arXiv subjects

Jacek Dziarmaga

Publications and source records attributed to Jacek Dziarmaga.

At least 37 records · Page 2Linked to original sources

arXiv2022

Quantum phase transition dynamics in the two-dimensional transverse-field Ising model

The quantum Kibble-Zurek mechanism (QKZM) predicts universal dynamical behavior near the quantum phase transitions (QPTs). It is now well understood for the one-dimensional quantum matter. Higher-dimensional systems, however, remain a challenge, complicated by the fundamentally different character of the associated QPTs and their underlying conformal field theories. In this work, we take the first steps toward theoretical exploration of the QKZM in two dimensions for interacting quantum matter. We study the dynamical crossing of the QPT in the paradigmatic Ising model by a joint effort of modern state-of-the-art numerical methods, including artificial neural networks and tensor networks. As a central result, we quantify universal QKZM behavior close to the QPT. We also note that, upon traversing further into the ferromagnetic regime, deviations from the QKZM prediction appear. We explain the observed behavior by proposing an {\it extended QKZM} taking into account spectral information as well as phase ordering. Our work provides a testing platform for higher-dimensional quantum simulators.

cond-mat.str-el↗

arXiv2022

Kink correlations, domain size distribution, and the emptiness formation probability after the Kibble-Zurek quench in the quantum Ising chain

Linear quench of the transverse field drives the quantum Ising chain across a quantum critical point from the paramagnetic to the ferromagnetic phase. We focus on normal and anomalous quadratic correlators between fermionic kink creation and annihilation operators. They depend not only on the Kibble-Zurek (KZ) correlation length but also on a dephasing length scale, which differs from the KZ length by a logarithmic correction. Additional slowing down of the ramp in the ferromagnetic phase further increases the dephasing length and suppresses the anomalous correlator. The quadratic correlators enter Pfaffians that yield experimentally relevant kink correlation functions, the probability distribution of ferromagnetic domain sizes, and, closely related, emptiness formation probability. The latter takes the form of a Pfaffian of a block Toeplitz matrix that allows for some analytic asymptotes. Finally, we obtain further insight into the structure of the state at the end of the ramp by interpreting it as a paired state of fermionic kinks characterized by its pair wave function. All those quantities are sensitive to quantum coherence between eigenstates with different numbers of kinks, thus making them a convenient probe of the quantumness of a quantum simulator platform.

quant-ph↗

arXiv2022

Time evolution of an infinite projected entangled pair state: a gradient tensor update in the tangent space

Time evolution of an infinite 2D many body quantum lattice system can be described by the Suzuki-Trotter decomposition applied to the infinite projected entangled pair state (iPEPS). Each Trotter gate increases the bond dimension of the tensor network, $D$, that has to be truncated back in a way that minimizes a suitable error measure. This paper goes beyond simplified error measures -- like the one used in the full update (FU), the simple update (SU), and their intermediate neighborhood tensor update (NTU) -- and directly maximizes an overlap between the exact iPEPS with the increased bond dimension and the new iPEPS with the truncated one. The optimization is performed in a tangent space of the iPEPS variational manifold. This gradient tensor update (GTU) is benchmarked by a simulation of a sudden quench of a transverse field in the 2D quantum Ising model and the quantum Kibble-Zurek mechanism in the same 2D system.

quant-ph↗

arXiv2021

Simulation of many body localization and time crystals in two dimensions with the neighborhood tensor update

The Heisenberg antiferromagnet with discrete disorder on an infinite square lattice is evolved in time from an initial Néel state. The simulation is performed with the neighborhood tensor update (NTU) algorithm for an infinite projected entangled pair state (iPEPS) [Phys. Rev. B 104, 094411 (2021)]. Ancillary spins are used to average over $2$ or $5$ discrete values of disorder. With a bond dimension up to $20$, evolution times are long enough to identify a many body localized regime for a strong enough disorder. Furthermore, the same Hamiltonian is subject to periodic spin flips. Simulations of the Floquet dynamics show that it can sustain a time crystalline stage for a strong enough disorder.

