We numerically study the system of rapidly rotating Bose atoms at the filling factor (ratio of particle number to vortex number) $ν=1$ with the dipolar interaction. A moderate dipolar interaction stabilizes the incompressible quantum liquid at $ν=1$. Further addition induces a collapse of it. The state after the collapse is a compressible state which has phases with stripes and bubbles.
There are two types of bubbles with a different array. We also investigate models constructed from truncated interactions and the models with the three-body contact interaction. They also have phases with stripes and bubbles.
Hartree-Fock theory predicted stripe or bubble phase in the third and higher Landau levels for two-dimensional electrons, and experimental evidences has been accumulated. In this paper theoretical confirmation of the stripe phase and bubble phase in higher Landau level is given by means of the density matrix renormalization group (DMRG) method, which can give essentially exact ground state for electron systems with up to 18 electrons. From the study of the pair correlation function, the stripe phase, bubble phase, and the Wigner crystal phase are identified, and phase diagram is obtained. The …
We launch surface acoustic waves (SAW) along both the $<110>$ and the $<1\bar{1}0>$ directions of a Hall bar and measure the anisotropic conductivity in a high purity GaAs 2-D electron system in the Quantum Hall regime of the stripe and the bubble phases. In the anisotropic stripe phase, SAW propagating along the "easy" $<110>$ direction sense a compressible behavior (finite resistance) which is seen in standard transport measurement only if current flows along the "hard" $<1\bar{1}0>$ direction. In the isotropic bubble phase, the SAW data show compressible behavior in both directions, in mark…
We experimentally investigate transport across the incompressible stripe at the sample edge in the fractional quantum Hall effect regime at bulk filling factors $ν=2/3$ and $ν=2/5$. We obtain the dependence of the equilibration length, that is a phenomenological characteristics of the transport, on the voltage imbalance and the temperature, at high voltage imbalances. These dependencies are found to be of the power-law form, which is a strong evidence for the Luttigger liquid density of states.
Much of the present day qualitative phenomenology of the fractional quantum Hall effect can be understood by neglecting the interactions between composite fermions altogether. For example the fractional quantum Hall effect at $ν=n/(2pn\pm 1)$ corresponds to filled composite-fermion Landau levels,and the compressible state at $ν=1/2p$ to the Fermi sea of composite fermions. Away from these filling factors, the residual interactions between composite fermions will determine the nature of the ground state. In this article, a model is constructed for the residual interaction between composite ferm…
We present a brief survey of the charge density wave phases of a two-dimensional electron liquid in moderate to weak magnetic fields where several higher Landau levels are occupied. The review follows the chronological development of this new and emerging field: from the ideas that led to the original theoretical prediction of the novel ground states, to their dramatic experimental discovery, to the currently pursued directions and open questions.
The physics of itinerant electrons in condensed matter is by and large governed by repulsive Coulomb forces. However, rare cases exist where local attractive interactions emerge and prevail in determining the ground state of the system despite the dominant Coulomb repulsion. The most notable example is no doubt electron pairing, which leads to superconductivity and is mediated by electron phonon coupling or more intricate mechanisms such as antiferromagnetic spin order in high-temperature superconductors. The interplay of attractive and repulsive interaction components may also instigate spont…
We present recent experimental results confirming previously predicted strong asymmetries of the current distribution in narrow Hall bars under the conditions of the integer quantum Hall effect (IQHE). Using a previously developed self-consistent screening and transport theory of the IQHE, we investigate how these asymmetries, which are due to a non-linear feedback effect of the imposed current on the electron distribution in the sample, depend on relevant parameters, such as the strength of the imposed current, the magnetic field, the temperature, and the collision broadening of the Landau-qu…
Two-dimensional (2D) topological electronic insulators are known to give rise to gapless edge modes, which underlie low energy dynamics, including electrical and thermal transport. This has been thoroughly investigated in the context of quantum Hall phases, and time-reversal invariant topological insulators. Here we study the edge of a 2D, topologically trivial insulating phase, as a function of the strength of the electronic interactions and the steepness of the confining potential. For sufficiently smooth confining potentials, alternating compressible and incompressible stripes appear at the…
