arXiv subjects

Fang Xie

Publications and source records attributed to Fang Xie.

52 records · Page 3Linked to original sources

arXiv2022

Twisted Bilayer Graphene V: Exact Analytic Many-Body Excitations in Twisted Bilayer Graphene Coulomb Hamiltonians: Charge Gap, Goldstone Modes and Absence of Cooper Pairing

We find exact analytic expressions for the energies and wavefunctions of the charged and neutral excitations above the exact ground states (at rational filling per unit cell) of projected Coulomb Hamiltonians in twisted bilayer graphene. Our exact expressions are valid for any form of the Coulomb interaction and any form of $AA$ and $AB/BA$ tunneling. The single charge excitation energy is a convolution of the Coulomb potential with a quantum geometric tensor of the TBG bands. The neutral excitations are (high-symmetry group) magnons, and their dispersion is analytically calculated in terms of the form factors of the active bands in TBG. The two-charge excitation energy and wavefunctions are also obtained, and a sufficient condition on the graphene eigenstates for obtaining a Cooper-pair from Coulomb interactions is obtained. For the actual TBG bands at the first magic angle, we can analytically show that the Cooper pair binding energy is zero in all such projected Coulomb models, implying that either phonons and/or non-zero kinetic energy are needed for superconductivity. Since the [Phys. Rev. Lett. 122, 246401] showed that the kinetic energy bounds on the superexchange energy are less $10^{-3}$ in Coulomb units, the phonon mechanism becomes then very likely. If nonetheless the superconductivity is due to kinetic terms which render the bands non-flat, one prediction of our theory is that the highest $T_c$ would not occur at the highest DOS.

cond-mat.str-el↗

arXiv2022

Twisted bilayer graphene.VI. An Exact Diagonalization Study of Twisted Bilayer Graphene at Non-Zero Integer Fillings

Using exact diagonalization, we study the projected Hamiltonian with Coulomb interaction in the 8 flat bands of first magic angle twisted bilayer graphene. Employing the U(4) (U(4)$\times$U(4)) symmetries in the nonchiral (chiral) flat band limit, we reduced the Hilbert space to an extent which allows for study around $ν=\pm 3,\pm2,\pm1$ fillings. In the first chiral limit $w_0/w_1=0$ where $w_0$ ($w_1$) is the $AA$ ($AB$) stacking hopping, we find that the ground-states at these fillings are extremely well-described by Slater determinants in a so-called Chern basis, and the exactly solvable charge $\pm1$ excitations found in [arXiv:2009.14200] are the lowest charge excitations up to system sizes $8\times8$ (for restricted Hilbert space) in the chiral-flat limit. We also find that the Flat Metric Condition (FMC) used in [arXiv:2009.11301,2009.11872,2009.12376,2009.13530,2009.14200] for obtaining a series of exact ground-states and excitations holds in a large parameter space. For $ν=-3$, the ground state is the spin and valley polarized Chern insulator with $ν_C=\pm1$ at $w_0/w_1\lesssim0.9$ (0.3) with (without) FMC. At $ν=-2$, we can only numerically access the valley polarized sector, and we find a spin ferromagnetic phase when $w_0/w_1\gtrsim0.5t$ where $t\in[0,1]$ is the factor of rescaling of the actual TBG bandwidth, and a spin singlet phase otherwise, confirming the perturbative calculation [arXiv:2009.13530]. The analytic FMC ground state is, however, predicted in the intervalley coherent sector which we cannot access [arXiv:2009.13530]. For $ν=-3$ with/without FMC, when $w_0/w_1$ is large, the finite-size gap $Δ$ to the neutral excitations vanishes, leading to phase transitions. Further analysis of the ground state momentum sectors at $ν=-3$ suggests a competition among (nematic) metal, momentum $M_M$ ($π$) stripe and $K_M$-CDW orders at large $w_0/w_1$.

cond-mat.str-el↗

arXiv2022

Evidence for a Monolayer Excitonic Insulator

The interplay between topology and correlations can generate a variety of quantum phases, many of which remain to be explored. Recent advances have identified monolayer WTe2 as a promising material for doing so in a highly tunable fashion. The ground state of this two-dimensional (2D) crystal can be electrostatically tuned from a quantum spin Hall insulator (QSHI) to a superconductor. However, much remains unknown about the gap-opening mechanism of the insulating state. Here we report evidence that the QSHI is also an excitonic insulator (EI), arising from the spontaneous formation of electron-hole bound states (excitons). We reveal the presence of an intrinsic insulating state at the charge neutrality point (CNP) in clean samples and confirm the correlated nature of this charge-neutral insulator by tunneling spectroscopy. We provide evidence against alternative scenarios of a band insulator or a localized insulator and support the existence of an EI phase in the clean limit. These observations lay the foundation for understanding a new class of correlated insulators with nontrivial topology and identify monolayer WTe2 as a promising candidate for exploring quantum phases of ground-state excitons.

