arXiv subjects

Santanu Dey

Publications and source records attributed to Santanu Dey.

At least 19 recordsLinked to original sources

arXiv2026

Distributed Linear Programming on GPU Clusters at Extreme Scale

Large linear programs can exceed the memory of a single compute node. Although first-order methods replace sparse factorizations with GPU-suited matrix-vector products, other solver phases can reintroduce a single-node memory limit. We present SHARDLP, a distributed GPU LP solver that keeps the matrix and primal-dual state partitioned from sharded input through solution output. On the Google PDLP benchmark, SHARDLP reaches the published criterion on nine of eleven instances, compared with eight in the published CPU PDLP study. On the largest benchmark, eight H200 GPUs solve a 1.185-billion-variable, 6.338-billion-nonzero LP in 9.9 minutes; the published CPU experiment reports 21.06 hours on different hardware. Beyond this benchmark, separately checked multi-node solves reach up to 13.604 billion variables and 40.807 billion nonzeros, while validated executions span up to 76 GPUs across 29 compute nodes. For column-partitioned solves, support-aware communication skips GPUs that store no coefficients for a row; on an LP with 2.76 billion nonzeros, it cuts modelled communication by 92.97% and improves solver time by 1.27x-1.52x

math.OC↗

arXiv2026

The Skew Commutators of Toeplitz and Hankel operators in vector-valued Hardy Space

In this article, we characterize when a Toeplitz operator and a Hankel operator on the vector-valued Hardy space are skew commutators of each other, and determine necessary and sufficient conditions under which their product is self-adjoint. These characterizations extend the results in \cite{LZD}. In addition, we completely classify the skew commutators of the unilateral shift $S$, its adjoint $S^*$ and $S\oplus S^*.$ We also characterize the class of bounded linear operators that have $S$, $S^*$ and $S\oplus S^*$ as skew-commutators.

math.FA↗

arXiv2024

On the rank of extremal marginal states

Let $\rho_1$ and $\rho_2$ be two states on $\mathbb{C}^{d_1}$ and $\mathbb{C}^{d_2}$ respectively. The marginal state space, denoted by $\mathcal{C}(\rho_1,\rho_2)$, is the set of all states $\rho$ on $\mathbb{C}^{d_1}\otimes \mathbb{C}^{d_2}$ with partial traces $\rho_1, \rho_2$. K. R. Parthasarathy established that if $\rho$ is an extreme point of $\mathcal{C}(\rho_1,\rho_2)$, then the rank of $\rho$ does not exceed $\sqrt{d_1^2+d_2^2-1}$. Rudolph posed a question regarding the tightness of this bound. In 2010, Ohno gave an affirmative answer by providing examples in low-dimensional matrix algebras $\mathbb{M}_3$ and $\mathbb{M}_4$. This article aims to provide a positive answer to the Rudolph question in various matrix algebras. Our approaches, to obtain the extremal marginal states with tight upper bound, are based on Choi-Jamio\l kowski isomorphism and tensor product of extreme points.

math.OA↗

arXiv2024

Hyperbolic Spin Liquids

Hyperbolic lattices present a unique opportunity to venture beyond the conventional paradigm of crystalline many-body physics and explore correlated phenomena in negatively curved space. As a theoretical benchmark for such investigations, we extend Kitaev's spin-1/2 honeycomb model to hyperbolic lattices and exploit their non-Euclidean space-group symmetries to solve the model exactly. We elucidate the ground-state phase diagram on the $\{8,3\}$ lattice and find a gapped $\mathbb{Z}_2$ spin liquid with Abelian anyons, a gapped chiral spin liquid with non-Abelian anyons and chiral edge states, and a Majorana metal whose finite low-energy density of states is dominated by non-Abelian Bloch states.

cond-mat.str-el↗

arXiv2024

Simulating Holographic Conformal Field Theories on Hyperbolic Lattices

We demonstrate how table-top settings combining hyperbolic lattices with nonlinear dynamics universally encode aspects of the bulk-boundary-correspondence between gravity in anti-de-Sitter (AdS) space and conformal field theory (CFT). Our concrete and broadly applicable holographic toy model simulates gravitational self-interactions in the bulk and features an emergent CFT with nontrivial correlations on the boundary. We measure the CFT data contained in the two- and three-point functions and clarify how a thermal CFT is simulated through an effective black hole geometry. As a concrete example, we propose and simulate an experimentally feasible protocol to measure the holographic CFT using electrical circuits.

