Hopf insulators are exotic topological states of matter outside the standard ten-fold way classification based on discrete symmetries. Its topology is captured by an integer invariant that describes the linking structures of the Hamiltonian in the three-dimensional momentum space. In this paper, we investigate the quantum dynamics of Hopf insulators across a sudden quench and show that the quench dynamics is characterized by a $\mathbb{Z}_2$ invariant $ν$ which reveals a rich interplay between quantum quench and static band topology. We construct the $\mathbb{Z}_2$ topological invariant using the loop unitary operator, and prove that $ν$ relates the pre- and post-quench Hopf invariants through $ν=(\mathcal{L}-\mathcal{L}_0)\bmod 2$. The $\mathbb{Z}_2$ nature of the dynamical invariant is in sharp contrast to the $\mathbb{Z}$ invariant for the quench dynamics of Chern insulators in two dimensions. The non-trivial dynamical topology is further attributed to the emergence of $π$-defects in the phase band of the loop unitary. These $π$-defects are generally closed curves in the momentum-time space, for example, as nodal rings carrying Hopf charge.
A quantum quench is a nonequilibrium dynamics governed by the unitary evolution. We propose a two-band model whose quench dynamics is characterized by an arbitrary Hopf number belonging to the homotopy group $π_{3}(S^{2})=\mathbb{Z}$. When we quench a system from an insulator with the Chern number $C_{i}\in π_{2}(S^{2})=\mathbb{Z}$ to another insulator with the Chern number $C_{f} $, the preimage of the Hamiltonian vector forms links having the Hopf number $C_{f}-C_{i}$. We also investigate a quantum-quench dynamics for a four-band model carrying an arbitrary second-Chern number $N\in π_{4}(S^…
We present a topological characterization of time-periodically driven two-band models in 2+1 dimensions as Hopf insulators. The intrinsic periodicity of the Floquet system with respect to both time and the underlying two-dimensional momentum space constitutes a map from a three dimensional torus to the Bloch sphere. As a result, we find that the driven system can be understood by appealing to a Hopf map that is directly constructed from the micromotion of the drive. Previously found winding numbers are shown to correspond to Hopf invariants, which are associated with linking numbers describing…
In a recent experimental work [Z. Wu et al., Science 354, 83 (2016)], the PKU-USTC group realized a two-dimensional two-band quantum anomalous Hall model on a square Raman lattice. By quenching the atom-laser detuning of such a Raman lattice, the time-dependent Bloch vectors for each quasimomentum points define a Hopf mapping from quasimomentum-time space $(k_x,k_y,t)\in T^3$ to the Bloch sphere $S^2$. The Hopf links between the preimages of any two Bloch vectors on $S^2$ can be measured experimentally through Bloch state tomography using spin-resolved time-of-flight measurement together with …
We propose to uncover the topology of a pseudo-Hermitian Chern insulator by quantum quench dynamics. The Bloch Hamiltonian of the pseudo-Hermitian Chern insulator is defined in the basis of the q-deformed Pauli matrices, which are related to the representation of the deformed algebras. We show the bulk-surface duality of the pseudo-Hermitian phases, then further build a concrete relation between the static band topology and quench dynamics, in terms of the time-averaged spin textures. The results are also generalized into a fully nonequilibrium case where the postquench evolution is governed b…
We predict the existence of a novel Floquet topological insulator in three-dimensional two-band systems, the Floquet Hopf insulator, which possesses two distinct topological invariants. One is the Hopf $\mathbb{Z}$ invariant, a linking number characterizing the (non-driven) Hopf topological insulator. The second invariant is an intrinsically Floquet $\mathbb{Z}_2$ invariant, and represents a condensed matter realization of the topology underlying the Witten anomaly in particle physics. Both invariants arise from topological defects in the system's time-evolution, subject to a process in which …
A dynamical Hopf insulator is experimentally synthesized with a quenched two-dimensional quantum anomalous Hall system on a square Raman lattice. The quench dynamics for the quasimomentum-time-dependent Bloch vectors defines a Hopf map from $(q_x,q_y,t)\in T^3$ to the Bloch sphere $S^2$. In this Hopf map, a dynamical Hopf number can be defined, and it exactly equals the Chern number of the post-quench Hamiltonian. We experimentally measure the Hopf link between the fibers for the North and South Poles on $S^2$, which are the trajectories in $(q_x,q_y,t)$ space with maximal spin polarization, t…
The Hopf insulator is a weak topological insulator characterized by an insulating bulk with conducting edge states protected by an integer-valued linking number invariant. The state exists in three-dimensional two-band models. We demonstrate that the Hopf insulator can be naturally realized in lattices of dipolar-interacting spins, where spin exchange plays the role of particle hopping. The long-ranged, anisotropic nature of the dipole-dipole interactions allows for the precise detail required in the momentum-space structure, while different spin orientations ensure the necessary structure of …
