arXiv subjects

Somendra M. Bhattacharjee

Publications and source records attributed to Somendra M. Bhattacharjee.

At least 19 recordsLinked to original sources

arXiv2026

A Qubit as a Bridge Between Statistical Mechanics and Quantum Dynamics

This work presents a unified perspective on thermal equilibrium and quantum dynamics by examining the simplest quantum system, a qubit, as a minimal model. We show that both the thermal partition function and the Loschmidt amplitude can be understood as extensions of a single analytic function along different paths in the complex plane. The zeros of Loschmidt amplitude encode dynamical features such as orthogonality, rate function singularities, and quantum speed limits, in analogy with the role of partition function zeros in equilibrium statistical mechanics. We further establish, through the Cauchy-Riemann equations, that the high-temperature specific heat corresponds to early-time evolution. The discussion follows a pedagogical progression from a single qubit to an interacting spin chain, all with finite dimensional Hilbert spaces.

quant-ph↗

arXiv2026

Julia Set in Quantum Evolution: The case of Dynamical Quantum Phase Transitions

Dynamical quantum phase transitions (DQPTs) are a class of non-equilibrium phase transitions that occur in many-body quantum systems during real-time evolution, rather than through parameter tuning as in conventional phase transitions. This paper presents an exact analytical approach to studying DQPTs by combining complex dynamics with the real-space renormalization group (RG). RG transformations are interpreted as iterated maps on the complex plane, establishing a connection between DQPTs and the Julia set, the boundary separating the basins of attraction of the stable fixed points. This framework is applied to a quantum quench in the one-dimensional transverse field Ising model, where we examine the sensitivity of DQPTs to variations in boundary conditions. We show that altering the topology of the spin chain can suppress DQPTs and provide a qualitative explanation based on quantum speed limits.

cond-mat.stat-mech↗

arXiv2025

DNA Unzipping Transition

This review focuses on the force-induced unzipping transition of double-stranded DNA. It begins with a brief history of DNA melting, which emerged alongside the growth of the field of molecular biology, juxtaposed with the advancements in physics during the same post-World War II period. The earlier theories of melting of DNA were based on the Ising model and its modifications, but gradually moved towards polymer-based models. The idea of force-induced unzipping was first introduced in 1999 as a cooperative mechanism for breaking base pairs without the need for temperature changes. The paper discusses several subsequent developments addressing different aspects of the unzipping transition.

cond-mat.soft↗

arXiv2024

Large bubble drives melting in circular DNA

We investigate the melting transition of non-supercoiled circular DNA of different lengths, employing Brownian dynamics simulation. In the absence of supercoiling, we find that melting of circular DNA is driven by a large bubble, which agrees with the previous predictions of circular DNA melting in the presence of supercoiling. By analyzing sector-wise changes in average base-pair distance, our study reveals that the melting behavior of circular DNA closely resembles that of linear DNA. Additionally, we find a marked difference in the thermal stability of circular DNA over linear DNA at very short length scales, an effect that diminishes as the length of circular DNA increases. The stability of smaller circular DNA is linked to the occurrence of transient small bubbles, characterized by a lower probability of growth.

cond-mat.soft↗

arXiv2024

Complex dynamics approach to dynamical quantum phase transitions: the Potts model

This paper introduces complex dynamics methods to study dynamical quantum phase transitions in the one- and two-dimensional quantum 3-state Potts model. The quench involves switching off an infinite transverse field. The time-dependent Loschmidt echo is evaluated by an exact renormalization group (RG) transformation in the complex plane where the thermal Boltzmann factor is along the positive real axis, and the quantum time evolution is along the unit circle. One of the characteristics of the complex dynamics constituted by repeated applications of RG is the Julia set, which determines the phase transitions. We show that special boundary conditions can alter the nature of the transitions, and verify the claim for the one-dimensional system by transfer matrix calculations. In two dimensions, there are alternating symmetry-breaking and restoring transitions, both of which are first-order, despite the criticality of the Curie point. In addition, there are finer structures because of the fractal nature of the Julia set. Our approach can be extended to multi-variable problems, higher dimensions, and approximate RG transformations expressed as rational functions.