cond-mat.dis-nn↗

arXiv2021

Quantum Kibble-Zurek mechanism: Kink correlations after a quench in the quantum Ising chain

The transverse field in the quantum Ising chain is linearly ramped from the para- to the ferromagnetic phase across the quantum critical point at a rate characterized by a quench time $τ_Q$. We calculate a connected kink-kink correlator in the final state at zero transverse field. The correlator is a sum of two terms: a negative (anti-bunching) Gaussian that depends on the Kibble-Zurek (KZ) correlation length only and a positive term that depends on a second longer scale of length. The second length is made longer by dephasing of the state excited near the critical point during the following ramp across the ferromagnetic phase. This interpretation is corroborated by considering a linear ramp that is halted in the ferromagnetic phase for a finite waiting time and then continued at the same rate as before the halt. The extra time available for dephasing increases the second scale of length that asymptotically grows linearly with the waiting time. The dephasing also suppresses magnitude of the second term making it negligible for waiting times much longer than $τ_Q$. The same dephasing can be obtained with a smooth ramp that slows down in the ferromagnetic phase. Assuming sufficient dephasing we obtain also higher order kink correlators and the ferromagnetic correlation function.

quant-ph↗

arXiv2021

Time evolution of an infinite projected entangled pair state: a neighborhood tensor update

The simple update (SU) and full update (FU) are the two paradigmatic time evolution algorithms for a tensor network known as the infinite projected entangled pair state (iPEPS). They differ by an error measure that is either, respectively, local or takes into account full infinite tensor environment. In this paper we test an intermediate neighborhood tensor update (NTU) accounting for the nearest neighbor environment. This small environment can be contracted exactly in a parallelizable way. It provides an error measure that is Hermitian and non-negative down to machine precision. In the 2D quantum Ising model NTU is shown to yield stable unitary time evolution following a sudden quench. It also yields accurate thermal states despite correlation lengths that reach up to 20 lattice sites. The latter simulations were performed with a manifestly Hermitian purification of a thermal state. Both were performed with reduced tensors that do not include physical (and ancilla) indices. This modification naturally leads to two other schemes: a local SVD update (SVDU) and a full tensor update (FTU) being a variant of FU.

quant-ph↗

arXiv2021

Variational methods for characterizing matrix product operator symmetries

We present a method of extracting information about topological order from the ground state of a strongly correlated two-dimensional system represented by an infinite projected entangled pair state (iPEPS). As in Phys. Rev. B 101, 041108 (2020) and 102, 235112 (2020) we begin by determining symmetries of the iPEPS represented by infinite matrix product operators (iMPO) that map between the different iPEPS transfer matrix fixed points, to which we apply the fundamental theorem of MPS to find zipper tensors between products of iMPO's that encode fusion properties of the anyons. The zippers can be combined to extract topological $F$-symbols of the underlying fusion category, which unequivocally identify the topological order of the ground state. We bring the $F$-symbols to the canonical gauge, as well as compute the Drinfeld center of this unitary fusion category to extract the topological $S$ and $T$ matrices encoding mutual- and self-statistics of the emergent anyons. The algorithm is applied to Abelian toric code, double semion and twisted quantum double of $Z_3$, as well as to non-Abelian double Fibonacci, double Ising, and quantum double of $S_3$ and ${\rm Rep}(S_3)$ string net models.

cond-mat.str-el↗

arXiv2021

Is there a correlation length in a model with long-range interactions?