We study a linear ramp of the nearest-neighbor tunneling rate in the Bose-Hubbard model driving the system from the Mott insulator state into the superfluid phase. We employ the truncated Wigner approximation to simulate linear quenches of a uniform system in 1,2, and 3 dimensions, and in a harmonic trap in 3 dimensions. In all these setups the excitation energy decays like one over third root of the quench time. The -1/3 scaling arises from an impulse-adiabatic approximation - a variant of the Kibble-Zurek mechanism - describing a crossover from non-adiabatic to adiabatic evolution when the s…
Dynamical quantum phase transitions (DQPTs) represent a counterpart in non-equilibrium quantum time evolution of thermal phase transitions at equilibrium, where real time becomes analogous to a control parameter such as temperature. In quenched quantum systems, recently the occurrence of DQPTs has been demonstrated, both with theory and experiment, to be intimately connected to changes of topological properties. Here, we contribute to broadening the systematic understanding of this relation between topology and DQPTs to multi-orbital and disordered systems. Specifically, we provide a detailed …
Dynamical quantum phase transitions (DQPTs) are topologically characterized in quantum quench dynamics in topological systems. In this paper, we study Loschmidt amplitudes and DQPTs in quantum quenches in mirror-symmetric topological phases. Based on the topological classification of mirror-symmetric insulators, we show that mirror symmetry creates symmetry-protected DQPTs. If mirror symmetry is present, topologically robust DQPTs can occur in quantum quenches, even in high-dimensional time-reversal invariant systems. Then, we also show that symmetry-protected DQPTs occur in quenches in two-di…
In this work we study a quench between a Mott insulator and a repulsive Lieb-Liniger liquid. We find explicitly the stationary state when a long time has passed after the quench. It is given by a GGE density matrix which we completely characterize, calculating the quasiparticle density describing the system after the quench. In the long time limit we find an explicit form for the local three body density density density correlation function and the asymptotic long distance limit of the density density correlation function. The later is shown to have a Gaussian decay at large distances.
We study quenches across the Bose-Hubbard Mott-insulator-to-superfluid quantum phase transition using an ultra-cold atomic gas trapped in an optical lattice. Quenching from the Mott insulator to superfluid phase is accomplished by continuously tuning the ratio of Hubbard tunneling to interaction energy. Excitations of the condensate formed after the quench are measured using time-of-flight imaging. We observe that the degree of excitation is proportional to the fraction of atoms that cross the phase boundary, and that the quantity of excitations and energy produced during the quench have a pow…
The existing paradigm for topological insulators asserts that an energy gap separates conduction and valence bands with opposite topological invariants. Here, we propose that \textit{equal}-energy bands with opposite Chern invariants can be \textit{spatially} separated -- onto opposite facets of a finite crystalline Hopf insulator. On a single facet, the number of curvature quanta is in one-to-one correspondence with the bulk homotopy invariant of the Hopf insulator -- this originates from a novel bulk-to-boundary flow of Berry curvature which is \textit{not} a type of Callan-Harvey anomaly in…
We propose a simple scheme for tomography of band-insulating states in one- and two-dimensional optical lattices with two sublattice states. In particular, the scheme maps out the Berry curvature in the entire Brillouin zone and extracts topological invariants such as the Chern number. The measurement relies on observing---via time-of-flight imaging---the time evolution of the momentum distribution following a sudden quench in the band structure. We consider two examples of experimental relevance: the Harper model with $π$-flux and the Haldane model on a honeycomb lattice. Moreover, we illustr…
Mott insulator to superfluid quenches have been used by recent experiments to generate exotic superfluid phases. While the final Hamiltonian following the sudden quench is that of a superfluid, it is not a priori clear how close the final state of the system approaches the ground state of the superfluid Hamiltonian. To understand the nature of the final state we calculate the temporal evolution of the momentum distribution following a Mott insulator to superfluid quench. Using the numerical infinite time-evolving block decimation approach and the analytical rotor model approximation we establi…
We apply the theory of inhomogeneous Kibble-Zurek mechanism to understand quench dynamics from the Mott insulator to the superfluid in a cold Bose gases confined in both a two-dimensional optical lattice and a harmonic trap. The local quench time and the freeze-out region associated with the nonadiabatic transition take a nontrivial positional dependence due to the Mott-lobe structure of the ground state phase diagram of the Bose-Hubbard model. We demonstrate that the quench dynamics through the time-dependent Gutzwiller simulations, revealing inhomogeneous properties of the growth of the supe…
