arXiv subjects

Xin Wan

Publications and source records attributed to Xin Wan.

At least 55 records · Page 3Linked to original sources

arXiv2019

Edge Induced Topological Phase Transition of the Quantum Hall state at Half Filling

We show that in quantum Hall systems at half-filling, edge potentials alone can drive transitions between the Pfaffian and anti-Pfaffian topological phases. We conjecture this is true in realistic systems even in the presence of weak bulk interactions that break the particle-hole symmetry. The strong effects of edge potentials could be understood from different topological shifts of competing phases, manifested on the disk geometry as the variation of orbital numbers at fixed number of particles. In particular, we show analytically particle-hole conjugation of Hamiltonians on the disk is equivalent to the tuning of edge potentials, which allows us to explicitly demonstrate the phase transition numerically. The importance of edge potentials in various experimental contexts, including the recently discovered particle-hole symmetric phase is also discussed.

cond-mat.str-el↗

arXiv2019

Extracting Critical Exponent by Finite-Size Scaling with Convolutional Neural Networks

Machine learning has been successfully applied to identify phases and phase transitions in condensed matter systems. However, quantitative characterization of the critical fluctuations near phase transitions is lacking. In this study we propose a finite-size scaling approach based on a convolutional neural network and analyze the critical behavior of a quantum Hall plateau transition. The localization length critical exponent learned by the neural network is consistent with the value obtained by conventional approaches. We show that the general-purposed method can be used to extract critical exponents in models with drastically different physics and input data, such as the two-dimensional Ising model and 4-state Potts model.

cond-mat.dis-nn↗

arXiv2018

Localization length exponent in two models of quantum Hall plateau transitions

Motivated by the recent numerical studies on the Chalker-Coddington network model that found a larger-than-expected critical exponent of the localization length characterizing the integer quantum Hall plateau transitions, we revisited the exponent calculation in the continuum model and in the lattice model, both projected to the lowest Landau level or subband. Combining scaling results with or without the corrections of an irrelevant length scale, we obtain $ν= 2.48 \pm 0.02$, which is larger but still consistent with the earlier results in the two models, unlike what was found recently in the network model. The scaling of the total number of conducting states, as determined by the Chern number calculation, is accompanied by an effective irrelevant length scale exponent $y = 4.3$ in the lattice model, indicating that the irrelevant perturbations are insignificant in the topology number calculation.

cond-mat.dis-nn↗

arXiv2018

Identifying Product Order with Restricted Boltzmann Machines

Unsupervised machine learning via a restricted Boltzmann machine is an useful tool in distinguishing an ordered phase from a disordered phase. Here we study its application on the two-dimensional Ashkin-Teller model, which features a partially ordered product phase. We train the neural network with spin configuration data generated by Monte Carlo simulations and show that distinct features of the product phase can be learned from non-ergodic samples resulting from symmetry breaking. Careful analysis of the weight matrices inspires us to define a nontrivial machine-learning motivated quantity of the product form, which resembles the conventional product order parameter.

cond-mat.dis-nn↗

arXiv2018

Perrin-Riou's main conjecture for elliptic curves at supersingular primes

In 1987, B. Perrin-Riou formulated a Heegner point main conjecture for elliptic curves at primes of ordinary reduction. In this paper, we formulate an analogue of Perrin-Riou's main conjecture for supersingular primes. We then prove this conjecture under mild hypotheses, and deduce from this result a $Λ$-adic extension of Kobayashi's $p$-adic Gross-Zagier formula, new cases of B.-D. Kim's doubly-signed main conjectures, and a strengthened version of Skinner's converse to the Gross-Zagier-Kolyvagin theorem for supersingular primes.

math.NT↗

arXiv2016

Emergent infinite-randomness fixed points from the extensive random bipartitions of the spin-1 Affleck-Kennedy-Lieb-Tasaki topological state

Quantum entanglement under an extensive bipartition can reveal the critical boundary theory of a topological phase in the parameter space. In this study we demonstrate that the infinite-randomness fixed point for spin-1/2 degrees of freedom can emerge from an extensive random bipartition of the spin-1 Affleck-Kennedy-Lieb-Tasaki chain. The nested entanglement entropy of the ground state of the reduced density matrix exhibits a logarithmic scaling with an effective central charge $\tilde{c} = 0.72 \pm 0.02 \approx \ln 2$. We further discuss, in the language of bulk quantum entanglement, how to understand all phase boundaries and the surrounding Griffiths phases for the antiferromagnetic Heisenberg spin-1 chain with quenched disorder and dimerization.

