such an individual magnetic TI thin film, the top and bottom surfaces are coupled and play a joint role, so this case is described by the Chalker–Coddington model with two channels 37 , 38 . In other words, the magnetization reversal during the QAH plateau-to-plateau transition occurs on both the top and bottom surfaces. This may be the reason why a different κ ~ 0.6 was found for the QAH to Anderson insulator transition 39 and the QAH plateau-to-plateau transition 40 . However, the magnetization reversal in magnetic TI sandwich heterostructures during the QAH insulator to the axion insulator transition occurs on only one surface.

A recent study on the quantum phase transition between a Chern insulator to an axion insulator driven by the external magnetic field in even number layers of topological antiferromagnet MnBi 2 Te 4 reports a critical exponent κ ~ 0.47 41 . In the QH plateau-to-plateau transition, values of κ ranging from 0.15 to ~0.8 have been reported in the two-dimensional electron/hole gas samples 2 , 5 , 42 , 43 , 44 , 45 , 46 , 47 , 48 , 49 . Several factors can cause larger κ , e.g., the high concentration of ionized impurities or clustered impurities 4 . If the quantum percolation is turning into a classical percolation, κ is expected to eventually increase to 0.75 31 , 50 . It is noteworthy that the determination of κ is also sensitive to experimental issues, e.g., the small temperature regime where all curves crossing each other as shown in Fig. 2c . It is not appropriate to include the high-temperature data in our scaling analysis, which can cause a larger and nonreliable κ (Supplementary Note 4 ). We speculate that the appropriate amount of disorder in our sample might make κ as close as to the expected value based on the recent high-precision numerical results of the correlation length exponent ν ~ 2.6 22 , 23 , 24 , 25 .

As noted above, the AC two-terminal measurement is not reliable when R 12 > 10 3 h/e 2 . To measure R 12 accurately in the most insulating regime, we employed the quasi-DC measurement method. The quasi-DC method uses a low frequency ( f = 25.3 mHz) square wave as output, which can eliminate the influence of parasitic capacitance. The thermal voltage can be removed by taking the difference of the square wave generated current. Therefore, the quasi-DC method can provide more accurate resistance values when R 12 is between ~25.8 MΩ (~ 10 3 h/e 2 ) and 2 GΩ. By varying the magnetic field B , the most insulating state R 12,max ~ 1.3 GΩ (~5.0 × 10 4 h/e 2 ) was found at B = 0.40 T at T = 50 mK (Supplementary Note 5 ). The lowest R xx in QAH state measured at 50 mK is ~20 Ω, so this suggests an order of ~10 8 ON/OFF resistance ratio in our magnetic TI sandwich samples.

Figure 3b shows the maximum two-terminal resistance R 12,max of the axion insulator state measured at different temperatures between 50 mK and 220 mK. The logarithmic R 12,max shows a linear dependence on T −1 with a slope ~333 mK below 95 mK and a steeper slope ~478 mK above 95 mK (Supplementary Note 5 ). We note that for T ≥ 100 mK, the scaling behavior starts to disappear (inset of Fig. 3a ). Therefore, the insulating behavior at the high-temperature regime might involve the contribution from the classical thermal activation, whereas the low-temperature regime reflects solely the property of an unconventional insulating quantum state (i.e., axion insulator) favoring zero-temperature limit. This further validates our choice to carry out scaling behavior study between 45 mK and 80 mK.

To summarize, we analyzed the critical scaling behavior of the quantum phase transition from a QAH insulator to an axion insulator state in magnetic TI sandwich samples. We found that the derivative of the longitudinal resistance R xx at the critical magnetic field follows a power-law dependence on temperature. Our experimental results combined with theoretical studies on the Chalker–Coddington model demonstrated that the QAH to axion insulator phase transition shares the same universality class with the QH plateau-to-plateau transition instead of the QAH plateau-to-plateau transition. Our work opens the door for further explorations of the critical behaviors of quantum phase transitions in topological materials.