We provide a rigorous derivation of Gutzwiller mean-field dynamics for lattice bosons, showing that it is exact on fully connected lattices. We apply this formalism to quenches in the interaction parameter from the Mott insulator to the superfluid state. Although within mean-field the Mott insulator is a steady state, we show that a dynamical critical interaction $U_d$ exists, such that for final interaction parameter $U_f>U_d$ the Mott insulator is exponentially unstable towards emerging long-range superfluid order, whereas for $U_f<U_d$ the Mott insulating state is stable. We discuss the imp…
We explore the non-equilibrium response of Chern insulators. Focusing on the Haldane model, we study the dynamics induced by quantum quenches between topological and non-topological phases. A notable feature is that the Chern number, calculated for an infinite system, is unchanged under the dynamics following such a quench. However, in finite geometries, the initial and final Hamiltonians are distinguished by the presence or absence of edge modes. We study the edge excitations and describe their impact on the experimentally-observable edge currents and magnetization. We show that, following a …
We study the coherent non-equilibrium dynamics of interacting two-dimensional systems after a quench from a trivial to a topological Chern insulator phase. While the many-body wavefunction is constrained to remain topologically trivial under local unitary evolution, we find that the Hall response of the system can dynamically approach a thermal value of the post-quench Hamiltonian, even though the efficiency of this thermalization process is shown to strongly depend on the microscopic form of the interactions. Quite remarkably, the effective temperature of the steady state Hall response can be…
We investigate slow quenches in Chern insulators in ribbon geometry. We consider the Qi-Wu-Zhang model and slowly ramp the parameters (large time of the quench $τ$) from a non-topological (Chern number = 0) to a topological regime (Chern number $\ne$ 0). In contrast to the Haldane model considered in [L. Privitera and G. E. Santoro, Phys. Rev. B 93, 241406(R) (2016)] earlier, the in-gap state degeneracy point is pinned to an inversion symmetric momentum, which changes the behavior drastically. The density of excitations in the in-gap states scales with the quench time as $τ^{-1/2}$ as the ramp…
We present a quantitative, near-term experimental blueprint for the quantum simulation of topological insulators using lattice-trapped ultracold polar molecules. In particular, we focus on the so-called Hopf insulator, which represents a three-dimensional topological state of matter existing outside the conventional tenfold way and crystalline-symmetry-based classifications of topological insulators. Its topology is protected by a \emph{linking number} invariant, which necessitates long-range spin-orbit coupled hoppings for its realization. While these ingredients have so far precluded its rea…
We classify the topology of quench dynamics by homotopy groups. A relation between the topological invariant of a post-quench order parameter and the topological invariant of a static Hamiltonian is shown in one, two and three dimensions. The mid-gap states in the entanglement spectrum of the post-quench state reveal its topological nature. When a trivial quantum state under a sudden quench to a Chern insulator, the mid-gap states in entanglement spectrum form rings. These rings are analogous to the boundary Fermi rings in the Hopf insulators. Finally, we show a post-quench state in 3+1 dimens…
We discuss the topological invariant in the (2+1)-dimensional quench dynamics of a two-dimensional two-band Chern insulator starting from a topological initial state (i.e., with a nonzero Chern number $c_i$), evolved by a post-quench Hamiltonian (with Chern number $c_f$). In contrast to the process with $c_i=0$ studied in previous works, this process cannot be characterized by the Hopf invariant that is described by the sphere homotopy group $π_3(S^2)=\mathbb{Z}$. It is possible, however, to calculate a variant of the Chern-Simons integral with a complementary part to cancel the Chern number o…
Recent experiments began to explore the topological properties of quench dynamics, i.e. the time evolution following a sudden change in the Hamiltonian, via tomography of quantum gases in optical lattices. In contrast to the well established theory for static band insulators or periodically driven systems, at present it is not clear whether, and how, topological invariants can be defined for a general quench of band insulators. Previous work solved a special case of this problem beautifully using Hopf mapping of two-band Hamiltonians in two dimensions. But it only works for topologically trivi…
In this letter we show how the topological number of a static Hamiltonian can be measured from a dynamical quench process. We focus on a two-band Chern insulator in two-dimension, for instance, the Haldane model, whose dynamical process can be described by a mapping from the $[k_x,k_y,t]$ space to the Bloch sphere, characterized by the Hopf invariant. Such a mapping has been constructed experimentally by measurements in cold atom systems. We show that, taking any two constant vectors on the Bloch sphere, their inverse images of this mapping are two trajectories in the $[k_x,k_y,t]$ space, and …
We predict the existence of novel collapse and revival oscillations that are a distinctive signature of the short-range off-diagonal coherence associated with particle-hole pairs in Mott insulator states. Starting with an atomic Mott state in a one-dimensional optical lattice, suddenly raising the lattice depth freezes the particle-hole pairs in place and induces phase oscillations. The peak of the quasi-momentum distribution, revealed through time of flight interference, oscillates between a maximum occupation at zero quasi-momentum (the $Γ$ point) and the edge of the Brillouin zone. We show …
