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Arnab Sen

Publications and source records attributed to Arnab Sen.

At least 73 records · Page 4Linked to original sources
arXiv2016

Parisi formula, disorder chaos and fluctuation for the ground state energy in the spherical mixed p-spin models

We show that the limiting ground state energy of the spherical mixed p-spin model can be identified as the infimum of certain variational problem. This complements the well-known Parisi formula for the limiting free energy in the spherical model. As an application, we obtain explicit formulas for the limiting ground state energy in the replica symmetry, one level of replica symmetry breaking and full replica symmetry breaking phases at zero temperature. In addition, our approach leads to new results on disorder chaos in spherical mixed even p-spin models. In particular, we prove that when there is no external field, the location of the ground state energy is chaotic under small perturbations of the disorder. We also establish that in the spherical mixed even p-spin model, the ground state energy superconcentrates in the absence of external field, while it obeys a central limit theorem if the external field is present.

math.PR↗
arXiv2016

Level spectroscopy in a two-dimensional quantum magnet: Linearly dispersing spinons at the deconfined quantum critical point

We study the level structure of excitations at the "deconfined" critical point separating antiferromagnetic and valence-bond-solid phases in two-dimensional quantum spin systems using the $J$-$Q$ model as an example. Energy gaps in different spin ($S$) and momentum (${\bf k}$) sectors are extracted from imaginary-time correlation functions obtained in quantum Monte Carlo simulations. We find strong quantitative evidence for deconfined linearly dispersing spinons with gapless points at ${\bf k}=(0,0)$, $(π,0)$, $(0,π)$, and $(π,π)$, as inferred from two-spinon excitations ($S=0$ and $S=1$ states) around these points. We also observe a duality between singlet and triplet excitations at the critical point and inside the ordered phases, in support of an enhanced symmetry, possibly SO(5).

cond-mat.str-el↗
arXiv2016

Entanglement generation in periodically driven integrable systems: dynamical phase transitions and steady state

We study a class of periodically driven $d-$dimensional integrable models and show that after $n$ drive cycles with frequency $ω$, pure states with non-area-law entanglement entropy $S_n(l) \sim l^{α(n,ω)}$ are generated, where $l$ is the linear dimension of the subsystem, and $d-1 \le α(n,ω) \le d$. We identify and analyze the crossover phenomenon from an area ($S \sim l^{ d-1}$ for $d\geq1$) to a volume ($S \sim l^{d}$) law and provide a criterion for their occurrence which constitutes a generalization of Hastings' theorem to driven integrable systems in one dimension. We also find that $S_n$ generically decays to $S_{\infty}$ as $(ω/n)^{(d+2)/2}$ for fast and $(ω/n)^{d/2}$ for slow periodic drives; these two dynamical phases are separated by a topological transition in the eigensprectrum of the Floquet Hamiltonian. This dynamical transition manifests itself in the temporal behavior of all local correlation functions and does not require a critical point crossing during the drive. We find that these dynamical phases show a rich re-entrant behavior as a function of $ω$ for $d=1$ models, and also discuss the dynamical transition for $d>1$ models. Finally, we study entanglement properties of the steady state and show that singular features (cusps and kinks in $d=1$) appear in $S_{\infty}$ as a function of $ω$ whenever there is a crossing of the Floquet bands. We discuss experiments which can test our theory.

cond-mat.str-el↗
arXiv2016

A fractionalised "$\mathbb{Z}_2$" classical Heisenberg spin liquid

Quantum spin systems are by now known to exhibit a large number of different classes of spin liquid phases. By contrast, for \textit{classical} Heisenberg models, only one kind of fractionalised spin liquid phase, the so-called Coulomb or $U(1)$ spin liquid, has until recently been identified: this exhibits algebraic spin correlations and impurity moments, `orphan spins', whose size is a fraction of that of the underlying microscopic degrees of freedom. Here, we present two Heisenberg models exhibiting fractionalisation in combination with exponentially decaying correlations. These can be thought of as a classical continuous spin version of a $\mathbb{Z}_2$ spin liquid. Our work suggests a systematic search and classification of classical spin liquids as a worthwhile endeavour.