Fabrications of magnetic TI sandwich Hall bar device

The magnetic TI sandwich heterostructure growth was carried out using a commercial MBE system with a vacuum ~2 × 10 −10 mbar. The heat-treated insulating SrTiO 3 (111) substrates were outgassed at ~530 °C for 1 h before the growth of the TI sandwich heterostructures. High-purity Bi (99.999%), Sb (99.9999%), Cr (99.999%), and Te (99.9999%) were evaporated from Knudsen effusion cells. During the growth of the TI, the substrate was maintained at ~240 °C. The flux ratio of Te per (Bi + Sb) was set to be >10 to prevent Te deficiency in the samples. To avoid possible contamination, a 10 nm-thick Te layer is deposited at room temperature on top of the magnetic TI sandwich heterostructures prior to their removal from the MBE chamber. Our prior study has demonstrated that the Te capping layer is much more insulating than the axion insulator in the low-temperature regime and plays a negligible effect in our electrical transport measurements 51 . We next scratched these magnetic TI heterostructure samples into a Hall bar geometry (Supplementary Note 6 ) using a computer-controlled probe station. The effective area of the Hall bar devices is ~1 mm × 0.5 mm. The electrical ohmic-contacts for dilution measurements were made by pressing indium spheres on the Hall bar. The bottom-gate electrode was prepared through an indium foil on the back side of the SrTiO 3 substrate.

Transport measurements

Transport measurements were performed using two dilution refrigerators (Leiden 6 mK/14 T and 8 mK/9 T) with the magnetic field applied perpendicular to the film plane. The bottom-gate voltage was applied using a Keithley 2400. The four-terminal measurements are obtained by applying current from 1 to 2 (Fig. 1b ), and R xx = V 34 /I 12 while R xy = V 35 /I 12 . The excitation current in the dilution refrigerator measurements is ≤0.3 nA to suppress the heating effect. The lowest trusted electron temperature in our dilution transport measurements is ~45 mK (Supplementary Note 2 ). More details about the dilution measurements can be found in the Supplementary Information .

Effective model for theoretical analysis

In the Chalker–Coddington network model, we consider a square lattice of nodes and there are two incoming and two outgoing channels at each node. Here, the wave function for a particle on each link is represented by the probability amplitudes Z i = 1,…,4 , where a random phase φ i ∈ [0, 2π] is accumulated along with the link. Because of current conservation, $|Z_1|^2 + |Z_3|^2 = |Z_2|^2 + |Z_4|^2$ . Therefore, the two incoming channels (labeled by Z i = 1,3 ) and two outgoing channels (labeled by Z i = 2,4 ) at the node can be related by a 2 × 2 scattering matrix S

where α is a real parameter characterizing the tunneling. If α is identical for all nodes, the network model is critical at α = α c = π/4. In the experiment, the critical states are described by a random Dirac Hamiltonian with randomness in the mass Δ t ( r ) and the scalar potential V t ( r ): $H_{\mathrm{t}} = \hbar v_{\mathrm{F}}( {s_xk_y\: - \:s_yk_x} ) + {\Delta}_{\mathrm{t}}\left( {\mathrm{r}} \right)s_z + V_t(r)$ . Here, the links in the network model are related to the chiral edge modes along contours of zero-mass domain walls, whereas the scattering parameter α at the nodes can be obtained by solving the eigenvalue equation of H t . In this Hamiltonian, s x,y,z are Pauli matrices acting on spin space, k x,y are wave vectors along x and y directions, and v F is the Fermi velocity.

Data availability

The data that support the findings of this study are available from X.L. or C.-Z. Chang upon reasonable request.

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Acknowledgements

We thank C. X. Liu for helpful discussion. The work in China is supported by the NSFC (11674009), the National Key Research and Development Program of China (2017YFA0303301), NSFC (11534001 and 110801719), the Beijing Natural Science Foundation (JQ18002), the Strategic Priority Research Program of Chinese Academy of Sciences (Grant XDB28000000), and the Priority Academic Program Development of Jiangsu Higher Education Institutions and NSFC of Jiangsu province (BK20190813). C.-Z. Chang and M.H.W.C. acknowledge the support from the DOE grant (DE-SC0019064). N.S. acknowledges the support from the Penn State 2DCC-MIP under NSF grant DMR-1539916. C.-Z. Chang acknowledges the support from ARO Young Investigator Program Award (W911NF1810198) and the Gordon and Betty Moore Foundation’s EPiQS Initiative (Grant GBMF9063 to C.-Z. Chang).

