Distinguishing XY from Ising Electron Nematics

TL;DR

This paper investigates the nature of electron nematics in quantum Hall systems, showing that their behavior aligns with the Ising universality class due to lattice interactions, and proposes tests to distinguish universality classes.

Contribution

It demonstrates that including lattice interactions explains the temperature dependence of electron nematic anisotropy, establishing the Ising universality class for these systems.

Findings

Electron nematic behavior deviates from XY model at low temperatures.

Inclusion of 4-fold lattice symmetry breaking explains temperature dependence.

Quantum Hall electron nematic belongs to the Ising universality class.

Abstract

At low temperatures in ultraclean GaAs-AlGaAs heterojunctions, high fractional Landau levels break rotational symmetry, leading to increasingly anisotropic transport properties as temperature is lowered below $\sim$150mK. While the onset of transport anisotropy is well described by an XY model of an electron nematic in the presence of a weak uniform symmetry-breaking term, the low temperature behavior deviates significantly from this model. We find that inclusion of interactions between the electron nematic and the underlying crystalline lattice in the form of a 4-fold symmetry breaking term is sufficient to describe the entire temperature dependence of the transport anisotropy. This implies that the quantum Hall electron nematic is in the Ising universality class. We propose new experimental tests that can distinguish whether any two-dimensional electron nematic is in the XY or Ising universality class.

Equations

$$H=$$

$$H=-J\sum _{⟨i,j⟩}{\sigma }_i{\sigma }_j-h\sum _j{\sigma }_i.$$

$$M(T,h=0)=\bigg{[}1-sinh^{-4}\bigg{(}\frac{2J}{T}\bigg{)}\bigg{]}^{\frac{1}{8}}\ ,\ T<T_{c}~{}.$$

Full text

Strong electron correlations can drive systems into a variety of novel electronic phases of matter. Electronic liquid crystalsKivelson et al. (1998); Fradkin and Kivelson (1999); Kivelson et al. (2003) form when electronic degrees of freedom partially break the symmetries of the host crystal. Like their molecular counterparts, electron nematic phases break rotational symmetry, while retaining liquidity. Such oriented electronic liquids have been observed in a variety of systems, including strontium ruthenatesBorzi et al. (2007), iron superconductorsFang et al. (2008); Xu et al. (2008); Mazin and Johannes (2009), cuprate superconductorsAndo et al. (2002); Hinkov et al. (2008), and high fractional Landau levels. The key signature in the quantum Hall regime is a pronounced transport anisotropy that develops at low temperature.Fradkin et al. (2010); Lilly et al. (1999); Du et al. (1999); You et al. (2014, 2016)

At high fractional Landau levels, uniform quantum Hall phases are unstable to the formation of stripe and bubble phases, with the stripe phases being preferred near high half-filling.Fogler et al. (1996) Several stripe phases are possible, including (insulating) stripe crystals, as well as (compressible) electronic liquid crystal phases like nematic or smectic.Fradkin and Kivelson (1999) The quantum Hall state in GaAs-AlGaAs heterojunctions at filling $\nu=9/2$Lilly et al. (1999); Du et al. (1999) has been identified as a nematic.Fradkin et al. (2000) Fradkin et al.Fradkin et al. (2000) developed an order parameter theory of the nematic to describe the temperature evolution of the resistivity anisotropy as it develops. Using symmetry to map the resistivity anisotropy to the nematic order parameter, they showed that the temperature evolution of the resistivity anisotropy in the $\nu=9/2$ state is well described by a classical 2D XY model, with a weak uniform symmetry-breaking term, through the onset of the resistivity anisotropy as temperature is lowered below $\sim$150mK, with deviations from the theory beginning below $\sim$55mK. This model places the transition in the BKT universality classBerezinskii (1971); Kosterlitz and Thouless (2001).

One difficulty with this identification is that a true BKT transition does not break symmetry, and in fact in that model long-range order of a nematic is forbidden at finite temperature. However, as stressed in Ref. Fradkin et al., 2000, the nematic susceptibility is sufficiently strong in the BKT phase that net nematicity can develop anyway in the presence of even a weak uniform orienting field. Note that without the development of net nematicity, the resistivity anisotropy would be zero.

