Quantum melting a Wigner crystal into Hall liquids

TL;DR

This paper demonstrates through variational Monte Carlo that a magnetic field can melt a Wigner crystal into an integer quantum Hall liquid, driven by quantum oscillations at specific densities.

Contribution

It provides a theoretical explanation for how magnetic fields induce melting of Wigner crystals into quantum Hall liquids, highlighting the role of quantum oscillations.

Findings

Magnetic field can induce melting of Wigner crystals into quantum Hall liquids.

Quantum oscillations cause energy cusps at integer filling factors.

The study identifies density ranges for the quantum melting transition.

Abstract

Recent experiments have shown that, counterintuitively, applying a magnetic field to a Wigner crystal can induce quantum Hall effects. In this Letter, we use variational Monte Carlo to show that a magnetic field can melt zero-field Wigner crystals into integer quantum Hall liquids. This melting originates from quantum oscillations in the liquid's ground state energy, which develops downward cusps at integer filling factors due to incompressibility. Our calculations establish a range of densities in which this quantum melting transition occurs.

Equations

$$\begin{array}{rl}H& =\sum _i\frac{{\mathbf{\pi }}_i^2}{2m}+\sum _{i<j}\frac{e^2}{|{\mathbf{r}}_i-{\mathbf{r}}_j|}\end{array}$$

$$\kappa =\frac{e^2}{\ell \hslash {\omega }_c}=r_s\sqrt{\frac{\nu }{2}}$$

$$\begin{array}{rl}\Psi (R)& =\det \left[{\psi }_i({\mathbf{r}}_j)\right]\exp \left(\sum _{i<j}u({\mathbf{r}}_i-{\mathbf{r}}_j)\right)\end{array}$$

$$\begin{array}{r}|{\psi }_{\mathbf{k}m}⟩=\sum _{n=0}^{n_{\text{max}}}{U}_{mn}^{\mathbf{k}}|{\varphi }_{\mathbf{k}n}⟩\end{array}$$

$$\begin{array}{r}{S}_{\mathbf{q}}=\frac{1}{N}\sum _{i,j}⟨\Psi |e^{i\mathbf{q}\cdot ({\mathbf{r}}_i-{\mathbf{r}}_j)}|\Psi ⟩,\end{array}$$

$$\begin{array}{rl}g(\mathbf{r})& =\frac{A}{N^2}\sum _{i\ne j}⟨\Psi |{\delta }^2(\mathbf{r}-{\mathbf{r}}_i+{\mathbf{r}}_j)|\Psi ⟩,\end{array}$$

$$\begin{array}{rl}{\omega }_p(q)& =\sqrt{{\omega }_c^2+\frac{2\pi e^{2}n}{m}q}.\end{array}$$

$$\begin{array}{rl}{S}_q& =\frac{\hslash q^2}{2m{\omega }_p(q)}=\frac{{\ell }^2}{2}{q}^2-\frac{\kappa \nu {\ell }^3}{4}{q}^3+\mathcal{O}(q^4)\end{array}$$

Full text

Introduction. When Coulomb interactions overwhelm kinetic energy, an electron gas inevitably crystallizes [1, 2]. According to quantum Monte Carlo calculations [3, 4, 5, 6, 7, 8], this zero-temperature freezing transition occurs at an interaction strength of $r_{s}\approx 37$ in two dimensions. Evidence for Wigner crystals has been observed in several material systems at both strong [9, 10, 11, 12, 13, 14, 15, 16] and zero [17, 18, 19, 20, 21, 22, 23] magnetic fields.

How does an out-of-plane magnetic field influence electron crystallization? Applying a magnetic field $B$ to a Wigner crystal monotonically reduces its electrons’ quantum zero-point motion [24, 25]. In contrast, applying a magnetic field to a Fermi liquid induces quantum Hall states accompanied by quantum oscillations that are periodic in $1/B$. Because electron liquids and crystals respond differently to magnetic fields, the liquid-crystal phase boundary should depend strongly on field strength, calling for theoretical and experimental study.

