arXiv subjects

Arnab Sen

Publications and source records attributed to Arnab Sen.

At least 91 records · Page 5Linked to original sources

arXiv2013

Coulomb phase diagnostics as function of temperature, interaction range and disorder

The emergent gauge field characteristic of the Coulomb phase of spin ice betrays its existence via pinch points in the spin structure factor ${\cal{S}}$ in reciprocal space which takes the form of a transverse projector ${\cal{P}}$ at low temperature: ${\cal{S}} (q) \sim {\cal{P}} \sim q^2_\bot/q^2$. We develop a theory which establishes the fate of the pinch points at low and high temperature, for hard and soft spins, for short- and long-ranged (dipolar) interactions, as well as in the presence of disorder. We find that their detailed shape can be used to read off the relative sizes of entropic and magnetic Coulomb interactions of monopoles in spin ice, and we resolve the question why pinch points have been experimentally observed for Ho$_{1.7}$Y$_{0.3}$Ti$_2$O$_7$ even at high temperature in the presence of strong disorder.

cond-mat.str-el↗

arXiv2012

Phase transitions in the frustrated Ising model on the square lattice

We consider the thermal phase transition from a paramagnetic to stripe-antiferromagnetic phase in the frustrated two-dimensional square-lattice Ising model with competing interactions J1<0 (nearest neighbor, ferromagnetic) and J2 >0 (second neighbor, antiferromagnetic). The striped phase breaks a Z4 symmetry and is stabilized at low temperatures for g=J2/|J1|>1/2. Despite the simplicity of the model, it has proved difficult to precisely determine the order and the universality class of the phase transitions. This was done convincingly only recently by Jin et al. [PRL 108, 045702 (2012)]. Here, we further elucidate the nature of these transitions and their anomalies by employing a combination of cluster mean-field theory, Monte Carlo simulations, and transfer-matrix calculations. The J1-J2 model has a line of very weak first-order phase transitions in the whole region 1/2 1/2. Most of our results are based on Monte Carlo calculations, while the cluster mean-field and transfer-matrix results provide useful methodological bench-marks for weakly first-order behaviors and Ashkin-Teller criticality.

cond-mat.stat-mech↗

arXiv2012

On reverse hypercontractivity

We study the notion of reverse hypercontractivity. We show that reverse hypercontractive inequalities are implied by standard hypercontractive inequalities as well as by the modified log-Sobolev inequality. Our proof is based on a new comparison lemma for Dirichlet forms and an extension of the Strook-Varapolos inequality. A consequence of our analysis is that {\em all} simple operators $L=Id-\E$ as well as their tensors satisfy uniform reverse hypercontractive inequalities. That is, for all $q<p<1$ and every positive valued function $f$ for $t \geq \log \frac{1-q}{1-p}$ we have $\| e^{-tL}f\|_{q} \geq \| f\|_{p}$. This should be contrasted with the case of hypercontractive inequalities for simple operators where $t$ is known to depend not only on $p$ and $q$ but also on the underlying space. The new reverse hypercontractive inequalities established here imply new mixing and isoperimetric results for short random walks in product spaces, for certain card-shufflings, for Glauber dynamics in high-temperatures spin systems as well as for queueing processes. The inequalities further imply a quantitative Arrow impossibility theorem for general product distributions and inverse polynomial bounds in the number of players for the non-interactive correlation distillation problem with $m$-sided dice.

math.PR↗

arXiv2012

Geometric Influences II: Correlation Inequalities and Noise Sensitivity

In a recent paper, we presented a new definition of influences in product spaces of continuous distributions, and showed that analogues of the most fundamental results on discrete influences, such as the KKL theorem, hold for the new definition in Gaussian space. In this paper we prove Gaussian analogues of two of the central applications of influences: Talagrand's lower bound on the correlation of increasing subsets of the discrete cube, and the Benjamini-Kalai-Schramm (BKS) noise sensitivity theorem. We then use the Gaussian results to obtain analogues of Talagrand's bound for all discrete probability spaces and to reestablish analogues of the BKS theorem for biased two-point product spaces.

math.PR↗

arXiv2012

Geometric influences

We present a new definition of influences in product spaces of continuous distributions. Our definition is geometric, and for monotone sets it is identical with the measure of the boundary with respect to uniform enlargement. We prove analogs of the Kahn-Kalai-Linial (KKL) and Talagrand's influence sum bounds for the new definition. We further prove an analog of a result of Friedgut showing that sets with small "influence sum" are essentially determined by a small number of coordinates. In particular, we establish the following tight analog of the KKL bound: for any set in $\mathbb{R}^n$ of Gaussian measure $t$, there exists a coordinate $i$ such that the $i$th geometric influence of the set is at least $ct(1-t)\sqrt{\log n}/n$, where $c$ is a universal constant. This result is then used to obtain an isoperimetric inequality for the Gaussian measure on $\mathbb{R}^n$ and the class of sets invariant under transitive permutation group of the coordinates.