We use a fully self-consistent laterally resolved Hartree-Fock approximation for numerically addressing the electron configurations at higher Landau levels in the quantum Hall regime for near-macroscopic sample sizes. Our results give microscopic details of stripe- and bubble-like charge density modulations and show how these emerge depending on the filling factor. We find that there exists a region at the boundaries of the stripes and bubbles with a density modulation that corresponds to a filling factor around half filling. The microscopic details of these boundary regions determine the geom…
Transport experiments provide conflicting evidence on the possible existence of fractional order within integer quantum Hall systems. In fact integer edge states sometimes behave as monolithic objects with no inner structure, while other experiments clearly highlight the role of fractional substructures. Recently developed low-temperature scanning probe techniques offer today an opportunity for a deeper-than-ever investigation of spatial features of such edge systems. Here we use scanning gate microscopy and demonstrate that fractional features were unambiguously observed in every integer quan…
We consider the two lowest Landau levels at half filling. In the higher Landau level (nu =5/2), we find a first order phase transition separating a compressible striped phase from a paired quantum Hall state, which is identified as the Moore-Read state. The critical point is very near the Coulomb potential and the transition can be driven by increasing the width of the electron layer. We find a much weaker transition (either second order or a crossover) from pairing to the composite fermion Fermi liquid behavior. A very similar picture is obtained for the lowest Landau level but the transition…
The well-known Landauer-Buttiker (LB) picture used to explain the quantum Hall effect uses the concept of (chiral) edge states that carry the current. In their seminal 1992 article, Chklovskii, Shklovskii and Glazman (CSG) showed that the LB picture does not account for some very basic properties of the gas, such as its density profile, as it lacks a proper treatment of the electrostatic energy. They showed that, instead, one should consider alternated stripes of compressible and incompressible phases. In this letter, we revisit this issue using a full solution of the quantum-electrostatic pro…
Optimizing magnetic thin films for nanotechnologies often requires imaging nanoscale magnetic domain patterns via magnetic microscopy. The finite size of the image may however significantly affect the characterization of the observed magnetic states. We evaluated finite image size effects on the characterization of a variety of stripe and bubble domain patterns exhibited by ferromagnetic Co/Pt multilayers with perpendicular magnetic anisotropy, where the domain size (stripe width and bubble diameter) is around 100 nm. If the image size is too small, below ~ 5 microns, it may cause a significan…
I propose an effective theory of zero-temperature phases of the quantum Hall stipes: a smectic phase where the stripes are static and a novel quantum nematic phase where the positional order is destroyed by quantum fluctuations. The nematic is viewed as a Bose condensate of dislocations whose interactions are mediated by a U(1) gauge field. Collective mode spectrum and the dynamical structure factor in the two phases are calculated.
We show that dipolar interactions have dramatic effects on the groundstates of rotating atomic Bose gases in the weak interaction limit. With increasing dipolar interaction (relative to the net contact interaction), the mean-field, or high filling fraction, groundstate undergoes a series of transitions between vortex lattices of different symmetries: triangular, square, ``stripe'', and ``bubble'' phases. We also study the effects of dipolar interactions on the quantum fluids at low filling fractions. We show that the incompressible Laughlin state at filling fraction $ν=1/2$ is replaced by comp…
Stripe-like and bubble-like patterns spontaneously form in numerous physical, chemical, and biological systems when competing long-range and short-range interactions banish uniformity. Stripe-like and the related nematic morphology are also under intense scrutiny in various strongly correlated electron systems. In contrast, the electronic bubble morphology is rare. Some of the most intriguing electron bubbles develop in the two-dimensional electron gas subjected to a perpendicular magnetic field. However, in contrast to bubbles forming in classical systems such as the Turing activator-inhibito…
Competing interactions produce finite-size textures in myriad condensed matter systems, typically forming elongated stripe or round bubble domains. Transitions between stripe and bubble phases, driven by field or temperature, are expected to be reversible in nature. Here we report on the distinct character of the analogous transition for nanoscale spin textures in chiral Co/Pt-based multilayer films, known to host Néel skyrmions, using microscopy, magnetometry, and micromagnetic simulations. Upon increasing field, individual stripes fission into multiple skyrmions, and this transition exhibits…
Interaction of domain walls (DWs) in ferromagnetic stripes is studied with relevance to the formation of stable complexes of many domains. Two DW system is described with the Landau-Lifshitz-Gilbert equation including regimes of narrow and wide stripes which correspond the presence of transverse and vortex DWs. The DWs of both kinds are characterized with their chiralities (the direction of the magnetization rotation in the stripe plane) and polarities (the magnetization orientation in the center of a vortex and/or halfvortices), hence, their interactions are analyzed with dependence on these …
