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Ranjan Modak

Publications and source records attributed to Ranjan Modak.

At least 19 recordsLinked to original sources
arXiv2026

Entangling Power Dynamics: Ergodicity and Mixing

We study quantum dynamics through the lens of entanglement generation and characterize the underlying unitary evolution by the distinct signatures it imprints on the time-dependent entangling power. For a unitary operator, we characterize ergodicity by the equality between its long-time-averaged entangling power and the Haar-averaged linear entropy. We define mixing more stringently as the convergence of the time-dependent entangling power itself to the Haar value at long times. Within this framework, we establish the ergodic hierarchy of dynamical behavior, showing in particular that mixing implies ergodicity, whereas ergodicity does not necessarily imply mixing. As an application, we find that two-qubit unitary gates are neither ergodic nor mixing: their long-time-averaged entangling power can take only four discrete values, none of which coincides with the Haar average. We then investigate many-body dynamics using the kicked Ising chain and find that the long-time-averaged entangling power converges to the Haar value in both integrable and nonintegrable cases, indicating ergodicity. Remarkably, however, the nonintegrable chain exhibits mixing, whereas the integrable chain, despite being ergodic, is demonstrably nonmixing. We also introduce a Lyapunov-like exponent to characterize the rate at which the time-dependent entangling power approaches its saturation value. We find that this exponent increases systematically with the degree of integrability breaking in the many-body system. Our results establish entanglement generation as a useful framework for characterizing dynamical systems and reveal qualitatively different signatures of integrability beyond conventional diagnostics.

quant-ph↗
arXiv2026

Anderson localization via Peierls phase modulation

We investigate a two leg ladder system subjected to an external magnetic field. In the absence of a magnetic field, the system is described by a clean tight binding model, with no disorder in either the onsite potential or the hopping amplitudes. The effect of magnetic field in this system is studied by introducing the Peierls phases in the hopping amplitudes along a leg (appropriate when the Landau gauge is chosen). For a uniform magnetic field, characterized by a constant Peierls phase, we find that all eigenstates remain delocalized. In contrast, random Peierls phases, representing a random magnetic field, lead to complete localization of the eigenstates. We further show that a quasiperiodic modulation of the Peierls phase can drive a transition from a fully delocalized to a fully localized phase upon tuning the quasiperiodicity. For a two parameter quasiperiodic Peierls phase, varying analogously to a generalized Aubry Andre type potential, we construct the phase diagram of the system. The phase diagram exhibits regions of delocalized and localized phases, separated by intermediate regimes of mixed phase. We also perform a semiclassical analysis that qualitatively yields a similar phase diagram, capturing the localization transition. Our results demonstrate a mechanism for controlling transport properties via the Peierls phase engineering.

cond-mat.dis-nn↗
arXiv2026

Quantum-to-semiclassical Husimi dynamics of non-Hermitian localization transitions

The localization transition in the Hermitian Aubry-Andr\'e model is known to have a clear classical origin, with the critical point being exactly predictable from an analysis of classical phase-space trajectories. Motivated by this correspondence, we investigate whether a similar classical origin exists for localization transitions in non-Hermitian quasiperiodic Hamiltonians. Using semiclassical Husimi dynamics together with a detailed phase-space stability analysis, we show that localization transitions persist even in the semiclassical limit of such non-Hermitian models. However, in sharp contrast to the Hermitian Aubry-Andr\'e case, the transition point inferred from classical phase-space analysis does not coincide with the quantum critical point. Instead, we find that the semiclassical transition depends sensitively on the choice of the irrational parameter defining the quasiperiodic potential, indicating the absence of a universal classical-quantum correspondence for the localization transition in the non-Hermitian setting. Nonetheless, we identify a suitable parameter regime in which the classical dynamics can faithfully mimic the quantum dynamics over a finite but appreciable time window.