math.PR↗

arXiv2012

Vacancy-induced spin textures and their interactions in a classical spin liquid

Motivated by experiments on the archetypal frustrated magnet SrCr$_{9p}$Ga$_{12-9p}$O$_{19}$ (SCGO), we study the classical Heisenberg model on the pyrochlore slab (Kagomé bilayer) lattice with site-dilution $x=1-p$. This allows us to address generic aspects of the physics of non-magnetic vacancies in a classical spin liquid. We explicitly demonstrate that the pure ($x=0$) system remains a spin-liquid down to the lowest temperatures, with an unusual {\em non-monotonic} temperature dependence of the susceptibility, which even turns diamagnetic for the apical spins between the two kagome layers. For $x> 0$ but small, the low temperature magnetic response of the system is most naturally described in terms of the properties of spatially extended spin textures that cloak an "orphan" $S=3/2$ Cr$^{3+}$ spin in direct proximity to a pair of missing sites belonging to the same triangular simplex. In the $T \rightarrow 0$ limit, these orphan-texture complexes each carry a net magnetization that is exactly half the magnetic moment of an individual spin of the undiluted system. Furthermore, we demonstrate that they interact via an entropic {\em temperature dependent} {\em pair-wise exchange interaction} $J_{eff}(T,\vec{r}) \sim T {\mathcal J} (\vec{r} \sqrt{T})$ that has a logarithmic form at short-distances and decays exponentially beyond a thermal correlation length $ξ(T) \sim 1/\sqrt{T}$. The sign of $J_{eff}$ depends on whether the two orphan spins belong to the same Kagome layer or not. We provide a detailed analytical account of these properties using an effective field theory approach specifically tailored for the problem at hand. These results are in quantitative agreement with large-scale Monte Carlo numerics.

cond-mat.str-el↗

arXiv2012

Coalescing systems of non-Brownian particles

A well-known result of Arratia shows that one can make rigorous the notion of starting an independent Brownian motion at every point of an arbitrary closed subset of the real line and then building a set-valued process by requiring particles to coalesce when they collide. Arratia noted that the value of this process will be almost surely a locally finite set at all positive times, and a finite set almost surely if the initial value is compact: the key to both of these facts is the observation that, because of the topology of the real line and the continuity of Brownian sample paths, at the time when two particles collide one or the other of them must have already collided with each particle that was initially between them. We investigate whether such instantaneous coalescence still occurs for coalescing systems of particles where either the state space of the individual particles is not locally homeomorphic to an interval or the sample paths of the individual particles are discontinuous. We give a quite general criterion for a coalescing system of particles on a compact state space to coalesce to a finite set at all positive times almost surely and show that there is almost sure instantaneous coalescence to a locally finite set for systems of Brownian motions on the Sierpinski gasket and stable processes on the real line with stable index greater than one.

math.PR↗

arXiv2012

Stochastic population growth in spatially heterogeneous environments

Classical ecological theory predicts that environmental stochasticity increases extinction risk by reducing the average per-capita growth rate of populations. To understand the interactive effects of environmental stochasticity, spatial heterogeneity, and dispersal on population growth, we study the following model for population abundances in $n$ patches: the conditional law of $X_{t+dt}$ given $X_t=x$ is such that when $dt$ is small the conditional mean of $X_{t+dt}^i-X_t^i$ is approximately $[x^iμ_i+\sum_j(x^j D_{ji}-x^i D_{ij})]dt$, where $X_t^i$ and $μ_i$ are the abundance and per capita growth rate in the $i$-th patch respectivly, and $D_{ij}$ is the dispersal rate from the $i$-th to the $j$-th patch, and the conditional covariance of $X_{t+dt}^i-X_t^i$ and $X_{t+dt}^j-X_t^j$ is approximately $x^i x^j σ_{ij}dt$. We show for such a spatially extended population that if $S_t=(X_t^1+...+X_t^n)$ is the total population abundance, then $Y_t=X_t/S_t$, the vector of patch proportions, converges in law to a random vector $Y_\infty$ as $t\to\infty$, and the stochastic growth rate $\lim_{t\to\infty}t^{-1}\log S_t$ equals the space-time average per-capita growth rate $\sum_iμ_i\E[Y_\infty^i]$ experienced by the population minus half of the space-time average temporal variation $\E[\sum_{i,j}σ_{ij}Y_\infty^i Y_\infty^j]$ experienced by the population. We derive analytic results for the law of $Y_\infty$, find which choice of the dispersal mechanism $D$ produces an optimal stochastic growth rate for a freely dispersing population, and investigate the effect on the stochastic growth rate of constraints on dispersal rates. Our results provide fundamental insights into "ideal free" movement in the face of uncertainty, the persistence of coupled sink populations, the evolution of dispersal rates, and the single large or several small (SLOSS) debate in conservation biology.

