Artificial neural networks have been recently introduced as a general ansatz to compactly represent many- body wave functions. In conjunction with Variational Monte Carlo, this ansatz has been applied to find Hamil- tonian ground states and their energies. Here we provide extensions of this method to study properties of ex- cited states, a central task in several many-body quantum calculations. First, we give a prescription that allows to target eigenstates of a (nonlocal) symmetry of the Hamiltonian. Second, we give an algorithm that allows to compute low-lying excited states without symmetries. We demonstrate our approach with both Restricted Boltzmann machines states and feedforward neural networks as variational wave-functions. Results are shown for the one-dimensional spin-1/2 Heisenberg model, and for the one-dimensional Bose-Hubbard model. When comparing to available exact results, we obtain good agreement for a large range of excited-states energies. Interestingly, we also find that deep networks typically outperform shallow architectures for high-energy states.
Non-equilibrium physics including many-body localization (MBL) has attracted increasing attentions, but theoretical approaches of reliably studying non-equilibrium properties remain quite limited. In this Letter, we propose a systematic approach to probe MBL phases via the excited-state variational quantum eigensolver (VQE) and demonstrate convincing results of MBL on a quantum hardware, which we believe paves a promising way for future simulations of non-equilibrium systems beyond the reach of classical computations in the noisy intermediate-scale quantum (NISQ) era. Moreover, the MBL probing…
The eigenstates of many-body localized (MBL) Hamiltonians exhibit low entanglement. We adapt the highly successful density-matrix renormalization group method, which is usually used to find modestly entangled ground states of local Hamiltonians, to find individual highly excited eigenstates of many body localized Hamiltonians. The adaptation builds on the distinctive spatial structure of such eigenstates. We benchmark our method against the well studied random field Heisenberg model in one dimension. At moderate to large disorder, we find that the method successfully obtains excited eigenstate…
We study the properties of excited states in one-dimensional many-body localized (MBL) systems using a matrix product state algorithm. First, the method is tested for a large disordered non-interacting system, where for comparison we compute a quasi-exact reference solution via a Monte Carlo sampling of the single-particle levels. Thereafter, we present extensive data obtained for large interacting systems of L~100 sites and large bond dimensions chi~1700, which allows us to quantitatively analyze the scaling behavior of the entanglement S in the system. The MBL phase is characterized by a log…
We introduce approximate, functional renormalization group based schemes to obtain correlation functions in pure excited eigenstates of large fermionic many-body systems at arbitrary energies. The algorithms are thouroughly benchmarked and their strengths and shortcomings are documented using a one-dimensional interacting tight-binding chain as a prototypical testbed. We study two `toy applications' from the world of Luttinger liquid physics: the survival of power laws in lowly-excited states as well as the spectral function of high-energy `block' excitations which feature several single-parti…
Optimization of quantum states using the variational principle has recently seen an upsurge due to developments of increasingly expressive wave functions. In order to improve on the accuracy of the ansätze, it is a time-honored strategy to impose the systems' symmetries. We present an ansatz where global non-abelian symmetries are inherently embedded in its structure. We extend the model to incorporate lattice symmetries as well. We consider the prototypical example of the frustrated two-dimensional $J_1$-$J_2$ model on a square lattice, for which eigenstates have been hard to model variationa…
We present a fully many-body formulation of topological invariants for various topological phases of fermions protected by antiunitary symmetry, which does not refer to single particle wave functions. For example, we construct the many-body $\mathbb{Z}_2$ topological invariant for time-reversal symmetric topological insulators in two spatial dimensions, which is a many-body counterpart of the Kane-Mele $\mathbb{Z}_2$ invariant written in terms of single-particle Bloch wave functions. We show that an important ingredient for the construction of the many-body topological invariants is a fermioni…
We compute concurrence, a measure of bipartite entanglement, of the first excited state of the $1$-D Heisenberg frustrated $J_1$-$J_2$ spin-chain and observe a sudden change in the entanglement of the eigen state near the coupling strength $α=J_2/J_1\approx0.241$, where a quantum phase transition from spin-fluid phase to dimer phase has been previously reported. We numerically observe this phenomena for spin-chain with $8$ sites to $16$ sites, and the value of $α$ at which the change in entanglement is observed asymptotically tends to a value $α_c\approx0.24116$. We have calculated the finite-…
