We explore a class of “open” quantum circuit models with local decoherence (“noise”) and local projective measurements, each respecting a global ${\mathbb{Z}}_2$ symmetry. The model supports a spin-glass phase where the ${\mathbb{Z}}_2$ symmetry is spontaneously broken [not possible in an equilibrium one-dimensional (1D) system], a paramagnetic phase characterized by a divergent susceptibility, and an intermediate “trivial” phase. All three phases are also stable to ${\mathbb{Z}}_2$-symmetric local unitary gates, and the dynamical phase transitions between the phases are in the percolation universality class. The open circuit dynamics can be purified by explicitly introducing a bath with its own “scrambling” dynamics, which does not change any of the universal physics. Within the spin-glass phase the circuit dynamics can be interpreted as a quantum repetition code, with each stabilizer of the code measured stochastically at a finite rate, and the decoherences as effective bit-flip errors. Motivated by the geometry of the spin-glass phase, we devise a decoding algorithm for recovering an arbitrary initial qubit state in the code space, assuming knowledge of the history of the measurement outcomes, and the ability of performing local Pauli measurements and gates on the final state. For a circuit with $L^d$ qubits running for time $T$, the time needed to execute the decoder scales as $O\left(L^{d}T\right)$ (with dimensionality $d$). With this decoder in hand, we find that the information of the initial encoded qubit state can be retained (and then recovered) for a time logarithmic in $L$ for a 1D circuit, and for a time at least linear in $L$ in two dimensions below a finite-error threshold. For both the repetition and toric codes, we compare and contrast our decoding algorithm with earlier algorithms that map the error model to the random bond Ising model.

In positive muon spin rotation and relaxation (${\mu }^{+}\mathrm{SR}$) spectroscopy, positive muons (${\mu }^{+}$) implanted into solid oxides are conventionally treated as immobile spin probes at interstitial sites below room temperature. This is because each ${\mu }^{+}$ is thought to be tightly bound to an oxygen atom in the host lattice to form a muonic analog of the hydroxy group. On the basis of this concept, anomalies in ${\mu }^{+}\mathrm{SR}$ spectra observed in oxides have been attributed in most cases to the intrinsic properties of host materials. On the other hand, global ${\mu }^{+}$ diffusion with an activation energy of $\sim 0.1\mathrm{eV}$ has been reported in some chemically substituted perovskite oxides at cryogenic temperatures, although the reason for the small activation energy despite the formation of the strong $\mathrm{O}\mu$ bond has not yet been quantitatively understood. In this study, we investigated interstitial ${\mu }^{+}$ diffusion in the perovskite oxide lattice using ${\mathrm{KTaO}}_3$ cubic perovskite as a model system. We used the ${\mu }^{+}\mathrm{SR}$ method and density functional theory calculations along with the harmonic transition state theory to study this phenomenon both experimentally and theoretically. Experimental activation energies for global ${\mu }^{+}$ diffusion obtained below room temperature were less than a quarter of the calculated classical potential barrier height for a bottleneck ${\mu }^{+}$ transfer path. The reduction in the effective barrier height could be explained by the harmonic transition state theory with a zero-point energy correction; a significant difference in zero-point energies for ${\mu }^{+}$ at the positions in the $\mathrm{O}\mu$ bonding equilibrium state and a bond-breaking transition state was the primary cause of the reduction. This suggests that the assumption of immobile ${\mu }^{+}$ in solid oxides is not always satisfied since such a significant decrease in diffusion barrier height can also occur in other oxides.

We provide a prescription to construct Rényi and von Neumann entropy of a system of interacting fermions from a knowledge of its correlation functions. We show that Rényi entanglement entropy of interacting fermions in arbitrary dimensions can be represented by a Schwinger-Keldysh free energy on replicated manifolds with a current between the replicas. The current is local in real space and is present only in the subsystem which is not integrated out. This allows us to construct a diagrammatic representation of entanglement entropy in terms of connected correlators in the standard field theory with no replicas. This construction is agnostic to how the correlators are calculated, and one can use calculated, simulated, or measured values of the correlators in this formula. Using this diagrammatic representation, one can decompose entanglement into contributions which depend on the one-particle correlator, two-particle correlator, and so on. We provide an analytic formula for the one-particle contribution and a diagrammatic construction for higher-order contributions. We show how this construction can be extended for von Neumann entropy through analytic continuation. For a practical implementation of a quantum state, where one usually has information only about few-particle correlators, this provides an approximate way of calculating entanglement commensurate with the limited knowledge about the underlying quantum state.

