We investigate the phenomenon of disorder-free localization in quantum systems with global permutation symmetry. We use permutation group theory to systematically construct permutation-symmetric many-fermion Hamiltonians and interpret them as generators of continuous-time quantum walks. When the number of fermions is very large we find that all the canonical basis states localize at all times, without the introduction of any disorder coefficients. This time-independent localization is not the result of any emergent disorder, distinguishing it from existing mechanisms for disorder-free localization. Next we establish the conditions under which the localization is preserved. We find that interactions that preserve and break the global permutation symmetry sustain localization. Furthermore, the basis states of systems with reduced permutation symmetry localize even for a small number of fermions when the symmetry-reducing parameters are tuned accordingly. We show that similar localization also occurs for a permutation-symmetric Heisenberg spin chain and permutation-symmetric bosonic systems, implying that the localization is independent of the superselected symmetry. Finally, we make connections of the Hamiltonians studied here to the adjacency matrices of graphs and use this to propose a prescription for disorder-free localization in continuous-time quantum walk systems. Many of the models proposed here feature all-to-all connectivity and can be potentially realized on superconducting quantum circuits, trapped ion systems, and ultracold atoms.
We investigate the phenomenon of disorder-free localization in quantum systems with global permutation symmetry. We use permutation group theory to systematically construct permutation-symmetric many-fermion Hamiltonians and interpret them as generators of continuous-time quantum walks. When the number of fermions is very large we find that all the canonical basis states localize at all times, without the introduction of any disorder coefficients. This time-independent localization is not the result of any emergent disorder, distinguishing it from existing mechanisms for disorder-free localization. Next we establish the conditions under which the localization is preserved. We find that interactions that preserve and break the global permutation symmetry sustain localization. Furthermore, the basis states of systems with reduced permutation symmetry localize even for a small number of fermions when the symmetry-reducing parameters are tuned accordingly. We show that similar localization also occurs for a permutation-symmetric Heisenberg spin chain and permutation-symmetric bosonic systems, implying that the localization is independent of the superselected symmetry. Finally, we make connections of the Hamiltonians studied here to the adjacency matrices of graphs and use this to propose a prescription for disorder-free localization in continuous-time quantum walk systems. Many of the models proposed here feature all-to-all connectivity and can be potentially realized on superconducting quantum circuits, trapped ion systems, and ultracold atoms.
For quantum search via the continuous-time quantum walk, the evolution of the whole system is usually limited in a small subspace. In this paper, we discuss how the symmetries of the graphs are related to the existence of such an invariant subspace, which also suggests a dimensionality reduction method based on group representation theory. We observe that in the one-dimensional subspace spanned by each desired basis state which assembles the identically evolving original basis states, we always get a trivial representation of the symmetry group. So, we could find the desired basis by exploiting the projection operator of the trivial representation. Besides being technical guidance in this type of problem, this discussion also suggests that all the symmetries are used up in the invariant subspace and the asymmetric part of the Hamiltonian is very important for the purpose of quantum search.
Symmetries are a key tool in understanding quantum systems, and, among many other things, can be exploited to increase the efficiency of numerical simulations of quantum dynamics. Disordered systems usually feature reduced symmetries and additionally require averaging over many realizations, making their numerical study computationally demanding. However, when studying quantities linear in the time-evolved state, i.e., expectation values of observables, one can apply the averaging procedure to the time evolution operator, resulting in an effective dynamical map, which restores symmetry at the level of superoperators. In this work, we develop schemes for efficiently constructing symmetric sectors of the disorder-averaged dynamical map using short-time and weak-disorder expansions. To benchmark the method, we apply it to an Ising model with random all-to-all interactions in the presence of a transverse field. After disorder averaging, this system becomes effectively permutation-invariant, and thus the size of the symmetric subspace scales polynomially in the number of spins, allowing for the simulation of large systems.