math.PR↗

arXiv2012

Ashkin-Teller criticality and pseudo first-order behavior in a frustrated Ising model on the square lattice

We study the challenging thermal phase transition to stripe order in the frustrated square-lattice Ising model with couplings J1<0 (nearest-neighbor, ferromagnetic) and J2>0 (second-neighbor, antiferromagnetic) for g=J2/|J1|>1/2. Using Monte Carlo simulations and known analytical results, we demonstrate Ashkin-Teller criticality for g>= g*, i.e., the critical exponents vary continuously between those of the 4-state Potts model at g=g* and the Ising model for g -> infinity. Thus, stripe transitions offer a route to realizing a related class of conformal field theories with conformal charge c=1 and varying exponents. The transition is first-order for g<g*= 0.67(1), much lower than previously believed, and exhibits pseudo first-order behavior for g* < g < 1.

cond-mat.stat-mech↗

arXiv2011

Absolute continuity of the limiting eigenvalue distribution of the random Toeplitz matrix

We show that the limiting eigenvalue distribution of random symmetric Toeplitz matrices is absolutely continuous with density bounded by 8, partially answering a question of Bryc, Dembo and Jiang (2006). The main tool used in the proof is a spectral averaging technique from the theory of random Schrödinger operators. The similar question for Hankel matrices remains open.

math.PR↗

arXiv2011

Spectra of large random trees

We analyze the eigenvalues of the adjacency matrices of a wide variety of random trees. Using general, broadly applicable arguments based on the interlacing inequalities for the eigenvalues of a principal submatrix of a Hermitian matrix and a suitable notion of local weak convergence for an ensemble of random trees, we show that the empirical spectral distributions for each of a number of random tree models converge to a deterministic (model dependent) limit as the number of vertices goes to infinity. We conclude for ensembles such as the linear preferential attachment models, random recursive trees, and the uniform random trees that the limiting spectral distribution has a set of atoms that is dense in the real line. We obtain precise asymptotics on the mass assigned to zero by the empirical spectral measures via the connection with the cardinality of a maximal matching. Moreover, we show that the total weight of a weighted matching is asymptotically equivalent to a constant multiple of the number of vertices when the edge weights are independent, identically distributed, non-negative random variables with finite expected value. We greatly extend a celebrated result obtained by Schwenk for the uniform random trees by showing that, under mild conditions, with probability converging to one, the spectrum of a realization is shared by at least one other tree. For the the linear preferential attachment model with parameter $a > -1$, we show that the suitably rescaled $k$ largest eigenvalues converge jointly.

math.PR↗

arXiv2010

Scaling Limits for Width Two Partially Ordered Sets: The Incomparability Window

We study the structure of a uniformly randomly chosen partial order of width 2 on n elements. We show that under the appropriate scaling, the number of incomparable elements converges to the height of a one dimensional Brownian excursion at a uniformly chosen random time in the interval [0,1], which follows the Rayleigh distribution.

math.PR↗

arXiv2010

Example of a first-order Néel to Valence-Bond-Solid transition in two-dimensions

We consider the $S=1/2$ Heisenberg model with nearest-neighbor interaction $J$ and an additional multi-spin interaction $Q_3$ on the square lattice. The $Q_3$ term consists of three bond-singlet projectors and is chosen to favor the formation of a valence-bond solid (VBS) where the valence bonds (singlet pairs) form a staggered pattern. The model exhibits a quantum phase transition from the Néel state to the VBS as a function of $Q_3/J$. We study the model using quantum Monte Carlo (stochastic series expansion) simulations. The Néel-VBS transition in this case is strongly first-order in nature, in contrast to similar previously studied models with continuous transitions into columnar VBS states. The qualitatively different transitions illustrate the important role of an emerging U(1) symmetry in the latter case, which is not possible in the present model due to the staggered VBS pattern (which does not allow local fluctuations necessary to rotate the local VBS order parameter).

cond-mat.str-el↗

arXiv2010

Fractional spin textures in the frustrated magnet SrCr$_{9p}$Ga$_{12-9p}$O$_{19}$