cond-mat.mes-hall↗

arXiv2024

Block quantum dynamical semigroups of completely positive definite kernels

Kolmogorov decomposition for a given completely positive definite kernel is a generalization of Paschke's GNS construction for the completely positive map. Using Kolmogorov decomposition, to every quantum dynamical semigroup (QDS) for completely positive definite kernels over a set $S$ on given $C^*$-algebra $\mathcal{A},$ we shall assign an inclusion system $F = (F_s)_{s\ge 0}$ of Hilbert bimodules over $\mathcal{A}$ with a generating unit $\xi^{\sigma}=(\xi^{\sigma}_s)_{s\ge 0}.$ Consider a von Neumann algebra $\mathcal{B}$, and let $\mathfrak{T}=(\mathfrak{T}_s)_{s\ge 0}$ be a QDS over a set $S$ on the algebra $M_2(\mathcal{B})$ with $\mathfrak{T}_s=\begin{pmatrix}\mathfrak{K}_{s,1} & \mathfrak{L}_s\\\mathfrak{L}_s^*& \mathfrak{K}_{s,2} \end{pmatrix}$ which acts block-wise. Further, suppose that $(F^i_s )_{s\ge 0}$ is the inclusion system affiliated to the diagonal QDS $(\mathfrak{K}_{s,i})_{s\ge 0}$ along with the generating unit $(\xi^{\sigma}_{s,i} )_{s\ge 0},$ $\sigma\in S,i\in \{1,2\}$, then we prove that there exists a unique contractive (weak) morphism $V = (V_s)_{s\ge 0}:F^2_s \to F^1_s$ such that $\mathfrak{L}_s^{\sigma,\sigma'}(b)=\langle \xi_{s,1}^{\sigma},V_s b\xi_{s,2}^{\sigma'}\rangle$ for every $\sigma',\sigma\in S$ and $b\in \mathcal{B}.$ We also study the semigroup version of a factorization theorem for $\mathfrak{K}$-families.

math.OA↗

arXiv2023

Field-driven transition from quantum spin liquid to magnetic order in triangular-lattice antiferromagnets

Recently several triangular-lattice magnets with delafossite structure have been found to display spin-liquid behavior down to the lowest temperatures. Remarkably, applying a magnetic field destroys the spin liquid which then gives way to symmetry-breaking states, identified as semiclassical coplanar states including a magnetization plateau at 1/3 total magnetization. Here we provide a theoretical approach rationalizing this dichotomy, utilizing a Schwinger-boson theory that captures both ordered and disordered magnetic phases. We show that a zero-field spin liquid, driven by strong frustration, is naturally destabilized in a magnetic field via spinon condensation. Symmetry-breaking order akin to the standard triangular-lattice Heisenberg model then arises via an order-by-disorder mechanism. We discuss implications for pertinent experiments.

cond-mat.str-el↗

arXiv2023

A Boosted Machine Learning Framework for the Improvement of Phase and Crystal Structure Prediction of High Entropy Alloys Using Thermodynamic and Configurational Parameters

The reason behind the remarkable properties of High-Entropy Alloys (HEAs) is rooted in the diverse phases and the crystal structures they contain. In the realm of material informatics, employing machine learning (ML) techniques to classify phases and crystal structures of HEAs has gained considerable significance. In this study, we assembled a new collection of 1345 HEAs with varying compositions to predict phases. Within this collection, there were 705 sets of data that were utilized to predict the crystal structures with the help of thermodynamics and electronic configuration. Our study introduces a methodical framework i.e., the Pearson correlation coefficient that helps in selecting the strongly co-related features to increase the prediction accuracy. This study employed five distinct boosting algorithms to predict phases and crystal structures, offering an enhanced guideline for improving the accuracy of these predictions. Among all these algorithms, XGBoost gives the highest accuracy of prediction (94.05%) for phases and LightGBM gives the highest accuracy of prediction of crystal structure of the phases (90.07%). The quantification of the influence exerted by parameters on the model's accuracy was conducted and a new approach was made to elucidate the contribution of individual parameters in the process of phase prediction and crystal structure prediction.

