Unusual Metal to Marginal-Metal Transition in Two-Dimensional Ferromagnetic Electron Gases

TL;DR

This paper investigates the quantum criticality and phase transitions in two-dimensional ferromagnetic electron gases with disorder and spin-orbit interactions, revealing a novel marginal metal phase and unorthodox scaling laws.

Contribution

It uncovers a new marginal metal phase with critical states and demonstrates unconventional scaling laws near quantum phase transitions in 2D ferromagnetic electron gases.

Findings

Identification of a marginal metal phase with critical states

Unusual exponential divergence of correlation lengths near transitions

Universal fractal dimension of critical states D=1.90±0.02

Abstract

Two-dimensional ferromagnetic electron gases subject to random scalar potentials and Rashba spin-orbit interactions exhibit a striking quantum criticality. As disorder strength $W$ increases, the systems undergo a transition from a normal diffusive metal consisting of extended states to a marginal metal consisting of critical states at a critical disorder $W_{c,1}$. Further increase of $W$, another transition from the marginal metal to an insulator occurs at $W_{c,2}$. Through highly accurate numerical procedures based on the recursive Green's function method and the exact diagonalization, we elucidate the nature of the quantum criticality and the properties of the pertinent states. The intrinsic conductances follow an unorthodox single-parameter scaling law: They collapse onto two branches of curves corresponding to diffusive metal phase and insulating phase with correlation lengths diverging exponentially as $\xi\propto\exp[\alpha/\sqrt{|W-W_c|}]$ near transition points. Finite-size analysis of inverse participation ratios reveals that the states within the critical regime $[W_{c,1},W_{c,2}]$ are fractals of a universal fractal dimension $D=1.90\pm0.02$ while those in metallic (insulating) regime spread over the whole system (localize) with $D=2$ ($D=0$). A phase diagram in the parameter space illuminates the occurrence and evolution of diffusive metals, marginal metals, and the Anderson insulators.

Equations

$$\begin{array}{c}H=\sum _i{c}_i^{†}{ϵ}_i{c}_i-\sum _{⟨i,j⟩}{c}_i^{†}{R}_{ij}{c}_j,\end{array}$$

$$\begin{array}{c}{ϵ}_i={ϵ}_0{\sigma }_0-\Delta {\sigma }_z+V_i{\sigma }_0.\end{array}$$

$$\begin{array}{c}{R}_x=\frac{1}{2}(t{\sigma }_0+i\tilde{\alpha }{\sigma }_y)\text{and}{R}_y=\frac{1}{2}(t{\sigma }_0-i\tilde{\alpha }{\sigma }_x).\end{array}$$

$$\begin{array}{c}h(\mathbf{k})=({ϵ}_0-\cos k_x-\cos k_y){\sigma }_0-\Delta {\sigma }_z\\ +\tilde{\alpha }(\sin k_x{\sigma }_y-\sin k_y{\sigma }_x).\end{array}$$

$$\begin{array}{c}{g}_L=f(L/\xi ),\end{array}$$

$$\begin{array}{c}{p}_2(E,W)=\sum _i|{\psi }_i(E,W){|}^4\end{array}$$

$$\begin{array}{c}{p}_2(E,W)\propto L^{-D}\end{array}$$

Full text

Anderson localization is a long-lasting fundamental concept in condensed matter physics pwanderson1 ; sjohn1 ; dswiersma1 ; mstorzer1 ; aachabanov1 ; psheng1 ; hhu1 ; tschwartz1 ; cwang1 and keeps bringing us surprising, especially in its critical dimensionality of two palee1 ; bkramer1 ; fevers1 . In the early time, the orthodox view is the absence of diffusion of an initially localized wave packet at an arbitrary weak disorder in one- and two-dimensional electron gases (2DEGs) while metallic states and Anderson localization transitions (ALTs) can occur in 3D eabrahams1 ; dfriedan1 . Later more careful renormalization group calculations shikami1 and numerical simulations snevangelou1 ; rmerkt1 ; yasada1 ; pmarkos1 ; gorso1 , together with experiments bkramer1 , show that intrinsic degrees of freedom can alter the results in 2D: Half-integer spin particle systems can also support ALTs when the spin rotational symmetry is broken through spin-orbit interactions (SOIs), regardless whether the time reversal symmetry is preserved (symplectic class) shikami1 ; snevangelou1 ; rmerkt1 ; yasada1 ; pmarkos1 ; gorso1 or not (unitary class) cwang2 ; ysu1 ; cwang3 . The most unquestionable examples would be quantum Hall effects of both non-interacting djthouless1 ; ammpruisken1 ; bhuckestein2 and interacting dctsui1 ; rblaughlin1 2DEGs in strong perpendicular magnetic fields. On the other hand, all states of disordered non-interacting integer-spin particle systems must be localized rsepehrinia1 ; ysu2 .