We consider the archetypal frustrated antiferromagnet SrCr$_{9p}$Ga$_{12-9p}$O$_{19}$ in its well-known spin-liquid state, and demonstrate that a Cr$^{3+}$ spin $S=3/2$ ion in direct proximity to a pair of vacancies (in disordered $p<1$ samples) is cloaked by a spatially extended spin texture that encodes the correlations of the parent spin-liquid. In this spin-liquid regime, the combined object has a magnetic response identical to a classical spin of length $S/2 = 3/4$, which dominates over the small intrinsic susceptibility of the pure system. This fractional-spin texture leaves an unmistakable imprint on the measured $^{71}$Ga nuclear magnetic resonance (NMR) lineshapes, which we compute using Monte-Carlo simulations and compare with experimental data.

cond-mat.str-el↗

arXiv2009

Large dimensional random k circulants

Consider random k-circulants A_{k,n} with n tends to infinity, k=k(n) and whose input sequence \{a_l\}_{l \ge 0} is independent with mean zero and variance one and \sup_n n^{-1}\sum_{l=1}^n \E |a_l|^{2+δ}< \infty for some δ> 0. Under suitable restrictions on the sequence \{k(n)\}_{n \ge 1}, we show that the limiting spectral distribution (LSD) of the empirical distribution of suitably scaled eigenvalues exists and identify the limits. In particular, we prove the following: Suppose g \ge 1 is fixed and p_1 is the smallest prime divisor of g. Suppose P_g=\prod_{j=1}^g E_j where \{E_j\}_{1 \le j \le g} are i.i.d. exponential random variables with mean one. (i) If k^g = -1+ s n where s=1 if g=1 and s = o(n^{p_1 -1}) if g>1, then the empirical spectral distribution of n^{-1/2}A_{k,n} converges weakly in probability to U_1P_g^{1/2g} where U_1 is uniformly distributed over the (2g)th roots of unity, independent of P_g. (ii) If g \ge 2 and k^g = 1+ s n with s = o(n^{p_1-1}) then the empirical spectral distribution of n^{-1/2}A_{k,n} converges weakly in probability to U_2P_g^{1/2g} where U_2 is uniformly distributed over the unit circle in \mathbb R^2, independent of P_g. On the other hand, if k \ge 2, k= n^{o(1)} with \gcd(n,k) = 1, and the input is i.i.d. standard normal variables, then F_{n^{-1/2}A_{k,n}} converges weakly in probability to the uniform distribution over the circle with center at (0,0) and radius r = \exp(\E [ \log \sqrt E_1]). We also show that when n=k^2+1\to \infty, and the input is i.i.d. with finite (2+δ) moment, then the spectral radius, with appropriate scaling and centering, converges to the Gumbel distribution.

math.PR↗

arXiv2009

Branching Process approach for 2-SAT thresholds

It is well known that, as $n$ tends to infinity, the probability of satisfiability for a random 2-SAT formula on $n$ variables, where each clause occurs independently with probability $α/2n$, exhibits a sharp threshold at $α=1$. We study a more general 2-SAT model in which each clause occurs independently but with probability $α_i/2n$ where $i \in \{0,1,2\}$ is the number of positive literals in that clause. We generalize branching process arguments by Verhoeven(99) to determine the satisfiability threshold for this model in terms of the maximum eigenvalue of the branching matrix.

math.PR↗

arXiv2008

Semiclassical spin liquid state of easy axis Kagome antiferromagnets

Motivated by recent experiments on Nd-langasite, we consider the effect of strong easy axis single-ion anisotropy $D$ on $S > 3/2$ spins interacting with antiferromagnetic exchange $J$ on the Kagome lattice. When $T \ll DS^2$, the collinear low energy states selected by the anisotropy map on to configurations of the classical Kagome lattice Ising antiferromagnet. However, the low temperature limit is quite different from the cooperative Ising paramagnet that obtains classically for $T \ll JS^2$. We find that sub-leading ${\mathcal O}(J^3S/D^2)$ multi-spin interactions arising from the transverse quantum dynamics result in a crossover from an intermediate temperature classical cooperative Ising paramagnet to a semiclassical spin liquid with distinct short-ranged correlations for $T \ll J^3S/D^2$.

cond-mat.str-el↗

arXiv2008

Collinear ordering of easy-axis triangular lattice antiferromagnets

Antiferromagnetically coupled moments on the frustrated triangular lattice typically order in a coplanar state at low temperature. Here, we demonstrate that the presence of not-very-large easy axis single ion anisotropy leads to an interesting orientationally ordered collinear state in triangular lattice antiferromagnets with moments $S \ge 3/2$. This ordered state breaks the symmetry of $π/3$ rotations about a lattice site, while leaving intact the translational symmetry of the lattice.

cond-mat.str-el↗