arXiv subjects

Igor Ermakov

Publications and source records attributed to Igor Ermakov.

15 recordsLinked to original sources

arXiv2025

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↗

arXiv2025

Symbolic operator growth and recursion method for spin-S Ising and q-states Potts models

Operator growth is well understood for systems with a two-dimensional local Hilbert space, and much less so beyond them, where an operator can grow not only spatially but also in depth, inside the local algebra of each site. We compute the moments of infinite-temperature autocorrelation functions for the spin-$S$ Ising model and the $q$-state Potts model exactly and symbolically in the Hamiltonian parameters, in one, two and three dimensions. For the Ising model the moments are symbolic in the spin magnitude as well, so that a single computation covers arbitrary $S$. From them we obtain the Lanczos coefficients, rigorous Taylor bounds on the initial decay of the autocorrelation function, and its intermediate-time behavior via the recursion method. Our results support the Universal Operator Growth Hypothesis beyond local dimension two. Namely, the Lanczos coefficients grow linearly in the non-integrable cases and exhibit a clean square-root growth in the integrable one-dimensional Potts chain. For the spin-$S$ Ising model we show that every moment converges with growing spin to the corresponding moment of the classical spin model. The Lanczos coefficients therefore approach a limiting sequence with $1/S^2$ corrections, so that classicalization occurs at the level of the whole Lanczos sequence rather than of a single observable. This provides a justification for quasiclassical methods in infinite-temperature spin dynamics. All the results are exact, symbolic and obtained directly in the thermodynamic limit, and the computed moments are available in a public repository.

quant-ph↗

arXiv2025

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↗

arXiv2024

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↗

arXiv2024

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↗

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↗

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

Generalized Almost Complete Revivals in quantum spin chains

The conception of almost complete revivals has been introduced recently. In a quantum many-body system local observable may exhibit an almost complete revival to its maximal value at the predetermined moment of time. In this paper we extend the original procedure such that the revival may be from an arbitrary point on the Bloch sphere to the arbitrary point. Furthermore in the proposed procedure the reviving and collapsing sites are not necessarily the same. We also demonstrate that for spins $S$ higher than $1/2$ almost complete revivals are suppressed as $1/S$.

quant-ph↗

arXiv2020

Almost complete revivals in quantum many-body systems

Revivals of initial non-equilibrium states is an ever-present concern for the theory of dynamic thermalization in many-body quantum systems. Here we consider a nonintegrable lattice of interacting spins 1/2 and show how to construct a quantum state such that a given spin 1/2 is maximally polarized initially and then exhibits an almost complete recovery of the initial polarization at a predetermined moment of time. An experimental observation of such revivals may be utilized to benchmark quantum simulators with a measurement of only one local observable. We further propose to utilize these revivals for a delayed disclosure of a secret.

quant-ph↗

arXiv2020

Testing eigenstate decoherence hypothesis in a model of collisional decoherence

The eigenstate decoherence hypothesis (EDH) asserts that each individual eigenstate of a large closed system is locally classical-like. We test this hypothesis for a heavy particle interacting with a gas of light particles. This system is paradigmatic for studies of the quantum-to-classical transition: The reduced state of the heavy particle is widely believed to rapidly loose any nonclassical features due to the interaction with the gas. Yet, we find numerical evidence that the EDH is violated: certain eigenstates of this model are manifestly non-classical. Only the weak version of EDH referring to the majority (instead of the totality) of eigenstates holds.

quant-ph↗

arXiv2019

Cooper pair polaritons in cold fermionic atoms within a cavity

We formulate a Bardeen-Cooper-Schriffer (BCS) theory of quasiparticles in a degenerate Fermi gas strongly coupled to photons in a optical cavity. The elementary photonic excitations of the system are cavity polaritons, which consist of a cavity photon and an excitation of an atom within the Fermi sea. The excitation of the atom out of the Fermi sea leaves behind a hole, which together results in a loosely bound Cooper pair, allowing for the system to be written by a BCS wavefunction. As the density of the excitations is increased, the excited atom and hole become more strongly bound, crossing over into the molecular regime. This thus realizes an alternative BCS to BEC crossover scenario, where the participating species are quasiparticle excitations in a Fermi sea consisting of excited atoms and holes.

cond-mat.quant-gas↗

arXiv2019

Time dynamics of Bethe ansatz solvable models

We develop a method for finding the time evolution of exactly solvable models by Bethe ansatz. The dynamical Bethe wavefunction takes the same form as the stationary Bethe wavefunction except for time varying Bethe parameters and a complex phase prefactor. From this, we derive a set of first order nonlinear coupled differential equations for the Bethe parameters, called the dynamical Bethe equations. We find that this gives the exact solution to particular types of exactly solvable models, including the Bose-Hubbard dimer and Tavis-Cummings model. These models go beyond the Gaudin class, and offers an interesting possibility for performing time evolution in exactly solvable models.

quant-ph↗

arXiv2017

High accuracy energy formulas for the attractive two-site Bose-Hubbard model

The attractive two-site Bose-Hubbard model is studied within the framework of the analytical solution obtained by the application of Quantum Inverse Scattering Method. The structure of the ground and excited states is analyzed in terms of solutions of Bethe equations, and an approximate solution for the Bethe roots are given. This yields approximate formulas for the ground state and for the first excited state energies. The obtained formulas work with remarkable precision for a wide range of parameters of the model, and confirmed numerically. An expansion of the Bethe state vectors into a Fock space is also provided for evaluation of expectation values, although this does not have the similar accuracy to the energies.

quant-ph↗

arXiv2017

Quantum coherence of planar spin models with Dzyaloshinsky-Moriya interaction

The quantum coherence of one dimensional planar spin models with the Dzyaloshinsky-Moriya interaction is investigated. The anisotropic XY model, the isotropic XX model and the transverse field model are studied in the large N-limit using the two qubit reduced density matrices and the two point correlation functions. From our investigations we find that the coherence as measured using the Jensen-Shannon divergence can be used to detect the quantum phase transitions and the quantum critical points. The derivative of coherence shows non-analytic behavior at the critical points leading to the conclusions that these transitions are of second order. Further we show that the presence of the Dzyaloshinsky-Moriya coupling suppresses the phase transition due to the residual ferromagnetism which is caused by spin canting.

quant-ph↗

arXiv2017

Dynamic correlation funtions of the generalized Tavis-Cummings model

The model describing interaction between two-level atoms and a single mode field in an optical cavity enclosed by a medium with Kerr nonlinearity is considered in our paper. We study the model within the framework of the analytical solution obtained by application of the quantum inverse method (QIM). The dynamic correlation functions are calculated and transition elements of photons are represented in the determinant form. The obtained answers depend on the solutions of Bethe equations. We provide the numerical solutions of these equations for the different parameters of the model and study the behaviour of certain dynamical correlation functions.

quant-ph↗