cond-mat.str-el↗

arXiv2016

Iwasawa Main Conjecture for Heegner Points: Supersingular Case

In this paper we propose and prove an anticyclotomic Iwasawa main conjecture for Heegner points on supersingular elliptic curves with $a_p=0$. The result has a "$\pm$" nature in the sense of Kobayashi. The proof uses a recent work of the second author on one divisibility in the Iwasawa--Greenberg main conjecture for Rankin-Selberg $p$-adic $L$-functions, together with an argument of Howard (adapted to our "$\pm$"-situation). As a byproduct, we also obtain an improvement of Skinner's result on a converse to the Gross--Zagier--Kolyvagin theorem.

math.NT↗

arXiv2015

The Birch and Swinnerton-Dyer Formula for Elliptic Curves of Analytic Rank One

Let $E/\mathbb{Q}$ be a semistable elliptic curve such that $\mathrm{ord}_{s=1}L(E,s) = 1$. We prove the $p$-part of the Birch and Swinnerton-Dyer formula for $E/\mathbb{Q}$ for each prime $p \geq 5$ of good reduction such that $E[p]$ is irreducible: $$ \mathrm{ord}_p \left (\frac{L'(E,1)}{Ω_E\cdot\mathrm{Reg}(E/\mathbb{Q})} \right ) = \mathrm{ord}_p \left (\#\mathrm{Sha}(E/\mathbb{Q})\prod_{\ell\leq \infty} c_\ell(E/\mathbb{Q}) \right ). $$ This formula also holds for $p=3$ provided $a_p(E)=0$ if $E$ has supersingular reduction at $p$.

math.NT↗

arXiv2015

Scaling analysis of the quasiparticle tunneling in the ${\mathbb Z}_k$ parafermion states

Quasiparticle tunneling between two counter-propagating edges through point contacts could provide information on the statistics of the quasiparticles. Previous study on a disk found a scaling behavior by varying the tunneling distance. It was found that in the limit with zero tunneling distance, the Abelian quasiparticles tunneling obey the scaling analysis while the non-Abelian quasiparticles exhibit some non-trivial behaviors on the scaling exponents. Because of the limitation of disk geometry, we put the fractional quantum Hall (FQH) state on the surface of a cylinder which has a larger tunable tunneling distance than that on disk by varying the aspect ratio $γ$. We analyze the scaling behavior of the quasiholes, especially the non-Abelian quasiholes in the Read-Rezayi ${\mathbb Z}_k$ parafermion states. We aim to address the existance of the anomalous correction of the scaling parameter in the long tunneling distance.

cond-mat.str-el↗

arXiv2015

Striped quantum Hall state in a half-filled Landau level

Nature of the fractional quantum Hall state at Landau level filling factor 5/2 remains elusive despite intensive experimental and theoretical work. While the leading theoretical candidates are Moore-Read Pfaffian (Pf) and its particle-hole conjugate anti-Pfaffian (APf), neither received unambiguous experimental support. We show that a state that is intermediate between them, made of alternating stripes of Pf and APf in the bulk, is a viable candidate. Such a state is shown to be incompressible and thus a charge insulator in the bulk, but a heat conductor due to the presence of gapless neutral bulk modes. We argue that properties of such a state is consistent with existing numerical and experimental work, and discuss possible experimental probes of its presence.

cond-mat.mes-hall↗

arXiv2015

Schrödinger type operators on generalized Morrey spaces

In this paper we introduce a class of generalized Morrey spaces associated with Schrödinger operator $L=-Δ+V$. Via a pointwise estimate, we obtain the boundedness of the operators $V^{β_{2}}(-Δ+V)^{-β_{1}}$ and their dual operators on these Morrey spaces.

math.CA↗

arXiv2015

Emerging criticality in the disordered three-color Ashkin-Teller model

We study the effects of quenched disorder on the first-order phase transition in the two-dimensional three-color Ashkin-Teller model by means of large-scale Monte Carlo simulations. We demonstrate that the first-order phase transition is rounded by the disorder and turns into a continuous one. Using a careful finite-size-scaling analysis, we provide strong evidence for the emerging critical behavior of the disordered Ashkin-Teller model to be in the clean two-dimensional Ising universality class, accompanied by universal logarithmic corrections. This agrees with perturbative renormalization-group predictions by Cardy. As a byproduct, we also provide support for the strong-universality scenario for the critical behavior of the two-dimensional disordered Ising model. We discuss consequences of our results for the classification of disordered phase transitions as well as generalizations to other systems.

