arXiv subjects

Uma Divakaran

Publications and source records attributed to Uma Divakaran.

At least 37 records · Page 2Linked to original sources

arXiv2014

Non-equilibrium quantum relaxation across a localization-delocalization transition

We consider the one-dimensional $XX$-model in a quasi-periodic transverse-field described by the Harper potential, which is equivalent to a tight-binding model of spinless fermions with a quasi-periodic chemical potential. For weak transverse field (chemical potential), $h h_c$. We study the non-equilibrium relaxation of the system by applying two protocols: a sudden change of $h$ (quench dynamics) and a slow change of $h$ in time (adiabatic dynamics). For a quench into the delocalized (localized) phase, the entanglement entropy grows linearly (saturates) and the order parameter decreases exponentially (has a finite limiting value). For a critical quench the entropy increases algebraically with time, whereas the order parameter decreases with a stretched-exponential. The density of defects after an adiabatic field change through the critical point is shown to scale with a power of the rate of field change and a scaling relation for the exponent is derived.

cond-mat.dis-nn↗

arXiv2014

Fidelity susceptibility and Loschmidt echo for generic paths in a three spin interacting transverse Ising model

We study the effect of presence of different types of critical points such as ordinary critical point, multicritical point and quasicritical point along different paths on the Fidelity susceptibility and Loschmidt echo of a three spin interacting transverse Ising chain using a method which does not involve the language of tensors. We find that the scaling of fidelity susceptibility and Loschmidt echo with the system size at these special critical points of the model studied, is in agreement with the known results, thus supporting our method.

cond-mat.stat-mech↗

arXiv2013

The three site interacting spin chain in staggered field: Fidelity vs Loschmidt echo

We study the the ground state fidelity and the ground state Loschmidt echo of a three site interacting XX chain in presence of a staggered field which exhibits special types of quantum phase transitions due to change in the topology of the Fermi surface, apart from quantum phase transitions from gapped to gapless phases. We find that on one hand, the fidelity is able to detect only the boundaries separating the gapped from the gapless phase; it is completely insensitive to the phase transition from two Fermi points region to four Fermi points region lying within this gapless phase. On the other hand, Loschmidt echo shows a dip only at a special point in the entire phase diagram and hence fails to detect any quantum phase transition associated with the present model. We provide appropriate arguments in support of this anomalous behavior.

cond-mat.stat-mech↗

arXiv2012

The scaling of the decoherence factor of a qubit coupled to a spin chain driven across quantum critical points

We study the scaling of the decoherence factor of a qubit (spin-1/2) using the central spin model in which the central spin (qubit) is globally coupled to a transverse XY spin chain. The aim here is to study the non-equilibrium generation of decoherence when the spin chain is driven across (along) quantum critical points (lines) and derive the scaling of the decoherence factor in terms of the driving rate and some of the exponents associated with the quantum critical points. Our studies show that the scaling of logarithm of decoherence factor is identical to that of the defect density in the final state of the spin chain following a quench across isolated quantum critical points for both linear and non-linear variations of a parameter even if the defect density may not satisfy the standard Kibble-Zurek scaling. However, one finds an interesting deviation when the spin chain is driven along a critical line. Our analytical predictions are in complete agreement with numerical results. Our study, though limited to integrable two-level systems, points to the existence of a universality in the scaling of the decoherence factor which is not necessarily identical to the scaling of the defect density.

cond-mat.stat-mech↗

arXiv2011

Non-equilibrium quantum dynamics after local quenches

We study the quantum dynamics resulting from preparing a one-dimensional quantum system in the ground state of initially two decoupled parts which are then joined together (local quench). Specifically we focus on the transverse Ising chain and compute the time-dependence of the magnetization profile, m_l(t), and correlation functions at the critical point, in the ferromagnetically ordered phase and in the paramagnetic phase. At the critical point we find finite size scaling forms for the nonequilibrium magnetization and compare predictions of conformal field theory with our numerical results. In the ferromagnetic phase the magnetization profiles are well matched by our predictions from a quasi-classical calculation.

cond-mat.stat-mech↗

arXiv2010

Landau-Zener problem with waiting at the minimum gap and related quench dynamics of a many-body system

