arXiv subjects

Oleg Lychkovskiy

Publications and source records attributed to Oleg Lychkovskiy.

At least 19 recordsLinked to original sources

arXiv2026

Symbolic recursion method for strongly correlated fermions in two and three dimensions

We present a symbolic implementation of the recursion method for dynamical correlations and transport in fermionic systems on one-, two-, and three-dimensional lattices. The implementation is applicable in the strongly correlated regime, yields results directly in the thermodynamic limit, and covers all time scales, from short times through the thermalization time to the late-time asymptotics. Focusing on two paradigmatic models -- interacting spinless fermions and the Hubbard model -- we confirm the universal operator growth hypothesis in the fermionic case, compute infinite-temperature current-current autocorrelation functions, and determine the charge diffusion constant and the high-temperature conductivity. The diffusion constant is obtained in both the perturbative and the nonperturbative regime of the interaction strength, and we locate the boundary between them. We compare our approach with the Majorana propagation method, which converges up to intermediate times and agrees with ours where converged. Our results highlight a symbolic computational paradigm in which the most expensive step is performed once, producing reusable symbolic output that yields physical observables for arbitrary model parameters.

cond-mat.str-el↗

arXiv2026

QCommute: a tool for symbolic computation of nested commutators in quantum many-body spin-1/2 systems

We present QCommute, a software tool implemented in C++ for symbolic computation of nested commutators between a Hamiltonian and local observables in quantum many-body spin-1/2 systems on one-, two-, and three-dimensional hypercubic lattices. The computation is performed algebraically directly in the thermodynamic limit, and the Hamiltonian parameters are kept symbolic. Importantly, this way the entire parameter space is covered in a single run. The implementation supports extensive parallelization to achieve high computational performance. QCommute can serve as a computational backend for Heisenberg-picture approaches to quantum dynamics in strongly correlated regimes, ranging from direct Taylor expansion in time to advanced techniques such as the recursion method.

cond-mat.str-el↗

arXiv2026

Recursion method for quench dynamics: strengths and limitations

The recursion method, which solves coupled Heisenberg equations in a Lanczos operator basis, has recently emerged as a powerful nonperturbative tool for computing dynamical correlation functions in strongly correlated two- and three-dimensional quantum many-body systems. Motivated by this success, we investigate whether the method can be extended to expectation values of observables following a quantum quench. We find that such an extension encounters an obstacle absent in the computation of dynamical correlation functions. The latter are fully determined by the Lanczos coefficients $b_n$, which in generic systems exhibit universal behavior, enabling reliable extrapolation from the first few dozens of explicitly computed coefficients. In contrast, quench dynamics additionally requires "quench coefficients" $c_n$, defined as overlaps of Lanczos basis operators with the initial state. We show that, unlike the Lanczos coefficients, the quench coefficients exhibit no universal structure and cannot be reliably extrapolated, thereby limiting the time up to which the method yields accurate results. The behavior of quench coefficients is highly state-dependent, ranging from decaying to irregular or even growing sequences; typically, the less regular the sequence $c_n$, the shorter the accessible timescale. Nevertheless, for favorable initial states, the method remains competitive with state-of-the-art approaches. Moreover, its symbolic implementation allows a single computation to be reused across different Hamiltonian parameters and initial states, making it particularly advantageous in studies requiring extensive scans over Hamiltonian parameters or initial states.

cond-mat.str-el↗

arXiv2026

Duality between open systems and closed bilayer systems: Thermofield double states as quantum many-body scars

We establish a duality between open many-body systems governed by the Gorini-Kossakowski-Sudarshan-Lindblad (GKSL) equation and satisfying the detailed balance condition on the one side, and closed bilayer systems with a self-adjoint Hamiltonian on the other side. Under this duality, the identity operator on the open system side maps to a quantum many-body scar of the dual Hamiltonian $\mathcal H$. This scar eigenstate has a form of a thermofield double state for a single-body conserved quantity entering the detailed balance conditions. A remarkable feature of this thermofield scar is a tunable single-layer entanglement entropy controlled by the reservoir temperature on the open system side. Further, we identify broad classes of many-body open systems with nontrivial explicit eigen operators $Q$ of the Lindbladian superoperator. The expectation values of the corresponding observables exhibit a simple exponential decay, $\langle Q\rangle_t=e^{-Γt} \langle Q \rangle_0$, irrespectively of the initial state. Under the above duality, these eigen operators give rise to additional (towers of) scars. Finally, we point out that more general superoperators (not necessarily of the GKSL form) can be mapped to self-adjoint Hamiltonians of bilayer systems harbouring scars, and provide an example thereof.