External monitoring of quantum many-body systems can give rise to a measurement-induced phase transition characterized by a change in behavior of the entanglement entropy from an area law to an unbounded growth. In this paper, we show that this transition extends beyond bipartite correlations to multipartite entanglement. Using the quantum Fisher information, we investigate the entanglement dynamics of a continuously monitored quantum Ising chain. Multipartite entanglement exhibits the same phase boundaries observed for the entropy in the postselected no-click trajectory. Instead, quantum jumps give rise to a more complex behavior that still features the transition, but adds the possibility of having a third phase with logarithmic entropy but bounded multipartiteness.

The breakdown of Lieb-Robinson bounds in local, non-Hermitian quantum systems opens up the possibility for a rich landscape of quantum many-body phenomenology. We elucidate this by studying information scrambling and quantum chaos in non-Hermitian variants of paradigmatic local quantum spin-chain models. We utilize a mixture of exact diagonalization and tensor network techniques for our numerical results and focus on three dynamical quantities: (i) out-of-time-ordered correlators (OTOCs), (ii) operator entanglement of the dynamics, and (iii) entanglement growth following a quench from product initial states. We show that while OTOCs fail to capture information scrambling in a simple, local, non-Hermitian transverse-field Ising model, the closely related operator entanglement is a robust measure of dynamical properties of interest. Moreover, we show that the short-time growth of operator entanglement can generically detect entanglement phase transitions in these systems while its long-time average is shown to be a reliable indicator of quantum chaos and entanglement phases. This allows us to extend operator entanglement-based diagnostics from previous works on closed and open quantum systems, to the new arena of monitored quantum dynamics. Finally, we remark on the efficacy of these dynamical quantities in detecting integrability/chaos in the presence of continuous monitoring.

The concept of “deep thermalization” has recently been introduced to characterize moments of an ensemble of pure states, resulting from projective measurements on a subsystem, which lie beyond the purview of conventional eigenstate thermalization hypothesis (ETH). In this paper, we study deep thermalization in systems with kinetic constraints, such as the quantum East and the PXP models, which have been known to weakly break ETH by slow dynamics and high sensitivity to the initial conditions. We demonstrate a sharp contrast in deep thermalization between the first and higher moments in these models by studying quench dynamics from initial product states in the computational basis: While the first moment shows good agreement with ETH, higher moments deviate from the uniform Haar ensemble at infinite temperature. We show that such behavior is caused by an interplay of time-reversal symmetry and an operator that anticommutes with the Hamiltonian. We formulate sufficient conditions for violating deep thermalization, even for systems that are otherwise “thermal” in the ETH sense. By appropriately breaking these properties, we illustrate how the PXP model fully deep thermalizes for all initial product states in the thermodynamic limit. Our results highlight the sensitivity of deep thermalization as a probe of physics beyond ETH in kinetically-constrained systems.

We consider a paradigmatic solvable model of topological order in two dimensions, Kitaev's honeycomb Hamiltonian, and turn it into a measurement-only dynamics consisting of stochastic measurements of two-qubit bond operators. We find an entanglement phase diagram that resembles that of the Hamiltonian problem in some ways, while being qualitatively different in others. When one type of bond is dominantly measured, we find area-law entangled phases that protect two topological qubits (on a torus) for a time exponential in system size. This generalizes the recently proposed idea of Floquet codes, where logical qubits are dynamically generated by a time-periodic measurement schedule, to a stochastic setting. When all types of bonds are measured with comparable frequency, we find a critical phase with a logarithmic violation of the area law, which sharply distinguishes it from its Hamiltonian counterpart. The critical phase has the same set of topological qubits, as diagnosed by the tripartite mutual information, but protects them only for a time polynomial in system size. Furthermore, we observe an unusual behavior for the dynamical purification of mixed states, characterized at late times by the dynamical exponent $z=\frac{1}{2}$, a superballistic dynamics made possible by measurements.