We use dynamic scanning capacitance microscopy (DSCM) to image compressible and incompressible strips at the edge of a Hall bar in a two-dimensional electron gas (2DEG) in the quantum Hall effect (QHE) regime. This method gives access to the complex local conductance, Gts, between a sharp metallic tip scanned across the sample surface and ground, comprising the complex sample conductance. Near integer filling factors we observe a bright stripe along the sample edge in the imaginary part of Gts. The simultaneously recorded real part exhibits a sharp peak at the boundary between the sample inter…
The ground state phase diagram of 2D electrons in a high Landau level (index N=2) is studied by the density matrix renormalization group method. Pair correlation functions are systematically calculated for various filling factors from v=1/8 to 1/2. It is shown that the ground state phase diagram consists of three different CDW states called stripe-phase, bubble-phase, and Wigner crystal. The boundary between the stripe and the bubble phases is determined to be v_c = 0.38, and that for the bubble phase and Wigner crystal is v_c = 0.24. Each transition is of first order.
In recent experiments, external anisotropy has been a useful tool to tune different phases and study their competitions. In this paper, we look at the quantum Hall charge density wave states in the $N=2$ Landau level. Without anisotropy, there are two first-order phase transitions between the Wigner crystal, the $2$-electron bubble phase, and the stripe phase. By adding mass anisotropy, our analytical and numerical studies show that the $2$-electron bubble phase disappears and the stripe phase significantly enlarges its domain in the phase diagram. Meanwhile, a regime of stripe crystals that m…
In Landau levels N > 1, the ground state of the two-dimensional electron gas (2DEG) in a perpendicular magnetic field evolves from a Wigner crystal for small filling of the partially filled Landau level, into a succession of bubble states with increasing number of guiding centers per bubble as the filling increases, to a modulated stripe state near half filling. In this work, we show that these first-order phase transitions between the bubble states lead to measurable discontinuities in several physical quantities such as the density of states and the magnetization of the 2DEG. We discuss in d…
In this report we summarize a recent progress in exploration of correlated two-dimensional electron states in partially filled high Landau levels. At a mean-field Hartree-Fock level they can be described as charge-density waves, either unidirectional (stripes) or with the symmetry of the triangular lattice (bubbles). Thermal and quantum fluctuations have a profound effect on the stripe density wave and give rise to novel phases, which are quite similar to smectic and nematic liquid crystals. We discuss the effective theories for these phases, their collective modes, and phase transitions betwe…
We investigate the details of pattern formation and transitions between different modulated phases in ultra-thin Fe films on Cu(001). At high temperature, the transitions between the uniform saturated state, the bubble state and the striped state are completely reversible, while at low temperature the bubble phase is avoided. The observed non-equilibrium behavior can be qualitatively explained by considering the intrinsic energy barriers appearing in the system due to the competition between the short-ranged exchange and the long-ranged dipolar interactions. Our experiments suggest that the he…
Stripe phases, in which the rotational symmetry of charge density is spontaneously broken, occur in many strongly correlated systems with competing interactions. One representative example is the copper-oxide superconductors, where stripe order is thought to be relevant to the mechanism of high-temperature superconductivity. Identifying and studying the stripe phases in conventional strongly correlated systems are, however, challenging due to the complexity and limited tunability of these materials. Here we uncover stripe phases in WSe2/WS2 moiré superlattices with continuously gate-tunable ch…
The transition between the stripe state and the liquid state in a high magnetic field is studied by the density-matrix renormalization-group (DMRG) method. Systematic analysis on the ground state of two-dimensional electrons in the lowest Landau level shows that the transition from the stripe state to the liquid state at v=3/8 is caused by a reduction of repulsive interaction around r=3. The same reduction of the interaction also stabilizes the incompressible liquid states at v=1/3 and 2/5, which shows a similarity between the two liquid states at v=3/8 and 1/3. It is also shown that the stron…
We propose a hydrodynamics theory of collective quantum Hall states, which describes incompressible liquids, hexatic liquid crystals, a bubble solid and a Wigner crystal states within a unified framework. The structure of the theory is uniquely determined by the space-time symmetry, and a symmetry with respect to static shear deformations. In agreement with recent experiments the theory predicts two gapped collective modes for incompressible liquids. We argue that the presence of the above two modes is a universal property of a magnetized two-dimensional collective liquid.