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C.-Z. Chang and X.L. conceived and designed the experiment. D.X. and L.Z. grew the sandwich heterostructure samples with the help of M.H.W.C., N.S., and C.-Z. Chang. X.W. and J.S. performed the dilution refrigerator measurements with the help of L.X. C.-Z. Chen and X.C.X. provided theoretical support and did all theoretical calculations. X.W., C.-Z. Chen, X.L., and C.-Z. Chang analyzed the data and wrote the manuscript with contributions from all authors.

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Wu, X., Xiao, D., Chen, CZ. et al. Scaling behavior of the quantum phase transition from a quantum-anomalous-Hall insulator to an axion insulator. Nat Commun 11 , 4532 (2020). https://doi.org/10.1038/s41467-020-18312-z

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AB - Molecules and atoms that are normally gaseous at room temperature and pressure will liquefy or solidify as their temperature reduces or their pressure increases. In the apparatus detailed here a commercial cell contains pressurised SF6 so that it exists in both gaseous and liquid states, with the density of both being similar at room temperature. The critical point on the phase diagram can then be investigated, by elevating the temperature of the cell so the densities of liquid and gas equalize. Under these conditions light passing through the cell is highly scattered by density fluctuations in the fluid. By measuring the transmitted light as a function of both temperature and wavelength, the critical exponent can be derived and the effects of light scattering with wavelength investigated.

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Keith slevin and tomi ohtsuki, phys. rev. b 80 , 041304(r) – published 9 july 2009, see synopsis: a coincidence of errors.

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We report an estimate ν = 2.593 [2.587,2.598] of the critical exponent of the Chalker-Coddington model of the integer quantum Hall effect that is significantly larger than previous numerical estimates and in disagreement with experiment. This suggests that models of noninteracting electrons cannot explain the critical phenomena of the integer quantum Hall effect.

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A coincidence of errors

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In integer quantum Hall systems, do theory and experiment agree for the wrong reasons?

