Calculating the spectral function of two dimensional systems is arguably one of the most pressing challenges in modern computational condensed matter physics. While efficient techniques are available in lower dimensions, two dimensional systems prese
nt insurmountable hurdles, ranging from the sign problem in quantum Monte Carlo (MC), to the entanglement area law in tensor network based methods. We hereby present a variational approach based on a Chebyshev expansion of the spectral function and a neural network representation for the wave functions. The Chebyshev moments are obtained by recursively applying the Hamiltonian and projecting on the space of variational states using a modified natural gradient descent method. We compare this approach with a modified approximation of the spectral function which uses a Krylov subspace constructed from the Chebyshev wave-functions. We present results for the one-dimensional and two-dimensional Heisenberg model on the square lattice, and compare to those obtained by other methods in the literature.
We present a method to reduce the variance of stochastic trace estimators used in quantum typicality (QT) methods via a randomized low-rank approximation of the finite-temperature density matrix $e^{-beta H}$. The trace can be evaluated with higher a
ccuracy in the low-rank subspace while using the QT estimator to approximate the trace in the complementary subspace. We present two variants of the trace estimator and demonstrate their efficacy using numerical experiments. The experiments show that the low-rank approximation outperforms the standard QT trace estimator for moderate- to low-temperature. We argue this is due to the low-rank approximation accurately represent the density matrix at low temperatures, allowing for accurate results for the trace.
We study a generalized quantum spin ladder with staggered long range interactions that decay as a power-law with exponent $alpha$. Using the density matrix renormalization group (DMRG) method and exact diagonalization, we show that this model undergo
es a transition from a rung-dimer phase characterized by a non-local string order parameter, to a symmetry broken Neel phase at $alpha_csim 2.1$. We find evidence that the transition is second order with a dynamic critical exponent $z=1$ and $ uapprox 1.2$. In the magnetically ordered phase, the spectrum exhibits gapless modes, while excitations in the gapped phase are well described in terms of triplons -- bound states of spinons across the legs. We obtained the momentum resolved spin dynamic structure factor numerically and found that the triplon band is well defined at high energies and adiabatically connected to the magnon dispersion. However, at low energies it emerges as the lower edge of continuum of excitations that shifts to high energies across the transition. We further discuss the possibility of deconfined criticality in this model.
We discuss quantum annealing of the two-dimensional transverse-field Ising model on a D-Wave device, encoded on $L times L$ lattices with $L le 32$. Analyzing the residual energy and deviation from maximal magnetization in the final classical state,
we find an optimal $L$ dependent annealing rate $v$ for which the two quantities are minimized. The results are well described by a phenomenological model with two powers of $v$ and $L$-dependent prefactors to describe the competing effects of reduced quantum fluctuations (for which we see evidence of the Kibble-Zurek mechanism) and increasing noise impact when $v$ is lowered. The same scaling form also describes results of numerical solutions of a transverse-field Ising model with the spins coupled to noise sources. We explain why the optimal annealing time is much longer than the coherence time of the individual qubits.
Theoretical studies of quantum phase transitions have suggested critical points with higher symmetries than those of the underlying Hamiltonian. Here we demonstrate a surprising emergent symmetry of the coexistence state at a strongly discontinuous p
hase transition between two ordered ground states. We present a quantum Monte Carlo study of a two-dimensional $S=1/2$ quantum magnet hosting the antiferromagnetic (AFM) and plaquette-singlet solid (PSS) states recently detected in SrCu$_2$(BO$_3$)$_2$. We observe that the O(3) symmetric AFM order and the Z$_2$ symmetric PSS order form an O(4) vector at the transition. The control parameter $g$ (a coupling ratio) rotates the vector between the AFM and PSS sectors and there are no energy barriers between the two at the transition point $g_c$. This phenomenon may be observable in SrCu$_2$(BO$_3$)$_2$.