cond-mat.stat-mech↗

arXiv2026

Classical periodic trajectories and quantum scars in many-spin systems

We numerically investigate the stability of exceptional periodic classical trajectories in rather generic chaotic many-body systems and explore a possible connection between these trajectories and exceptional nonthermal quantum eigenstates known as "quantum many-body scars". The systems considered are chaotic spin chains with short-range interactions, both classical and quantum. On the classical side, the chosen periodic trajectories are such that all spins instantaneously point in the same direction, which evolves as a function of time. We find that the largest Lyapunov exponents characterising the stabillity of these trajectories have surprisingly strong and nontrivial dependencies on the interaction constants and chain lengths. In particular, we identify rather long spin chains, where the above periodic trajectories are Lyapunov-stable on many-body energy shells overwhelmingly dominated by chaotic motion. We also find that instabilities around periodic trajectories in modestly large spin chains develop into a transient nearly quasiperiodic non-ergodic regime. In some cases, the lifetime of this regime is extremely long, which we interpret as a manifestation of Arnold diffusion in the vicinity of integrable dynamics. On the quantum side, we numerically investigate the dynamics of quantum states starting with all spins initially pointing in the same direction: these are the quantum counterparts of the initial conditions for the above periodic classical trajectories. Our investigation reveals the existence of quantum many-body scars for numerically accessible finite chains of spins 3/2 and higher. The dynamic thermalisation process dominated by quantum scars is shown to exhibit a slowdown in comparison with generic thermalisation at the same energy. Finally, we identify quantum signatures of the proximity to a classical separatrix of the periodic motion.

quant-ph↗

arXiv2025

Exactly solvable models for universal operator growth

Quantum observables of generic many-body systems exhibit a universal pattern of growth in the Krylov space of operators. This pattern becomes particularly manifest in the Lanczos basis, where the evolution superoperator assumes the tridiagonal form. According to the universal operator growth hypothesis, the nonzero elements of the superoperator, known as Lanczos coefficients, grow asymptotically linearly. We introduce and explore broad families of Lanczos coefficients that are consistent with the universal operator growth and lead to the exactly solvable dynamics. Within these families, the subleading terms of asymptotic expansion of the Lanczos sequence can be controlled and fine-tuned to produce diverse dynamical patterns. For one of the families, the Krylov complexity is computed exactly.

quant-ph↗

arXiv2025

Polynomially restricted operator growth in dynamically integrable models

We provide a framework to determine the upper bound to the complexity of a computing a given observable with respect to a Hamiltonian. By considering the Heisenberg evolution of the observable, we show that each Hamiltonian defines an equivalence relation, causing the operator space to be partitioned into equivalence classes. Any operator within a specific class never leaves its equivalence class during the evolution. We provide a method to determine the dimension of the equivalence classes and evaluate it for various models, such as the $ XY $ chain and Kitaev model on trees. Our findings reveal that the complexity of operator evolution in the $XY$ model grows from the edge to the bulk, which is physically manifested as suppressed relaxation of qubits near the boundary. Our methods are used to reveal several new cases of simulable quantum dynamics, including a $XY$-$ZZ$ model which cannot be reduced to free fermions.

quant-ph↗

arXiv2025

Pseudomode expansion of many-body correlation functions

We present an expansion of a many-body correlation function in a sum of pseudomodes -- exponents with complex frequencies that encompass both decay and oscillations. The pseudomode expansion emerges in the framework of the Heisenberg version of the recursion method. This method essentially solves Heisenberg equations in a Lanczos tridiagonal basis constructed in the Krylov space of a given observable. To obtain pseudomodes, we first add artificial dissipation satisfying the dissipative generalization of the universal operator growth hypothesis, and then take the limit of the vanishing dissipation strength. Fast convergence of the pseudomode expansion is facilitated by the localization in the Krylov space, which is generic in the presence of dissipation and can survive the limit of the vanishing dissipation strength. As an illustration, we present pseudomode expansions of infinite-temperature autocorrelation functions in the quantum Ising and $XX$ spin-$1/2$ models on the square lattice. It turns out that it is enough to take a few first pseudomodes to obtain a good approximation to the correlation function.