In this theoretical Letter, we examine the influence of dimensionality, size reduction, and heat-transport direction on the phonon-drag contribution to the Seebeck coefficient of silicon nanostructures. Phonon-drag contribution, which arises from the momentum transfer between out-of-equilibrium phonon populations and charge carriers, significantly enhances the thermoelectric coefficient. Our implementation of the phonon-drag term accounts for the anisotropy of nanostructures, such as thin films and nanowires through the boundary- and momentum-resolved phonon lifetime. Our approach also takes into account the spin-orbit coupling, which turns out to be crucial for hole transport. We reliably quantify the phonon-drag contribution at various doping levels, temperatures, and nanostructure geometries for both electrons and holes in silicon nanostructures. Our results support the recent experimental findings, showing that a part of phonon-drag contribution survives in 100-nm silicon nanostructures.

Direct experimental detection of anyonic exchange statistics in fractional quantum Hall systems by braiding the excitations and measuring the wave-function phase is an enormous challenge. Here, we use a small, noisy quantum computer to emulate direct braiding within the framework of a simplified model applicable to a thin cylinder geometry and measure the topological phase. Our algorithm first prepares the ground state with two quasiholes. It then applies a unitary operation controlled by an ancilla, corresponding to a sequence of adiabatic evolutions that take one quasihole around the other. We finally extract the phase of the wave function from measuring the ancilla with a compound error mitigation strategy. Our results point out an avenue for studying braiding statistics in fractional Hall states.

The quantum dynamics of spin systems with uniform all-to-all interaction are often studied in the totally symmetric space (TSS) of maximal total spin. However, the TSS states are atypical in the full many-body Hilbert space. In this paper, we explore several aspects of the all-to-all quantum dynamics away from the TSS, and reveal surprising features of the “deep Hilbert space” (DHS). We study the out-of-time order correlator (OTOC) in the infinite-temperature ensemble of the full Hilbert space. We derive a phase-space representation of the DHS OTOC and show that the OTOC can have a superexponential initial growth in the large $N$ limit, due to the fast dynamics in an unbounded phase space (in finite systems, we observe numerically that the superexponential growth ends precociously and gives way to a power-law one until saturation). By a similar mechanism, the Krylov complexity grows explosively. We also study the entanglement growth in a quantum quench from a DHS product state, i.e., one of nonaligned spins that resemble the DHS infinite-temperature ensemble with respect to the statistics of the collective spins. Using a field-theoretical method, We exactly calculate the entanglement entropy in the large $N$ limit. We show that, in the DHS, fast OTOC growth does not imply fast entanglement growth, in contrast to the Zurek-Paz relation derived in the TSS.

We explore the statistical properties of energy transfer in ensembles of doubly driven random-matrix Floquet Hamiltonians based on universal symmetry arguments. The energy-pumping efficiency distribution $P\left(\overline{E}\right)$ is associated with the Hamiltonian parameter ensemble and the eigenvalue statistics of the Floquet operator. For specific Hamiltonian ensembles, $P\left(\overline{E}\right)$ undergoes a transition which cannot be associated with a symmetry breaking of the instantaneous Hamiltonian. The Floquet eigenvalue spacing distribution indicates the considered ensembles constitute generic nonintegrable Hamiltonian families. As a step towards Hamiltonian engineering, we develop a machine-learning classifier to understand the relative parameter importance in resulting high-conversion efficiency. We propose random Floquet Hamiltonians as a general framework to investigate frequency conversion effects in a class of generic dynamical processes beyond adiabatic pumps.

Barium zirconate $\left({\mathrm{BaZrO}}_3\right)$ is one of the very few perovskites that is claimed to retain an average cubic structure down to $0\mathrm{K}$, while being energetically very close to an antiferrodistortive phase obtained by condensation of a soft phonon mode at the $R$ point of the Brillouin zone boundary. In this work, we report a combined experimental and theoretical study of the temperature dependence of this soft phonon mode. Inelastic neutron and x-ray scattering measurements on single crystals show that it softens substantially from $9.4\mathrm{meV}$ at room temperature to $5.6\mathrm{meV}$ at $2\mathrm{K}$. In contrast, the acoustic mode at the same $R$ point is nearly temperature independent. The effect of the anharmonicity on the lattice dynamics is investigated nonperturbatively using direct dynamic simulations as well as a first-principles-based self-consistent phonon theory, including quantum fluctuations of the atomic motion. By adding cubic and quartic anharmonic force constants, quantitative agreement with the neutron data for the temperature dependence of the antiferrodistortive mode is obtained. The quantum fluctuations of the atomic motion are found to be important to obtain the proper temperature dependence at low temperatures. The mean-squared displacements of the different atoms are determined as function of temperature and are shown to be consistent with available experimental data. Adding anharmonicity to the computed fluctuations of the Ba-O distances also improves the comparison with available extended x-ray absorption fine structure data at $300\mathrm{K}$.