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Through a combination of numerical and renormalization group arguments we resolve the contradiction between theory and experiment and demonstrate how the 'universal window' for critical exponents observed in experiment arises from a competition between marginal operators.
PDF Experimental Critical-Exponent Values for Fluids
Consensus was reached at this conference that experimental asymptotic critical exponents of classical fluids and fluids mix-tures are in agreement with the exponent values for 3D Ising-like systems, since the cause of the apparent deviations of the series-expansions estimates from experiment [36, 37] and from theory [38, 40] had been found.
Critical Exponent
The various critical exponents are then defined as follows. The manner in which m 0 → 0, as T → T c from below, defines the exponent β: (1) m 0 ∼ ( T c − T) β ( h → 0, T ≲ T c). The manner in which the low-field susceptibility χ 0 diverges, as T → T c from above (or from below), defines the exponent γ (or γ ′ ): in the gas ...
Ising critical exponents
Ising critical exponents This article lists the critical exponents of the ferromagnetic transition in the Ising model. In statistical physics, the Ising model is the simplest system exhibiting a continuous phase transition with a scalar order parameter and symmetry. The critical exponents of the transition are universal values and characterize the singular properties of physical quantities ...
Critical Exponent
A modest attempt to evaluate the critical exponents is made in Section 12.9, which yields results that are inconsistent with the experiment but teaches us quite a few lessons about the shortcomings of the so-called classical approaches.
Quantum Kibble-Zurek mechanism and critical dynamics on a ...
The length-scaling exponent, μ, found experimentally sets constraints to the possible combinations of the critical exponents z and ν at a given interaction strength.
Universal window for two-dimensional critical exponents
Through a combination of numerical and renormalization group arguments we resolve the contradiction between theory and experiment and demonstrate how the 'universal window' for critical exponents observed in experiment arises from a competition between marginal operators. Two-dimensional condensed matter is realized in increasingly diverse ...
Ising critical exponents in real fluids: An experiment
Abstract We report precise optical measurements of the equations of state of Xe, SF_6, and CO2 very near their critical points (|T - T_c|/T_c < 5 x 10^-5). We find that the critical exponents of these fluids in this region are close to the exponents calculated from the three-dimensional Ising model. Citation Physical Review Letters Volume 37 ...
Experimental Critical-Exponent Values for Fluids
The critical exponent β calculated theoretically for the 3-dimensional Ising universality class is 0.326 ±0.002 [ 35 , 36 ]. Experimental values of the critical exponent β for a number of one ...
Critical exponent $\\ensuremath{\\gamma}$ for Ni, as measured by
Abstract A review of theoretical predictions for the critical exponent in the Heisenberg model indicates that . Available measurements on Ni, all using the bulk technique, show a range of values between 1.28 and 1.35, with the best measurement falling at . In order to investigate this apparent discrepancy between theory and experiment, we have done a new measurement of via perturbed angular ...
PDF Critical exponent for the quantum Hall transition
The most important characteristics of a continuous phase transition are its critical exponents. The critical exponents are a quantitative characteristic of the critical fixed point. Agreement of theory with experiment implies that the critical fixed point studied theoretically is the same fixed point ob-served in the experiment.
Scaling behavior of the quantum phase transition from a ...
As a consequence, the critical exponent κ = p /2 ν∼ 0.38, if we assume p = 2 is in good agreement with the κ- value obtained in our experiment.
Why can't experimentalists agree on the superconducting critical exponents?
Crystals. The experimentally-accessible critical regimes of the cuprate superconductors [1] excited a great interest in experiments on the normal-superconducting phase transition in these materials. Scaling theory predicts that the phase transition is governed by the static and dynamic critical exponents ν and z, respectively [2]. This theory ...
An undergraduate laboratory experiment to determine the critical point
By measuring the transmitted light as a function of both temperature and wavelength, the critical exponent can be derived and the effects of light scattering with wavelength investigated. ", T1 - An undergraduate laboratory experiment to determine the critical point of SF6 using light scattering at selected wavelengths.
Edge State, Localization Length, and Critical Exponent from Survival
This length derived from the survival probability is compared with the distance from the transition point, yielding a critical exponent of at the phase boundary. Our experiment provides an alternative route to characterizing topological phase transitions and extracting their key physical quantities. Received 30 May 2022 Revised 5 August 2022
The 'Rings Of Power' Situation Is F**king Insane
I love lord of the rings and i recently did a middle-earth lore dive so today we collectively cry about what modern hollywood did to rings of power season 2 ...
Ukraine to EU: You can still store gas with us despite ...
More broadly, Moscow has stepped up its barrage against the energy network, with Kyiv warning it faces the toughest winter in its history as a result of power shortages. "The most important priority is strengthening our air defense systems to protect Ukraine's energy infrastructure from enemy attacks," said Chernyshov.
Critical exponent for the quantum Hall transition
We report an estimate [2.587,2.598] of the critical exponent of the Chalker-Coddington model of the integer quantum Hall effect that is significantly larger than previous numerical estimates and in disagreement with experiment. This suggests that models of noninteracting electrons cannot explain the critical phenomena of the integer quantum ...
Ukraine's power sector in turmoil over claims of ...
The network is facing a crisis as a result of Russian strikes on critical infrastructure. In a missive released later Tuesday, Kudrytsky said he had called for a meeting of Ukraine's parliament to dispute the decision, saying the stated reasoning — that he had failed to maintain the network's security in the face of Moscow's drones and ...

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Experimental Critical-Exponent Values for Fluids

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Ising critical exponents in real fluids: An experiment

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An undergraduate laboratory experiment to determine the critical point of SF6 using light scattering at selected wavelengths.

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Physical Review B

Critical exponent for the quantum Hall transition

A coincidence of errors

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