However, recent numerical studies xcxie1 ; yavishai1 ; gxiong1 ; cwang2 ; ysu1 ; cwang3 ; jbang1 showed that the current understanding of ALTs in non-interacting 2DEGs is far from completed when SOIs are involved. For example, in contrast to the predictions based on the nonlinear $\sigma$ model that claims only localized states are allowed in the unitary ensemble shikami1 , a band of extended states together with an ALT or critical states usually accompanied by a Kosterlitz-Thouless (KT) transition could exist in 2DEGs without time-reversal symmetry, depending on the form of SOIs and the strength of magnetic field cwang2 ; ysu1 ; cwang3 . In this work, we observe an anomalous phase transition from a normal metal to a marginal metal yyzhang1 , consisting of a band of metallic critical states, in a ferromagnetic 2DEG on a square lattice subject to a Rashba SOI and random on-site potentials. The statements are supported by argumentations based on two independent highly-accurate numerical approaches: the finite-size scaling analysis of two-terminal conductances and the inverse participation ratios (IPRs) analysis of wave functions obtained from the exact diagonalization. The unaccustomed marginal metal (MM) phase exists between a diffusive metal (DM) phase at weak disorders and an Anderson insulator (AI) phase at strong disorders. Scaling analyses of IPRs show that wave functions of states in the MMs are of fractals of dimension $D=1.90\pm 0.02$, a feature reminiscent of a band of critical states in the random SU(2) model subject to strong magnetic fields cwang2 .

Our main result is the new (marginal metallic) phase whose $\beta$-function (symbols and black line) defined as $\beta(\ln g_{L})=d\ln g_{L}/d\ln L$ describes an unconventional DM-MM-AI transition, as summarized in Fig. 1. In comparison, $\beta$ for other types of phase transitions in 2D are also included (elucidate in the caption). Here $g_{L}$ and $L$ are dimensionless conductance and system size, respectively. For a large (small) conductance, i.e., $g_{L}>g_{c,1}$ ($g_{L}<g_{c,2}$), $\beta$ is positive (negative), indicating a metallic (insulating) phase. While between two critical conductances $g_{c,1}$ and $g_{c,2}$, $\beta=0$ shows a MM phase in which conductances are through critical states that are size independent. Different from an ALT at a fixed point $g_{c}$ snevangelou1 ; rmerkt1 ; yasada1 ; pmarkos1 ; gorso1 ; ysu1 ; cwang3 , the MM phase between $[g_{c,1},g_{c,2}]$ is a fixed line in which the system does not flow away when its size is scaled. Furthermore, near both DM-MM and MM-AI transition points, correlation lengths $\xi$ locating on the DM and AI sides, respectively, diverge with disorder strength $W$ as $\xi(W)\propto\exp[\alpha/\sqrt{|W-W_{c}|}]$, a similar finite-size scaling law in KT transitions (green line) cwang2 ; yyzhang1 ; xcxie1 .

