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arXiv:2002.09356 · Physical review. B./Physical review. B

Interplay between fractional quantum Hall liquid and crystal phases at low filling

Zheng-Wei Zuo, Ajit Coimbatore Balram, Songyang Pu, Jianyun Zhao, Th. Jolicœur, Arkadiusz Wójs, Jainendra K. Jain

Published Aug 19, 2020• 30 citations
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Abstract

The nature of the state at low Landau-level filling factors has been a long-standing puzzle in the field of the fractional quantum Hall effect (FQHE). While theoretical calculations suggest that a crystal is favored at filling factors $\ensuremath{\nu}\ensuremath{\lesssim}\frac{1}{6}$, experiments show, at somewhat elevated temperatures, minima in the longitudinal resistance that are associated with fractional quantum Hall effect at $\ensuremath{\nu}=\frac{1}{7}, \frac{2}{11}, \frac{2}{13}, \frac{3}{17}, \frac{3}{19}, \frac{1}{9}, \frac{2}{15}$, and $\frac{2}{17}$, which belong to the standard sequences $\ensuremath{\nu}=n/(6n\ifmmode\pm\else\textpm\fi{}1)$ and $n/(8n\ifmmode\pm\else\textpm\fi{}1)$. To address this paradox, we investigate the nature of some of the low-$\ensuremath{\nu}$ states, specifically $\ensuremath{\nu}=\frac{1}{7}, \frac{2}{13}$, and $\frac{1}{9}$, by variational Monte Carlo, density matrix renormalization group, and exact diagonalization methods. We conclude that in the thermodynamic limit, these are likely to be incompressible fractional quantum Hall liquids, albeit with strong short-range crystalline correlations. This suggests a natural explanation for the experimentally observed behavior and a rich phase diagram that admits, in the low-disorder limit, a multitude of crystal-FQHE liquid transitions as the filling factor is reduced.

