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Alioscia Hamma

Publications and source records attributed to Alioscia Hamma.

At least 91 records · Page 5Linked to original sources
arXiv2013

Probing topological order with Rényi entropy

We present an analytical study of the quantum phase transition between the topologically ordered toric-code-model ground state and the disordered spin-polarized state. The phase transition is induced by applying an external magnetic field, and the variation in topological order is detected via two non-local quantities: the Wilson loop and the topological Renyi entropy of order 2. By exploiting an equivalence with the transverse-field Ising model and considering two different variants of the problem, we investigate the field dependence of these quantities by means of an exact treatment in the exactly solvable variant and complementary perturbation theories around the limits of zero and infinite fields in both variants. We find strong evidence that the phase transition point between topological order and disorder is marked by a discontinuity in the topological Renyi entropy and that the two phases around the phase transition point are characterized by its different constant values. Our results therefore indicate that the topological Renyi entropy is a proper topological invariant: its allowed values are discrete and can be used to distinguish between different phases of matter.

quant-ph↗
arXiv2012

Topological order, entanglement, and quantum memory at finite temperature

We compute the topological entropy of the toric code models in arbitrary dimension at finite temperature. We find that the critical temperatures for the existence of full quantum (classical) topological entropy correspond to the confinement-deconfinement transitions in the corresponding Z_2 gauge theories. This implies that the thermal stability of topological entropy corresponds to the stability of quantum (classical) memory. The implications for the understanding of ergodicity breaking in topological phases are discussed.

quant-ph↗
arXiv2012

Ensembles of physical states and random quantum circuits on graphs

In this paper we continue and extend the investigations of the ensembles of random physical states introduced in A. Hamma et al. [ http://prl.aps.org/abstract/PRL/v109/i4/e040502 Phys. Rev. Lett. 109, 040502 (2012)]. These ensembles are constructed by finite-length random quantum circuits (RQC) acting on the (hyper)edges of an underlying (hyper)graph structure. The latter encodes for the locality structure associated with finite-time quantum evolutions generated by physical i.e., local, Hamiltonians. Our goal is to analyze physical properties of typical states in these ensembles, in particular here we focus on proxies of quantum entanglement as purity and $α$-Renyi entropies. The problem is formulated in terms of matrix elements of superoperators which depend on the graph structure, choice of probability measure over the local unitaries and circuit length. In the $α=2$ case these superoperators act on a restricted multi-qubit space generated by permutation operators associated to the subsets of vertices of the graph. For permutationally invariant interactions the dynamics can be further restricted to an exponentially smaller subspace. We consider different families of RQCs and study their typical entanglement properties for finite-time as well as their asymptotic behavior. We find that area law holds in average and that the volume law is a typical property (that is, it holds in average and the fluctuations around the average are vanishing for the large system) of physical states. The area law arises when the evolution time is O(1) with respect to the size $L$ of the system, while the volume law arises as typical when the evolution time scales like O(L).

quant-ph↗
arXiv2012

Quantum entanglement in random physical states

Most states in the Hilbert space are maximally entangled. This fact has proven useful to investigate - among other things - the foundations of statistical mechanics. Unfortunately, most states in the Hilbert space of a quantum many body system are not physically accessible. We define physical ensembles of states by acting on random factorized states by a circuit of length k of random and independent unitaries with local support. We study the typicality of entanglement by means of the purity of the reduced state. We find that for a time k=O(1) the typical purity obeys the area law. Thus, the upper bounds for area law are actually saturated {\em in average}, with a variance that goes to zero for large systems. Similarly, we prove that by means of local evolution a subsystem of linear dimensions $L$ is typically entangled with a volume law when the time scales with the size of the subsystem. Moreover, we show that for large values of k the reduced state becomes very close to the completely mixed state.

