Order-from-quantum disorder effects and Zeeman field tuned quantum phase transitions in a bosonic quantum anomalous Hall system

We study possible many body phenomena in the recent experimentally realized weakly interacting quantum anomalous Hall system of spinor bosons by Wu, et.al, Science 354, 83-88 (2016). At a zero Zeeman field $h=0$, by incorporating order from quantum disorder effects, we determine the quantum ground state to be a $N=2$ XY-antiferromagnetic superfluid state and also evaluate its excitation spectra. At a finite small $h$, the competition between the Zeeman energy and the effective potential generated by the order from the quantum disorder leads to a canted antiferromagnetic superfluid state, then drives a second order transition to a spin-polarized superfluid state along the $z$ direction. The transition is in the same universality class as the zero density superfluid to Mott transition. Scaling behaviours of various physical quantities are derived. The ongoing experimental efforts to detect these novel phenomena are discussed.Comment: 5 pages main text + 4 pages supplementary material

Worst-case Complexity of Cyclic Coordinate Descent: O(n2)O(n^2) Gap with Randomized Version

This paper concerns the worst-case complexity of cyclic coordinate descent (C-CD) for minimizing a convex quadratic function, which is equivalent to Gauss-Seidel method and can be transformed to Kaczmarz method and projection onto convex sets (POCS). We observe that the known provable complexity of C-CD can be $O(n^2)$ times slower than randomized coordinate descent (R-CD), but no example was rigorously proven to exhibit such a large gap. In this paper we show that the gap indeed exists. We prove that there exists an example for which C-CD takes at least $O(n^4 \kappa_{\text{CD}} \log\frac{1}{\epsilon})$ operations, where $\kappa_{\text{CD}}$ is related to Demmel's condition number and it determines the convergence rate of R-CD. It implies that in the worst case C-CD can indeed be $O(n^2)$ times slower than R-CD, which has complexity $O( n^2 \kappa_{\text{CD}} \log\frac{1}{\epsilon})$. Note that for this example, the gap exists for any fixed update order, not just a particular order. Based on the example, we establish several almost tight complexity bounds of C-CD for quadratic problems. One difficulty with the analysis is that the spectral radius of a non-symmetric iteration matrix does not necessarily constitute a \textit{lower bound} for the convergence rate. An immediate consequence is that for Gauss-Seidel method, Kaczmarz method and POCS, there is also an $O(n^2)$ gap between the cyclic versions and randomized versions (for solving linear systems). We also show that the classical convergence rate of POCS by Smith, Solmon and Wager [1] is always worse and sometimes can be infinitely times worse than our bound.Comment: 47 pages. Add a few tables to summarize the main convergence rates; add comparison with classical POCS bound; add discussions on another exampl

Rigidity of Graph Joins and Hendrickson's Conjecture

Whiteley \cite{wh} gives a complete characterization of the infinitesimal flexes of complete bipartite frameworks. Our work generalizes a specific infinitesimal flex to include joined graphs, a family of graphs that contain the complete bipartite graphs. We use this characterization to identify new families of counterexamples, including infinite families, in $\R^5$ and above to Hendrickson's conjecture on generic global rigidity

Quantum spin liquids in a square lattice subject to an Abelian flux

We report that a possible Z2 quantum spin liquid (QSL) can be observed in a new class of frustrated system: spinor bosons subject to a pi flux in a square lattice. We construct a new class of Ginsburg-Landau (GL) type of effective action to classify possible quantum or topological phases at any coupling strengths. It can be used to reproduce the frustrated SF with the 4 sublattice $90^{\circ}$ coplanar spin structure plus its excitations in the weak coupling limit and the FM Mott plus its excitations in the strong coupling limit achieved in our previous work. It also establishes deep and intrinsic connections between the GL effective action and the order from quantum disorder (OFQD) phenomena in the weak coupling limit. Most importantly, it predicts two possible new phases at intermediate couplings: a FM SF phase or a frustrated magnetic Mott phase. We argue that the latter one is more likely and melts into a $Z_2$ quantum spin liquid (QSL) phase. If the heating issue can be under a reasonable control at intermediate couplings $U/t \sim 1$, the topological order of the $Z_2$ QSL maybe uniquely probed by the current cold atom or photonic experimental techniques.Comment: 13 pages, 3 figure

Periodic Table of SYK and supersymmetric SYK

We develop a systematic and unified random matrix theory to classify Sachdev-Ye-Kitaev (SYK) and supersymmetric (SUSY) SYK models and also work out the structure of the energy levels in one periodic table. The SYK with even $q$- and SUSY SYK with odd $q$-body interaction, $N$ even or odd number of Majorana fermions are put on the same footing in the minimal Hilbert space, $N\pmod 8$ and $q\pmod 4$ double Bott periodicity are identified. Exact diagonalizations are performed to study both the bulk energy level statistics and hard edge behaviours. A new moment ratio of the smallest positive eigenvalue is introduced to determine hard edge index efficiently. Excellent agreements between the ED results and the symmetry classifications are demonstrated. Our complete and systematic methods can be transformed to map out more complicated periodic tables of SYK models with more degree of freedoms, tensor models and symmetry protected topological phases. Possible classification of charge neutral quantum black holes are hinted.Comment: One Table, 3 Figures, 22 page