cond-mat.stat-mech↗

arXiv2023

Ordering and topological defects in social wasps' nests

Social insects have evolved a variety of architectural formations. Bees and wasps are well known for their ability to achieve compact structures by building hexagonal cells. Polistes wattii, an open nesting paper wasp species, builds planar hexagonal structures. Here, using the pair correlation function approach, we show that their nests exhibit short-range hexagonal order no long-range order akin to amorphous materials. Hexagonal orientational order was well preserved globally. We also show the presence of topological defects such as dislocations (pentagon-heptagon disclination pairs) and Stone-Wales quadrupoles, and discuss how these defects were organized in the nest, thereby restoring order. Furthermore, we suggest the possible role of such defects in shaping nesting architectures of other social insect species.

physics.bio-ph↗

arXiv2023

A sheet-like structure in the proximity of compact DNA

We determine the phase diagram of DNA with inter- and intra-strand native-pair interactions that mimic the compaction of DNA. We show that DNA takes an overall sheet-like structure in the region where an incipient transition to a compact phase would have occurred. The stability of this phase is due to the extra entropy from the folding of the sheet, which is absent in the remaining polymer-like states of the phase diagram.

cond-mat.soft↗

arXiv2021

Efimov-DNA Phase diagram: three stranded DNA on a cubic lattice

We define a generalised model for three-stranded DNA consisting of two chains of one type and a third chain of a different type. The DNA strands are modelled by random walks on the three-dimensional cubic lattice with different interactions between two chains of the same type and two chains of different types. This model may be thought of as a classical analogue of the quantum three-body problem. In the quantum situation it is known that three identical quantum particles will form a triplet with an infinite tower of bound states at the point where any pair of particles would have zero binding energy. The phase diagram is mapped out, and the different phase transitions examined using finite-size scaling. We look particularly at the scaling of the DNA model at the equivalent Efimov point for chains up to 10000 steps in length. We find clear evidence of several bound states in the finite-size scaling. We compare these states with the expected Efimov behaviour.

cond-mat.stat-mech↗

arXiv2020

Softening of DNA near Melting as Disappearance of an Emergent Property

Near the melting transition the bending elastic constant, $κ$, an emergent property of double-stranded DNA (dsDNA), is shown not to follow the rod-like scaling for small length $N$. The reduction in $κ$ with temperature is determined by the denatured bubbles for a continuous transition, e.g., when the two strands are gaussian, but by the broken bonds near the open end in a Y-like configuration for a first-order transition as for strands with excluded volume interactions. In the latter case, a lever rule is operational implying a phase coexistence although dsDNA is known to be a single phase.

cond-mat.soft↗

arXiv2019

Boundaries and unphysical fixed points in Dynamical Quantum Phase Transitions

We show that dynamic quantum phase transitions (DQPT) in many situations involve renormalization group (RG) fixed points that are unphysical in the context of thermal phase transitions. In such cases, boundary conditions are shown to become relevant to the extent of even completely suppressing the bulk transitions. We establish these by performing exact RG analysis of the quantum Ising model on scale-invariant lattices of different dimensions, and by analyzing the zeros of the Loschmidt amplitude. Further corroboration of boundaries affecting the bulk transition comes from the three-state quantum Potts chain, for which we also show that the DQPT corresponds to a pair of period-2 fixed points.

cond-mat.stat-mech↗

arXiv2019

Zeros of partition function for Continuous Phase Transitions using cumulants

This paper explores the use of a cumulant method to determine the zeros of partition functions for continuous phase transitions. Unlike a first-order transition, with a uniform density of zeros near the transition point, a continuous transition is expected to show a power law dependence of the density with a nontrivial slope for the line of zeros. Different types of models and methods of generating cumulants are used as a testing ground for the method. These include exactly solvable DNA melting problem on hierarchical lattices, heterogeneous DNA melting with randomness in sequence, Monte Carlo simulations for the well-known square lattice Ising model. The method is applicable for closest zeros near the imaginary axis, as these are needed for dynamical quantum phase transitions. In all cases, the method is found to provide the basic information about the transition, and most importantly, avoids root finding methods.

cond-mat.stat-mech↗

arXiv2018

Global Density Profile for Asymmetric Simple Exclusion Process from Renormalization Group Flows

The totally asymmetric simple exclusion process along with particle adsorption and evaporation kinetics is a model of boundary-induced nonequilibrium phase transition. In the continuum limit, the average particle density across the system is described by a singular differential equation involving multiple scales which lead to the formation of boundary layers (BL) or shocks. A renormalization group analysis is developed here by using the location and the width of the BL as the renormalization parameters. It not only allows us to cure the large distance divergences in the perturbative solution for the BL but also generates, from the BL solution, an analytical form for the global density profile. The predicted scaling form is checked against numerical solutions for finite systems.

cond-mat.stat-mech↗

arXiv2016

What is dimension?