quant-ph↗
arXiv2012

Phase diagram and quench dynamics of the Cluster-XY spin chain

We study the complete phase space and the quench dynamics of an exactly solvable spin chain, the cluster-XY model. In this chain, the cluster term and the XY couplings compete to give a rich phase diagram. The phase diagram is studied by means of the quantum geometric tensor. We study the time evolution of the system after a critical quantum quench using the Loschmidt echo. The structure of the revivals after critical quantum quenches presents a non trivial behavior depending on the phase of the initial state and the critical point.

quant-ph↗
arXiv2012

Optimal correlations in many-body quantum systems

Information and correlations in a quantum system are closely related through the process of measurement. We explore such relation in a many-body quantum setting, effectively bridging between quantum metrology and condensed matter physics. To this aim we adopt the information-theory view of correlations, and study the amount of correlations after certain classes of Positive-Operator-Valued Measurements are locally performed. As many-body system we consider a one-dimensional array of interacting two-level systems (a spin chain) at zero temperature, where quantum effects are most pronounced. We demonstrate how the optimal strategy to extract the correlations depends on the quantum phase through a subtle interplay between local interactions and coherence.

quant-ph↗
arXiv2012

Universality and robustness of revivals in the transverse field XY model

We study the structure of the revivals in an integrable quantum many-body system, the transverse field XY spin chain, after a quantum quench. The time evolutions of the Loschmidt echo, the magnetization, and the single spin entanglement entropy are calculated. We find that the revival times for all of these observables are given by integer multiples of T_rev \sim L / v_max where L is the linear size of the system and v_max is the maximal group velocity of quasiparticles. This revival structure is universal in the sense that it does not depend on the initial state and the size of the quench. Applying non-integrable perturbations to the XY model, we observe that the revivals are robust against such perturbations: they are still visible at time scales much larger than the quasiparticle lifetime. We therefore propose a generic connection between the revival structure and the locality of the dynamics, where the quasiparticle speed v_max generalizes into the Lieb$-$Robinson speed v_LR.

quant-ph↗
arXiv2011

Trapped surfaces and emergent curved space in the Bose-Hubbard model

A Bose-Hubbard model on a dynamical lattice was introduced in previous work as a spin system analogue of emergent geometry and gravity. Graphs with regions of high connectivity in the lattice were identified as candidate analogues of spacetime geometries that contain trapped surfaces. We carry out a detailed study of these systems and show explicitly that the highly connected subgraphs trap matter. We do this by solving the model in the limit of no back-reaction of the matter on the lattice, and for states with certain symmetries that are natural for our problem. We find that in this case the problem reduces to a one-dimensional Hubbard model on a lattice with variable vertex degree and multiple edges between the same two vertices. In addition, we obtain a (discrete) differential equation for the evolution of the probability density of particles which is closed in the classical regime. This is a wave equation in which the vertex degree is related to the local speed of propagation of probability. This allows an interpretation of the probability density of particles similar to that in analogue gravity systems: matter inside this analogue system sees a curved spacetime. We verify our analytic results by numerical simulations. Finally, we analyze the dependence of localization on a gradual, rather than abrupt, fall-off of the vertex degree on the boundary of the highly connected region and find that matter is localized in and around that region.

hep-th↗
arXiv2011

Quantum discord in a spin system with symmetry breaking

We analyze the quantum discord Q throughout the low-temperature phase diagram of the quantum XY model in transverse field. We first focus on the T=0 order-disorder quantum phase transition both in the symmetric ground state and in the symmetry broken one. Besides it, we highlight how Q displays clear anomalies also at a non critical value of the control parameter inside the ordered phase, where the ground state is completely factorized. We evidence how the phenomenon is in fact of collective nature and displays universal features. We also study Q at finite temperature. We show that, close to the quantum phase transition, Q exhibits quantum-classical crossover of the system with universal scaling behavior. We evidence a non trivial pattern of thermal correlations resulting from the factorization phenomenon.