In this Letter, we apply variational Monte Carlo (VMC) to determine the quantum phase diagram of the fully spin-polarized two-dimensional electron gas (2DEG) at integer Landau level fillings $\nu$. We use a unified Slater-Jastrow wavefunction to describe both electron crystal and Hall liquid phases. At $\nu=1$ and $2$, we find phase transitions between integer quantum Hall states and electron crystals at critical interaction strengths of $r_{s}\approx 47$ and $\approx 38$, significantly larger than the value $r_{s}\approx 33$ for the transition between the fully spin-polarized Fermi liquid and Wigner crystal at $B=0$ [8, 6]. Thus, our study reveals that quantum Hall states penetrate into the Wigner crystal region of the $n-B$ plane, as shown in Fig. 1(b).

Our theory explains the puzzling observation in ZnO-based 2DEGs that applying a magnetic field to a Wigner crystal can induce quantum Hall effects [21].

Additionally, we show that, at long wavelengths, the static structure factor is directly tied to the magnetoplasmon dispersion relation and shows universal behavior across the Hall liquid and Wigner crystal phases.

Before presenting detailed calculations, in Fig. 1(a) we sketch the ground state energy $E$ of liquid and crystal phases as a function of $B$ at a density slightly below the onset of Wigner crystallization at $B=0$. The liquid’s energy oscillates as a function of $\nu\propto 1/B$ with downward cusps at integer $\nu$ due to the formation of incompressible quantum Hall states. In contrast, the crystal’s energy is expected to increase smoothly as a function of $B$ due to an increase in zero-point energy. As a result of the different behaviors in $E(B)$, the liquid’s energy dips below the crystal’s in a range of magnetic fields near integer $\nu$, causing a transition from crystal to liquid states. This implies a density window in which a Wigner crystal present at $B=0$ melts into quantum Hall liquids, as illustrated in Fig. 1(b).

Model. We consider a system of spinless electrons in two dimensions interacting through a Coulomb potential in a homogeneous, out-of-plane magnetic field. This system has the Hamiltonian

[TABLE]

where $\bm{\pi}_{i}=\bm{p}_{i}+\frac{e}{c}\bm{A}(\bm{r}_{i})$ and $\bm{\nabla}\times\bm{A}=-B\hat{z}$. Full spin polarization is likely to occur at densities and magnetic field strengths of interest.

Our system involves three independent energy scales: the kinetic energy $\hbar^{2}n/m$ (where $n$ is the electron density), the interaction energy $e^{2}\sqrt{n}$, and the cyclotron energy $\hbar\omega_{c}$ with cyclotron frequency $\omega_{c}=\frac{eB}{mc}$. The ratio of the interaction and kinetic energies defines the dimensionless interaction strength parameter: $r_{s}=1/\sqrt{\pi a_{B}^{2}n}$ where $a_{B}=\frac{\hbar^{2}}{e^{2}m}$ is the Bohr radius. Similarly, the ratio of interaction and cyclotron energies defines the Landau level mixing parameter

[TABLE]

where $\ell=\sqrt{\frac{\nu}{2\pi n}}=\sqrt{\frac{\hbar c}{eB}}$ is the magnetic length and $\nu=2\pi\ell^{2}n$ is the Landau level filling factor. The two dimensionless parameters $r_{s}$ and $\nu$ determine our system’s phase diagram. We emphasize that, unlike at fractional filling where crystallization can, in principle, occur at infinitesimally weak interaction [26, 27, 28, 29, 30, 31, 32, 33, 34], crystallization at integral $\nu$ requires strong interactions and Landau level mixing.

Hartree-Fock and Lindemann analysis. Before presenting our VMC analysis, we summarize two simpler approaches to the problem at hand: Hartree-Fock theory and the Lindemann melting criterion. Hartree-Fock theory predicts a liquid-to-crystal transition at $r_{s}\approx 2.7$ when $B=0$ [35] and $r_{s}\approx 9.6$ when $\nu=1$ [36, 20]. However, both of these values are much smaller than the estimates of $r_{s}\approx 33$ from diffusion Monte Carlo calculations at $B=0$ [8] and $r_{s}\approx 47$ from our VMC calculations at $\nu=1$. We present a Hartree-Fock phase diagram for $\nu=1,2,...,7$ in the Supplemental Material, which predicts that the critical $r_{s}$ increases as $\nu$ decreases. The Lindemann melting criterion asserts that the Wigner crystal melts when $\gamma=\sqrt{\langle\bm{u}^{2}\rangle}/a_{\text{WC}}$ exceeds a critical value $\gamma^{*}$, where $a_{\text{WC}}$ is the lattice constant and $\bm{u}$ is the displacement of an electron’s position from its lattice site. This heuristic fails to capture magnetic-field-induced melting because, within the harmonic approximation, $\sqrt{\langle\bm{u}^{2}\rangle}$ monotonically decreases as a function of $B$ [36, 25]. These theories’ shortcomings call for a more accurate approach.