quant-ph↗
arXiv2025

Subsystem localization in a two-leg ladder system

We consider a ladder system where one leg, referred to as the ``bath", is governed by an Aubry-Andr\'{e} (AA) type Hamiltonian, while the other leg, termed the ``subsystem", follows a standard tight-binding Hamiltonian. We investigate the localization properties in the subsystem induced by its coupling to the bath. For the coupling strength larger than a critical value ($t'>t'_c$), the analysis of the static properties shows that there are three distinct phases as the AA potential strength $V$ is varied: a fully delocalized phase at low $V$, a localized phase at intermediate $V$, and a weakly delocalized (fractal) phase at large $V$. The fractal phase also appears in a narrow region along the boundary between the delocalized and localized phases. An analysis of the projected wavepacket dynamics in the subsystem shows that the delocalized phase exhibits a ballistic behavior, whereas the weakly delocalized phase is subdiffusive. Interestingly, the narrow fractal phase shows a super- to subdiffusive behavior as we go from the delocalized to localized phase. When $t'<t'_c$, the intermediate localized phase disappears, and we find a delocalized (ballistic) phase at low $V$ and a weakly delocalized (subdiffusive) phase at large $V$. Between those two phases, there is also an anomalous crossover regime where the system can be super- or subdiffusive. Beyond the ballistic phase observed at low $V$, we also identify a superdiffusive regime emerging in the limit $t'/V \ll 1$, which continuously approaches the ballistic behavior as $t' \to 0$. Finally, in some limiting scenario, we also establish a mapping between our ladder system and a well-studied one-dimensional generalized Aubry-Andr\'{e} (GAA) model.

cond-mat.stat-mech↗
arXiv2025

Emergence of Hermitian topology from non-Hermitian knots

The non-Hermiticity of the system gives rise to a distinct knot topology in the complex eigenvalue spectrum, which has no counterpart in Hermitian systems. In contrast, the singular values of a non-Hermitian (NH) Hamiltonian are always real by definition, meaning that they can also be interpreted as the eigenvalues of some underlying Hermitian Hamiltonian. In this work, we demonstrate that if the singular values of an NH Hamiltonian are treated as eigenvalues of prototype translational invariant Hermitian models that undergo a topological phase transition between two distinct topological phases, the complex eigenvalues of the NH Hamiltonian will also undergo a {\it{first order knot transition}} between different knot structures. Unlike the usual knot transition, this transition is not accompanied by an Exceptional point (EP); in contrast, the real and complex parts of the eigenvalues of the NH Hamiltonian show a discrete jump at the transition point. We emphasize that the choice of an NH Hamiltonian whose singular values match the eigenvalues of a Hermitian model is not unique. However, our study suggests that this connection between the NH and Hermitian models remains robust as long as the periodicity in lattice momentum is the same for both. Furthermore, we provide an example showing that a change in the topology of the Hermitian model implies a transition in the underlying NH knot topology, but a change in knot topology does not necessarily signal a topological transition in the Hermitian system.

quant-ph↗
arXiv2025

Topological properties of curved spacetime extended Su-Schrieffer-Heeger model

The Su-Schrieffer-Heeger (SSH) model, a prime example of a one-dimensional topologically nontrivial insulator, has been extensively studied in flat space-time. In recent times, many studies have been conducted to understand the properties of the low-dimensional quantum matter in curved spacetime, which can mimic the gravitational event horizon and black hole physics. However, the impact of curved spacetime on the topological properties of such systems remains unexplored. In this work, we investigate the curved spacetime (CST) version of the extended SSH model, by introducing a position-dependent hopping parameter. The extended SSH model already exhibits topological phases and the associated phase transitions. Different topological markers suggest that for the same choice of parameters, the CST version of the model retains the imprint of the same topological phases and transitions. Furthermore, the topologically non-trivial phase of the CST model hosts zero-energy edge modes, which are spatially asymmetric in contrast to those of the conventional SSH model. We find that at the topological transition points between phases with different winding numbers, a critical slowdown takes place for zero-energy wave packets near the boundary, indicating the presence of a horizon, and interestingly, if one moves even a slight distance away from the topological transition points, wave packets start bouncing back and reverse direction before reaching the boundary. Moreover, we have also quantified the time scale of the critical slowdown of the wavepacket across different winding-number transition phases. A semiclassical description of the wave packet trajectories also supports these results.