Van Roosbroeck's equations constitute a versatile tool to determine the dynamics of electrons under time- and space-dependent perturbations. Extensively utilized in ordinary semiconductors, their potential to model devices made from topological materials remains untapped. Here, we adapt van Roosbroeck's equations to theoretically study the bulk response of a Weyl semimetal to an ultrafast and spatially localized light pulse in the presence of a quantizing magnetic field. We predict a transient oscillatory photovoltage that originates from the chiral anomaly. The oscillations take place at the plasma frequency (THz range) and are damped by intervalley scattering and dielectric relaxation. Our results illustrate the ability of van Roosbroeck's equations to unveil the interplay between electronic band topology and fast carrier dynamics in microelectronic devices.

For families of Hamiltonians defined by parts that are local, the most general definition of a symmetry algebra is the commutant algebra, i.e., the algebra of operators that commute with each local part. Thinking about symmetry algebras as commutant algebras allows for the treatment of conventional symmetries and unconventional symmetries (e.g., those responsible for weak ergodicity breaking phenomena) on equal algebraic footing. In this work, we discuss two methods for numerically constructing this commutant algebra starting from a family of Hamiltonians. First, we use the equivalence of this problem to that of simultaneous block diagonalization of a given set of local operators and discuss a probabilistic method that has been found to work with probability 1 for both Abelian and non-Abelian symmetries or commutant algebras. Second, we map this problem onto the problem of determining frustration-free ground states of certain Hamiltonians, and we use ideas from tensor network algorithms to efficiently solve this problem in one dimension. These numerical methods are useful in detecting standard and nonstandard conserved quantities in families of Hamiltonians, which includes examples of regular symmetries, Hilbert space fragmentation, and quantum many-body scars, and we show many such examples. In addition, they are necessary for verifying several conjectures on the structure of the commutant algebras in these cases, which we have put forward in earlier works. Finally, we also discuss similar methods for the inverse problem of determining local operators with a given symmetry or commutant algebra, which connects to existing methods in the literature. A special case of this construction reduces to well-known “Eigenstate to Hamiltonian” methods for constructing Hermitian local operators that have a given state as an eigenstate.

We describe two approximation methods designed to capture the leading behavior of the low-energy tail of the momentum-dependent spectral density $A(\mathbf{k},E)$ and the tunneling density of states $D\left(E\right)$ for an injected particle, such as an electron or an exciton, interacting with a bath of phonons at a nonzero initial temperature $T$, including quantum corrections due to the nonzero frequencies of the relevant phonons. In our imaginary-time-dependent Hartree (ITDH) approximation, we consider a situation in which the particle is injected into a specified coherent state of the phonon system, and we show how one can use the ITDH approximation to obtain the correlation function $C\left(\tau \right)$ for that initial state. The thermal average $C\left(\tau \right)$ is obtained, in principle, by integrating the result over all possible initial phonon coherent states, weighted by a thermal distribution. However, in the low-energy tail, one can obtain a good first approximation by considering only initial states near the one that maximizes the integrand. Our second approximation, the fixed-wave-function (FWF) approximation, assumes that the wave function of the injected particle evolves instantaneously to a wave function which then is independent of time, while the phonon system continues to evolve due to interaction with the particle. We discuss how to invert the Laplace transform and how to obtain $A(\mathbf{k},E)$ as well as $D\left(E\right)$ from the imaginary-time analysis. The FWF approximation is used to calculate $D\left(E\right)$ for a one-dimensional continuum model of a particle interacting with acoustic phonons, and effects due to the quantum motion of phonons are observed. In the classical phonon limit, where the nuclear mass is taken to infinity while the elastic constants and other parameters are held fixed, the dominant behaviors of both the ITDH and FWF approximations in the low-energy tail reduce to that found in the past for a particle in a random potential with a Gaussian statistical distribution.