cs.LG↗

arXiv2023

Characteristic Functions and Colligations

The characteristic function of row contractions and the characteristic function of liftings of row contractions are multi-analytic operators which are complete invariants up to unitary equivalence for row contractions and liftings of row contractions, respectively. We provide alternate proofs for these properties of characteristic functions using colligations. Co-isometric observable colligations with certain class of basic operators are characterized. Blaschke factor based transformations of the characteristic function of lifting are studied.

math.FA↗

arXiv2022

Factorization of Characteristic Functions of Iterated Liftings

We obtain a factorization of the characteristic function of a contractive two-step iterated lifting in terms of the characteristic functions of constituent liftings of the iterated lifting and the Julia-Halmos matrix. We also give an expression for the characteristic function of the minimal part of a contractive two-step iterated lifting as a restriction of the product of the characteristic functions of constituent liftings of the iterated lifting.

math.FA↗

arXiv2022

Quantum-critical electrodynamics of Luttinger fermions

We study the quantum electrodynamics of Luttinger fermions with quadratic band-crossing dispersion in three dimensions. The model can be viewed as the low-energy effective theory of a putative $U(1)$ quantum spin liquid with fermionic Luttinger spinons, or as an extension of the Luttinger-Abrikosov-Beneslavskii (LAB) model that accounts for transverse gauge fluctuations with finite photon velocity. Aided by a renormalization group analysis below four dimensions, we elucidate the presence and stability of quantum critical phenomena in this model. We find that the non-Fermi liquid LAB phase is stable against gauge fluctuations, and can thus also be viewed as a $U(1)$ spin liquid with gapless Luttinger spinons. We discover a multicritical point with Lifshitz scaling that corresponds to a time-reversal symmetry-breaking quantum phase transition from the LAB state to a chiral spin liquid with spinon Landau levels and birefringent emergent photons. This multicritical point is characterized by a finite fermion-photon coupling in the infrared and can be viewed as a fermionic analog of the Rokhsar-Kivelson point in three-dimensional quantum dimer models.

cond-mat.str-el↗

arXiv2020

Using Submodularity within Column Generation to Solve the Flight-to-Gate Assignment Problem

In this paper, we provide a column generation-based approach for solving the airport flight-to-gate assignment problem, where the goal is to minimize the on-ground portion of arrival delays by optimally assigning each scheduled flight to a compatible gate. Specifically, we use a set covering formulation for the master problem and decompose the pricing problem such that each gate is the basis for an independent pricing problem to be solved for assignment patterns with negative reduced costs. We use a combination of an approximation algorithm based on the submodularity of the underlying set and dynamic programming algorithms to solve the independent pricing problems. To the best of our knowledge, this is the first use of submodularity property to efficiently solve pricing problems and improve the performance of column generation algorithm. We show that the dynamic programming algorithm is pseudo-polynomial when there are integer inputs. We also design and employ a rolling horizon method and block decomposition algorithm to solve large-sized instances. Finally, we perform extensive computational experiments to validate the performance of our approach.

math.OC↗

arXiv2020

Destabilization of U(1) Dirac spin-liquids on two dimensional non-bipartite lattices by quenched disorder

The stability of the Dirac spin-liquid on two-dimensional lattices has long been debated. It was recently demonstrated [Nature Commun. 10, 4254 (2019) and Phys. Rev. B 93, 144411 (2016)] that the staggered $\pi$-flux Dirac spin-liquid phase on the non-bipartite triangular lattice may be stable in the clean limit. However, quenched disorder plays a crucial role in determining whether such a phase is experimentally viable. For SU(2) spin systems, the effective zero-temperature, low-energy description of Dirac spin-liquids in $(2+1)$ dimensions is given by the compact quantum electrodynamics ($\rm cQED_{2+1}$) which admits monopoles. It is already known that generic quenched random perturbations to the non-compact version of $\rm QED_{2+1}$ (where monopoles are absent) lead to strong-coupling instabilities. In this paper we study $\rm cQED_{2+1}$ in the presence of a class of time-reversal invariant quenched disorder perturbations. We show that in this model, random non-Abelian vector potentials make the symmetry-allowed monopole operators more relevant. The disorder-induced underscreening of monopoles, thus, generically makes the gapless spin-liquid phase fragile.