cond-mat.mes-hall↗

arXiv2021

TSTG II: Projected Hartree-Fock Study of Twisted Symmetric Trilayer Graphene

The Hamiltonian of the magic-angle twisted symmetric trilayer graphene (TSTG) can be decomposed into a TBG-like flat band Hamiltonian and a high-velocity Dirac fermion Hamiltonian. We use Hartree-Fock mean field approach to study the projected Coulomb interacting Hamiltonian of TSTG developed in Călugăru et al. [Phys. Rev. B 103, 195411 (2021)] at integer fillings $ν=-3, -2, -1$ and $0$ measured from charge neutrality. We study the phase diagram with $w_0/w_1$, the ratio of $AA$ and $AB$ interlayer hoppings, and the displacement field, which introduces an interlayer potential $U$ and hybridizes the TBG-like bands with the Dirac bands. At small $U$, we find the ground states at all fillings $ν$ are in the same phases as the tensor products of a Dirac semimetal with the filling $ν$ TBG insulator ground states, which are spin-valley polarized at $ν=-3$, and fully (partially) intervalley coherent at $ν=-2,0$ ($ν=-1$) in the flat bands. An exception is $ν=-3$ with $w_0/w_1 \gtrsim 0.7$, which possibly become a metal with competing orders at small $U$ due to charge transfers between the Dirac and flat bands. At strong $U$ where the bandwidths exceed interactions, all the fillings $ν$ enter a metal phase with small or zero valley polarization and intervalley coherence. Lastly, at intermediate $U$, semimetal or insulator phases with zero intervalley coherence may arise for $ν=-2,-1,0$. Our results provide a simple picture for the electron interactions in TSTG systems, and reveal the connection between the TSTG and TBG ground states.

cond-mat.mes-hall↗

arXiv2021

Open Momentum Space Method for Hofstadter Butterfly and the Quantized Lorentz Susceptibility

We develop a generic $\mathbf{k}\cdot \mathbf{p}$ open momentum space method for calculating the Hofstadter butterfly of both continuum (Moiré) models and tight-binding models, where the quasimomentum is directly substituted by the Landau level (LL) operators. By taking a LL cutoff (and a reciprocal lattice cutoff for continuum models), one obtains the Hofstadter butterfly with in-gap spectral flows. For continuum models such as the Moiré model for twisted bilayer graphene, our method gives a sparse Hamiltonian, making it much more efficient than existing methods. The spectral flows in the Hofstadter gaps can be understood as edge states on a momentum space boundary, from which one can determine the two integers ($t_ν,s_ν$) of a gap $ν$ satisfying the Diophantine equation. The spectral flows can also be removed to obtain a clear Hofstadter butterfly. While $t_ν$ is known as the Chern number, our theory identifies $s_ν$ as a dual Chern number for the momentum space, which corresponds to a quantized Lorentz susceptibility $γ_{xy}=eBs_ν$.

cond-mat.mes-hall↗

arXiv2021

Aggregating Knockoffs for False Discovery Rate Control with an Application to Gut Microbiome Data

Recent discoveries suggest that our gut microbiome plays an important role in our health and wellbeing. However, the gut microbiome data are intricate; for example, the microbial diversity in the gut makes the data high-dimensional. While there are dedicated high-dimensional methods, such as the lasso estimator, they always come with the risk of false discoveries. Knockoffs are a recent approach to control the number of false discoveries. In this paper, we show that knockoffs can be aggregated to increase power while retaining sharp control over the false discoveries. We support our method both in theory and simulations, and we show that it can lead to new discoveries on microbiome data from the American Gut Project. In particular, our results indicate that several phyla that have been overlooked so far are associated with obesity.

stat.ME↗

arXiv2021

TSTG I: Single-Particle and Many-Body Hamiltonians and Hidden Non-local Symmetries of Trilayer Moiré Systems with and without Displacement Field