Considering an example of the long-range Kitaev model, we are looking for a correlation length in a model with long range interactions whose correlation functions away from a critical point have power-law tails instead of the usual exponential decay. It turns out that quasiparticle spectrum depends on a distance from the critical point in a way that allows to identify the standard correlation length exponent, $ν$. The exponent implicitly defines a correlation length $ξ$ that diverges when the critical point is approached. We show that the correlation length manifests itself also in the correlation function but not in its exponential tail because there is none. Instead $ξ$ is a distance that marks a crossover between two different algebraic decays with different exponents. At distances shorter than $ξ$ the correlator decays with the same power law as at the critical point while at distances longer than $ξ$ it decays faster, with a steeper power law. For this correlator it is possible to formulate the usual scaling hypothesis with $ξ$ playing the role of the scaling distance. The correlation length also leaves its mark on the subleading anomalous fermionic correlator but, interestingly, there is a regime of long range interactions where its short distance critical power-law decay is steeper than its long distance power law tail.

cond-mat.str-el↗

arXiv2021

Non-adiabatic dynamics across a first order quantum phase transition: Quantized bubble nucleation

Metastability is a quintessential feature of first order quantum phase transitions, which is lost either by dynamical instability or by nucleating bubbles of a true vacuum through quantum tunneling. By considering a drive across the first order quantum phase transition in the quantum Ising chain in the presence of both transverse and longitudinal fields, we reveal multiple regions in the parameter space where the initial metastable state loses its metastability in successive stages. The mechanism responsible is found to be semi-degenerate resonant tunnelings to states with specific bubble sizes. We show that such dynamics of quantized bubble nucleations can be understood in terms of Landau-Zener transitions, which provide quantitative predictions of nucleation probabilities for different bubble sizes.

cond-mat.stat-mech↗

arXiv2021

Tensor network study of the $m=1/2$ magnetization plateau in the Shastry-Sutherland model at finite temperature

The two-dimensional infinite projected entangled pair states tensor network is evolved in imaginary time with the full update (FU) algorithm to simulate the Shastry-Sutherland model in a magnetic field at finite temperature directly in the thermodynamic limit. We focus on the phase transition into the $m=1/2$ magnetization plateau, which was observed in experiments on SrCu$_2$(BO$_3$)$_2$. For the largest simulated bond dimension, the early evolution in the high-temperature regime is simulated with the simple update (SU) scheme and then, as the correlation length increases, continued with the FU scheme towards the critical regime. We apply a small symmetry-breaking bias field and then extrapolate towards zero bias using a simple scaling theory in the bias field. The combined SU + FU scheme provides an accurate estimate of the critical temperature, even though the results could not be fully converged in the bond dimension in the vicinity of the transition. The critical temperature estimate is improved with a generalized scaling theory that combines two divergent length scales: One due to the bias, and the other due to the finite bond dimension. The obtained results are consistent with the transition being in the universality class of the two-dimensional classical Ising model. The estimated critical temperature is $3.5(2)$ K, which is well above the temperature $2.1$ K used in the experiments.

cond-mat.str-el↗

arXiv2020

Inhomogeneity induced shortcut to adiabaticity in Ising chains with long-range interactions

Driving a homogeneous system across a quantum phase transition in a quench-time $τ_Q$ generates excitations on wavelengths longer than the Kibble-Zurek (KZ) length $\hatξ\proptoτ_Q^{ν/(1+zν)}$ within the KZ time window $\hat t\proptoτ_Q^{zν/(1+zν)}$, where $z$ and $ν$ are the critical exponents. Quenches designed with local time-dependent inhomogeneity can introduce a gap in the spectrum. For a variety of setups with short-range interactions, they have been shown to suppress excitations if the spatial velocity of the inhomogenous front is below the characteristic KZ velocity $\hat v \propto \hatξ/\hat t$. Ising-like models with long-range interactions can have no sonic horizon, spreading information instantaneously across the system. Usually, this should imply that inhomogenous transitions will render the dynamics adiabatic regardless of the front velocity. However, we show that we get an adiabatic transition with no defects only when the inhomogeneous front moves slower than a characteristic crossover velocity $\tilde v \propto θ^{(z-1)ν/(1+ν)}$, where $θ$ is the slope of the inhomogeneous front at the critical point. The existence of this crossover velocity and adiabaticity of the model is a consequence of the energy gap in the quasiparticle spectrum that is opened by the inhomogeneity. This effect can be employed for efficient adiabatic quantum state preparation in systems with long-range interactions.