cond-mat.mes-hall↗
arXiv2025

Dependence of Krylov complexity on the initial operator and state

Krylov complexity, a quantum complexity measure which uniquely characterizes the spread of a quantum state or an operator, has recently been studied in the context of quantum chaos. However, the definitiveness of this measure as a chaos quantifier is in question in light of its strong dependence on the initial condition. This article clarifies the connection between the Krylov complexity dynamics and the initial operator or state. We find that the Krylov complexity depends monotonically on the inverse participation ratio (IPR) of the initial condition in the eigenbasis of the Hamiltonian. We explain the reversal of the complexity saturation levels observed in \href{https://doi.org/10.1103/PhysRevE.107.024217}{ Phys.Rev.E.107,024217, 2023} using the initial spread of the operator in the Hamiltonian eigenbasis. IPR dependence is present even in the fully chaotic regime, where popular quantifiers of chaos, such as out-of-time-ordered correlators and entanglement generation, show similar behavior regardless of the initial condition. Krylov complexity averaged over many initial conditions still does not characterize chaos.

quant-ph↗
arXiv2024

Measurement-induced phase transition in periodically driven free-fermionic systems

It is well known that unitary evolution tends to increase entanglement, whereas continuous monitoring counteracts this growth by pinning the wavefunction trajectories to the eigenstates of the measurement operators. In this work, we investigate the fate of the measurement-induced phase transition in a periodically driven free-fermionic quantum system, where the hopping amplitude is modulated periodically in time using a square pulse. In the high-frequency limit, a renormalization group analysis of the non-Hermitian quantum sine-Gordon model [as proposed in {Phys. Rev. X 11, 041004 (2021)}] reveals that if the hopping amplitude is varied symmetrically around zero, the system always favors the area-law phase, where the steady-state entanglement entropy is independent of subsystem size. In contrast, asymmetry in the drive amplitudes tends to promote entanglement growth. Furthermore, numerical evidence for the system sizes accessible to us suggests that decreasing the drive frequency typically favors entanglement growth. For such driven systems, at least for reasonably small frequency regimes, as a function of measurement strength, we observe a potential signature of a Berezinskii-Kosterlitz-Thouless (BKT) phase transition between a gapless critical phase, characterized by logarithmic growth of entanglement entropy with subsystem size, and a gapped area-law phase. However, it is almost impossible to rule out the possibility that the transition observed here is not an actual thermodynamic transition, but a finite-size crossover between logarithmic to area law entanglement phase. Even in that scenario, the critical length scale beyond which the area law phase prevails increases with the increasing time period of driving. On the other hand, for a symmetric drive, the system consistently exhibits an area-law phase, regardless of the driving frequency.

cond-mat.stat-mech↗
arXiv2023

Probing quantum phase transition via quantum speed limit

Quantum speed limit (QSL) is the lower bound on the time required for a state to evolve to a desired final state under a given Hamiltonian evolution. Three well-known QSLs exist Mandelstam-Tamm (MT), Margolus-Levitin (ML), and dual ML (ML$^*$) bounds. We consider one-dimensional systems that undergoes delocalization-localization transition in the presence of quasiperiodic and linear potential. By performing sudden quenches across the phase boundary, we find that the exact dynamics get captured very well by QSLs. We show that the MT bound is always tighter in the short time limit for any arbitrary state, while the optimal bound for the time of orthogonalization (time required to reach the orthogonal state) depends on the choice of the initial state. Further, for extreme quenches, we prove that the MT bound remains tighter for the time of orthogonalization, and it can qualitatively describe the non-analyticity in free energy for dynamical quantum phase transition (DQPT). Finally, we also demonstrate that the localization-delocalization transition point can be exactly identified from QSLs, whose computation cost is much less compared to many other diagnostic tools.