*Variational Monte Carlo methods.*To model both liquid and crystal states, we use a unified Slater-Jastrow wavefunction with a homogeneous two-body Jastrow factor:

[TABLE]

where $R=(\bm{r}_{1},...,\bm{r}_{N})$ and $N$ is the number of electrons. We impose the magnetic quasi-periodic boundary condition $t_{i}(\bm{L}_{1,2})\Psi(R)=\Psi(R)$ where $\bm{L}_{1}$ and $\bm{L}_{2}$ are primitive supercell lattice vectors and $t_{i}$ is a magnetic translation operator for particle $i$ [37, 38, 39]. To satisfy this boundary condition, we expand each single-particle orbital $\ket{\psi_{\bm{k}n}}$ in the Slater determinant as a linear combination of Landau level orbitals $\ket{\phi_{\bm{k}n}}$, which comprise an orthogonal basis for the boundary condition. Concretely, we write

[TABLE]

where $U^{\bm{k}}_{mn}$ is a unitary orbital rotation matrix [36]. Here $n$ is a Landau level index and $\bm{k}$ is a wavevector in the magnetic Brillouin zone. Each row of the Slater matrix $\psi_{i}(\bm{r}_{j})$ corresponds to one of the orbitals $\ket{\psi_{\bm{k}m}}$, for $0\leq m<\nu$ and every $\bm{k}$. This construction yields the most general Slater determinant, up to the Landau level cutoff $n_{\text{max}}$, consistent with the magnetic unit cell that we choose to match the Wigner crystal.

The variational parameters of our wavefunction are contained in $u(\bm{r})$ and the orbital rotation matrix $U^{\bm{k}}_{mn}$. Using the stochastic reconfiguration algorithm [40, 41], we optimize these parameters to minimize the energy expectation value of $\Psi(R)$. For each $r_{s}$, we initialize and constrain the parameters in two ways. For the liquid phase, we optimize only $u(\bm{r})$ and set $U^{\bm{k}}_{mn}=\delta_{mn}$ so that the determinant is the non-interacting ground state. For the Wigner crystal phase, we generate crystal “seed” orbitals by optimizing a randomly initialized determinant with a Jastrow factor at large $r_{s}$. For other values of $r_{s}$, we initialize the orbitals to this seed and then optimize them together with the Jastrow factor. In the Supplemental Material, we benchmark our results with exact diagonalization on small system sizes and with a Slater-Jastrow wavefunction of Gaussian orbitals, which we find performs similarly.

Variational Monte Carlo phase diagram. In Fig. 2(a), we show variational energies obtained from these two Ansätze at $\nu=1$ and several values of $r_{s}$. When $r_{s}\geq 50$, the crystal has lower energy, while for $r_{s}\leq 45$, the liquid is favored. A linear interpolation of the liquid and crystal energies between $r_{s}=45$ and $50$ locates the melting transition at $r_{s}\approx 47$. We show analogous data at $\nu=2$ that indicate a phase transition near $r_{s}\approx 38$ in the Supplemental Material [36].

Fig. 2(b) shows that the variational ground state for $r_{s}\gtrapprox 50$ has Bragg peaks, indicating crystallization. Our ansatz assumes magnetic translation symmetry with respect to the Wigner crystal lattice and is therefore incapable of describing microemulsion or other intermediate phases in which translation symmetry is broken at a longer length scale [42, 43, 44]. Bearing this caveat in mind, the cusp in the ground state energy and sharp discontinuity in the ground state Bragg peak height at the phase boundary are consistent with a first-order melting transition.