We present a major update to QuSpin, SciPostPhys.2.1.003 -- an open-source Python package for exact diagonalization and quantum dynamics of arbitrary boson, fermion and spin many-body systems, supporting the use of various (user-defined) symmetries i
n one and higher dimension and (imaginary) time evolution following a user-specified driving protocol. We explain how to use the new features of QuSpin using seven detailed examples of various complexity: (i) the transverse-field Ising chain and the Jordan-Wigner transformation, (ii) free particle systems: the Su-Schrieffer-Heeger (SSH) model, (iii) the many-body localized 1D Fermi-Hubbard model, (iv) the Bose-Hubbard model in a ladder geometry, (v) nonlinear (imaginary) time evolution and the Gross-Pitaevskii equation on a 1D lattice, (vi) integrability breaking and thermalizing dynamics in the translationally-invariant 2D transverse-field Ising model, and (vii) out-of-equilibrium Bose-Fermi mixtures. This easily accessible and user-friendly package can serve various purposes, including educational and cutting-edge experimental and theoretical research. The complete package documentation is available under http://weinbe58.github.io/QuSpin/index.html.
We study the Neel-paramagnetic quantum phase transition in two-dimensional dimerized $S=1/2$ Heisenberg antiferromagnets using finite-size scaling of quantum Monte Carlo data. We resolve the long standing issue of the role of cubic interactions arisi
ng in the bond-operator representation when the dimer pattern lacks a certain symmetry. We find non-monotonic (monotonic) size dependence in the staggered (columnar) dimerized model, where cubic interactions are (are not) present. We conclude that there is an irrelevant field in the staggered model that is not present in the columnar case, but, at variance with previous claims, it is not the leading irrelevant field. The new exponent is $omega_2 approx 1.25$ and the prefactor of the correction $L^{-omega_2}$ is large and comes with a different sign from that of the formally leading conventional correction with exponent $omega_1 approx 0.78$. Our study highlights the possibility of competing scaling corrections at quantum critical points.
We study the problem of preparing a quantum many-body system from an initial to a target state by optimizing the fidelity over the family of bang-bang protocols. We present compelling numerical evidence for a universal spin-glass-like transition cont
rolled by the protocol time duration. The glassy critical point is marked by a proliferation of protocols with close-to-optimal fidelity and with a true optimum that appears exponentially difficult to locate. Using a machine learning (ML) inspired framework based on the manifold learning algorithm t-SNE, we are able to visualize the geometry of the high-dimensional control landscape in an effective low-dimensional representation. Across the transition, the control landscape features an exponential number of clusters separated by extensive barriers, which bears a strong resemblance with replica symmetry breaking in spin glasses and random satisfiability problems. We further show that the quantum control landscape maps onto a disorder-free classical Ising model with frustrated nonlocal, multibody interactions. Our work highlights an intricate but unexpected connection between optimal quantum control and spin glass physics, and shows how tools from ML can be used to visualize and understand glassy optimization landscapes.
We reveal a continuous dynamical heating transition between a prethermal and an infinite-temperature stage in a clean, chaotic periodically driven classical spin chain. The transition time is a steep exponential function of the drive frequency, showi
ng that the exponentially long-lived prethermal plateau, originally observed in quantum Floquet systems, survives the classical limit. Even though there is no straightforward generalization of Floquets theorem to nonlinear systems, we present strong evidence that the prethermal physics is well described by the inverse-frequency expansion. We relate the stability and robustness of the prethermal plateau to drive-induced synchronization not captured by the expansion. Our results set the pathway to transfer the ideas of Floquet engineering to classical many-body systems, and are directly relevant for photonic crystals and cold atom experiments in the superfluid regime.
We analyze the physics of optimal protocols to prepare a target state with high fidelity in a symmetrically coupled two-qubit system. By varying the protocol duration, we find a discontinuous phase transition, which is characterized by a spontaneous
breaking of a $mathbb{Z}_2$ symmetry in the functional form of the optimal protocol, and occurs below the quantum speed limit. We study in detail this phase and demonstrate that even though high-fidelity protocols come degenerate with respect to their fidelity, they lead to final states of different entanglement entropy shared between the qubits. Consequently, while globally both optimal protocols are equally far away from the target state, one is locally closer than the other. An approximate variational mean-field theory which captures the physics of the different phases is developed.