cond-mat.str-el↗

arXiv2024

Fermionic transport through a driven quantum point contact: breakdown of Floquet thermalization beyond a critical driving frequency

We study a quantum system that consists of two fermionic chains coupled by a driven quantum point contact (QPC). The QPC contains a bond with a periodically varying tunneling amplitude. Initially the left chain is packed with fermions while the right one is empty. We numerically track the evolution of the system and demonstrate that, at frequencies above a critical one, the current through the QPC halts, and the particle imbalance between the chains remains forever. This implies a spectacular breakdown of the Floquet version of the eigenstate thermalization hypothesis which predicts a homogeneous particle density profile at large times. We confirm the effect for various driving protocols and interparticle interactions.

cond-mat.str-el↗

arXiv2024

One-dimensional Fermi polaron after a kick: two-sided singularity of the momentum distribution, Bragg reflection and other exact results

A mobile impurity particle immersed in a quantum fluid forms a polaron - a quasiparticle consisting of the impurity and a local disturbance of the fluid around it. We ask what happens to a one-dimensional polaron after a kick, i.e. an abrupt application of a force that instantly delivers a finite impulse to the impurity. In the framework of an integrable model describing an impurity in a one-dimensional gas of fermions or hard-core bosons, we calculate the distribution of the polaron momentum established when the post-kick relaxation is over. A remarkable feature of this distribution is a two-sided power-law singularity. It emerges due to one of two processes. In the first process, the whole impulse is transferred to the polaron, without creating phonon-like excitations of the fluid. In the second process, the impulse is shared between the polaron and the center-of-mass motion of the fluid, again without creating any fluid excitations. The latter process is, in fact, a Bragg reflection at the edge of the emergent Brillouin zone. We carefully analyze the conditions for each of the two processes. The asymptotic form of the distribution in the vicinity of the singularity is derived.

cond-mat.quant-gas↗

arXiv2024

Exact dynamics of quantum dissipative $XX$ models: Wannier-Stark localization in the fragmented operator space

We address dissipative dynamics of the one-dimensional nearest-neighbour $XX$ spin-$1/2$ chain governed by the Gorini-Kossakowski-Sudarshan-Lindblad (GKSL) equation. In the absence of dissipation the model is integrable. We identify a broad class of dissipative terms that generically destroy integrability but leave the operator space of the model fragmented into an extensive number of dynamically disjoint subspaces of varying dimensions. In sufficiently small subspaces the GKSL equation in the Heisenberg representation can be easily solved, sometimes in a closed analytical form. We provide an example of such an exact solution for a specific choice of dissipative terms. It is found that observables experience the Wannier-Stark localization in the corresponding operator subspace. As a result, the expectation values of the observables are linear combinations of essentially a few discrete decay modes, the long time dynamics being governed by the slowest mode. We examine the complex Liouvillian eigenvalue corresponding to this latter mode as a function of the dissipation strength. We find an exceptional point at a critical dissipation strength that separates oscillating and non-oscillating decay. We also describe a different type of dissipation that leads to a single decay mode in the whole operator subspace. Finally, we point out that our exact solutions of the GKSL equation entail exact solutions of the Schrödinger equation describing the quench dynamics in closed spin ladders dual to the dissipative spin chains.

cond-mat.str-el↗

arXiv2024

Unified framework for efficiently computable quantum circuits

Quantum circuits consisting of Clifford and matchgates are two classes of circuits that are known to be efficiently simulatable on a classical computer. We introduce a unified framework that shows in a transparent way the special structure that allows these circuits can be efficiently simulatable. The approach relies on analyzing the operator spread within a network of basis operators during the evolution of quantum circuit. Quantifying the complexity of a calculation by the number of operators with amplitude above a threshold value, we show that there is a generic form of the complexity curve involving an initial exponential growth, saturation, then exponential decay in the presence of decoherence. Our approach is naturally adaptable into a numerical procedure, where errors can be consistently controlled as a function of the complexity of the simulation.

quant-ph↗

arXiv2024

Quantum dynamics in one and two dimensions via recursion method

We report an implementation of the recursion method that addresses quantum many-body dynamics in the nonperturbative regime. The method essentially amounts to constructing a Lanczos basis in the space of operators and solving coupled Heisenberg equations in this basis. The reported implementation has two key ingredients: a computer-algebraic routine for symbolic calculation of nested commutators and a procedure to extrapolate the sequence of Lanczos coefficients according to the universal operator growth hypothesis. We apply the method to calculate infinite-temperature correlation functions for spin-$1/2$ systems on one- and two-dimensional lattices. In two dimensions the accessible timescale is large enough to essentially embrace the relaxation to equilibrium. The method allows one to accurately calculate transport coefficients. As an illustration, we compute the diffusion constant for the transverse-field Ising model on a square lattice.