We compute the dynamics of localized excitations produced by a quantum quench in the spin 1/2 XXZ chain. Using numerics combining the density matrix renormalization group and exact time evolution, as well as analytical arguments, we show that fractionalization due to interactions in the pre-quench state gives rise to "ultrarelativistic" density waves that travel at the maximum band velocity. The system is initially prepared in the ground state of the chain within the gapless XY phase, which admits a Luttinger liquid (LL) description at low energies and long wavelengths. The Hamiltonian is then…
We study the Loschmidt echo for quenches in open one-dimensional lattice models with symmetry protected topological phases. For quenches where dynamical quantum phase transitions do occur we find that cusps in the bulk return rate at critical times tc are associated with sudden changes in the boundary contribution. For our main example, the Su-Schrieffer-Heeger model, we show that these sudden changes are related to the periodical appearance of two eigenvalues close to zero in the dynamical Loschmidt matrix. We demonstrate, furthermore, that the structure of the Loschmidt spectrum is linked to…
We consider the steady state behavior of observables in the Su-Schrieffer-Heeger model and in the one dimensional transverse field quantum Ising model after a sudden quantum quench of the parameter controlling the gap. In the thermodynamic limit, and for times $t\rightarrow\infty$, we find non-analyticities even in simple local observables as a function of the quench parameter, that is, a non-equilibrium quantum phase transition. We trace the appearance of this non-equilibrium quantum phase transition to an effective metal-insulator transition which occurs on the level of the generalized Gibbs…
The nonequilibrium ultracold bosonic quantum dynamics in finite optical lattices of unit filling following a linear interaction quench from a superfluid to a Mott insulator state and vice versa is investigated. The resulting dynamical response consists of various inter and intraband tunneling modes. We find that the competition between the quench rate and the interparticle repulsion leads to a resonant dynamical response, at moderate ramp times, being related to avoided crossings in the many-body eigenspectrum with varying interaction strength. Crossing the regime of weak to strong interaction…
We study the dynamical process of equilibration of topological properties in quantum many-body systems undergoing a parameter quench between two topologically inequivalent Hamiltonians. This scenario is motivated by recent experiments on ultracold atomic gases, where a trivial initial state is prepared before the Hamiltonian is ramped into a topological insulator phase. While the many-body wave function must stay topologically trivial in the coherent post-quench dynamics, here we show how the topological properties of the single particle density matrix dynamically change and equilibrate in the…
We discuss a scheme to measure the many-body entanglement growth during quench dynamics with bosonic atoms in optical lattices. By making use of a 1D or 2D setup in which two copies of the same state are prepared, we show how arbitrary order Renyi entropies can be extracted using tunnel-coupling between the copies and measurement of the parity of on-site occupation numbers, as has been performed in recent experiments. We illustrate these ideas for a Superfluid-Mott insulator quench in the Bose-Hubbard model, and also for hard-core bosons, and show that the scheme is robust against imperfection…
We study the Hall conductance of a Chern insulator after a global quench of the Hamiltonian. The Hall conductance in the long time limit is obtained by applying the linear response theory to the diagonal ensemble. It is expressed as the integral of the Berry curvature weighted by the occupation number over the Brillouin zone. We identify a topologically driven nonequilibrium phase transition, which is indicated by the nonanalyticity of the Hall conductance as a function of the energy gap m_f in the post-quench Hamiltonian H_f. The topological invariant for the quenched state is the winding num…
The effects of downfolding a Brillouin zone can open gaps and quench the kinetic energy by flattening bands. Quasiperiodic systems are extreme examples of this process, which leads to new phases and critical eigenstates. We analytically and numerically investigate these effects in a two dimensional topological insulator with a quasiperiodic potential and discover a complex phase diagram. We study the nature of the resulting eigenstate quantum phase transitions; a quasiperiodic potential can make a trivial insulator topological and induce topological insulator-to-metal phase transitions through…
We study the quasiadiabatic dynamics of a one-dimensional system of ultracold bosonic atoms loaded in an optical superlattice. Focusing on a slow linear variation in time of the superlattice potential, the system is driven from a conventional Mott insulator phase to a superlattice-induced Mott insulator, crossing in between a gapless critical superfluid region. Due to the presence of a gapless region, a number of defects depending on the velocity of the quench appear. Our findings suggest a power-law dependence similar to the Kibble-Zurek mechanism for intermediate values of the quench rate. F…