Here, we propose a model of the quantum Hall nematic which solves both the problem of the deviation of the low temperature resistivity anisotropy data from the order parameter theory, as well as the issue of long-range order. The nematic order parameter is a headless vector, which depends on overall orientation but not the direction. (That is, it is symmetric with respect to a 180o rotation.) If the interaction between the electron nematic and the host crystal is sufficiently weak, the nematic is free to form in any direction, and an XY model of the development of nematicity is appropriate.Fradkin et al. (2000) We first consider this case, but in the presence of lattice effects which ultimately at low temperature lock the nematic to a crystalline axis:

[TABLE]

where J gives the interaction between neighboring regions, h is an orienting field, and the $V$ term captures the four-fold symmetry of the underlying lattice. The angle the orienting field $h$ makes with the crystal field is $\phi$ (Fig. 1). The net orienting field $h$ can include intrinsic orienting effects, or be tuned via, e.g., an applied orienting field such as an in-plane magnetic field or strain, among other things.Carlson and Dahmen (2011); Cooper (2003) The “nematicity” (order parameter of the nematic) in this model is $\mathcal{N}=\left<e^{2i\theta}\right>$.Fradkin et al. (2000) Because the (normalized) macroscopic transport anisotropy $\rho_{a}$ transforms under rotations in the same way as the nematicity, the two are related as \rho_{a}\equiv\big{[}(r+1)/(r-1)\big{]}\big{(}\rho_{xx}-\rho_{yy}\big{)}/\big{(}\rho_{xx}+\rho_{yy}\big{)}=f(\mathcal{N}) where $f(\mathcal{N})$ is an odd function of $\mathcal{N}$, and $r\equiv\rho_{xx}(\mathcal{N}\rightarrow 1)/\rho_{yy}(\mathcal{N}\rightarrow 1)$ is what the ratio of macroscopic resistivities would be in a fully oriented state. For small $\mathcal{N}$, $f(\mathcal{N})=\mathcal{N}$.Fradkin et al. (2000); Carlson et al. (2006)

Results for the model of Eqn. 1 are shown in Fig. 2(a). As shown in Fig. 2 of Ref. Fradkin et al., 2000, the experimental data of Ref. Lilly et al., 1999 can be matched reasonably well for $T\gtrsim 55$mK in the presence of a weak uniform orienting field $h$ and $V=0$, but with significant deviation below $55$mK. We find that the entire temperature evolution can be captured in the presence of both nonzero $h$ and nonzero $V$, as shown in Fig. 2(a). In the Figure, we use uniform orientational field $h=.15J$ along with four-fold symmetry breaking term $V=6J$, and $J=35.3$mK. Smaller values of $V$ have too steep of a slope at low temperature. For larger values of $V$, the higher temperature behavior ($100-150$mK) can no longer be captured. For the parameters of Fig. 2(a), the absolute strength of the interaction $J$ is about half that of Ref. Fradkin et al., 2000. Because the value of $V$ that we use is not small with respect to $J$, the universality class of the transition is now Ising, not XY. For a pure XY model with $h=0$ and $V=0$, the transition temperature is $T_{\rm KT}=.89J$.Kosterlitz and Thouless (2001), but in Fig. 2(a) the onset of nematicity is happening closer to the (2D) Ising transition temperature of $T_{c}=2.27J$, consistent with this shift of universality class. (See Supplementary Information for results with other values of parameters.)

The effect of rotating the uniform orienting field $h$ away from a crystalline axis is explored in Fig. 3, where the angle $\phi$ between $h$ and the crystalline axes is varied. Note that up until $\phi\approx 30^{o}$ the impact on the temperature evolution is negligible. However, at the high symmetry point $\phi=45^{o}$, there is a true symmetry breaking transition, and the temperature onset is quite sudden.