Disorder-free localization is a recently discovered phenomenon of nonergodicity that can emerge in quantum many-body systems hosting gauge symmetries when the initial state is prepared in a superposition of gauge superselection sectors.Thermalization is then prevented up to all accessible evolution times despite the model being nonintegrable and translation invariant.In a recent work [Halimeh et al., arXiv:2111.02427(2021)], it has been shown that terms linear in the gauge-symmetry generator stabilize disorder-free localization in U(1) gauge theories against gauge errors that couple different superselection sectors.Here, we show in the case of Z 2 gauge theories that disorder-free localization can not only be stabilized, but also enhanced by the addition of translation-invariant terms linear in a local Z 2 pseudogenerator that acts identically to the full generator in a single superselection sector, but not necessarily outside of it.We show analytically and numerically how this leads through the quantum Zeno effect to the dynamical emergence of a renormalized gauge theory with an enhanced local symmetry, which contains the Z 2 gauge symmetry of the ideal model, associated with the Z 2 pseudogenerator.The resulting proliferation of superselection sectors due to this dynamically emergent gauge theory creates an effective disorder greater than that in the original model, thereby enhancing disorder-free localization.We demonstrate the experimental feasibility of the Z 2 pseudogenerator by providing a detailed readily implementable experimental proposal for the observation of disorder-free localization in a Rydberg setup.
Localization due to the presence of disorder has proven crucial for our current understanding of relaxation in isolated quantum systems. The many-body localized phase constitutes a robust alternative to the thermalization of complex interacting systems, but recently the importance of disorder has been brought into question. A number of disorder-free localization mechanisms have been put forward connected to local symmetries of lattice gauge theories. Here, starting from translationally invariant (1+1)-dimensional quantum electrodynamics, we modify the dynamics of the gauge field which allows us to construct a lattice model with a U(1) local gauge symmetry revealing a mechanism of disorder-free localization. We consider two different discretizations of the continuum model resulting in a free-fermion soluble model in one case and an interacting model in the other. We diagnose the localization of our translationally invariant model in the far-from-equilibrium dynamics following a global quantum quench.
Strongly interacting quantum many-body systems are expected to thermalize, however, some evade thermalization due to symmetries. Quantum synchronization provides one such example of ergodicity breaking, but previous studies have focused on open systems. Here, motivated by the problem of ergodicity breaking in closed systems and the study of non-trivial dynamics, we investigate synchronization in a closed disordered Heisenberg spin chain. In the presence of large random disorder, strongly breaking the permutation symmetry of the system, we observe the emergence of spatial synchronization, where spins lock into locally synchronized patches. This behavior can be interpreted as a fragmentation of the global dynamical symmetry S+ into a collection of local dynamical symmetries, each characterized by a distinct frequency. In the weak-disorder regime, still without permutation symmetry, we show that the synchronization mechanism can be understood perturbatively within Krylov space. In the absence of disorder, the Krylov space associated with the dynamical symmetry S+ is two-dimensional. Introducing disorder couples this subspace to the remainder of the Krylov space. This coupling leads only to a second-order correction to the frequency of the dynamical symmetry, thereby preserving coherent oscillations despite the presence of small disorder. At stronger disorder, the perturbation modifies S+ so that it acquires a finite lifetime, providing an example of a transient dynamical symmetry.
We analyse a continuous-time quantum walk on a chimera graph, which is a graph of choice for designing quantum annealers, and we discover beautiful quantum walk features such as localization that starkly distinguishes classical from quantum behaviour. Motivated by technological thrusts, we study continuous-time quantum walk on enhanced variants of the chimera graph and on diminished chimera graph with a random removal of vertices. We explain the quantum walk by constructing a generating set for a suitable subgroup of graph isomorphisms and corresponding symmetry operators that commute with the quantum walk Hamiltonian; the Hamiltonian and these symmetry operators provide a complete set of labels for the spectrum and the stationary states. Our quantum walk characterization of the chimera graph and its variants yields valuable insights into graphs used for designing quantum-annealers.