cond-mat.str-el↗
arXiv2016

Signatures and conditions for phase band crossings in periodically driven integrable systems

We present generic conditions for phase band crossings for a class of periodically driven integrable systems represented by free fermionic models subjected to arbitrary periodic drive protocols characterized by a frequency $ω_D$. These models provide a representation for the Ising and $XY$ models in $d=1$, the Kitaev model in $d=2$, several kinds of superconductors, and Dirac fermions in graphene and atop topological insulator surfaces. Our results demonstrate that the presence of a critical point/region in the system Hamiltonian (which is traversed at a finite rate during the dynamics) may change the conditions for phase band crossings that occur at the critical modes. We also show that for $d>1$, phase band crossings leave their imprint on the equal-time off-diagonal fermionic correlation functions of these models; the Fourier transforms of such correlation functions, $F_{\vec k_0}( ω_0)$, have maxima and minima at specific frequencies which can be directly related to $ω_D$ and the time at which the phase bands cross at $\vec k = \vec k_0$. We discuss the significance of our results in the contexts of generic Hamiltonians with $N>2$ phase bands and the underlying symmetry of the driven Hamiltonian.

cond-mat.str-el↗
arXiv2016

Eigenstate Gibbs Ensemble in Integrable Quantum Systems

The Eigenstate Thermalization Hypothesis implies that for a thermodynamically large system in one of its eigenstates, the reduced density matrix describing any finite subsystem is determined solely by a set of {\it relevant} conserved quantities. In a generic system, only the energy plays that role and hence eigenstates appear locally thermal. Integrable systems, on the other hand, possess an extensive number of such conserved quantities and hence the reduced density matrix requires specification of an infinite number of parameters (Generalized Gibbs Ensemble). However, here we show by unbiased statistical sampling of the individual eigenstates with a given finite energy density, that the local description of an overwhelming majority of these states of even such an integrable system is actually Gibbs-like, i.e. requires only the energy density of the eigenstate. Rare eigenstates that cannot be represented by the Gibbs ensemble can also be sampled efficiently by our method and their local properties are then shown to be described by appropriately {\it truncated} Generalized Gibbs Ensembles. We further show that the presence of these rare eigenstates differentiates the model from the generic (non-integrable) case and leads to the system being described by a Generalized Gibbs Ensemble at long time under a unitary dynamics following a sudden quench, even when the initial state is a Gibbs-like eigenstate of the pre-quench Hamiltonian.

cond-mat.stat-mech↗
arXiv2016

Double roots of random polynomials with integer coefficients

We consider random polynomials whose coefficients are independent and identically distributed on the integers. We prove that if the coefficient distribution has bounded support and its probability to take any particular value is at most $\tfrac12$, then the probability of the polynomial to have a double root is dominated by the probability that either $0$, $1$, or $-1$ is a double root up to an error of $o(n^{-2})$. We also show that if the support of coefficient distribution excludes $0$ then the double root probability is $O(n^{-2})$. Our result generalizes a similar result of Peled, Sen and Zeitouni for Littlewood polynomials.

math.PR↗
arXiv2015

Coalescing Brownian flows: A new approach

The coalescing Brownian flow on $\mathbb{R}$ is a process which was introduced by Arratia [Coalescing Brownian motions on the line (1979) Univ. Wisconsin, Madison] and Tóth and Werner [Probab. Theory Related Fields 111 (1998) 375-452], and which formally corresponds to starting coalescing Brownian motions from every space-time point. We provide a new state space and topology for this process and obtain an invariance principle for coalescing random walks. This result holds under a finite variance assumption and is thus optimal. In previous works by Fontes et al. [Ann. Probab. 32 (2004) 2857-2883], Newman et al. [Electron. J. Probab. 10 (2005) 21-60], the topology and state-space required a moment of order $3-\varepsilon$ for this convergence to hold. The proof relies crucially on recent work of Schramm and Smirnov on scaling limits of critical percolation in the plane. Our approach is sufficiently simple that we can handle substantially more complicated coalescing flows with little extra work - in particular similar results are obtained in the case of coalescing Brownian motions on the Sierpinski gasket. This is the first such result where the limiting paths do not enjoy the noncrossing property.