The most recent diffusion Monte Carlo calculations available estimate the Fermi-liquid-to-Wigner-crystal transition in the fully spin-polarized 2DEG at $B=0$ to occur at $r_{s}\approx 33$ [8] (lower than $r_{s}\approx 37$ in the presence of spin [7, 8]). To obtain a fairer comparison with our VMC results at $B\neq 0$, we perform a VMC calculation at $B=0$ and find $r_{s}\approx 33$ [36], in agreement with Ref. [8]. This is significantly lower than the values $r_{s}\approx 47$ and $38$ we find at $\nu=1$ and $2$. Therefore, there exists a density window where a Wigner crystal forms at $B=0$ but melts into quantum Hall liquids at small integer fillings.

Density correlations and collective excitations. Having established our system’s phase diagram, we now examine the properties of its liquid and crystal phases. In Fig. 3(a,b) we show density correlation functions of the optimized wavefunction at $r_{s}=30$. The static structure factor,

[TABLE]

peaks smoothly around the reciprocal lattice vector $Q_{\text{WC}}=\sqrt{\frac{8\pi^{2}}{\sqrt{3}}n}$ of a Wigner crystal. Similarly, the pair correlation function,

[TABLE]

peaks smoothly around the lattice constant $a_{\text{WC}}=\sqrt{\frac{2}{\sqrt{3}n}}$ of a Wigner crystal. These features are typical of a strongly correlated liquid. In contrast, the density correlation functions at $r_{s}=60$ shown in Fig. 3(c,d) reveal crystallization. The real-space pair correlation function exhibits periodic modulation throughout the system. This long-range order appears as Bragg peaks in the static structure factor at the reciprocal lattice vector $Q_{\text{WC}}\hat{x}$ and its $\pi/3$ rotations.

We now relate our system’s long-wavelength density-density correlations to its collective excitations. Classically, a two-dimensional electron liquid in an out-of-plane magnetic field interacting through a Coulomb potential $v(q)=2\pi e^{2}/q$ has a longitudinal collective excitation mode, the magnetoplasmon, with a dispersion relation

[TABLE]

Quantum mechanically, the magnetoexciton of the integer quantum Hall state and longitudinal phonon of the Wigner crystal follow this dispersion relation at small $q$, as can be shown rigorously in the limits of small and large $r_{s}$ [45, 46, 47]. In the limit $B\rightarrow 0$, the plasmon dispersion of a 2D Coulomb gas, $\omega_{p}(q)=\sqrt{\frac{2\pi e^{2}n}{m}q}$, is recovered.

Let us assume that this collective mode exists in the quantum system and exhausts the $f$-sum rule at generic $r_{s}$. Under this assumption, the static structure factor obeys

[TABLE]

at small $q$. The leading $\mathcal{O}(q^{2})$ term is an exact property of the quantum system guaranteed by Kohn’s theorem [48] – that is, the optical spectral weight is completely exhausted by the cyclotron mode due to the decoupling between electrons’ center-of-mass and relative degrees of freedom. In the quantum Hall phase, the property $\lim_{q\rightarrow 0}S_{q}/q^{2}\geq\frac{\ell^{2}}{2}$ is also required by the system’s topologically quantized Hall conductance $\sigma_{H}=\nu e^{2}/h$ [49, 50]. The subleading $\mathcal{O}(q^{3})$ term, which is non-analytic as a function of $\bm{q}$, is a consequence of the Coulomb interaction in gapped two-dimensional systems.

In Fig. 4, we test this assumption by comparing $\hbar\omega_{p}(q)$ with the single-mode approximation dispersion, $\Delta(q)\equiv\frac{\hbar^{2}q^{2}}{2mS_{q}}$, across a wide range of interaction strengths [51]. $\hbar\omega_{p}(q)$ and $\Delta(q)$ converge as $q\rightarrow 0$ in both the liquid and Wigner crystal phases. This supports the existence of a magnetoplasmon mode that exhausts the $f$-sum rule in the limit $q\rightarrow 0$ for all $r_{s}$. It also shows that our wavefunctions respect Kohn’s theorem. We emphasize that this behavior is not enforced by construction but rather obtained through optimization, signaling that our wavefunctions are accurate. Additionally, when $r_{s}$ is large, $\Delta(q)$ shows a roton minimum near $Q_{\text{WC}}$. This rather generic feature of strongly interacting quantum liquids is a direct consequence of the peak in $S_{q}$ and reflects strong short-range correlations [51, 52].