Our model is a tight-binding Hamiltonian on a square lattice of size $L^{2}$ with unit lattice constant,

[TABLE]

where $c^{\dagger}_{i}=(c^{\dagger}_{i,\uparrow},c^{\dagger}_{i,\downarrow})$ and $c_{i}$ are, respectively, the single electron creation and annihilation operators on site $i=(x_{i},y_{i})$ with $x_{i},y_{i}$ being integers and $1\leq x_{i},y_{i}\leq L$. $\langle ij\rangle$ denotes $i$ and $j$ as the nearest-neighbor sites. The first term stands for on-site energy:

[TABLE]

Here $\sigma_{0}$ and $\sigma_{x,y,z}$ are the 2-by-2 identity and the Pauli matrices respectively. $\epsilon_{0}$ is a constant energy term. $-\Delta\sigma_{z}$ introduces a ferromagnetic term that breaks the time-reversal symmetry with $\Delta$ quantifying the mean-field exchange splitting nnagaosa1 . Disorders are modelled by $V_{i}\sigma_{0}$ with uncorrelated random numbers $V_{i}$ following the normal distribution of zero mean and the variance of $W^{2}$. Thus, $W$ measures the degree of randomness. A Rashba SOI is encoded in the hopping matrices $R_{ij}$, parameterized by matrices $R_{x}$ and $R_{y}$ along the $x-$ and $y-$directions, respectively,

[TABLE]

where $t$ is the energy unit and $\tilde{\alpha}$ measures the strength of SOIs. In the clean limit, model (1) can be blocked diagonalized in the momentum space as $H=\sum_{\bm{k}}c^{\dagger}_{\bm{k}}h(\bm{k})c_{\bm{k}}$ with

[TABLE]

Hereafter we fix $\epsilon_{0}=2$ such that the effective $\bm{k}\cdot\bm{p}$ Hamiltonian near the band edge reads $\frac{\bm{p}^{2}}{2}+\tilde{\alpha}(\bm{p}\times\bm{\sigma})\cdot\hat{z}$. This form of Hamiltonians has been widely employed to enlighten the intrinsic and extrinsic mechanism of the anomalous Hall effect, and possible physical realizations of model (1) include a large family of ferromagnetic semiconductors such as GaAs and other III-V host materials nnagaosa1 .

To investigate the localization properties of states of model (1), we employ the Landauer formula to calculate the dimensionless conductance of a disordered sample between two clean semi-infinite leads at a given Fermi level $E$, $\tilde{g}_{L}=\text{Tr}[TT^{\dagger}]$, where $T$ is the transmission matrix amackinnon1 . To exclude the contribution from contact resistances, we define the dimensionless conductance $g_{L}$ as, $1/g_{L}=1/\tilde{g}_{L}-1/N_{c}$, where $N_{c}$ is the number of propagating modes in the leads at Fermi energy $E$ sdatta1 . The determination of quantum phase transitions is based on the following criteria: (1) For the Fermi level in the DM (AI) phase, $dg_{L}/dL$ is positive (negative), while in the MM phase, $g_{L}$ is independent of $L$. (2) In the vicinity of phase transition points, there exists the one-parameter scaling hypothesis eabrahams1 such that $g_{L}$ of different system sizes merge into a universal smooth scaling function $f(x)$, i.e.,

[TABLE]

with the correlation length $\xi$ diverging at the transition points.

Two typical examples are shown in Fig. 2(a) and its inset that plot $g_{L}$ as a function of $W$ for $E=0.2$ (near band edge), $\tilde{\alpha}=0.2$, and $\Delta=0.01$ and [math] (inset), respectively. Clearly, in the absence of the ferromagnetic coupling $\Delta=0$ when model (1) belongs to the symplectic class, all curves cross at a single point $W_{c}$. $dg_{L}/dL$ is positive (negative) for $W$ smaller (larger) $W_{c}$. These features are concrete evidence of an ALT at $W_{c}$. Finite-size scaling analysis supp shows that $g_{L}$ for different sizes $L$ collapse to a single smooth scaling curve, and $\xi$ diverges as $|E-E_{c}|^{-\nu}$ with $\nu=2.8\pm 0.2$, consistent with previous estimates yasada1 ; pmarkos1 ; gorso1 .

