Dynamical potentials for non-equilibrium quantum many-body phases

Out of equilibrium phases of matter exhibiting order in individual eigenstates, such as many-body localised spin glasses and discrete time crystals, can be characterised by inherently dynamical quantities such as spatiotemporal correlation functions. In this work, we introduce dynamical potentials which act as generating functions for such correlations and capture eigenstate phases and order. These potentials show formal similarities to their equilibrium counterparts, namely thermodynamic potentials. We provide three representative examples: a disordered, many-body localised XXZ chain showing many-body localisation, a disordered Ising chain exhibiting spin-glass order and its periodically-driven cousin exhibiting time-crystalline order.Comment: 4+epsilon pages, 4 figures + supplementary materia

Psycholinguistic changes in the communication of adolescent users in a suicidal ideation online community during the COVID-19 pandemic

Since the outbreak of the COVID-19 pandemic, increases in suicidal ideation and suicide attempts in adolescents have been registered. Many adolescents experiencing suicidal ideation turn to online communities for social support. In this retrospective observational study, we investigated the communication—language style, contents and user activity—in 7975 unique posts and 51,119 comments by N = 2862 active adolescent users in a large suicidal ideation support community (SISC) on the social media website reddit.com in the onset period of the COVID-19 pandemic. We found significant relative changes in language style markers for hopelessness such as negative emotion words (+ 10.00%) and positive emotion words (− 3.45%) as well as for social disengagement such as social references (− 8.63%) and 2nd person pronouns (− 33.97%) since the outbreak of the pandemic. Using topic modeling with Latent Dirichlet Allocation (LDA), we identified significant changes in content for the topics Hopelessness (+ 23.98%), Suicide Methods (+ 17.11%), Social Support (− 14.91%), and Reaching Out to users (− 28.97%). Changes in user activity point to an increased expression of mental health issues and decreased engagement with other users. The results indicate a potential shift in communication patterns with more adolescent users expressing their suicidal ideation rather than relating with or supporting other users during the COVID-19 pandemic

Measuring complex-partition-function zeros of Ising models in quantum simulators

Studying the zeros of partition functions in the space of complex control parameters allows one to understand formally how critical behavior of a many-body system can arise in the thermodynamic limit despite various no-go theorems for finite systems. In this work we propose protocols that can be realized in quantum simulators to measure the location of complex-partition-function zeros of classical Ising models. The protocols are solely based on the implementation of simple two-qubit gates, local spin rotations, and projective measurements along two orthogonal quantization axes. Besides presenting numerical simulations of the measurement outcomes for an exemplary classical model, we discuss the effect of projection noise and the feasibility of the implementation on state of the art platforms for quantum simulation

The Kibble-Zurek mechanism at exceptional points

Exceptional points (EPs) are ubiquitous in non-hermitian systems, and represent the complex counterpart of critical points. By driving a system through a critical point at finite rate induces defects, described by the Kibble-Zurek mechanism, which finds applications in diverse fields of physics. Here we generalize this to a ramp across an EP. We find that adiabatic time evolution brings the system into an eigenstate of the final non-hermitian Hamiltonian and demonstrate that for a variety of drives through an EP, the defect density scales as $\tau^{-(d+z)\nu/(z\nu+1)}$ in terms of the usual critical exponents and $1/\tau$ the speed of the drive. Defect production is suppressed compared to the conventional hermitian case as the defect state can decay back to the ground state close to the EP. We provide a physical picture for the studied dynamics through a mapping onto a Lindblad master equation with an additionally imposed continuous measurement.Comment: 7 pages, 3 figure

The Kibble-Zurek mechanism at exceptional points

Exceptional points (EPs) are ubiquitous in non-hermitian systems, and represent the complex counterpart of critical points. By driving a system through a critical point at finite rate induces defects, described by the Kibble-Zurek mechanism, which finds applications in diverse fields of physics. Here we generalize this to a ramp across an EP. We find that adiabatic time evolution brings the system into an eigenstate of the final non-hermitian Hamiltonian and demonstrate that for a variety of drives through an EP, the defect density scales as $\tau^{-(d+z)\nu/(z\nu+1)}$ in terms of the usual critical exponents and $1/\tau$ the speed of the drive. Defect production is suppressed compared to the conventional hermitian case as the defect state can decay back to the ground state close to the EP. We provide a physical picture for the studied dynamics through a mapping onto a Lindblad master equation with an additionally imposed continuous measurement.Comment: 7 pages, 3 figure

Disorder-free localization transition in a two-dimensional lattice gauge theory

Disorder-free localization is a novel mechanism for ergodicity breaking which can occur in interacting quantum many-body systems such as lattice gauge theories (LGTs). While the nature of the quantum localization transition (QLT) is still debated for conventional many-body localization, here we provide the first comprehensive characterization of the QLT in two dimensions (2D) for a disorder-free case. Disorder-free localization can appear in homogeneous 2D LGTs such as the U(1) quantum link model (QLM) due to an underlying classical percolation transition fragmenting the system into disconnected real-space clusters. Building on the percolation model, we characterize the QLT in the U(1) QLM through a detailed study of the ergodicity properties of finite-size real-space clusters via level-spacing statistics and localization in configuration space. We argue for the presence of two regimes - one in which large finite-size clusters effectively behave non-ergodically, a result naturally accounted for as an interference phenomenon in configuration space and the other in which all large clusters behave ergodically. As one central result, in the latter regime we claim that the QLT is equivalent to the classical percolation transition and is hence continuous. Utilizing this equivalence we determine the universality class and critical behaviour of the QLT from a finite-size scaling analysis of the percolation problem.Comment: 4.5 pages, 4 figures; comments welcome. V2 resembles published versio