cond-mat.dis-nn↗

arXiv2014

Topologically distinct critical theories emerging from the bulk entanglement spectrum of integer quantum Hall states on a lattice

The critical theories for the topological phase transitions of integer quantum Hall states to a trivial insulating state with the same symmetry can be obtained by calculating the ground state entanglement spectrum under a symmetric checkerboard bipartition. In contrast to the gapless edge excitations under the left-right bipartition, a quantum network with bulk gapless excitations naturally emerges at the Brillouin zone center without fine tuning. On a large finite lattice, the resulting critical theory for the $ν=1$ state is the (2+1) dimensional relativistic quantum field theory characterized by a \textit{single} Dirac cone spectrum and a pair of \textit{fractionalized} zero-energy states, while for the $ν=2$ state the critical theory exhibits a parabolic spectrum and no sign of fractionalization in the zero-energy states. A triangular correspondence is established among the bulk topological theory, gapless edge theory, and the critical theory via the ground state entanglement spectrum.

cond-mat.str-el↗

arXiv2014

Heegner Point Kolyvagin System and Iwasawa Main Conjecture

In this paper we prove an anticyclotomic Iwasawa main conjecture proposed by Perrin-Riou for Heegner points when the global sign is -1, using a recent work of the author on one divisibility of Iwasawa-Greenberg main conjecture for Rankin-Selberg p-adic L-functions. As a byproduct we also prove the equality for the above mentioned main conjecture under some local conditions.

math.NT↗

arXiv2014

p-adic Eisenstein series and L-functions of certain cusp forms on definite unitary groups

We construct p-adic families of Klingen Eisenstein series and L-functions for cuspforms (not necessarily ordinary) unramified at an odd prime p on definite unitary groups of signature (r, 0) (for any positive integer r) for a quadratic imaginary field split at p. When r=2, we show that the constant term of the Klingen Eisenstein family is divisible by a certain p-adic L-function.

math.NT↗

arXiv2014

Critical entanglement spectrum of one-dimensional symmetry protected topological phases

Under an appropriate symmetric extensive bipartition in a one-dimensional symmetry protected topological (SPT) phase, a bulk critical entanglement spectrum can be obtained, resembling the excitation spectrum of the critical point separating the SPT phase from the trivial (vacuum) state. Such a critical point is beyond the standard Landau-Ginzburg-Wilson paradigm for symmetry breaking phase transitions. For the $S=1$ SPT (Haldane) phase with the Affleck-Kennedy-Lieb-Tasaki exact wave function, the resulting critical entanglement spectrum has a residual entropy per lattice site $s_{r}=0.67602$, showing a delocalized version of the edge excitations in the SPT phase. From the wave function corresponding to the lowest entanglement energy level, the central charge of the critical point can be extracted $c\approx 1.01\pm 0.01$. The critical theory can be identified as the same effective field theory as the spin-1/2 antiferromagnetic Heisenberg chain or the spin-1/2 Haldane-Shastry model with inverse square long-range interaction.

cond-mat.str-el↗

arXiv2014

The single-mode description of the integer quantum Hall state with dipole-dipole interaction

A topological phase can often be represented by a corresponding wavefunction (exact eigenstate of a model Hamiltonian) that has a higher underlying symmetry than necessary. When the symmetry is explicitly broken in the Hamiltonian, the model wavefunction fails to account for the change due to the lack of a variational parameter. Here we exemplify the case by an integer quantum Hall system with anisotropic interaction. We demonstrate the recovery of the variational parameter in a single-mode approximation, which is consistent with the recently proposed geometric consideration of the quantum Hall phases.

cond-mat.str-el↗

arXiv2014

A Memory of Majorana Fermions through Quantum Quench

We study the sudden quench of a one-dimensional p-wave superconductor through its topological signature in the entanglement spectrum. The long-time evolution of the system and its topological characterization depend on a pseudomagnetic field ${\bs R}_{\mbox{\text eff}}(k)$, which connects both the initial and the final Hamiltonians, hence exhibiting a memory effect. In particular, we explore the robustness of the Majorana zero-mode associated with the entanglement cut in the topologically nontrivial phase and identify the parameter space in which the mode can survive in the infinite-time limit.

cond-mat.mes-hall↗