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Register a new userCornering the universal shape of fluctuations
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Added by
William Witczak-Krempa
Publication date
2021
fields
Physics
and research's language is
English
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Understanding the fluctuations of observables is one of the main goals in science, be it theoretical or experimental, quantum or classical. We investigate such fluctuations when only a subregion of the full system can be observed, focusing on geometries with sharp corners. We report that the dependence on the opening angle is super-universal: up to a numerical prefactor, this function does not depend on anything, provided the system under study is uniform, isotropic, and correlations do not decay too slowly. The prefactor contains important physical information: we show in particular that it gives access to the long-wavelength limit of the structure factor. We illustrate our findings with several examples, including fractional quantum Hall states, scale invariant quantum critical theories, and metals. Finally, we discuss connections with quantum entanglement, extensions to three dimensions, as well as experiments to probe the geometry of fluctuations.
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Precise nature of MBL transitions in both random and quasiperiodic (QP) systems remains elusive so far. In particular, whether MBL transitions in QP and random systems belong to the same universality class or two distinct ones has not been decisively resolved. Here we investigate MBL transitions in one-dimensional ($d!=!1$) QP systems as well as in random systems by state-of-the-art real-space renormalization group (RG) calculation. Our real-space RG shows that MBL transitions in 1D QP systems are characterized by the critical exponent $ u!approx!2.4$, which respects the Harris-Luck bound ($ u!>!1/d$) for QP systems. Note that $ u!approx! 2.4$ for QP systems also satisfies the Harris-CCFS bound ($ u!>!2/d$) for random systems, which implies that MBL transitions in 1D QP systems are stable against weak quenched disorder since randomness is Harris irrelevant at the transition. We shall briefly discuss experimental means to measure $ u$ of QP-induced MBL transitions.
We report a systematic study of the fractional quantum Hall effect (FQHE) using the density-matrix renormalization group (DMRG) method on two different geometries: the sphere and the cylinder. We provide convergence benchmarks based on model Hamiltonians known to possess exact zero-energy ground states, as well as an analysis of the number of sweeps and basis elements that need to be kept in order to achieve the desired accuracy.The ground state energies of the Coulomb Hamiltonian at $ u=1/3$ and $ u=5/2$ filling are extracted and compared with the results obtained by previous DMRG implementations in the literature. A remarkably rapid convergence in the cylinder geometry is noted and suggests that this boundary condition is particularly suited for the application of the DMRG method to the FQHE.
Non-unitary evolution can give rise to novel steady states classified by their entanglement properties. In this work, we aim to understand its interplay with long-range hopping that decays with $r^{-alpha}$ in free-fermion systems. We first study two solvable Brownian models with long-range non-unitary dynamics: a large-$N$ SYK$_2$ chain and a single-flavor fermion chain and we show that they share the same phase diagram. When $alpha>0.5$, we observe two critical phases with subvolume entanglement scaling: (i) $alpha>1.5$, a logarithmic phase with dynamical exponent $z=1$ and logarithmic subsystem entanglement, and (ii) $0.5<alpha<1.5$, a fractal phase with $z=frac{2alpha-1}{2}$ and subsystem entanglement $S_Apropto L_A^{1-z}$, where $L_A$ is the length of the subsystem $A$. These two phases cannot be distinguished by the purification dynamics, in which the entropy always decays as $L/T$. We then confirm that the results are also valid for the static SYK$_2$ chain, indicating the phase diagram is universal for general free-fermion systems. We also discuss phase diagrams in higher dimensions and the implication in measurement-induced phase transitions.
We develop the Floquet-Magnus expansion for a classical equation of motion under a periodic drive that is applicable to both isolated and open systems. For classical systems, known approaches based on the Floquet theorem fail due to the nonlinearity and the stochasticity of their equations of motion (EOMs) in contrast to quantum ones. Here, employing their master equation, we successfully extend the Floquet methodology to classical EOMs to obtain their Floquet-Magnus expansions, thereby overcoming this difficulty. Our method has a wide range of application from classical to quantum as long as they are described by differential equations including the Langevin equation, the Gross-Pitaevskii equation, and the time-dependent Ginzburg-Landau equation. By analytically evaluating the higher-order terms of the Floquet-Magnus expansion, we find that it is, at least asymptotically, convergent and well approximates the relaxation to their prethermal or non-equilibrium steady states. To support these analytical findings, we numerically analyze two examples: (i) the Kapitza pendulum with friction and (ii) laser-driven magnets described by the stochastic Landau-Lifshitz-Gilbert equation. In both cases, the effective EOMs obtained from their Floquet-Magnus expansions correctly reproduce their exact time evolution for a long time up to their non-equilibrium steady states. In the example of driven magnets, we demonstrate the controlled generations of a macroscopic magnetization and a spin chirality by laser and discuss possible applications to spintronics.
The fractional quantum Hall (FQH) effect illustrates the range of novel phenomena which can arise in a topologically ordered state in the presence of strong interactions. The possibility of realizing FQH-like phases in models with strong lattice effects has attracted intense interest as a more experimentally accessible venue for FQH phenomena which calls for more theoretical attention. Here we investigate the physical relevance of previously derived geometric conditions which quantify deviations from the Landau level physics of the FQHE. We conduct extensive numerical many-body simulations on several lattice models, obtaining new theoretical results in the process, and find remarkable correlation between these conditions and the many-body gap. These results indicate which physical factors are most relevant for the stability of FQH-like phases, a paradigm we refer to as the geometric stability hypothesis, and provide easily implementable guidelines for obtaining robust FQH-like phases in numerical or real-world experiments.
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