quant-ph↗

arXiv2020

Determining non-Abelian topological order from infinite projected entangled pair states

We generalize the method introduced in Phys. Rev. B 101, 041108 (2020) of extracting information about topological order from the ground state of a strongly correlated two-dimensional system represented by an infinite projected entangled pair state (iPEPS) to non-Abelian topological order. When wrapped on a torus the unique iPEPS becomes a superposition of degenerate and locally indistinguishable ground states. We find numerically symmetries of the iPEPS, represented by infinite matrix product operators (MPO), and their fusion rules. The rules tell us how to combine the symmetries into projectors onto states with well defined anyon flux. A linear structure of the MPO projectors allows for efficient determination for each state its second Renyi topological entanglement entropy on an infinitely long cylinder directly in the limit of infinite cylinder's width. The same projectors are used to compute topological $S$ and $T$ matrices encoding mutual- and self-statistics of emergent anyons. The algorithm is illustrated by examples of Fibonacci and Ising non-Abelian string net models.

cond-mat.str-el↗

arXiv2020

Kibble-Zurek Dynamics in a Trapped Ultracold Bose Gas

The dynamical evolution of an inhomogeneous ultracold atomic gas quenched at different controllable rates through the Bose-Einstein condensation phase transition is studied numerically in the premise of a recent experiment in an anisotropic harmonic trap. Our findings based on the stochastic (projected) Gross-Pitaevskii equation are shown to be consistent at early times with the predictions of the homogeneous Kibble-Zurek mechanism. This is demonstrated by collapsing the early dynamical evolution of densities, spectral functions and correlation lengths for different quench rates, based on an appropriate characterization of the distance to criticality felt by the quenched system. The subsequent long-time evolution, beyond the identified dynamical critical region, is also investigated by looking at the behaviour of the density wavefront evolution and the corresponding phase ordering dynamics.

cond-mat.quant-gas↗

arXiv2020

Sonic horizons and causality in the phase transition dynamics

A system gradually driven through a symmetry-breaking phase transition is subject to the Kibble-Zurek mechanism (KZM). As a consequence of the critical slowing down, its state cannot follow local equilibrium, and its evolution becomes non-adiabatic near the critical point. In the simplest approximation, that stage can be regarded as "impulse" where the state of the system remains unchanged. It leads to the correct KZM scaling laws. However, such "freeze-out" might suggest that the coherence length of the nascent order parameter remains unchanged as the critical region is traversed. By contrast, the original causality-based discussion emphasized the role of the {\it sonic horizon}: domains of the broken symmetry phase can expand with a velocity limited by the speed of the relevant sound. This effect was demonstrated in the quantum Ising chain where the dynamical exponent $z=1$ and quasiparticles excited by the transition have a fixed speed of sound. To elucidate the role of the sonic horizon, in this paper we study two systems with $z>1$ where the speed of sound is no longer fixed, and the fastest excited quasiparticles set the size of the sonic horizon. Their effective speed decays with the increasing transition time. In the extreme case, the dynamical exponent $z$ can diverge such as in the Griffiths region of the random Ising chain where localization of excited quasiparticles freezes the growth of the correlation range when the critical region is traversed. Of particular interest is an example with $z<1$ -- the long-range extended Ising chain, where there is no upper limit to the velocity of excited quasiparticles with small momenta. Initially, the power-law tail of the correlation function grows adiabatically, but in the non-adiabatic stage it lags behind the adiabatic evolution -- in accord with a generalized Lieb-Robinson bound.

