arXiv subjects

D. Dhar

Publications and source records attributed to D. Dhar.

7 recordsLinked to original sources

arXiv2021

Phases of the hard-plate lattice gas on a three-dimensional cubic lattice

We study the phase diagram of a system of $2\times 2\times 1$ hard plates on the three dimensional cubic lattice, {\em i.e.} a lattice gas of plates that each cover an elementary plaquette of the cubic lattice and occupy its four vertices, with the constraint that no two plates occupy the same site of the cubic lattice. We focus on the isotropic system, with equal fugacities for the three orientations of plates. We show, using grand canonical Monte Carlo simulations, that the system undergoes two density-driven phase transitions with increasing density of plates: the first from a disordered fluid to a layered phase, and the second from the layered phase to a sublattice-ordered phase. In the layered phase, the system breaks up into disjoint slabs of thickness two along one spontaneously chosen cartesian direction. Plates with normals perpendicular to this layering direction are preferentially contained entirely within these slabs, while plates straddling two successive slabs have a lower density. Additionally the symmetry between the three types of plates is spontaneously broken, as plates with normal along the layering direction have a lower density than the other two types of plates. Intriguingly, the occupied slabs exhibit two-dimensional power-law columnar order even in the presence of a nonzero density of vacancies. In contrast, inter-slab correlations of the two-dimensional columnar order parameter decay exponentially with the separation between the slabs. In the sublattice-ordered phase, there is two-fold ($Z_2$) breaking of lattice translation symmetry along all three cartesian directions. We present numerical evidence that the disordered to layered transition is continuous and consistent with the three-dimensional $O(3)$ universality class, while the layered to sublattice transition is first-order in nature.

cond-mat.stat-mech↗

arXiv2021

Spontaneous layering and power-law order in the three-dimensional fully-packed hard-plate lattice gas

We obtain the phase diagram of fully-packed hard plates on a cubic lattice. Each plate covers an elementary plaquette of the cubic lattice and occupies its four vertices, with each vertex of the cubic lattice occupied by exactly one such plate. We consider the general case with fugacities $s_\mu$ for `$\mu$ plates', whose normal is the $\mu$ direction ($\mu = x,y,z$). At and close to the isotropic point, we find, consistent with previous work, a phase with long-range sublattice order. When two of the fugacities $s_{\rm \mu_1}$ and $s_{\mu_2}$ are comparable, and the third fugacity $s_{\mu_{3}}$ is much smaller, we find a spontaneously-layered phase. In this phase, the system breaks up into disjoint slabs of width two stacked along the $\mu_3$ axis. $\mu_1$ and $\mu_2$ plates are preferentially contained entirely within these slabs, while plates straddling two successive slabs have a lower density. In the opposite limit, with $\mu_3 \gg \mu_1 \sim \mu_2$, we find a phase with long-range columnar order, corresponding to simultaneous $Z_2$ symmetry breaking of lattice translation symmetry in directions $\mu_1$ and $\mu_2$. The spontaneously-layered phases display critical behaviour, with power-law decay of correlations in the $\mu_1$ and $\mu_2$ directions when the slabs are stacked in the $\mu_3$ direction, and represent examples of `floating phases' discussed earlier in the context of coupled Luttinger liquids and quasi-two-dimensional classical systems. We ascribe this remarkable behaviour to the constrained motion of defects in this phase, and develop a coarse-grained effective field theoretical understanding of the stability of power-law order in this unusual three-dimensional floating phase.

cond-mat.stat-mech↗

arXiv2000

Continuously varying exponents in a sandpile model with dissipation near surface

We consider the directed Abelian sandpile model in the presence of sink sites whose density f_t at depth t below the top surface varies as c~1/t^chi. For chi>1 the disorder is irrelevant. For chi<1, it is relevant and the model is no longer critical for any nonzero c. For chi=1 the exponents of the avalanche distributions depend continuously on the amplitude c of the disorder. We calculate this dependence exactly, and verify the results with simulations.

cond-mat.stat-mech↗

arXiv1996

Fractal Dimension of Backbone of Eden Trees

We relate the fractal dimension of the backbone, and the spectral dimension of Eden trees to the dynamical exponent z. In two dimensions, it gives fractal dimension of backbone equal to 4/3 and spectral dimension of trees equal to 5/4. In three dimensions, it provides us a new way to estimate z numerically. We get z=1.617 +/- 0.004.

cond-mat↗

arXiv1994

Driven Depinning in Anisotropic Media

We show that the critical behavior of a driven interface, depinned from quenched random impurities, depends on the isotropy of the medium. In anisotropic media the interface is pinned by a bounding (conducting) surface characteristic of a model of mixed diodes and resistors. Different universality classes describe depinning along a hard and a generic direction. The exponents in the latter (tilted) case are highly anisotropic, and obtained exactly by a mapping to growing surfaces. Various scaling relations are proposed in the former case which explain a number of recent numerical observations.

cond-mat↗

arXiv1994

Algebraic Aspects of Abelian Sandpile Models

The abelian sandpile models feature a finite abelian group G generated by the operators corresponding to particle addition at various sites. We study the canonical decomposition of G as a product of cyclic groups G = Z_{d_1} X Z_{d_2} X Z_{d_3}...X Z_{d_g}, where g is the least number of generators of G, and d_i is a multiple of d_{i+1}. The structure of G is determined in terms of toppling matrix. We construct scalar functions, linear in height variables of the pile, that are invariant toppling at any site. These invariants provide convenient coordinates to label the recurrent configurations of the sandpile. For an L X L square lattice, we show that g = L. In this case, we observe that the system has nontrivial symmetries coming from the action of the cyclotomic Galois group of the (2L+2)th roots of unity which operates on the set of eigenvalues of the toppling matrix. These eigenvalues are algebraic integers, whose product is the order |G|. With the help of this Galois group, we obtain an explicit factorizaration of |G|. We also use it to define other simpler, though under-complete, sets of toppling invariants.

cond-mat↗

arXiv1994

Algebraic Aspects of Abelian Sandpile Models

The abelian sandpile models feature a finite abelian group $G$ generated by the operators corresponding to particle addition at various sites. We study the canonical decomposition of $G$ as a product of cyclic groups $G = Z_{d_1} \times Z_{d_2} \times Z_{d_3} >... \times Z_{d_g}$ where $g$ is the least number of generators of $G$, and $d_i$ is a multiple of $d_{i+1}$. The structure of $G$ is determined in terms of the toppling matrix $Δ$. We construct scalar functions, linear in height variables of the pile, that are invariant under toppling at any site. These invariants provide convenient coordinates to label the recurrent configurations of the sandpile. For an $L \times L$ square lattice, we show that $g = L$. In this case, we observe that the system has nontrivial symmetries, transcending the obvious symmetries of the square, viz. those coming from the action of the cyclotomic Galois group Gal$_L$ of the $2(L+1)$--th roots of unity (which operates on the set of eigenvalues of $Δ$). We use Gal$_L$ to define other simpler, though under-complete, sets of toppling invariants.

cond-mat↗