We derive the Hamiltonian for trilayer moiré systems with the Coulomb interaction projected onto the bands near the charge neutrality point. Motivated by the latest experimental results, we focus on the twisted symmetric trilayer graphene (TSTG) with a mirror-symmetry with respect to the middle layer. We provide a full symmetry analysis of the non-interacting Hamiltonian with a perpendicular displacement field coupling the band structure made otherwise of the twisted bilayer graphene (TBG) and the high velocity Dirac fermions, and we identify a hidden non-local symmetry of the problem. In the presence of this displacement field, we construct an approximate single-particle model, akin to the tripod model for TBG, capturing the essence of non-interacting TSTG. We also derive more quantitative perturbation schemes for the low-energy physics of TSTG with displacement field, obtaining the corresponding eigenstates. This allows us to obtain the Coulomb interaction Hamiltonian projected in the active band TSTG wavefunctions and derive the full many-body Hamiltonian of the system. We also provide an efficient parameterization of the interacting Hamiltonian. Finally, we show that the discrete symmetries at the single-particle level promote the $\mathrm{U} \left( 2 \right) \times \mathrm{U} \left( 2 \right)$ spin-valley symmetry to enlarged symmetry groups of the interacting problem under different limits. The interacting part of the Hamiltonian exhibits a large $\mathrm{U} \left( 4 \right) \times \mathrm{U} \left( 4 \right) \times \mathrm{U} \left( 4 \right) \times \mathrm{U} \left( 4 \right)$ symmetry in the chiral limit. Moreover, by identifying a new symmetry which we dub spatial many-body charge conjugation, we show that the physics of TSTG is symmetric around charge neutrality.

cond-mat.str-el↗

arXiv2020

Statistical Guarantees for Regularized Neural Networks

Neural networks have become standard tools in the analysis of data, but they lack comprehensive mathematical theories. For example, there are very few statistical guarantees for learning neural networks from data, especially for classes of estimators that are used in practice or at least similar to such. In this paper, we develop a general statistical guarantee for estimators that consist of a least-squares term and a regularizer. We then exemplify this guarantee with $\ell_1$-regularization, showing that the corresponding prediction error increases at most sub-linearly in the number of layers and at most logarithmically in the total number of parameters. Our results establish a mathematical basis for regularized estimation of neural networks, and they deepen our mathematical understanding of neural networks and deep learning more generally.

cs.LG↗

arXiv2020

The Landau Level of Fragile Topology

We study the Hofstadter butterfly and Landau levels of the twisted bilayer graphene (TBG). We show that the nontrivial fragile topology of the lowest two bands near the charge neutral point makes their Hofstadter butterfly generically connected with higher bands, closing the gap between the first and second conduction (valence) bands at a certain magnetic flux per unit cell. We also develop a momentum space method for calculating the TBG Hofstadter butterfly, from which we identify three phases where the Hofstadter butterflies of the lowest two bands and the higher bands are connected in different ways. We show this leads to a crossing between the $ν=4$ Landau fan from the charge neutral point and the zero field band gap at one flux per Moiré unit cell, which corresponds to a magnetic field $25θ^2$T (twist angle $θ$ in degrees). This provides an experimentally testable feature of the fragile topology. In general, we expect it to be a generic feature that the Hofstadter butterfly of topological bands are connected with the Hofstadter spectra of other bands. We further show the TBG band theory with Zeeman splitting being the most sizable splitting could result in Landau fans at the charge neutral point and half fillings near the magic angle, and we predict their variations under an in-plane magnetic field.

cond-mat.mes-hall↗

arXiv2020

Tuning-free ridge estimators for high-dimensional generalized linear models

Ridge estimators regularize the squared Euclidean lengths of parameters. Such estimators are mathematically and computationally attractive but involve tuning parameters that can be difficult to calibrate. In this paper, we show that ridge estimators can be modified such that tuning parameters can be avoided altogether. We also show that these modified versions can improve on the empirical prediction accuracies of standard ridge estimators combined with cross-validation, and we provide first theoretical guarantees.

stat.ME↗

arXiv2020

Estimating the Penalty Level of $\ell_{1}$-minimization via Two Gaussian Approximation Methods

In this paper, we aim to give a theoretical approximation for the penalty level of $\ell_{1}$-regularization problems. This can save much time in practice compared with the traditional methods, such as cross-validation. To achieve this goal, we develop two Gaussian approximation methods, which are based on a moderate deviation theorem and Stein's method respectively. Both of them give efficient approximations and have good performances in simulations. We apply the two Gaussian approximation methods into three types of ultra-high dimensional $\ell_{1}$ penalized regressions: lasso, square-root lasso, and weighted $\ell_{1}$ penalized Poisson regression. The numerical results indicate that our two ways to estimate the penalty levels achieve high computational efficiency. Besides, our prediction errors outperform that based on the 10-fold cross-validation.