We have found that within an XY description (Eqn. 1), moderate values of $V/J$ are required to capture the entire temperature dependence of the resistivity anisotropy in the quantum Hall nematic at $\nu=9/2$. This naturally leads to the question of how well a simple Ising model can account for the data:

[TABLE]

Here, we make the assumption that the electron nematic, once it develops, tends to lock to a favorable lattice direction. In a crystal with 4-fold rotational symmetry, because the director of the nematic is a headless vector, the order parameter of the nematic is explicitly in the Ising universality class with the two possible orientations of the nematic being mapped to $\sigma=\pm 1$. A uniform orienting field (whether intrinsic or applied) is modeled by $h$.

The nematic order parameter in this case is $\mathcal{N}=(1/N)\sum_{i}\sigma_{i}$. As in the case of an XY model of an electron nematic, the (normalized) macroscopic resistivity anisotropy $\rho_{a}$ maps to the macroscopic order parameter in the Ising description as $\rho_{a}=g(\mathcal{N})$ where $g(\mathcal{N})$ is an odd function of $\mathcal{N}$ and to first order in $\mathcal{N}$, $g=f$.

Note that for $h=0$, the magnetization for 2D-Ising model is

[TABLE]

Unlike the XY model where the low temperature nematicity can only develop for nonzero $h$ and is linear in temperature $T$, the Ising magnetization can develop even with $h=0$ and is flat at low temperatures. Our comparison of the experimental resistivity anisotropy to an Ising model is shown in Fig.2(b). We find that the data can be well described throughout the entire temperature range within a simple Ising model, with $J=32.5mK$ and $h=0.1J$.

Remarkably, we find that the entire temperature range of the resistivity anisotropy $\rho_{a}$ can be captured quite well within an Ising model in the presence of a weak uniform orienting field. Within this context, the low temperature saturation of \big{(}\rho_{xx}-\rho_{yy}\big{)}/\big{(}\rho_{xx}+\rho_{yy}\big{)} to a value $\approx.818\neq 1$ could have several origins:Fradkin et al. (2000); Carlson et al. (2006) (i) Taken at face value, the saturation implies that the bare “nematogens” represented by each Ising variable have an intrinsic resistivity anisotropy which persists down to the lowest temperatures, $r=\rho_{xx}/\rho_{yy}\approx 10.$ This could be attributable to quantum fluctuations within a bare nematogen. (ii) Similar saturation effects could also arise from even a small amount of quenched disorder, since the critical (random field type) disorder strength is zero in a two-dimensional Ising model. (iii) Nonlinear terms in the function $g(\mathcal{N})$ can lead to $g\neq 1$ as $\mathcal{N}\rightarrow 1$ at low temperature.

Experimental test of universality class: We propose that low temperature hysteresis measurements can distinguish whether any electron nematic (including the quantum Hall nematic at high fractional filling discussed here) is in the Ising or XY universality class. Fig. 4 shows the equilibrium phase diagram for both models, as a function of temperature and total orienting field $h$. Note that because the nematic order parameter switches sign upon rotating 90o, the orienting field is related to the applied in-plane magnetic field by $h\propto B_{x}^{2}-B_{y}^{2}$. Carlson and Dahmen (2011) Other external perturbations also contribute to an orienting field, such as strain.Carlson and Dahmen (2011); Koduvayur et al. (2011) In both models, a phase transition only exists at zero orienting field, $h=0$. In the Ising case, the phase transition is into a low-temperature, long-range ordered nematic phase which spontaneously breaks rotational symmetry. For the 2D XY model, the phase transition is in the BKT universality class, and the low temperature phase is critical throughout the temperature range, with no long range order, and therefore no net nematicity $\mathcal{N}$, measurable by $\mathcal{N}\propto(\rho_{xx}-\rho_{yy})/(\rho_{xx}+\rho_{yy})$. Upon field cooling in any weak $h$, both models will develop a net nematicity below a crossover temperature which is close to the phase transition temperature, whether $T_{c}=2.27J$ in the Ising case, or $T_{KT}=.89J$ in the XY case.