cond-mat.str-el↗

arXiv2023

Effect of dephasing on the current through a periodically driven quantum point contact

We consider two one-dimensional quantum $XX$ magnets linked by a periodically driven quantum point contact (QPC). If magnets are initially polarized in opposite directions, one expects that a spin current through the QPC will establish. It has been shown recently [Phys. Rev. B 103, L041405 (2021)] that, in fact, when the driving frequency exceeds a critical value, the current halts completely, the QPC being effectively insulating. Here we enquire how this picture is affected by quantum dephasing. Our findings reveal that any non-zero dephasing restores the current.

cond-mat.mes-hall↗

arXiv2022

Time scale for adiabaticity breakdown in driven many-body systems and orthogonality catastrophe

The adiabatic theorem is a fundamental result established in the early days of quantum mechanics, which states that a system can be kept arbitrarily close to the instantaneous ground state of its Hamiltonian if the latter varies in time slowly enough. The theorem has an impressive record of applications ranging from foundations of quantum field theory to computational recipes in molecular dynamics. In light of this success it is remarkable that a practicable quantitative understanding of what "slowly enough" means is limited to a modest set of systems mostly having a small Hilbert space. Here we show how this gap can be bridged for a broad natural class of physical systems, namely many-body systems where a small move in the parameter space induces an orthogonality catastrophe. In this class, the conditions for adiabaticity are derived from the scaling properties of the parameter dependent ground state without a reference to the excitation spectrum. This finding constitutes a major simplification of a complex problem, which otherwise requires solving non-autonomous time evolution in a large Hilbert space. We illustrate our general results by analyzing conditions for the transport quantization in a topological Thouless pump.

cond-mat.quant-gas↗

arXiv2022

Out-of-equilibrium dynamics of the Kitaev model on the Bethe lattice via coupled Heisenberg equations

The Kitaev model on the honeycomb lattice, while being integrable via the spin-fermion mapping, has generally resisted an analytical treatment of the far-from-equilibrium dynamics due to the extensive number of relevant configurations of conserved charges. Here we study a close proxy of this model, the isotropic Kitaev spin-$1/2$ model on the Bethe lattice. Instead of relying on the spin-fermion mapping, we take a straightforward approach of solving Heisenberg equations for a tailored subset of spin operators. The simplest operator in this subset corresponds to the energy contribution of a single bond direction. As an example, we calculate the time-dependent expectation value of this observable for a factorized translation-invariant (or staggered-translation-invariant) initial state with arbitrary initial (staggered) polarization.

cond-mat.stat-mech↗

arXiv2021

Mobile impurity in a one-dimensional quantum gas: Exact diagonalization in the Bethe Ansatz basis

We consider a mobile impurity particle injected into a one-dimensional quantum gas. The time evolution of the system strongly depends on whether the mass of the impurity and the masses of the host particles are equal or not. For equal masses, the model is Bethe Ansatz solvable, but for unequal masses, the model is no longer integrable and the Bethe Ansatz technique breaks down. We construct a controllable numerical method of computing the spectrum of the model with a finite number of host particles, based on exact diagonalization of the Hamiltonian in the truncated basis of the Bethe Ansatz states. We illustrate our approach on a few-body system of 5+1 particles, and trace the evolution of the spectrum depending on the mass ratio of the impurity and the host particles.

cond-mat.quant-gas↗

arXiv2021

Closed hierarchy of Heisenberg equations in integrable models with Onsager algebra

Dynamics of a quantum system can be described by coupled Heisenberg equations. In a generic many-body system these equations form an exponentially large hierarchy that is intractable without approximations. In contrast, in an integrable system a small subset of operators can be closed with respect to commutation with the Hamiltonian. As a result, the Heisenberg equations for these operators can form a smaller closed system amenable to an analytical treatment. We demonstrate that this indeed happens in a class of integrable models where the Hamiltonian is an element of the Onsager algebra. We explicitly solve the system of Heisenberg equations for operators from this algebra. Two specific models are considered as examples: the transverse field Ising model and the superintegrable chiral 3-state Potts model.

cond-mat.stat-mech↗