This chapter explores the notion of "dimension" of a set. Various power laws by which an Euclidean space can be characterized are used to define dimensions, which then explore different aspects of the set. Also discussed are the generalization to multifractals, and discrete and continuous scale invariance with the emergence of complex dimensions. The idea of renormalization group flow equations can be introduced in this framework, to show how the power laws determined by dimensional analysis (engineering dimensions) get modified by extra anomalous dimensions. As an example of the RG flow equation, the scaling of conductance by disorder in the context of localization is used. A few technicalities, including the connection between entropy and fractal dimension, can be found in the appendices.

cond-mat.stat-mech↗

arXiv2016

Rigidity of Melting DNA

The temperature dependence of DNA flexibility is studied in the presence of stretching and unzipping forces. Two classes of models are considered. In one case the origin of elasticity is entropic due to the polymeric correlations, and in the other the double-stranded DNA is taken to have an intrinsic rigidity for bending. In both cases single strands are completely flexible. The change in the elastic constant for the flexible case due to thermally generated bubbles is obtained exactly. For the case of intrinsic rigidity, the elastic constant is found to be proportional to the square root of the bubble number fluctuation.

cond-mat.soft↗

arXiv2015

Efimov like phase of a three stranded DNA (Efimov-DNA) and the renormalization group limit cycle

A three-stranded DNA with short range base pairings only is known to exhibit a classical analog of the quantum Efimov effect, viz., a three chain bound state at the two chain melting point where no two are bound. By using a non-perturbative renormalization group method for a rigid duplex DNA and a flexible third strand, with base pairings and strand exchange, we show that the Efimov-DNA is associated with a limit cycle type behavior of the flow of an effective three chain interaction. The analysis also shows that thermally generated bubbles play an essential role in producing the effect. A toy model for the flow equations shows the limit cycle in an extended three dimensional parameter space of the two-chain coupling and a complex three chain interaction.

cond-mat.soft↗

arXiv2014

Thermodynamic relations for DNA phase transitions

The force induced unzipping transition of a double stranded DNA is considered from a purely thermodynamic point of view. This analysis provides us with a set of relations that can be used to test microscopic theories and experiments. The thermodynamic approach is based on the hypothesis of impenetrability of the force in the zipped state. The melting and the unzipping transitions are considered in the same framework and compared with the existing statistical model results. The analysis is then extended to a possible continuous unzipping transition.

cond-mat.soft↗

arXiv2014

Melting behavior and different bound states in three-stranded DNA models

Thermal denaturation of DNA is often studied with coarse-grained models in which native sequential base pairing is mimicked by the existence of attractive interactions only between monomers at the same position along strands (Poland and Scheraga models). Within this framework, the existence of a three strand DNA bound state in conditions where a duplex DNA would be in the denaturated state was recently predicted from a study of three directed polymer models on simplified hierarchical lattices ($d>2$) and in $1+1$ dimensions. Such phenomenon which is similar to the Efimov effect in nuclear physics was named Efimov-DNA. In this paper we study the melting of the three-stranded DNA on a Sierpinski gasket of dimensions $d<2$ by assigning extra weight factors to fork openings and closings, to induce a two-strand DNA melting. In such a context we can find again the existence of the Efimov-DNA-like state but quite surprisingly we discover also the presence of a different phase, to be called a mixed state, where the strands are pair-wise bound but without three chain contacts. Whereas the Efimov DNA turns out to be a crossover near melting, the mixed phase is a thermodynamic phase.

cond-mat.stat-mech↗

arXiv2013

Flory theory for Polymers

We review various simple analytical theories for homopolymers within a unified framework. The common guideline of our approach is the Flory theory, and its various avatars, with the attempt of being reasonably self-contained. We expect this review to be useful as an introduction to the topic at the graduate students level.

cond-mat.stat-mech↗