cond-mat.str-el↗

arXiv2020

Random-bond disorder in two-dimensional non-collinear XY antiferromagnets: From quasi-long-range order to spin glass

We study effects of quenched bond disorder in frustrated easy-plane antiferromagnets in two space dimensions, using a combination of analytical and numerical techniques. We consider local-moment systems which display non-collinear long-range order at zero temperature, with the antiferromagnetic triangular-lattice XY model as a prime example. We show that (i) weak bond disorder transforms the clean ground state into a state with quasi-long-range order, and (ii) strong bond disorder results in a short-range-ordered glassy ground state. Using extensive Monte Carlo simulations, we also track the fate of the quasi-long-range-ordered phase at finite temperature and the associated Kosterlitz-Thouless-like transition. Making contact with previous studies of disordered XY models, we discuss similarities and differences concerning the interplay of disorder and frustration. Our results are also of relevance to non-collinear Heisenberg magnets in an external magnetic field.

cond-mat.str-el↗

arXiv2019

Order Matters at Fanatics Recommending Sequentially Ordered Products by LSTM Embedded with Word2Vec

A unique challenge for e-commerce recommendation is that customers are often interested in products that are more advanced than their already purchased products, but not reversed. The few existing recommender systems modeling unidirectional sequence output a limited number of categories or continuous variables. To model the ordered sequence, we design the first recommendation system that both embed purchased items with Word2Vec, and model the sequence with stateless LSTM RNN. The click-through rate of this recommender system in production outperforms its solely Word2Vec based predecessor. Developed in 2017, it was perhaps the first published real-world application that makes distributed predictions of a single machine trained Keras model on Spark slave nodes at a scale of more than 0.4 million columns per row.

cs.LG↗

arXiv2019

Destruction of long-range order in non-collinear two-dimensional antiferromagnets by random-bond disorder

We consider frustrated Heisenberg antiferromagnets, whose clean-limit ground state is characterized by non-collinear long-range order with non-zero vector chirality, and study the effects of quenched bond disorder, i.e., random exchange couplings. A single bond defect is known to induce a dipolar texture in the spin background independent of microscopic details. Using general analytical arguments as well as large-scale simulations for the classical triangular-lattice Heisenberg model, we show that any finite concentration of such defects destroys long-range order for spatial dimension $d\leq 2$, in favor of a glassy state whose correlation length in $d=2$ is exponentially large for small randomness. Our results are relevant for a wide range of layered frustrated magnets.

cond-mat.str-el↗

arXiv2019

The order-$n$ minors of certain $(n+k) \times n$ matrices

We determine sufficient conditions for certain classes of $(n+k) \times n$ matrices $E$ to have all order-$n$ minors to be nonzero. For a special class of $(n+1) \times n$ matrices $E,$ we give the formula for the order-$n$ minors. As an application we construct subspaces of $\C^m \otimes \C^n$ of maximal dimension, which does not contain any vector of Schmidt rank less than $k$ and which has a basis of Schmidt rank $k$ for $k=2,3,4$.

math.FA↗

arXiv2018

Comparing Geometric Discord and Negativity for Bipartite States

The geometric discord $\mathcal{D}$ of a state is a measure of the quantumness of the state and the negativity $\mathcal{N}$ is a measure of the entanglement of a state. It was proved by D. Girolami and G. Adesso that for states on $\mathbb{C}^2\otimes\mathbb{C}^2$, the geometric discord is always greater than or equal to the square of the negativity and conjectured that this holds in general. S. Rana and P. Parashar showed that this relation does not hold for all states on $\mathbb{C}^2\otimes\mathbb{C}^n$ for $n>2$. We provide several analytic families of states on $\mathbb{C}^2\otimes\mathbb{C}^3$ violating this relation. Certain upper and lower bounds for $\mathcal{N}^2-\mathcal{D}$ are obtained for states on $\mathbb{C}^m\otimes\mathbb{C}^n$ for any $m, n\in\mathbb{N}$.

quant-ph↗