We discuss a technique for solving the Landau-Zener (LZ) problem of finding the probability of excitation in a two-level system. The idea of time reversal for the Schrodinger equation is employed to obtain the state reached at the final time and hence the excitation probability. Using this method, which can reproduce the well-known expression for the LZ transition probability, we solve a variant of the LZ problem which involves waiting at the minimum gap for a time t_w; we find an exact expression for the excitation probability as a function of t_w. We provide numerical results to support our analytical expressions. We then discuss the problem of waiting at the quantum critical point of a many-body system and calculate the residual energy generated by the time-dependent Hamiltonian. Finally we discuss possible experimental realizations of this work.

cond-mat.stat-mech↗

arXiv2010

Adiabatic dynamics in passage across quantum critical lines and gapless phases

It is well known that the dynamics of a quantum system is always non-adiabatic in passage through a quantum critical point and the defect density in the final state following a quench shows a power-law scaling with the rate of quenching. However, we propose here a possible situation where the dynamics of a quantum system in passage across quantum critical regions is adiabatic and the defect density decays exponentially. This is achieved by incorporating additional interactions which lead to quantum critical behavior and gapless phases but do not participate in the time evolution of the system. To illustrate the general argument, we study the defect generation in the quantum critical dynamics of a spin-1/2 anisotropic quantum XY spin chain with three spin interactions and a linearly driven staggered magnetic field.

cond-mat.stat-mech↗

arXiv2009

Quenching through Dirac and semi-Dirac points in optical Lattices: Kibble-Zurek scaling for anisotropic Quantum-Critical systems

We propose that Kibble-Zurek scaling can be studied in optical lattices by creating geometries that support, Dirac, Semi-Dirac and Quadratic Band Crossings. On a Honeycomb lattice with fermions, as a staggered on-site potential is varied through zero, the system crosses the gapless Dirac points, and we show that the density of defects created scales as $1/τ$, where $τ$ is the inverse rate of change of the potential, in agreement with the Kibble-Zurek relation. We generalize the result for a passage through a semi-Dirac point in $d$ dimensions, in which spectrum is linear in $m$ parallel directions and quadratic in rest of the perpendicular $(d-m)$ directions. We find that the defect density is given by $ 1 /{τ^{mν_{||}z_{||}+(d-m)ν_{\perp}z_{\perp}}}$ where $ν_{||}, z_{||}$ and $ν_{\perp},z_{\perp}$ are the dynamical exponents and the correlation length exponents along the parallel and perpendicular directions, respectively. The scaling relations are also generalized to the case of non-linear quenching.

cond-mat.quant-gas↗

arXiv2009

Defect production due to quenching through a multicritical point and along a gapless line

In this review, we study the quenching dynamics of a one-dimensional XY Hamiltonian in a transverse field under linear variation of different parameters of the Hamiltonian so that the system is driven through various critical points and gapless lines. It is observed that the density of defects falls off as a power law with the quenching rate $1/τ$ where the power law is determined by the dimensionality of the underlying lattice and the critical exponents associated with the quantum critical point as predicted by Kibble-Zurek scaling. However, we show that when the system is driven through a multicritical point or along a gapless line, Kibble Zurek scaling needs to be modified. We discuss generalized scaling forms of the defect density for the above two situations.

cond-mat.stat-mech↗

arXiv2009

Reverse quenching in a one-dimensional Kitaev model

We present an exact result for the non-adiabatic transition probability and hence the defect density in the final state of a one-dimensional Kitaev model following a slow quench of the parameter $J_-$, which estimates the anisotropy between the interactions, as $J_-(t)\sim -|t/τ|$. Here, time $t$ goes from $-\infty$ to $+\infty$ and $τ$ defines the rate of change of the Hamiltonian. In other words, the spin chain initially prepared in its ground state is driven by changing $J_-$ linearly in time up to the quantum critical point, which in the model considered here occurs at at $t=0$, reversed and then gradually decreased to its initial value at the same rate. We have thoroughly compared the reverse quenching with its counterpart forward quenching $i.e.,$ $J_-\sim t/τ$. Our exact calculation shows that the probability of excitations is zero for the wave vector at which the instantaneous energy gap is zero at the critical point $J_{-}=0$ as opposed to the maximum value of unity in the forward quenching. It is also shown that the defect density in the final state following a reverse quenching, we propose here, is nearly half of the defects generated in the forward quenching. We argue that the defects produced when the system reaches the quantum critical point gets redistributed in the wave vector space at the final time in case of reverse quenching whereas it keeps on increasing till the final time in the forward quenching. We study the entropy density and also the time evolution of the diagonal entropy density in the case of the reverse quenching and compare it with the forward case.

