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arXiv:2302.10010 · Journal of Statistical Mechanics Theory and Experiment

Properties of Laughlin states on fractal lattices

Mani Chandra Jha, Anne E. B. Nielsen

Published May 1, 2023• 3 citations
arXivHTMLPDF

Abstract

Abstract Laughlin states have recently been constructed on fractal lattices and have been shown to be topological in such systems. Some of their properties are, however, quite different from the two-dimensional case. On the Sierpinski triangle, for instance, the entanglement entropy shows oscillations as a function of particle number and does not obey the area law despite being topologically ordered, and the particle density is non-uniform in the bulk. Here, we investigate these deviant properties in greater detail on the Sierpinski triangle, and we also study the properties on the Sierpinski carpet and the T-fractal. We find that the density variations across the fractal are present for all the considered fractal lattices and for most choices of the number of particles. The size of anyons inserted into the lattice Laughlin state also varies with position on the fractal lattice. We observe that quasiholes and quasiparticles have similar sizes and that the size of the anyons typically increases with decreasing Hausdorff dimension. As opposed to periodic lattices in two dimensions, the Sierpinski triangle and carpet have inner edges. We construct trial states for both inner and outer edge states. We find that oscillations of the entropy as a function of particle number are present for the T-fractal, but not for the Sierpinski carpet. Finally, we observe deviations from the area law for several different bipartitions on the Sierpinski triangle.