math.PR↗
arXiv2015

Extinction window of mean field branching annihilating random walk

We study a model of growing population that competes for resources. At each time step, all existing particles reproduce and the offspring randomly move to neighboring sites. Then at any site with more than one offspring, the particles are annihilated. This is a nonmonotone model, which makes the analysis more difficult. We consider the extinction window of this model in the finite mean-field case, where there are $n$ sites but movement is allowed to any site (the complete graph). We show that although the system survives for exponential time, the extinction window is logarithmic.

math.PR↗
arXiv2015

Classical spin-liquid on the maximally frustrated honeycomb lattice

We show that the honeycomb Heisenberg antiferromagnet with $J_1/2=J_2=J_3$, where $J_{1/2/3}$ are first-, second- and third-neighbour couplings respectively, forms a classical spin liquid with pinch-point singularities in the structure factor at the Brillouin zone corners. Upon dilution with non-magnetic ions, fractionalised degrees of freedom carrying $1/3$ of the free moment emerge. Their effective description in the limit of low-temperature is that of spins randomly located on a triangular lattice, with a frustrated interaction of long-ranged logarithmic form. The XY version of this magnet exhibits nematic thermal order by disorder, which comes with a clear experimental diagnostic.

cond-mat.str-el↗
arXiv2015

Juntas in the $\ell^{1}$-grid and Lipschitz maps between discrete tori

We show that if $A \subset [k]^n$, then $A$ is $ε$-close to a junta depending upon at most $\exp(O(|\partial A|/(k^{n-1}ε)))$ coordinates, where $\partial A$ denotes the edge-boundary of $A$ in the $\ell^1$-grid. This is sharp up to the value of the absolute constant in the exponent. This result can be seen as a generalisation of the Junta theorem for the discrete cube, from [E. Friedgut, Boolean functions with low average sensitivity depend on few coordinates, Combinatorica 18 (1998), 27-35], or as a characterization of large subsets of the $\ell^1$-grid whose edge-boundary is small. We use it to prove a result on the structure of Lipschitz functions between two discrete tori; this can be seen as a discrete, quantitative analogue of a recent result of Austin [T. Austin, On the failure of concentration for the $\ell^{\infty}$-ball, preprint]. We also prove a refined version of our junta theorem, which is sharp in a wider range of cases.

math.CO↗
arXiv2015

Velocity of excitations in ordered, disordered and critical antiferromagnets

We test three different approaches, based on quantum Monte Carlo simulations, for computing the velocity $c$ of triplet excitations in antiferromagnets. We consider the standard $S=1/2$ one- and two-dimensional Heisenberg models, as well as a bilayer Heisenberg model at its critical point. Computing correlation functions in imaginary time and using their long-time behavior, we extract the lowest excitation energy versus momentum using improved fitting procedures and a generalized moment method. The velocity is then obtained from the dispersion relation. We also exploit winding numbers to define a cubic space-time geometry, where the velocity is obtained as the ratio of the spatial and temporal lengths of the system when all winding number fluctuations are equal. The two methods give consistent results for both ordered and critical systems, but the winding-number estimator is more precise. For the Heisenberg chain, we accurately reproduce the exactly known velocity. For the two-dimensional Heisenberg model, our results are consistent with other recent calculations, but with an improved statistical precision; $c=1.65847(4)$. We also use the hydrodynamic relation $c^2=ρ_s/χ_\perp(q\to 0)$ between $c$, the spin stiffness $ρ_s$, and the transversal susceptibility $χ_\perp$, using the smallest non-zero momentum $q=2π/L$. This method also is well controlled in two dimensions, but the cubic criterion for winding numbers delivers better numerical precision. In one dimension the hydrodynamic relation is affected by logarithmic corrections which make accurate extrapolations difficult. As an application of the winding-number method, for the quantum-critical bilayer model our high-precision determination of the velocity enables us to quantitatively test, at an unprecedented level, field-theoretic predictions for low-temperature scaling forms where $c$ enters.