DOI
10.1103/physrevb.102.075307

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The nature of the state at low Landau-level filling factors has been a long-standing puzzle in the field of the fractional quantum Hall effect (FQHE). While theoretical calculations suggest that a crystal is favored at filling factors \ensuremathν\ensuremath≲16\ensuremath{\nu}\ensuremath{\lesssim}\frac{1}{6}\ensuremathν\ensuremath≲61​, experiments show, at somewhat elevated temperatures, minima in the longitudinal resistance that are associated with fractional quantum Hall effect at \ensuremathν=17,211,213,317,319,19,215\ensuremath{\nu}=\frac{1}{7}, \frac{2}{11}, \frac{2}{13}, \frac{3}{17}, \frac{3}{19}, \frac{1}{9}, \frac{2}{15}\ensuremathν=71​,112​,132​,173​,193​,91​,152​, and 217\frac{2}{17}172​, which belong to the standard sequences \ensuremathν=n/(6n\ifmmode±\else\textpm\fi1)\ensuremath{\nu}=n/(6n\ifmmode\pm\else\textpm\fi{}1)\ensuremathν=n/(6n\ifmmode±\else\textpm\fi1) and n/(8n\ifmmode±\else\textpm\fi1)n/(8n\ifmmode\pm\else\textpm\fi{}1)n/(8n\ifmmode±\else\textpm\fi1). To address this paradox, we investigate the nature of some of the low-\ensuremathν\ensuremath{\nu}\ensuremathν states, specifically \ensuremathν=17,213\ensuremath{\nu}=\frac{1}{7}, \frac{2}{13}\ensuremathν=71​,132​, and 19\frac{1}{9}91​, by variational Monte Carlo, density matrix renormalization group, and exact diagonalization methods. We conclude that in the thermodynamic limit, these are likely to be incompressible fractional quantum Hall liquids, albeit with strong short-range crystalline correlations. This suggests a natural explanation for the experimentally observed behavior and a rich phase diagram that admits, in the low-disorder limit, a multitude of crystal-FQHE liquid transitions as the filling factor is reduced.
While it is generally believed that a crystal state is favored at filling factors ν<1/6\nu< 1/6ν<1/6 in the lowest Landau level, experiments show, at elevated temperatures, minima in the longitudinal resistance that are associated with fractional quantum Hall effect (FQHE) at the standard sequences ν=n/(6n±1)\nu=n/(6n\pm 1)ν=n/(6n±1) and ν=n/(8n±1)\nu=n/(8n\pm 1)ν=n/(8n±1). To address this paradox, we investigate the nature of the state at filling factors ν=1/7\nu=1/7ν=1/7, 1/9, 1/11 and 2/13 by several methods and find that the competition between the crystal and liquid states is more subtle than previously thought. Exact diagonalization study in the spherical geometry shows behavior consistent with an FQHE liquid for system sizes available. Density matrix renormalization group study in the cylindrical geometry shows that the nature of the state depends on the particle number, with crystal favored for certain particle numbers but not all; these studies do not allow us to make conclusive statements regarding the thermodynamic nature of the state. A variational study shows that the composite-fermion crystal and the renormalized FQHE liquid are very close in energy. Based on these studies, it is not possible to rule out, definitively, an FQHE liquid with strong short-range crystalline correlations at these filling factors.
The interplay between strongly correlated liquid and crystal phases for two-dimensional electrons exposed to a high transverse magnetic field is of fundamental interest. Through the non-perturbative fixed phase diffusion Monte Carlo method, we determine the phase diagram of the Wigner crystal in the ν−κ\nu-\kappaν−κ plane, where ν\nuν is the filling factor and κ\kappaκ is the strength of Landau level mixing. The phase boundary is seen to exhibit a striking ν\nuν dependence, with the states away from the magic filling factors ν=n/(2pn+1)\nu=n/(2pn+1)ν=n/(2pn+1) being much more susceptible to crystallization due to Landau level mixing than those at ν=n/(2pn+1)\nu=n/(2pn+1)ν=n/(2pn+1). Our results explain the qualitative difference between the experimental behaviors observed in n-doped and p-doped GaAs quantum wells, and, in particular, the existence of an insulating state for ν<1/3\nu<1/3ν<1/3 and also for 1/3<ν<2/51/3 <\nu< 2/51/3<ν<2/5 in low density p-doped systems. We predict that in the vicinity of ν=1/5\nu=1/5ν=1/5 and ν=2/9\nu=2/9ν=2/9, increasing LL mixing causes a transition not into an ordinary electron Wigner crystal but rather into a strongly correlated crystal of composite fermions carrying two vortices.
We present a detailed study of the ground-state entanglement in disordered fractional quantum Hall liquids. We consider electrons at various filling fractions fff in the lowest Landau level, with Coulomb interactions. At f=1/3,1/5f=1/3,1/5f=1/3,1/5, and 2/52/52/5 where an incompressible ground-state manifold exists at zero disorder, we observe a pronounced minimum in the derivative of entanglement entropy with respect to disorder. At each filling, the position of this minimum is stable against increasing system size, but its magnitude grows monotonically and appears to diverge in the thermodynamic limit. We consider this behavior of the entropy derivative as a compelling signal of the expected disorder-driven phase transition from a topological fractional quantum Hall phase to an insulating phase. On the contrary, at f=1/2f=1/2f=1/2 where a compressible composite fermion sea is present at zero disorder, the entropy derivative exhibits much greater, almost chaotic, finite-size effects, without a clear phase transition signal for system sizes within our exact diagonalization limit. However, the dependence of entanglement entropy with system size changes with increasing disorder, consistent with the expectation of a phase transition from a composite fermion sea to an insulator. Finally, we consider f=1/7f=1/7f=1/7 where compressible Wigner crystals are quite competitive at zero disorder, and analyze the level statistics of entanglement spectrum at f=1/3f=1/3f=1/3.
The interplay between strongly correlated liquid and crystal phases for two-dimensional electrons exposed to a high transverse magnetic field is of fundamental interest. Through the nonperturbative fixed-phase diffusion Monte Carlo method, we determine the phase diagram of the Wigner crystal in the ν-κ plane, where ν is the filling factor and κ is the strength of Landau-level (LL) mixing. The phase boundary is seen to exhibit a striking ν dependence, with the states away from the magic filling factors ν=n/(2pn+1) being much more susceptible to crystallization due to Landau-level mixing than those at ν=n/(2pn+1). Our results explain the qualitative difference between the experimental behaviors observed in n- and p-doped gallium arsenide quantum wells and, in particular, the existence of an insulating state for ν<1/3 and also for 1/3<ν<2/5 in low-density p-doped systems. We predict that, in the vicinity of ν=1/5 and ν=2/9, increasing LL mixing causes a transition not into an ordinary electron Wigner crystal, but rather into a strongly correlated crystal of composite fermions carrying two vortices.
Recent experiments indicate the presence of new anisotropic fractional quantum Hall states at regimes not anticipated before. These experiments raise many fundamental questions regarding the inner nature of the electronic system that leads to such anisotropic states. Interplay between electron mass anisotropy and electron-electron correlation effects in a magnetic field can create a rich variety of possibilities. Several anisotropic electronic states ranging from anisotropic quantum Hall liquids to anisotropic Wigner solids may stabilize due to such effects. The electron mass anisotropy in a two-dimensional electron gas effectively leads to an anisotropic Coulomb interaction potential between electrons. An anisotropic interaction potential may strongly influence the stability of various quantum phases that are close in energy since the overall stability of an electronic system is very sensitive to local order. As a result there is a possibility that various anisotropic electronic phases may emerge even in the lowest Landau level in regimes where one would not expect them. In this work we study the state with filling factor 1/6 in the lowest Landau level, a state which is very close to the critical filling factor where the liquid-solid transition takes place. We investigate whether an anisotropic Coulomb interaction potential is able to stabilize an anisotropic electronic liquid state at this filling factor. We describe such an anisotropic state by means of a liquid crystalline wave function with broken rotational symmetry which can be adiabatically connected to the actual wave function for the corresponding isotropic phase. We perform quantum Monte Carlo simulations in a disk geometry to study the properties of the anisotropic electronic liquid state under consideration. The findings indicate stability of liquid crystalline order in presence of an anisotropic Coulomb interaction potential. The results are consistent with the existence of an anisotropic electronic liquid state in the lowest Landau level.
The fractional quantum Hall (FQH) effect arises from strong electron correlations in a quantising magnetic field, and features exotic emergent phenomena such as electron fractionalisation. Using the diagrammatic Monte Carlo approach with the combinatorial summation (CoS) algorithm, we obtain results with controlled accuracy for the microscopic model of interacting electrons in the lowest Landau level (LLL) in the thermodynamic limit. Starting from the macroscopically degenerate LLL at finite temperature, including interactions order by order, and applying a controlled resummation to the resulting series, we observe the emergence of the incompressible 1/3-filled state as the temperature is lowered. By analysing the long-time decay of the Green's function, we find spectral properties consistent with an energy gap at 1/3-filling, whereas at 1/2-filling our results are consistent with the pseudogapped behaviour previously observed experimentally and suggested theoretically. Our work provides the first demonstration that fractionalised phases of matter can be reliably described with Feynman's diagrammatic technique in terms of the fundamental electronic degrees of freedom, while also showing applicability of expansions in the bare Coulomb potential for precision calculations.