Strikingly, once systems enter the unitary class by turning on the ferromagnetic coupling, say $\Delta=0.01$, we observe a MM phase in the window of $W\in[W_{c,1},W_{c,2}]$ ($W_{c,1}=0.38\pm 0.01$ and $W_{c,2}=0.43\pm 0.01$) within which $dg_{L}/dL=0$ for all $L$, while states for $W<W_{c,1}$ and $W>W_{c,2}$ are extended and localized, respectively. Two phase transitions are evident in Figs. 2 (b) and (c), which illustrate the enlargements near $W_{c,1}$ and $W_{c,2}$, respectively. The MM-AI transition at $W_{c,2}$, in addition to the zero-plateau of $\beta$ function shown in Fig. 1, is highly evocative of the KT criticality arising in another unitary ensemble with random SOIs cwang2 and the graphene with random fluxes or long-range impurities xcxie1 ; yyzhang1 . Nonetheless, the DM-MM transition at $W_{c,1}$ from a band of extended states to a band of critical states is highly nontrivial since both of them are of metallic phases in the sense that their wave functions spread over the whole lattice (illustrate later). To the best of our knowledge, this kind of disorder-driven metal-metal transitions has never been observed before in 2D materials, but in 3D semimetals pgoswami1 ; kkobayashi1 ; broy1 ; czchen1 ; bfu1 .

To substantiate the validity of one-parameter scaling hypothesis, we show that all curves in Figs. 2(b,c) collapse into two smooth functions $f(x=L/\xi)$ shown in the insets of the figures supp , which offer direct verifications of quantum phase transitions at $W_{c,1}$ and $W_{c,2}$. On the insulating side and near the phase transition point $W_{c,2}$, the correlation length is expected to diverge as $\xi\propto\exp[\alpha_{2}/\sqrt{|W-W_{c,2}|}]$ with $\alpha_{2}=8.0\pm 0.8$, a fingerprint of the KT transitions xcxie1 ; yyzhang1 ; cwang2 . Differently, there are no reliable analytical and numerical estimates for the divergence law near the DM-MM transition $W_{c,1}$. Scaling analysis also suggests $\xi\propto\exp[\alpha_{1}/\sqrt{|W-W_{c,1}|}]$ with the exponent $\alpha_{1}=9\pm 3$. Besides, a power-law divergence of correlation lengths for ALTs, i.e., $\xi\propto|W-W_{c}|^{-\nu}$, also fit the numerical data with $\nu=32$. However, the obtained $\nu$ is much larger than any known critical exponents in disordered 2D systems fevers1 ; bkramer1 . We thus argue that at the DM-MM transition the correlation lengths show the same scaling behavior as those for KT transitions, rather than the power-law divergence for ALTs.

So far, we have provided one example of the DM-MM-AI transition in model (1). Needless to say, many questions arise, and, among them, the most important one may be the proof of the universality of such a quantum phase transition. In Supplemental Materials supp , we show indications of universality by substantiating emergences of the three phases at different Fermi energies with the same divergence law of $\xi$, i.e., $\xi\propto\exp[\alpha/\sqrt{|W-W_{c}|}]$ at critical points. The same physics is observed if we choose $E$ as the scaling variable at a fixed disorder. Furthermore, simulations for a different form of disorders and a distinct SOI due to the Dresselhaus effect are both in qualitative agreement with Fig. 2. All these features indicate that the MM phase prevails in ferromagnetic 2DEGs with SOIs and favors the exponential divergence of $\xi$ at critical points.

Having established the universality of DM-MM-AI transitions, we further consider the nature of wave functions in three phases, especially the fractal structure of wave functions in MM phase. Wave functions at an isolated critical point of an ALT or in the critical band are known to have multifractal structures characterizing by a set of anomalous dimensions measuring how their moments scale with sizes fevers1 ; cwang2 ; yyzhang1 . Among them, the fractal dimension plays a pre-eminent role, which is related to the IPR defined as

[TABLE]

with $\psi_{i}(E,W)$ being the renormalized wave functions of energy $E$ and disorder $W$ at site $i$ for a specific realization. For large enough systems, the average IPR scales with size $L$ as xrwang1 ; mjanssen1 ; jhpixley1

[TABLE]

with $D$ being the fractal dimension. If the state is extended (localized), $D=d$ ($D=0$), where $d=2$ is the spatial dimension. While for a critical state, an anomalous scaling with $L$ is expected, i.e., $D\in[0,d]$. Thus, we expect that states in the MM phase have a universal fractal dimension such that their wave functions occupy much sparse space like fractals.