cond-mat.str-el↗
arXiv2015

Random Coulomb antiferromagnets: from diluted spin liquids to Euclidean random matrices

We study a disordered classical Heisenberg magnet with uniformly antiferromagnetic interactions which are frustrated on account of their long-range Coulomb form, {\em i.e.} $J(r)\sim -A\ln r$ in $d=2$ and $J(r)\sim A/r$ in $d=3$. This arises naturally as the $T\rightarrow 0$ limit of the emergent interactions between vacancy-induced degrees of freedom in a class of diluted Coulomb spin liquids (including the classical Heisenberg antiferromagnets on checkerboard, SCGO and pyrochlore lattices) and presents a novel variant of a disordered long-range spin Hamiltonian. Using detailed analytical and numerical studies we establish that this model exhibits a very broad paramagnetic regime that extends to very large values of $A$ in both $d=2$ and $d=3$. In $d=2$, using the lattice-Green function based finite-size regularization of the Coulomb potential (which corresponds naturally to the underlying low-temperature limit of the emergent interactions between orphan-spins), we only find evidence that freezing into a glassy state occurs in the limit of strong coupling, $A=\infty$, while no such transition seems to exist at all in $d=3$. We also demonstrate the presence and importance of screening for such a magnet. We analyse the spectrum of the Euclidean random matrices describing a Gaussian version of this problem, and identify a corresponding quantum mechanical scattering problem.

cond-mat.str-el↗
arXiv2015

Double roots of random Littlewood polynomials

We consider random polynomials whose coefficients are independent and uniform on {-1,1}. We prove that the probability that such a polynomial of degree n has a double root is o(n^{-2}) when n+1 is not divisible by 4 and asymptotic to $\frac{8\sqrt{3}}{πn^2}$ otherwise. This result is a corollary of a more general theorem that we prove concerning random polynomials with independent, identically distributed coefficients having a distribution which is supported on { -1, 0, 1} and whose largest atom is strictly less than 1/\sqrt{3}. In this general case, we prove that the probability of having a double root equals the probability that either -1, 0 or 1 are double roots up to an o(n^{-2}) factor and we find the asymptotics of the latter probability.

math.PR↗
arXiv2014

On Testing Independence and Goodness-of-fit in Linear Models

We consider a linear regression model and propose an omnibus test to simultaneously check the assumption of independence between the error and the predictor variables, and the goodness-of-fit of the parametric model. Our approach is based on testing for independence between the residual obtained from the parametric fit and the predictor using the Hilbert--Schmidt independence criterion (Gretton et al. (2008)). The proposed method requires no user-defined regularization, is simple to compute, based merely on pairwise distances between points in the sample, and is consistent against all alternatives. We develop distribution theory for the proposed test statistic, both under the null and the alternative hypotheses, and devise a bootstrap scheme to approximate its null distribution. We prove the consistency of the bootstrap scheme. A simulation study shows that our method has better power than its main competitors. Two real datasets are analyzed to demonstrate the scope and usefulness of our method.

stat.ME↗
arXiv2014

A topological spin glass in diluted spin ice

It is a salient experimental fact that a large fraction of candidate spin liquid materials freeze as the temperature is lowered. The question naturally arises whether such freezing is intrinsic to the spin liquid ("disorder-free glassiness") or extrinsic, in the sense that a topological phase simply coexists with standard freezing of impurities. Here, we demonstrate a surprising third alternative, namely that freezing and topological liquidity are inseparably linked. The topological phase reacts to the introduction of disorder by generating degrees of freedom of a new type (along with interactions between them), which in turn undergo a freezing transition while the topological phase supporting them remains intact.

cond-mat.str-el↗
arXiv2013

Mean quantum percolation

We study the spectrum of adjacency matrices of random graphs. We develop two techniques to lower bound the mass of the continuous part of the spectral measure or the density of states. As an application, we prove that the spectral measure of bond percolation in the two dimensional lattice contains a non-trivial continuous part in the supercritical regime. The same result holds for the limiting spectral measure of a supercritical Erdos-Renyi graph and for the spectral measure of a unimodular random tree with at least two ends. We give examples of random graphs with purely continuous spectrum.

math.PR↗