quant-ph↗

arXiv2020

Determining topological order from infinite projected entangled pair states

We present a method of extracting information about the topological order from the ground state of a strongly correlated two-dimensional system computed with the infinite projected entangled pair state (iPEPS). For topologically ordered systems, the iPEPS wrapped on a torus becomes a superposition of degenerate, locally indistinguishable ground states. Projectors in the form of infinite matrix product operators (iMPO) onto states with well-defined anyon flux are used to compute topological $S$ and $T$ matrices (encoding mutual- and self-statistics of emergent anyons). The algorithm is shown to be robust against a perturbation driving string-net toric code across a phase transition to a ferromagnetic phase. Our approach provides accurate results near quantum phase transition, where the correlation length is prohibitively large for other numerical methods. Moreover, we used numerically optimized iPEPS describing the ground state of the Kitaev honeycomb model in the toric code phase and obtained topological data in excellent agreement with theoretical prediction.

cond-mat.str-el↗

arXiv2020

Tensor-Network Approach for Quantum Metrology in Many-Body Quantum Systems

Identification of the optimal quantum metrological protocols in realistic many particle quantum models is in general a challenge that cannot be efficiently addressed by the state-of-the-art numerical and analytical methods. Here we provide a comprehensive framework exploiting matrix product operators (MPO) type tensor networks for quantum metrological problems. Thanks to the fact that the MPO formalism allows for an efficient description of short-range spatial and temporal noise correlations, the maximal achievable estimation precision in such models, as well as the optimal probe states in previously inaccessible regimes can be identified. Moreover, the application of infinite MPO (iMPO) techniques allows for a direct and efficient determination of the asymptotic precision of optimal protocols in the limit of infinite particle numbers. We illustrate the potential of our framework in terms of an atomic clock stabilization (temporal noise correlation) example as well as for magnetic field sensing in the presence of locally correlated magnetic field fluctuations (spatial noise correlations). As a byproduct, the developed methods for calculating the quantum Fisher information via MPOs may be used to calculate the fidelity susceptibility - a parameter widely used in many-body physics to study phase transitions.

quant-ph↗

arXiv2019

Tensor network simulation of the Kitaev-Heisenberg model at finite temperature

We investigate the Kitaev-Heisenberg (KH) model at finite temperature using the exact environment full update (eeFU), introduced in Phys. Rev. B 99, 035115 (2019), which represents the purification of a thermal density matrix on an infinite hexagonal lattice by an infinite projected entangled pair state (iPEPS). We show that thanks to a dynamical mapping from a hexagonal to a rhombic lattice, the eeFU on the hexagonal lattice is as efficient as the simple full update (FU) algorithm. Critical temperatures for coupling constants in the stripy and the antiferromagnetic phase are estimated. They are an order of magnitude less than the couplings in the Hamiltonian. By a duality transformation, these results can be mapped to, respectively, the ferromagnetic and zigzag phases. For the special case of the pure Kitaev model, which is tractable by quantum Monte-Carlo but the most challenging for tensor networks, the algorithm is benchmarked against the Monte-Carlo results. It recovers accurately the crossover to spin ordering and qualitatively the one to flux ordering.

cond-mat.str-el↗

arXiv2019

Symmetry breaking bias and the dynamics of a quantum phase transition

The Kibble-Zurek mechanism predicts the formation of topological defects and other excitations that quantify how much a quantum system driven across a quantum critical point fails to be adiabatic. We point out that, thanks to the divergent linear susceptibility at the critical point, even a tiny symmetry breaking bias can restore the adiabaticity. The minimal required bias scales like $τ_Q^{-βδ/(1+zν)}$, where $β,δ,z,ν$ are the critical exponents and $τ_Q$ is a quench time. We test this prediction by DMRG simulations of the quantum Ising chain. It is directly applicable to the recent emulation of quantum phase transition dynamics in the Ising chain with ultracold Rydberg atoms.

cond-mat.stat-mech↗