quant-ph↗
arXiv2023

Quest for optimal quantum resetting: protocols for a particle on a chain

In the classical context, it is well known that, sometimes, if the search does not find its target, it is better to start the process anew again, known as resetting. The quantum counterpart of resetting also indicates speeding up the detection process by eliminating the dark states, i.e., situations where the particle avoids detection. In this work, we introduce the most probable position resetting(MPR) protocol in which, at a given resetting step, resets are done with certain probabilities to the set of possible peak positions (where the probability of finding the particle is maximum) that could occur because of the previous resets and followed by uninterrupted unitary evolution, irrespective of which path was taken by the particle in previous steps. In a tight-binding lattice model, there exists a 2-fold degeneracy (left and right) of the positions of maximum probability. The survival probability with optimal restart rate approaches zero (detection probability approaches one) when the particle is reset with equal probability on both sides path independently. This protocol significantly reduces the optimal mean first-detected-passage time (FDT) and performs better even if the detector is far apart compared to the usual resetting protocols where the particle is brought back to the initial position. We propose a modified protocol, an adaptive two-stage MPR, by making the associated probabilities of going to the right and left a function of steps. In this protocol, we see a further reduction of the optimal mean FDT and improvement in the search process when the detector is far apart.

quant-ph↗
arXiv2023

Ergodic and mixing quantum channels: From two-qubit to many-body quantum systems

The development of classical ergodic theory has had a significant impact in the areas of mathematics, physics, and, in general, applied sciences. The quantum ergodic theory of Hamiltonian dynamics has its motivations to understand thermodynamics and statistical mechanics. Quantum channel, a completely positive trace-preserving map, represents a most general representation of quantum dynamics and is an essential aspect of quantum information theory and quantum computation. In this work, we study the ergodic theory of quantum channels by characterizing different levels of ergodic hierarchy from integrable to mixing. The quantum channels on single systems are constructed from the unitary operators acting on bipartite states and tracing out the environment. The interaction strength of these unitary operators measured in terms of operator entanglement provides sufficient conditions for the channel to be mixing. By using block diagonal unitary operators, we construct a set of non-ergodic channels. By using canonical form of two-qubit unitary operator, we analytically construct the channels on single qubit ranging from integrable to mixing. Moreover, we also study interacting many-body quantum systems that include the famous Sachdev-Ye-Kitaev (SYK) model and show that they display mixing within the framework of the quantum channel.

quant-ph↗
arXiv2023

Hellmann Feynman Theorem in Non-Hermitian system

We revisit the celebrated Hellmann-Feynman theorem (HFT) in the PT invariant non-Hermitian quantum physics framework. We derive a modified version of HFT by changing the definition of inner product and explicitly show that it holds good for both PT broken, unbroken phases and even at the exceptional point of the theory. The derivation is extremely general and works for even PT non-invariant Hamiltonian. We consider several examples of discrete and continuum systems to test our results. We find that if the eigenvalue goes through a real to complex transition as a function of the Hermiticity breaking parameter, both sides of the modified HFT expression diverge at that point. If that point turns out to be an EP of the PT invariant quantum theory, then one also sees the divergence at EP. Finally, we also derive a generalized Virial theorem for non-Hermitian systems using the modified HFT, which potentially can be tested in experiments.

quant-ph↗
arXiv2023

Engineering skin effect across a junction of Hermitian and non-Hermitian lattice

We study a system where the two edges of a non-Hermitian lattice with asymmetric nearest-neighbor hopping are connected with two Hermitian lattices with symmetric nearest-neighbor hopping. In the absence of those Hermitian lattices, the majority of the eigenstates of the system will be localized at the edges, the phenomena known as the non-Hermitian skin effect. We show that once we connect it with the Hermitian lattices, for open boundary conditions (OBC), the localized states exist at the junction of the non-Hermitian and Hermitian lattice; moreover, the spectrum shows mobility edges that separate delocalized and localized states. On the contrary, mobility edges vanish for periodic boundary conditions (PBC), and the delocalized phase turns into a scale-invariant localized phase, where the localized states are still peaked at the junctions. We also find that if the connected Hermitian lattices are thermodynamically large, in OBC, most of the states become delocalized, while in PBC, the system still shows the scale-invariant localized phase.