cond-mat.stat-mech↗

arXiv2009

Defect production due to quenching through a multicritical point

We study the generation of defects when a quantum spin system is quenched through a multicritical point by changing a parameter of the Hamiltonian as $t/τ$, where $τ$ is the characteristic time scale of quenching. We argue that when a quantum system is quenched across a multicritical point, the density of defects ($n$) in the final state is not necessarily given by the Kibble-Zurek scaling form $n \sim 1/τ^{d ν/(z ν+1)}$, where $d$ is the spatial dimension, and $ν$ and $z$ are respectively the correlation length and dynamical exponent associated with the quantum critical point. We propose a generalized scaling form of the defect density given by $n \sim 1/τ^{d/(2z_2)}$, where the exponent $z_2$ determines the behavior of the off-diagonal term of the $2 \times 2$ Landau-Zener matrix at the multicritical point. This scaling is valid not only at a multicritical point but also at an ordinary critical point.

cond-mat.stat-mech↗

arXiv2008

Quenching along a gapless line: A different exponent for defect density

We use a new quenching scheme to study the dynamics of a one-dimensional anisotropic $XY$ spin-1/2 chain in the presence of a transverse field which alternates between the values $h+\de$ and $h-\de$ from site to site. In this quenching scheme, the parameter denoting the anisotropy of interaction ($\ga$) is linearly quenched from $-\infty$ to $ +\infty$ as $\ga = t/τ$, keeping the total strength of interaction $J$ fixed. The system traverses through a gapless phase when $\ga$ is quenched along the critical surface $h^2 = \de^2 + J^2$ in the parameter space spanned by $h$, $\de$ and $\ga$. By mapping to an equivalent two-level Landau-Zener problem, we show that the defect density in the final state scales as $1/τ^{1/3}$, a behavior that has not been observed in previous studies of quenching through a gapless phase. We also generalize the model incorporating additional alternations in the anisotropy or in the strength of the interaction, and derive an identical result under a similar quenching. Based on the above results, we propose a general scaling of the defect density with the quenching rate $τ$ for quenching along a gapless critical line.

cond-mat.stat-mech↗

arXiv2008

A random fiber bundle with many discontinuities in the threshold distribution

We study the breakdown of a random fiber bundle model (RFBM) with $n$-discontinuities in the threshold distribution using the global load sharing scheme. In other words, $n+1$ different classes of fibers identified on the basis of their threshold strengths are mixed such that the strengths of the fibers in the $i-th$ class are uniformly distributed between the values $σ_{2i-2}$ and $σ_{2i-1}$ where $1 \leq i \leq n+1$. Moreover, there is a gap in the threshold distribution between $i-th$ and $i+1-th$ class. We show that although the critical stress depends on the parameter values of the system, the critical exponents are identical to that obtained in the recursive dynamics of a RFBM with a uniform distribution and global load sharing. The avalanche size distribution (ASD), on the other hand, shows a non-universal, non-power law behavior for smaller values of avalanche sizes which becomes prominent only when a critical distribution is approached. We establish that the behavior of the avalanche size distribution for an arbitrary $n$ is qualitatively similar to a RFBM with a single discontinuity in the threshold distribution ($n=1$), especially when the density and the range of threshold values of fibers belonging to strongest ($n+1$)-th class is kept identical in all the cases.

cond-mat.stat-mech↗

arXiv2008

The effect of the three-spin interaction and the next-nearest neighbor interaction on the quenching dynamics of a transverse Ising model