Discussion. We have shown that a magnetic field can melt a Wigner crystal into quantum Hall liquids at small integer Landau level fillings. Using the variational Monte Carlo method, we have computed the quantum phase diagram of the fully spin-polarized 2DEG at $\nu=1$ and $2$. We find phase transitions between integer quantum Hall and Wigner crystal states at $r_{s}\approx 47$ and $\approx 38$, respectively. These interaction strengths are larger than the $B=0$ transition points, $r_{s}\approx 37$ with spin and $r_{s}\approx 33$ without spin [7, 8]. This establishes a density window in which a Wigner crystal present at $B=0$ melts into a quantum Hall state under a magnetic field. Experimentally, this implies that at fixed density the system is strongly insulating at zero field but exhibits a quantized Hall effect with $R_{xx}=0$ at small integer $\nu$, as observed in Ref. [21]. Finally, we have shown that magnetoplasmon collective excitations control the system’s long-wavelength density correlations, which exhibit unified behavior across both liquid and crystal phases.

While we have focused on integer $\nu$, similar magnetic-field-induced transitions between liquid and crystal phases also occur at fractional Landau level filling. For example, at small $\nu<1$, electron crystal phases are interrupted by fractional quantum Hall liquids at special fractions such as $\frac{1}{3}$ or $\frac{1}{5}$, as has been observed in multiple material systems [53, 11, 14, 54, 55, 56, 57, 58, 59, 16, 60, 61] and studied theoretically [32, 33, 34]. When these liquid-crystal oscillations occur in higher Landau levels, they lead to “reentrant integer quantum Hall effects” [62, 63, 64, 65, 66, 67]. In some moiré systems at $B=0$, integer quantum anomalous Hall states are observed over a broad range of densities, except certain fractional band fillings where fractional quantum anomalous Hall states form [68, 69]. These phenomena likely also originate from downward cusps in the liquid’s energy due to incompressibility that bring it below the energy of competing crystal phases (see Fig. 1(b)).

The possible coexistence of quantized Hall effect with crystalline order in a Hall crystal phase has been proposed theoretically [70]. However, we find no evidence for such a state at $\nu=1$ given a standard Coulomb interaction, in agreement with the expectations of Ref. [70].

Our work invites further research in related directions. One natural extension is to explore the influence of a moiré potential, which can give rise to generalized Wigner crystals [71, 72, 73, 74, 75, 76, 77, 78]. Another compelling direction is to investigate interplay between magnetic fields and Wigner crystallization in few-layer graphene systems, which feature unconventional band dispersions and large Berry curvature [16]. Crystallization at $B=0$ in flat bands with Berry curvature is also an intriguing possibility to address numerically beyond the Hartree-Fock approximation [79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91]. More broadly, microscopic mechanisms for coexistence between crystallization and topology—whether at zero or finite magnetic field—remain to be explored [92, 70]. Finally, it may be possible to combine aspects of our wavefunction ansatz with neural networks to study competing liquid and crystal phases in quantum Hall systems and Chern bands [93, 94, 95, 96].

Acknowledgments. We thank Max Geier, Yaar Vituri, Bryan Clark, Shiwei Zhang, Agnes Valenti, David Dai, Patrick Ledwith, and Khachatur Nazaryan for helpful discussions. AR is particularly grateful to Ahmed Abouelkomsan for numerous helpful discussions and encouragement. The authors acknowledge the MIT SuperCloud and Lincoln Laboratory Supercomputing Center for providing computing resources that have contributed to the research results reported in this paper. This work was supported by the U.S. Army DEVCOM ARL Army Research Office through the MIT Institute for Soldier Nanotechnologies under Cooperative Agreement number W911NF-23-2-0121. LF was supported by a Simons Investigator Award from the Simons Foundation.

Data availability. The data and code that support the findings of this article are openly available [97].