math.ST↗

arXiv2020

Consistency of $\ell _{1}$ Penalized Negative Binomial Regressions

We prove the consistency of the $\ell_1$ penalized negative binomial regression (NBR). A real data application about German health care demand shows that the $\ell_1$ penalized NBR produces a more concise but more accurate model, comparing to the classical NBR.

math.ST↗

arXiv2019

Topology-Bounded Superfluid Weight In Twisted Bilayer Graphene

While regular flat bands are good for enhancing the density of states and hence the gap, they are detrimental to the superfluid weight. We show that the predicted nontrivial topology of the two lowest flat bands of twisted bilayer graphene plays an important role in the enhancement of the superfluid weight and hence of superconductivity. We derive the superfluid weight (phase stiffness) of the TBLG superconducting flat bands with a uniform pairing, and show that it can be expressed as an integral of the Fubini-Study metric of the flat bands. This mirrors results already obtained for nonzero Chern number bands even though the TBLG flat bands have zero Chern number. We further show the metric integral is lower bounded by the topological $C_{2z}T$ Wilson loop winding number of the TBLG flat bands, which renders the superfluid weight has a topological lower bound proportional to the pairing gap. In contrast, trivial flat bands have a zero superfluid weight. The superfluid weight is crucial in determining the BKT transition temperature of the superconductor. Based on the transition temperature measured in TBLG experiments, we estimate the topological contribution of the superfluid weight in TBLG.

cond-mat.supr-con↗

arXiv2018

Person-Job Fit: Adapting the Right Talent for the Right Job with Joint Representation Learning

Person-Job Fit is the process of matching the right talent for the right job by identifying talent competencies that are required for the job. While many qualitative efforts have been made in related fields, it still lacks of quantitative ways of measuring talent competencies as well as the job's talent requirements. To this end, in this paper, we propose a novel end-to-end data-driven model based on Convolutional Neural Network (CNN), namely Person-Job Fit Neural Network (PJFNN), for matching a talent qualification to the requirements of a job. To be specific, PJFNN is a bipartite neural network which can effectively learn the joint representation of Person-Job fitness from historical job applications. In particular, due to the design of a hierarchical representation structure, PJFNN can not only estimate whether a candidate fits a job, but also identify which specific requirement items in the job posting are satisfied by the candidate by measuring the distances between corresponding latent representations. Finally, the extensive experiments on a large-scale real-world dataset clearly validate the performance of PJFNN in terms of Person-Job Fit prediction. Also, we provide effective data visualization to show some job and talent benchmark insights obtained by PJFNN.

cs.IR↗

arXiv2017

Recruitment Market Trend Analysis with Sequential Latent Variable Models

Recruitment market analysis provides valuable understanding of industry-specific economic growth and plays an important role for both employers and job seekers. With the rapid development of online recruitment services, massive recruitment data have been accumulated and enable a new paradigm for recruitment market analysis. However, traditional methods for recruitment market analysis largely rely on the knowledge of domain experts and classic statistical models, which are usually too general to model large-scale dynamic recruitment data, and have difficulties to capture the fine-grained market trends. To this end, in this paper, we propose a new research paradigm for recruitment market analysis by leveraging unsupervised learning techniques for automatically discovering recruitment market trends based on large-scale recruitment data. Specifically, we develop a novel sequential latent variable model, named MTLVM, which is designed for capturing the sequential dependencies of corporate recruitment states and is able to automatically learn the latent recruitment topics within a Bayesian generative framework. In particular, to capture the variability of recruitment topics over time, we design hierarchical dirichlet processes for MTLVM. These processes allow to dynamically generate the evolving recruitment topics. Finally, we implement a prototype system to empirically evaluate our approach based on real-world recruitment data in China. Indeed, by visualizing the results from MTLVM, we can successfully reveal many interesting findings, such as the popularity of LBS related jobs reached the peak in the 2nd half of 2014, and decreased in 2015.

cs.AI↗

arXiv2017

Sparse Poisson Regression with Penalized Weighted Score Function

We proposed a new penalized method in this paper to solve sparse Poisson Regression problems. Being different from $\ell_1$ penalized log-likelihood estimation, our new method can be viewed as penalized weighted score function method. We show that under mild conditions, our estimator is $\ell_1$ consistent and the tuning parameter can be pre-specified, which shares the same good property of the square-root Lasso.

math.ST↗