DOI
10.1088/1742-5468/acd104

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Abstract Laughlin states have recently been constructed on fractal lattices and have been shown to be topological in such systems. Some of their properties are, however, quite different from the two-dimensional case. On the Sierpinski triangle, for instance, the entanglement entropy shows oscillations as a function of particle number and does not obey the area law despite being topologically ordered, and the particle density is non-uniform in the bulk. Here, we investigate these deviant properties in greater detail on the Sierpinski triangle, and we also study the properties on the Sierpinski carpet and the T-fractal. We find that the density variations across the fractal are present for all the considered fractal lattices and for most choices of the number of particles. The size of anyons inserted into the lattice Laughlin state also varies with position on the fractal lattice. We observe that quasiholes and quasiparticles have similar sizes and that the size of the anyons typically increases with decreasing Hausdorff dimension. As opposed to periodic lattices in two dimensions, the Sierpinski triangle and carpet have inner edges. We construct trial states for both inner and outer edge states. We find that oscillations of the entropy as a function of particle number are present for the T-fractal, but not for the Sierpinski carpet. Finally, we observe deviations from the area law for several different bipartitions on the Sierpinski triangle.
Laughlin states have recently been constructed on fractal lattices, and the charge and braiding statistics of the quasiholes were used to confirm that these states have Laughlin-type topology. Here we investigate density, correlation, and entanglement properties of the states on a fractal lattice derived from a Sierpinski triangle with the purpose of identifying similarities and differences compared to two-dimensional systems and with the purpose of investigating whether various probes of topology work for fractal lattices. Similarly to two-dimensional systems, we find that the connected particle-particle correlation function decays roughly exponentially with the distance between the lattice sites measured in the two-dimensional plane, but the values also depend on the local environment. Contrary to two-dimensional systems, we find that the entanglement entropy does not follow the area law if one defines the area to be the number of nearest-neighbor bonds that cross the edge of the selected subsystem. Considering bipartitions with two bonds crossing the edge, we find a close to logarithmic scaling of the entanglement entropy with the number of sites in the subsystem. This also means that the topological entanglement entropy cannot be extracted using the Kitaev-Preskill or the Levin-Wen methods. Studying the entanglement spectrum for different bipartitions, we find that the number of states below the entanglement gap is robust and the same as for Laughlin states on two-dimensional lattices.
There is a growing interest in searching for topology in fractal dimensions with the aim of finding different properties and advantages compared to the integer dimensional case. Here we construct a local Hamiltonian on a fractal lattice whose ground state exhibits topological braiding properties. The fractal lattice is obtained from a second-generation Sierpinski carpet with Hausdorff dimension 1.89. We use local potentials to trap and exchange anyons in the model, and the numerically obtained results for the exchange statistics of the anyons are close to the ideal statistics for quasiholes in a bosonic Laughlin state at half filling. For the considered system size, the energy gap is about three times larger for the fractal lattice than for a two-dimensional square lattice, and we find that the braiding results obtained on the fractal lattice are more robust against disorder. We propose a scheme to implement both fractal lattices and our proposed local Hamiltonian with ultracold atoms in optical lattices.
Anyons are mainly studied and considered in two spatial dimensions. For fractals, the scaling dimension that characterizes the system can be non integer and can take values between that of a standard one-dimensional or two-dimensional system. Generating Hamiltonians that meet locality conditions and support anyons is not a simple task. Here, we construct a local Hamiltonian on a fractal lattice which realizes physics similar to the fractional quantum Hall effect. The fractal lattice is obtained from a second generation Sierpinski carpet, which has 64 sites, and is characterized by a Hausdorff dimension of 1.89. We demonstrate that the proposed local Hamiltonian acting on the fractal geometry has Laughlin-type topological order by creating anyons and then studying their charge and braiding statistics. We also find that the energy gap between the ground state and the first excited state is approximately three times larger for the fractal lattice than for a standard square lattice with 64 sites, and the model on the fractal lattice is significantly more robust against disorder. We propose a scheme to implement fractal lattices and our proposed local Hamiltonian for ultracold atoms in optical lattices. The discussed scheme could also be utilized to study integer quantum Hall phases and the physics of other quantum systems on fractal lattices.
Random fractal lattices provide a geometrically disordered setting in which quantum correlations can be shaped by noninteger dimensionality rather than onsite randomness. We investigate the entanglement properties of noninteracting fermions on random fractal lattices generated by a stochastic growth algorithm. By varying the growth parameter and adding missing links with probability ppp, we tune the Hausdorff and spectral dimensions while keeping the system free of onsite disorder. For ground states at different fillings, we compute the bipartite entanglement entropy of subregions defined by graph distance and analyze its scaling with subsystem size. Over a broad parameter range, we find robust power-law behavior governed primarily by the Hausdorff dimension, consistent with a generalized area law and without the logarithmic enhancement familiar from Euclidean free fermions. We also study entanglement growth following a global quench from an uncorrelated checkerboard state and uncover an asymptotic scaling collapse in which the subsystem-size dependence is governed by the Hausdorff dimension, while the temporal evolution is governed by the spectral dimension. The resulting dynamics are logarithmically slow over an extended intermediate-time window. These results show that geometric randomness alone can generate both nontrivial ground-state entanglement structure and slow quantum-information spreading in free-fermion systems.
We numerically explore the interplay of fractal geometry and quantum entanglement by analyzing the von Neumann entropy (known as entanglement entropy) and the entanglement contour in the scaling limit. Adopting quadratic fermionic models on Sierpinski carpet, we uncover intriguing findings. For gapless ground states exhibiting a finite density of states at the chemical potential, we reveal a super-area law characterized by the presence of a logarithmic correction for area law in the scaling of entanglement entropy. This extends the well-established super-area law observed on translationally invariant Euclidean lattices where the Gioev-Klich-Widom conjecture regarding the asymptotic behavior of Toeplitz matrices holds significant influence. Furthermore, different from the fractal structure of the lattice, we observe the emergence of a self-similar and universal pattern termed an “entanglement fractal” in the entanglement contour data as we approach the scaling limit. Remarkably, this pattern bears resemblance to intricate Chinese paper-cutting designs. We provide general rules to artificially generate this fractal, offering insights into the universal scaling of entanglement entropy. Meanwhile, as the direct consequence of the entanglement fractal and beyond a single scaling behavior of entanglement contour in translation-invariant systems, we identify two distinct scaling behaviors in the entanglement contour of fractal systems. For gapped ground states, we observe that the entanglement entropy adheres to a generalized area law, with its dependence on the Hausdorff dimension of the boundary between complementary subsystems. Published by the American Physical Society 2024
I study the properties of the quantum critical point of the transverse-field quantum Ising model on various fractal lattices such as the Sierpiński carpet, Sierpiński gasket, and Sierpiński tetrahedron. Using a continuous-time quantum Monte Carlo simulation method and finite-size scaling analysis, I identify the quantum critical point and investigate its scaling properties. Among others, I calculate the dynamic critical exponent and find that it is greater than one for all three structures. The fact that it deviates from one is a direct consequence of the fractal structures not being integer-dimensional regular lattices. Other critical exponents are also calculated. The exponents are different from those of the classical critical point and satisfy the quantum scaling relation, thus confirming that I have indeed found the quantum critical point. I find that the Sierpiński tetrahedron, of which the dimension is exactly 2, belongs to a different universality class than that of the two-dimensional square lattice. I conclude that the critical exponents depend on more details of the structure than just the dimension and the symmetry.