Numerically, we use the exact diagonalization to find the eigenfunctions of model (1). In our scenario, we construct the tight-binding Hamiltonian by the Kwant package gwgroth1 in Python and solve the eigenequation $H\psi=E\psi$ by the Scipy library scipy for $L$ varying from 200 to 300 . The average logarithms of IPR as a function of $\ln L$ for $W=0.08$, 0.28, 0.42, 0.56, and 0.66 at $E=0.2$ (the same as Fig. 2(a)) are shown in Fig. 3(a). The corresponding curves are virtually straight lines, which provide strong evidence for the scaling law Eq. (7). The slopes (fractal dimensions) clearly decrease with the increasing of $W$ from the DM phase to the AI phase, and $D=1.90\pm 0.02$ for $W_{c,1}<W=0.42<W_{c,2}$ in the MM phase.

We further authenticate the universality of the fractal nature by displaying $D(W)$ at $E=0.2$ for a large range of disorders covering the three phases in Fig. 3(b). Apparently, a plateau of $D=1.90$ is observed in the MM phase determined by data in Fig. 2(a), indicating that the fractal dimension of the fixed line does not depend on $W$. For $W<W_{c,1}$ (DM), wave functions are not a fractal any more since $D\simeq d$, while, for $W\gg W_{c,2}$, IPRs are found to be independent of $L$, i.e., $D=0$, see the inset of Fig. 3(b), a typical feature for AIs. Noticeably, wave functions near MM-AI transitions and on the localized side, for example $W\in[0.44,0.64]>W_{c,2}$, have fractal structures as well, which can be contributed to the finite-size effect, rather than the criticality, since $D$ will decrease if we evaluate it by using larger system sizes supp .

It is also enlightening to compare the fractal dimensions of the MM phase in model (1) with those of critical states in other 2D materials. The fractal dimension of isolated critical levels for ALTs in the symplectic ensemble is found to be $1.66\pm 0.05$ hobuse1 while $D=1.75$ for the quantum Hall type criticality fevers2 . Thus, wave functions in the MM phase with $D=1.90\pm 0.02$ occupy a larger space than those critical states of random SU(2) model under strong magnetic fields cwang2 .

A more inclusive picture is procured by executing exhaustive simulations for different $\Delta$ at $E=0.2$. The fixed line persists at finite $\Delta$ as expected such that the three phases coexist, and model (1) always experiences the DM-MM-AI phase transitions at $\Delta-$dependent transition points $W_{c,1}$ and $W_{c,2}$. While, for $\Delta=0$, there is only one quantum critical state at which the system undergoes a normal ALT. Furthermore, it is numerically justified that the correlation lengths $\xi$ always diverge as $\xi\propto\exp[\alpha/\sqrt{|W-W_{c}|}]$ near $W_{c,1}$ (squares) and $W_{c,2}$ (circles) in Fig. 4, see clarifications in Supplemental Materials supp .

In conclusion, analyses of the dimensionless conductance and the IPR provide substantial evidence to the existence of an unusual marginal metallic phase between the diffusive metal and the Anderson insulator in ferromagnetic 2DEGs with SOIs. Such systems undergo a DM-MM-AI transition as either disorder strength or Fermi level varies. Near the transition points, the conductance can be described well by the one-parameter scaling hypothesis. The criticality of the DM-MM-AI transitions is consistent with universality description of a quantum phase transition in the sense that correlation lengths diverge as an exponential of an inverse square root of $|W-W_{c}|$ for all critical points. Besides, eigenfunctions in the MM phase are of fractals of dimension $D=1.90$, while extended states in the DM phase spread over the entire lattice. A schematic phase diagram in the $\Delta-W$ plane is presented.