However, hysteresis can clearly distinguish between these universality classes. The hysteresis protocol we propose (shown in Fig. 4) is the following: Cool in an orienting field $h>0$ such as in-plane magnetic field (see Ref. Carlson and Dahmen, 2011; Cooper, 2003 for a list of orienting fields), and go to low temperature, well within the nematic region. Then, reduce $h$ to zero, and sweep it to negative values $h<0$. Using, e.g., in-plane magnetic field as an orienting field, this is equivalent to cooling with an in-plane field configuration of $\vec{B}_{\rm in-plane}=(B_{x}>0,B_{y}=0)$, then holding the temperature fixed, decreasing $B_{x}$ to zero, then immediately increasing the field $B_{y}$ from zero while holding $B_{x}=0$ so as to end with an in-plane field configuration of $\vec{B}_{\rm in-plane}=(B_{x}=0,B_{y}>0)$. Indeed, quantum Hall stripes can be reoriented via application of in-plane field.Shi et al. (2016) At low temperature in the Ising case, there is hysteresis in the net nematicity $\mathcal{N}$ as the in-plane field is swept so as to take $h$ from positive to negative and back again, or vice versa. Therefore in the Ising case, the net nematicity should remain in an oriented state, until the coercive field strength $h_{c}\neq 0$ is reached.

However, in the XY case, there should be no hysteresis. This follows from the Mermin-Wagner-Hohenberg theorem, since decreasing an applied field $h$ so as to end on the critical phase at $h=0$ can leave no long range order, $\mathcal{N}(h\rightarrow 0)\rightarrow 0$ where $\mathcal{N}$ is the net nematicity. Because $h\rightarrow 0$ with $T<T_{\rm KT}$ is critical, $\mathcal{N}\propto h^{(1/\delta)}$ as field is swept, where the critical exponent $\delta=(4/\eta)-1$ varies from $\delta(T_{\rm KT})=15$ to $\delta(T\rightarrow 0)\rightarrow\infty$.Kosterlitz (1974) This case is shown in Fig. 5(b).

It should also be noted that the test is clearest in clean samples, since addition of random field effects in the presence of a net orienting field $h$ puts both models in the universality class of the random field Ising model,da Silveira and Kardar (1999) which has hysteresis at low temperature. Whereas hysteresis of a clean Ising model has a net macroscopic jump in the nematicity, hysteresis of a random field Ising model is smooth in two dimensions.Dahmen and Sethna (1996) At very weak but finite random field strength, the model predicts avalanches in the resistivity anisotropy around the hysteresis loop with power law behavior set by critical exponents characteristic of the 2D random field Ising model critical point.

Note that our simulations as well as those of Ref. Fradkin et al., 2000 indicate the presence of a weak intrinsic orienting field, $h_{\rm int}$ in the sample, on the order of $h_{\rm int}\approx 3-5$mK. This means that to achieve $h=0$ requires that some extrinsic orienting field, such as an in-plane magnetic field or uniaxial strain,Carlson and Dahmen (2011); Cooper (2003) must be applied to compensate. Assuming this could be achieved, then zero-field cooling (ZFC) with $h=h_{int}+h_{app}=0$ has stark differences in the two models. In the Ising case, ZFC gives rise to long-range order with net nematicity and macroscopic resistivity anisotropy, with Ising critical behavior at the onset of nematicity, and the direction of that nematicity can randomly switch upon repeated cooling at $h=0$. In the XY case, ZFC can’t produce long-range order or net nematicity, but the system would instead enter a topological phase with power-law nematic order, and accompanying critical phenomena.

In conclusion, we have shown that the entire temperature dependence of the observed resistivity anisotropy in a high fractional Landau level can be well described by taking into account the discrete rotational symmetry of the underlying crystal. Inclusion of such a symmetry-breaking term shifts the universality class of the electron nematic from the Kosterlitz-Thouless universality class of the two-dimensional XY model to the two-dimensional Ising universality class. We furthermore propose an experimental test for hysteresis that can clearly distinguish whether any 2D electron nematic is in the Ising or XY (Kosterlitz-Thouless) universality class.