cond-mat.dis-nn↗
arXiv2023

Non-Hermitian description of sharp quantum resetting

We study a non-interacting quantum particle, moving on a one-dimensional lattice, which is subjected to repetitive measurements. We investigate the consequence when such motion is interrupted and restarted from the same initial configuration, known as the quantum resetting problem. We show that such systems can be described by the time evolution under certain time-dependent non-Hermitian Hamiltonians. We construct two such Hamiltonians and compare the results with the exact dynamics. Using this effective non-Hermitian description we evaluate the timescale of the survival probability as well as the optimal resetting time for the system.

quant-ph↗
arXiv2023

Complexity growth for one-dimensional free-fermionic lattice models

Complexity plays a very important part in quantum computing and simulation where it acts as a measure of the minimal number of gates that are required to implement a unitary circuit. We study the lower bound of the complexity [Eisert, Phys. Rev. Lett. 127, 020501 (2021)] for the unitary dynamics of the one-dimensional lattice models of non-interacting fermions. We find analytically using quasiparticle formalism, the bound grows linearly in time and followed by a saturation for short-ranged tight-binding Hamiltonians. We show numerical evidence that for an initial Neel state the bound is maximum for tight-binding Hamiltonians as well as for the long-range hopping models. However, the increase of the bound is sub-linear in time for the later, in contrast to the linear growth observed for short-range models. The upper bound of the complexity in non-interacting fermionic lattice models is calculated, which grows linearly in time even beyond the saturation time of the lower bound, and finally, it also saturates.

quant-ph↗
arXiv2022

PT-Symmetry Breaking Transitions in Polymeric Systems

We show that classical DNA unzipping transition which is equivalently described by quantum mechanical localization-delocalization transition in the ground state of non-Hermitian single impurity Hatano-Nelson Hamiltonian is underpinned by generalized parity ({\bf P})-time reversal ({\bf T}) symmetry breaking transition. We also study the one-dimensional discretized version of Hatano-Nelson model in the presence of the single impurity and random disorder on a finite-size lattice. These discrete models are useful to study unzipping of a single adsorbed polymer from a surface. Our results show that the discrete models also undergo a phase transition from a PT unbroken phase to a broken phase. Interestingly, the generalized PT phase transition points coincide with the localization-delocalization transition for continuum as well as lattice models.

cond-mat.stat-mech↗
arXiv2022

Witnessing quantum chaos using observational entropy

We study observation entropy (OE) for the Quantum kicked top (QKT) model, whose classical counterpart possesses different phases: regular, mixed, or chaotic, depending on the strength of the kicking parameter. We show that OE grows logarithmically with coarse-graining length beyond a critical value in the regular phase, while OE growth is much faster in the chaotic regime. In the dynamics, we demonstrate that the short-time growth rate of OE acts as a measure of the chaoticity in the system, and we compare our results with out-of-time-ordered correlators (OTOC). Moreover, we show that in the deep quantum regime, the results obtained from OE are much more robust compared to OTOC results. Finally, we also investigate the long-time behaviour of OE to distinguish between saddle-point scrambling and true chaos, where the former shows large persistent fluctuations compared to the latter.

quant-ph↗
arXiv2022

One-dimensional L{\'e}vy Quasicrystal

Space-fractional quantum mechanics (SFQM) is a generalization of the standard quantum mechanics when the Brownian trajectories in Feynman path integrals are replaced by L{\'e}vy flights. We introduce L{\'e}vy quasicrystal by discretizing the space-fractional Schr$\ddot{\text{o}}$dinger equation using the Gr$\ddot{\text{u}}$nwald-Letnikov derivatives and adding on-site quasiperiodic potential. The discretized version of the usual Schr$\ddot{\text{o}}$dinger equation maps to the Aubry-Andr{\'e} Hamiltonian, which supports localization-delocalization transition even in one dimension. We find the similarities between L{\'e}vy quasicrystal and the Aubry-Andr{\'e} (AA) model with power-law hopping and show that the L{\'e}vy quasicrystal supports a delocalization-localization transition as one tunes the quasiperiodic potential strength and shows the coexistence of localized and delocalized states separated by mobility edge. Hence, a possible realization of SFQM in optical experiments should be a new experimental platform to test the predictions of AA models in the presence of power-law hopping.

cond-mat.stat-mech↗