We provide a systematic treatment of the tenfold way of classifying fermionic systems that naturally allows for the study of those with arbitrary $N$-body interactions. We identify four types of symmetries that such systems can possess, which consist of one ordinary type (usual unitary symmetries), and three non-ordinary symmetries (such as time reversal, charge conjugation and sublattice). Focusing on systems that possess no non-trivial ordinary symmetries, we demonstrate that the non-ordinary symmetries are strongly constrained. This approach not only leads very naturally to the tenfold clas…
We study the excited states of interacting fermions in one dimension with particle-hole symmetric disorder (equivalently, random-bond XXZ chains) using a combination of renormalization group methods and exact diagonalization. Absent interactions, the entire many-body spectrum exhibits infinite-randomness quantum critical behavior with highly degenerate excited states. We show that though interactions are an irrelevant perturbation in the ground state, they drastically affect the structure of excited states: even arbitrarily weak interactions split the degeneracies in favor of thermalization (w…
Exact diagonalization (ED) is one of the most reliable and established numerical methods of quantum many-body theory. The main limiting factor of the method is the exponential scaling of Hilbert space dimension with system size. Fortunately, by symmetry considerations the effective dimension can be reduced by multiple orders of magnitude. Here, we present lattice-symmetries, a package for working with such symmetry-adapted quantum many-body bases and operators. It supports bases for spin-1/2 particles with arbitrary user-defined symmetries and generic 1-, 2-, 3-, and 4-point operators. As an e…
We provide a guiding principle to generate entanglement of quantum many-body states by applying key ideas of the Higgs mechanism to systems without gauge structures. Unitary operators associated with the Higgs mechanism are constructed, named as mean-operators, and employed to prepare entangled many-body states out of a trivial state. We uncover a symmetry-protectedtopological state with two Ising symmetries on a square lattice and find entangled states with different symmetries and lattices. Plausible applications to quantum simulators such as Rydberg atoms and trapped ions, are also discusse…
We obtain multiple exact results on the entanglement of the exact excited states of non-integrable models we introduced in arXiv:1708.05021. We first discuss a general formalism to analytically compute the entanglement spectra of exact excited states using Matrix Product States and Matrix Product Operators and illustrate the method by reproducing a general result on single-mode excitations. We then apply this technique to analytically obtain the entanglement spectra of the infinite tower of states of the spin-$S$ AKLT models in the zero and finite energy density limits. We show that in the zer…
Deviating band symmetries and many-body interactions in a model hole doped iron pnictide superconductor
本文用角分辨光电子能谱研究了最佳掺杂的铁磷属元素化物超导体Ba $_{0.6}$ K $_{0.4}$ Fe $_2$ As $_2$（T $_c$ = 37 K）的低能能带结构.偏振对比测量被用来识别和跟踪所有三个低能量的类空穴带预测局域密度近似（LDA）计算。光发射的电子揭示了不一致的LDA预测的对称性沿$Γ$ -X高对称动量轴，由于电子动力学的意外强大的旋转各向异性。我们评估的多体效应，如莫特-哈伯德相互作用，可能是异常的基础，并讨论如何观察到的LDA能带结构的偏差影响铁磷属元素化物库珀配对的空穴掺杂制度的能量。
Highly excited states of quantum many-body systems are central objects in the study of quantum dynamics and thermalization that challenge classical computational methods due to their volume-law entanglement content. In this work, we explore the potential of variational quantum algorithms to approximate such states. We propose an adaptive variational algorithm, adaptive VQE-X, that self-generates a variational ansatz for arbitrary eigenstates of a many-body Hamiltonian $H$ by attempting to minimize the energy variance with respect to $H$. We benchmark the method by applying it to an Ising spin …
Weyl and Dirac (semi)metals in three dimensions have robust gapless electronic band structures. Their massless single-body energy spectra are protected by symmetries such as lattice translation, (screw) rotation and time reversal. In this manuscript, we discuss many-body interactions in these systems. We focus on strong interactions that preserve symmetries and are outside the single-body mean-field regime. By mapping a Dirac (semi)metal to a model based on a three dimensional array of coupled Dirac wires, we show (1) the Dirac (semi)metal can acquire a many-body excitation energy gap without …
Bootstrapping Symmetries in Quantum Many-Body Systems from the Cross Spectral Form Factor
Kavli Institute for Theoretical Sciences, University of Chinese Academy of Sciences（中国科学院大学理论科学研究所）；Department of Physics, Princeton University（普林斯顿大学物理系）；Philippe Meyer Institute, Physics Department, École Normale Supérieure (ENS), Université PSL（巴黎高等师范学校PSL大学物理系菲利普·迈耶研究所）
We examine the exact equation of motion for the relaxation of populations of strongly correlated electrons after a nonequilibrium excitation by a pulsed field, and prove that the populations do not change when the Green's functions have no average time dependence. We show how the average time dependence enters into the equation of motion to lowest order and describe what governs the relaxation process of the electron populations in the long-time limit. While this result may appear, on the surface, to be required by any steady-state solution, the proof is nontrivial, and provides new critical i…