We study the zero temperature quenching dynamics of various extensions of the transverse Ising model (TIM) when the transverse field is linearly quenched from $-\infty$ to $+\infty$ (or zero) at a finite and uniform rate. The rate of quenching is dictated by a characteristic scale given by $τ$. The density of kinks produced in these extended models while crossing the quantum critical points during the quenching process is calculated using a many body generalization of the Landau-Zener transition theory. The density of kinks in the final state is found to decay as $τ^{-1/2}$. In the first model considered here, the transverse Ising Hamiltonian includes an additional ferromagnetic three spin interaction term of strength $J_3$. For $J_3<0.5$, the kink density is found to increase monotonically with $J_3$ whereas it decreases with $J_3$ for $J_3>0.5$. The point $J_3=0.5$ and the transverse field $h=-0.5$is multicritical where the density shows a slower decay given by $τ^{-1/6}$. We also study the effect of ferromagnetic or antiferromagnetic next nearest neighbor (NNN) interactions on the dynamics of TIM under the same quenching scheme. In a mean field approximation, the transverse Ising Hamiltonians with NNN interactions are identical to the three spin Hamiltonian. The NNN interactions non-trivially modifies the dynamical behavior, for example an antiferromagnetic NNN interactions results to a larger number of kinks in the final state in comparison to the case when the NNN interaction is ferromagnetic.

cond-mat.stat-mech↗

arXiv2007

Quenching Dynamics of a quantum XY spin-1/2 chain in presence of a transverse field

We study the quantum dynamics of a one-dimensional spin-1/2 anisotropic XY model in a transverse field when the transverse field or the anisotropic interaction is quenched at a slow but uniform rate. The two quenching schemes are called transverse and anisotropic quenching respectively. Our emphasis in this paper is on the anisotropic quenching scheme and we compare the results with those of the other scheme. In the process of anisotropic quenching, the system crosses all the quantum critical lines of the phase diagram where the relaxation time diverges. The evolution is non-adiabatic in the time interval when the parameters are close to their critical values, and is adiabatic otherwise. The density of defects produced due to non-adiabatic transitions is calculated by mapping the many-particle system to an equivalent Landau-Zener problem and is generally found to vary as $1/\sqrtτ$, where $τ$ is the characteristic time scale of quenching, a scenario that supports the Kibble-Zurek mechanism. Interestingly, in the case of anisotropic quenching, there exists an additional non-adiabatic transition, in comparison to the transverse quenching case, with the corresponding probability peaking at an incommensurate value of the wave vector. In the special case in which the system passes through a multi-critical point, the defect density is found to vary as $1/τ^{1/6}$. The von Neumann entropy of the final state is shown to maximize at a quenching rate around which the ordering of the final state changes from antiferromagnetic to ferromagnetic.

cond-mat.stat-mech↗

arXiv2007

Effect of discontinuity in threshold distribution on the critical behaviour of a random fiber bundle

The critical behaviour of a Random Fiber Bundle Model with mixed uniform distribution of threshold strengths and global load sharing rule is studied with a special emphasis on the nature of distribution of avalanches for different parameters of the distribution. The discontinuity in the threshold strength distribution of fibers non-trivially modifies the critical stress as well as puts a restriction on the allowed values of parameters for which the recursive dynamics approach holds good. The discontinuity leads to a non-universal behaviour in the avalanche size distribution for smaller values of avalanche size. We observe that apart from the mean field behaviour for larger avalanches, a new behaviour for smaller avalanche size is observed as a critical threshold distribution is approached. The phenomenological understanding of the above result is provided using the exact analytical result for the avalanche size distribution. Most interestingly,the prominence of non-universal behaviour in avalanche size distribution depends on the system parameters.

cond-mat.stat-mech↗

arXiv2006

Critical behaviour of random fibers with mixed Weibull distribution

A random fiber bundle model with a mixed Weibull distribution is studied under the Global Load Sharing (GLS) scheme. The mixed model consists of two sets of fibers. The threshold strength of one set of fibers are randomly chosen from a Weibull distribution with a particular Weibull index, and another set of fibers with a different index. The mixing tunes the critical stress of the bundle and the variation of critical stress with the amount of mixing is determined using a probabilistic method where the external load is increased quasistatically. In a special case which we illustrate, the critical stress is found to vary linearly with the mixing parameter. The critical exponents and power law behaviour of burst avalanche size distribution is found to remain unaltered due to mixing.

cond-mat.stat-mech↗

arXiv2006

Fibers on a graph with local load sharing

We study a random fiber bundle model with tips of the fibers placed on a graph having co-ordination number 3. These fibers follow local load sharing with uniformly distributed threshold strengths of the fibers. We have studied the critical behaviour of the model numerically using a finite size scaling method and the mean field critical behaviour is established. The avalanche size distribution is also found to exhibit a mean field nature in the asymptotic limit.

cond-mat.stat-mech↗