Mean-field theory predicts that bilayer quantum Hall systems at odd integer total filling factors can have stripe ground states in which the top Landau level is occupied alternately by electrons in one of the two layers. We report on an analysis of the properties of these states based on a coupled Luttinger liquid description that is able to account for quantum fluctuations of charge-density and position along each stripe edge. The soft modes associated with the broken symmetries of the stripe state lead to an unusual coupled Luttinger liquid system with strongly enhanced low-temperature heat …
On the basis of energy calculations we investigate the competition between quantum-liquid and electron-solid phases in the Landau levels n=1,2, and 3 as a function of their partial filling factor. Whereas the quantum-liquid phases are stable only in the vicinity of quantized values 1/(2s+1) of the partial filling factor, an electron solid in the form of a triangular lattice of clusters with a few number of electrons (bubble phase) is energetically favorable between these fillings. This alternation of electron-solid phases, which are insulating because they are pinned by the residual impurities…
We study the ground states of rotating atomic Bose-Einstein condensates with dipolar interactions. We present the results of numerical studies on a periodic geometry which show vortex lattice ground states of various symmetries: triangular and square vortex lattices, "stripe crystal" and "bubble crystal". We present the phase diagram (for systems with a large number of vortices) as a function of the ratio of dipolar to contact interactions and of the chemical potential. We discuss the experimental requirements for observing transitions between vortex lattice groundstates via dipolar interactio…
We compare the energies of different electron solids, such as bubble crystals with triangular and square symmetry and stripe phases, to those of correlated quantum liquids in partially filled intermediate Landau levels. Multiple transitions between these phases when varying the filling of the top-most partially filled Landau level explain the observed reentrance of the integer quantum Hall effect. The phase transitions are identified as first-order. This leads to a variety of measurable phenomena such as the phase coexistence between a Wigner crystal and a two-electron bubble phase in a Landau…
Stabilizing Magnetic Bubble Domains in Epitaxial 2D Magnet/Topological Insulator Heterostructures through Interfacial Interactions
University of Southern California（南加州大学）；University of South Florida（南佛罗里达大学）；The Ohio State University（俄亥俄州立大学）；Max Planck Institute for Solid State Research（马克斯·普朗克固体研究所）；Helmholtz-Zentrum Berlin für Materialien und Energie（柏林赫尔姆霍兹材料能源中心）
The ground state of 2D electrons in high magnetic field is studied by the density matrix renormalization group method. The ground state energy, excitation gap, and pair correlation functions are systematically calculated at various fillings in the lowest and the second lowest Landau levels. The ground state phase diagram, which consists of incompressible liquid state, compressible liquid state, stripe state, pairing state, and Wigner crystal is determined.
The ground state phase diagram of two-dimensional electrons in high magnetic field is studied by the density matrix renormalization group (DMRG) method. The low energy excitations and pair correlation functions in Landau levels of N=0,1,2 are calculated for wide range of fillings. The obtained results for systems with up to 25 electrons confirm the existence of various electronic states in quantum Hall systems. The ground state phase diagram for N=0,1,2 consisting of incompressible liquids, compressible liquids, charge density waves called stripe, bubble and Wigner crystal is determined.
At high magnetic fields, where the Fermi level lies in the N=0 lowest Landau level (LL), a clean two-dimensional electron system (2DES) exhibits numerous incompressible liquid phases which display the fractional quantized Hall effect (FQHE) (Das Sarma and Pinczuk, 1997). These liquid phases do not break rotational symmetry, exhibiting resistivities which are isotropic in the plane. In contrast, at lower fields, when the Fermi level lies in the $N\ge2$ third and several higher LLs, the 2DES displays a distinctly different class of collective states. In particular, near half filling of these hig…