We explore the physics of a Chern insulator subjected to a two step Floquet drive. We analytically obtain the phase diagram and show that the system can exhibit different topological phases characterized by presence and chirality of edge-modes in the two bulk gaps of the Floquet quasienergy spectrum, around $0$ and $π$. We find that the phase of the system depends on the mean but not on the amplitude of the drive. The bulk topological invariants characterizing the phases can be extracted by mapping the unitary evolution within a time period to an energetically trivial but topologically non-tri…
Direct measurement of a bulk topological observable in topological phase of matter has been a long-standing issue. Recently, detection of bulk topology through quench dynamics has attracted growing interests. Here, we propose that topological characters of a quantum quadrupole insulator can be read out by quench dynamics. Specifically, we introduce a quantity, a quadrupole moment weighted by the eigenvalues of the chiral operator, which takes zero for the trivial phase and finite for the quadrupolar topological phase. By utilizing an efficient numerical method to track the unitary time evoluti…
We propose an innovative quantum emulator based on Moire superlattices showing that, by employing periodical modulation on each lattice site, one can create tunable, artificial gauge fields with imprinting Peierls phases on the hopping parameters and realize an analog of novel Haldane-like phase. As an application, we provide a methodology to directly quantify the topological invariant in such a system from a dynamical quench process. This design shows a robustly integrated platform which opens a new door to investigate topological physics.
We analyze the time evolution of the Bose-Hubbard model after a sudden quantum quench to a weakly interacting regime. Specifically, motivated by a recent experiment at Kyoto University, we numerically simulate redistribution of the kinetic and onsite-interaction energies at an early time, which was observed in non-equilibrium dynamics of ultracold Bose gases in a cubic optical lattice starting with a singly-occupied Mott-insulator state. In order to compute the short-time dynamics corresponding to the experimental situation, we apply the truncated-Wigner approximation (TWA) to the Bose-Hubbard…
We present an experimental study on non-equilibrium dynamics of a spinor condensate after it is quenched across a superfluid to Mott insulator (MI) phase transition in cubic lattices. Intricate dynamics consisting of spin-mixing oscillations at multiple frequencies are observed in time evolutions of the spinor condensate localized in deep lattices after the quantum quench. Similar spin dynamics also appear after spinor gases in the MI phase are suddenly moved away from their ground states via quenching magnetic fields. We confirm these observed spectra of spin-mixing dynamics can be utilized t…
Simulating the real-time evolution of quantum spin systems far out of equilibrium poses a major theoretical challenge, especially in more than one dimension. We experimentally explore the dynamics of a two-dimensional Ising spin system with transverse and longitudinal fields as we quench it across a quantum phase transition from a paramagnet to an antiferromagnet. We realize the system with a near unit-occupancy atomic array of over 200 atoms obtained by loading a spin-polarized band insulator of fermionic lithium into an optical lattice and induce short-range interactions by direct excitation…
Bosons and fermions, in the presence of frustration or background gauge fields, can form manybody ground states that support equilibrium 'charge' or 'spin' currents. Motivated by the experimental creation of frustration or artificial gauge fields in ultracold atomic systems, we propose a general scheme by which making a sudden anisotropic quench of the atom tunneling across the lattice and tracking the ensuing density modulations provides a powerful and gauge invariant route to visualizing diverse equilibrium current patterns. Using illustrative examples of trapped superfluid Bose and normal F…
Kibble-Zurek mechanism (KZM) uses critical scaling to predict density of topological defects and other excitations created in second order phase transitions. We point out that simply inserting asymptotic critical exponents deduced from the immediate vicinity of the critical point to obtain KZM predictions can lead to results that are inconsistent with a more careful KZM-like analysis based on causality -- on the comparison of the relaxation time of the order parameter with the time "distance" from the critical point. As a result, scaling of quench-generated excitations with quench rates can ex…
We study the dynamics of bosonic atoms on a two-dimensional square lattice, where atomic interactions are long-ranged with either a box or soft-core shape. The latter can be realized through laser dressing ground-state atoms to electronically excited Rydberg states. When the range of interactions is equal or larger than the lattice constant, the system is governed by an extended Bose-Hubbard model. We propose a quench process by varying the atomic hopping linearly across phase boundaries of the Mott insulator-supersolid and supersolid-superfluid phases. Starting from a Mott insulating state, t…