Making Trotterization adaptive and energy-self-correcting for NISQ devices and beyond

Simulation of continuous time evolution requires time discretization on both classical and quantum computers. A finer time step improves simulation precision, but it inevitably leads to increased computational efforts. This is particularly costly for today's noisy intermediate scale quantum computers, where notable gate imperfections limit the circuit depth that can be executed at a given accuracy. Classical adaptive solvers are well-developed to save numerical computation times. However, it remains an outstanding challenge to make optimal usage of the available quantum resources by means of adaptive time steps. Here, we introduce a quantum algorithm to solve this problem, providing a controlled solution of the quantum many-body dynamics of local observables. The key conceptual element of our algorithm is a feedback loop which self-corrects the simulation errors by adapting time steps, thereby significantly outperforming conventional Trotter schemes on a fundamental level and reducing the circuit depth. It even allows for a controlled asymptotic long-time error, where usual Trotterized dynamics is facing difficulties. Another key advantage of our quantum algorithm is that any desired conservation law can be included in the self-correcting feedback loop, which has potentially a wide range of applicability. We demonstrate the capabilities by enforcing gauge invariance which is crucial for a faithful and long-sought quantum simulation of lattice gauge theories. Our algorithm can be potentially useful on a more general level whenever time discretization is involved concerning, for instance, also numerical approaches based on time-evolving block decimation methods.Comment: 10+15 pages, 4+19 figure

Adaptive Trotterization for time-dependent Hamiltonian quantum dynamics using instantaneous conservation laws

Digital quantum simulation relies on Trotterization to discretize time evolution into elementary quantum gates. On current quantum processors with notable gate imperfections, there is a critical tradeoff between improved accuracy for finer timesteps, and increased error rate on account of the larger circuit depth. We present an adaptive Trotterization algorithm to cope with time-dependent Hamiltonians, where we propose a concept of instantaneous "conserved" quantities to estimate errors in the time evolution between two (nearby) points in time; these allow us to bound the errors accumulated over the full simulation period. They reduce to standard conservation laws in the case of time-independent Hamiltonians, for which we first developed an adaptive Trotterization scheme. We validate the algorithm for a time-dependent quantum spin chain, demonstrating that it can outperform the conventional Trotter algorithm with a fixed step size at a controlled error.Comment: 7 pages, 5 figure

Dynamical Signatures of Symmetry Broken and Liquid Phases in an S=1/2S=1/2 Heisenberg Antiferromagnet on the Triangular Lattice

We present the dynamical spin structure factor of the antiferromagnetic spin-$\frac{1}{2}$ $J_1-J_2$ Heisenberg model on a triangular lattice obtained from large-scale matrix-product state simulations. The high frustration due to the combination of antiferromagnetic nearest and next-to-nearest neighbour interactions yields a rich phase diagram. We resolve the low-energy excitations both in the $120^{\circ}$-ordered phase and in the putative spin liquid phase at $J_2/J_1 = 0.125$. In the ordered phase, we observe an avoided decay of the lowest magnon-branch, demonstrating the robustness of this phenomenon in the presence of gapless excitations. Our findings in the spin-liquid phase chime with the field-theoretical predictions for a gapless Dirac spin liquid, in particular the picture of low-lying monopole excitations at the corners of the Brillouin zone. We comment on possible practical difficulties of distinguishing proximate liquid and solid phases based on the dynamical structure factor

Making trotterization adaptive and energy-self-correcting for NISQ devices and beyond

Simulation of continuous-time evolution requires time discretization on both classical and quantum computers. A finer time step improves simulation precision but it inevitably leads to increased computational efforts. This is particularly costly for today’s noisy intermediate-scale quantum computers, where notable gate imperfections limit the circuit depth that can be executed at a given accuracy. Classical adaptive solvers are well developed to save numerical computation times. However, it remains an outstanding challenge to make optimal usage of the available quantum resources by means of adaptive time steps. Here, we introduce a quantum algorithm to solve this problem, providing a controlled solution of the quantum many-body dynamics of local observables. The key conceptual element of our algorithm is a feedback loop that self-corrects the simulation errors by adapting time steps, thereby significantly outperforming conventional Trotter schemes on a fundamental level and reducing the circuit depth. It even allows for a controlled asymptotic long-time error, where the usual Trotterized dynamics faces difficulties. Another key advantage of our quantum algorithm is that any desired conservation law can be included in the self-correcting feedback loop, which has a potentially wide range of applicability. We demonstrate the capabilities by enforcing gauge invariance, which is crucial for a faithful and long-sought-after quantum simulation of lattice gauge theories. Our algorithm can potentially be useful on a more general level whenever time discretization is involved also concerning, e.g., numerical approaches based on time-evolving block-decimation methods