Improved Algorithms for Exact and Approximate Boolean Matrix Decomposition

An arbitrary $m\times n$ Boolean matrix $M$ can be decomposed {\em exactly} as $M =U\circ V$, where $U$ (resp. $V$) is an $m\times k$ (resp. $k\times n$) Boolean matrix and $\circ$ denotes the Boolean matrix multiplication operator. We first prove an exact formula for the Boolean matrix $J$ such that $M =M\circ J^T$ holds, where $J$ is maximal in the sense that if any 0 element in $J$ is changed to a 1 then this equality no longer holds. Since minimizing $k$ is NP-hard, we propose two heuristic algorithms for finding suboptimal but good decomposition. We measure the performance (in minimizing $k$) of our algorithms on several real datasets in comparison with other representative heuristic algorithms for Boolean matrix decomposition (BMD). The results on some popular benchmark datasets demonstrate that one of our proposed algorithms performs as well or better on most of them. Our algorithms have a number of other advantages: They are based on exact mathematical formula, which can be interpreted intuitively. They can be used for approximation as well with competitive "coverage." Last but not least, they also run very fast. Due to interpretability issues in data mining, we impose the condition, called the "column use condition," that the columns of the factor matrix $U$ must form a subset of the columns of $M$.Comment: DSAA201

Weighted stationary phase of higher orders

An $n$th-order first derivative test for oscillatoric integrals is established. When the phase has a single stationary point, an $n$th-order asymptotic expansion of a weighted stationary phase integral is proved for arbitrary $n\geq1$. This asymptotic expansion sharpened the classical result for $n=1$ by Huxley. Possible applications include analysis and analytic number theory.Comment: arXiv admin note: substantial text overlap with arXiv:1510.0121

Improved subconvexity bounds for GL(2)xGL(3) and GL(3) L-functions by weighted stationary phase

Let $f$ be a fixed self-contragradient Hecke-Maass form for $SL(3,\mathbb Z)$, and $u$ an even Hecke-Maass form for $SL(2,\mathbb Z)$ with Laplace eigenvalue $1/4+k^2$, $k>0$. A subconvexity bound $O\big(k^{4/3+\varepsilon}\big)$ in the eigenvalue aspect is proved for the central value at $s=1/2$ of the Rankin-Selberg $L$-function $L(s,f\times u)$. Meanwhile, a subconvexity bound $O\big((1+|t|)^{2/3+\varepsilon}\big)$ in the $t$ aspect is proved for $L(1/2+it,f)$. These bounds improved corresponding subconvexity bounds proved by Xiaoqing Li (Annals of Mathematics, 2011). The main technique in the proof, other than those used by Li, is an $n$th-order asymptotic expansion of a weighted stationary phase integral, for arbitrary $n\geq1$. This asymptotic expansion sharpened the classical result for $n=1$ by Huxley.Comment: Published in Transactions of the American Mathematical Society online in December, 201

Towards gauge unified, supersymmetric hidden strong dynamics

We consider a class of models with extra complex scalars that are charged under both the Standard Model and a hidden strongly coupled $SU(N)_H$ gauge sector, and discuss the scenarios where the new scalars are identified as the messenger fields that mediate the spontaneously broken supersymmetries from the hidden sector to the visible sector. The new scalars are embedded into 5-plets and 10-plets of an $SU(5)_V$ gauge group that potentially unifies the Standard Model gauge groups. The Higgs bosons remain as elementary particles. In the supersymmetrized version of this class of models, vector-like fermions whose left-handed components are superperpartners of the new scalars are introduced. Due to the hidden strong force, the new low-energy scalars hadronize before decaying and thus evade the common direct searches of the supersymmetric squarks. This can be seen as a gauge mediation scenario with the scalar messenger fields forming low-energy bound states. We also discuss the possibility that among the tower of bound states formed under hidden strong dynamics (at least the TeV scale) one of them is the dark matter candidate, as well as the collider signatures (e.g. diphoton, diboson or dijet) of the models that may show up in the near future.Comment: 33 pages, 6 figures. Expanded the SUSY part. Modified the collider phenomenology chapte

Ultrafast Manipulation of Valley Pseudospin

The coherent manipulation of spin and pseudospin underlies existing and emerging quantum technologies, including NMR, quantum communication, and quantum computation. Valley polarization, associated with the occupancy of degenerate, but quantum mechanically distinct valleys in momentum space, closely resembles spin polarization and has been proposed as a pseudospin carrier for the future quantum electronics. Valley exciton polarization has been created in the transition metal dichalcogenide (TMDC) monolayers using excitation by circularly polarized light and has been detected both optically and electrically. In addition, the existence of coherence in the valley pseudospin has been identified experimentally. The manipulation of such valley coherence has, however, remained out of reach. Here we demonstrate an all-optical control of the valley coherence by means of the pseudomagnetic field associated with the optical Stark effect. Using below-bandgap circularly polarized light, we experimentally rotate the valley exciton pseudospin in monolayer WSe2 on the femtosecond time scale. Both the direction and speed of the rotation can be optically manipulated by tuning the dynamic phase of excitons in opposite valleys. This study completes the generation-manipulation-detection paradigm for valley pseudospin, enabling the platform of excitons in 2D materials for the control of this novel degree of freedom in solids.Comment: 9 pages, 4 figure