Ab initio Green-Kubo (aiGK) simulations of heat transport in solids allow for assessing lattice thermal conductivity in anharmonic or complex materials from first principles. In this work, we present a detailed account of their practical application and evaluation with an emphasis on noise reduction and finite-size corrections in semiconductors and insulators. To account for such corrections, we propose strategies in which all necessary numerical parameters are chosen based on the dynamical properties displayed during molecular dynamics simulations in order to minimize manual intervention. This paves the way for applying the aiGK method in semiautomated and high-throughput frameworks. The proposed strategies are presented and demonstrated for computing the lattice thermal conductivity at room temperature in the mildly anharmonic periclase MgO, and for the strongly anharmonic marshite CuI.

Quantum pathways inducing coherent lattice dynamics are studied in bismuth (Bi). A crystalline Bi film is excited by femtosecond midinfrared pulses and transient intensity changes on the (111) Bragg reflection are probed by hard x-ray pulses. A fast decrease and coherent oscillations of the diffracted intensity display up to 50% and 10% intensity change, respectively. The oscillation frequency of 2.6 THz is independent of pump intensity. Midinfrared excitation opens different quantum pathways for electron-hole generation, such as field-driven carrier tunneling at the $L$ points, which reduces the crystal symmetry and leads to optical phonon excitation at the $X$ point with the strongest electron-phonon coupling.

High-order sideband generation (HSG), as an analog of the interband processes in high-harmonic generation (HHG) in solids, is a nonperturbative nonlinear optical phenomenon in semiconductors that are simultaneously driven by a relatively weak near-infrared (NIR) laser and a sufficiently strong terahertz (THz) field. We derive an explicit formula for sideband polarization vectors in a prototypical two-band model based on the saddle-point method. Our formula connects the sideband amplitudes with the laser-field parameters, electronic structures, and nonequilibrium dephasing rates in a highly nontrivial manner. Our results indicate the possibility of extracting information on band structures and dephasing rates from high-order sideband generation experiments with simple algebraic calculations. We also expect our approach to be useful on the quantitative understanding of the interband HHG.

The electric field of an intense laser pulse can directly modify the electronic properties of materials via electromodulation up to the petahertz regime. In this regime, the energy of the quiver motion of the electron-hole pair is comparable with the photon energy, which results in complex nonadiabatic dynamics. This regime opens opportunities to probe the electronic structure of materials on the attosecond timescale. Here, we show how the quasistatic electromodulation spectroscopy based on the Franz-Keldysh effect (FKE) connects with its nonadiabatic limit, which we find to be determined by resonant transitions between Floquet-Bloch states. This insight can be applied to measure the effective mass, ponderomotive and binding energies of the electron-hole pair on a few-femtosecond timescale. We demonstrate this by experimentally investigating laser-field-driven fused silica, a prototypical material for light-wave electronics, with extreme-ultraviolet attosecond pulse trains. We reproduce the experimental transient absorption spectra with an effective band structure and a dynamical Franz-Keldysh model, offering a simple parametrization for a theoretically challenging but technologically abundant material. Ab initio calculations in α-quartz highlight the contributions of specific bands, symmetry, and crystal orientation that are hidden in the experimental data due to randomized crystallographic orientation and finite temporal and spatial coherence. We show that the dynamical FKE can be explained as a third-order nonlinear process in the weak-field regime. The delay-dependent position of the absorption maxima and minima has a minimum tilt angle, determined by transitions between the underlying Floquet-Bloch states. In our analysis, we discuss the main experimental observables and show their connection to the parameters of the solid, providing the basis for nonadiabatic electromodulation spectroscopy.

A strongly interacting plasma of linearly dispersing electron and hole excitations in two spatial dimensions (2D), also known as a Dirac fluid, can be captured by relativistic hydrodynamics and shares many universal features with other quantum critical systems. We propose a one-dimensional (1D) model to capture key aspects of the 2D Dirac fluid while including lattice effects and being amenable to nonperturbative computation. When interactions are added to the Dirac-like 1D dispersion without opening a gap, we show that this kind of irrelevant interaction is able to preserve Fermi-liquid-like quasiparticle features while relaxing a zero-momentum charge current via collisions between particle-hole excitations, leading to resistivity that is linear in temperature via a mechanism previously discussed for large-diameter metallic carbon nanotubes. We further provide a microscopic lattice model and obtain numerical results via density-matrix renormalization group simulations, which support the above physical picture. The limits on such fast relaxation at strong coupling are of considerable interest because of the ubiquity of bad metals in experiments.