cond-mat.stat-mech↗
arXiv2011

Quantum phase transition between cluster and antiferromagnetic states

We study a Hamiltonian system describing a three spin-1/2 cluster-like interaction competing with an Ising-like exchange. We show that the ground state in the cluster phase possesses symmetry protected topological order. A continuous quantum phase transition occurs as result of the competition between the cluster and Ising terms. At the critical point the Hamiltonian is self-dual. The geometric entanglement is also studied. Our findings in one dimension corroborate the analysis of the two dimensional generalization of the system, indicating, at a mean field level, the presence of a direct transition between an antiferromagnetic and a valence bond solid ground state.

quant-ph↗
arXiv2011

Ground state factorization and correlations with broken symmetry

We show how the phenomenon of factorization in a quantum many body system is of collective nature. To this aim we study the quantum discord $Q$ in the one dimensional XY model in a transverse field. We analyze the behavior of $Q$ at both the critical point and at the non critical factorizing field. The factorization is found to be governed by an exponential scaling law for $Q$. We also address the thermal effects fanning out from the anomalies occurring at zero temperature. Close to the quantum phase transition, $Q$ exhibits a finite-temperature crossover with universal scaling behavior, while the factorization phenomenon results in a non trivial pattern of correlations present at low temperature.

quant-ph↗
arXiv2010

Background independent condensed matter models for quantum gravity

A number of recent proposals for a quantum theory of gravity are based on the idea that spacetime geometry and gravity are derivative concepts and only apply at an approximate level. There are two fundamental challenges to any such approach. At the conceptual level, there is a clash between the "timelessness" of general relativity and emergence. Second, the lack of a fundamental spacetime makes difficult the straightforward application of well-known methods of statistical physics to the problem. We recently initiated a study of such problems using spin systems based on evolution of quantum networks with no a priori geometric notions as models for emergent geometry and gravity. In this article we review two such models. The first is a model of emergent (flat) space and matter and we show how to use methods from quantum information theory to derive features such as speed of light from a non-geometric quantum system. The second model exhibits interacting matter and geometry, with the geometry defined by the behavior of matter. This model has primitive notions of gravitational attraction which we illustrate with a toy black hole, and exhibits entanglement between matter and geometry and thermalization of the quantum geometry.

gr-qc↗
arXiv2010

A quantum Bose-Hubbard model with evolving graph as toy model for emergent spacetime

We present a toy model for interacting matter and geometry that explores quantum dynamics in a spin system as a precursor to a quantum theory of gravity. The model has no a priori geometric properties, instead, locality is inferred from the more fundamental notion of interaction between the matter degrees of freedom. The interaction terms are themselves quantum degrees of freedom so that the structure of interactions and hence the resulting local and causal structures are dynamical. The system is a Hubbard model where the graph of the interactions is a set of quantum evolving variables. We show entanglement between spatial and matter degrees of freedom. We study numerically the quantum system and analyze its entanglement dynamics. We analyze the asymptotic behavior of the classical model. Finally, we discuss analogues of trapped surfaces and gravitational attraction in this simple model.

gr-qc↗
arXiv2010

Entanglement and area law with a fractal boundary in a topologically ordered phase

Quantum systems with short range interactions are known to respect an area law for the entanglement entropy: the von Neumann entropy $S$ associated to a bipartition scales with the boundary $p$ between the two parts. Here we study the case in which the boundary is a fractal. We consider the topologically ordered phase of the toric code with a magnetic field. When the field vanishes it is possible to analytically compute the entanglement entropy for both regular and fractal bipartitions $(A,B)$ of the system, and this yields an upper bound for the entire topological phase. When the $A$-$B$ boundary is regular we have $S/p =1$ for large $p$. When the boundary is a fractal of Hausdorff dimension $D$, we show that the entanglement between the two parts scales as $S/p=γ\leq1/D$, and $γ$ depends on the fractal considered.

quant-ph↗
arXiv2010

Topological Entanglement Renyi Entropy and Reduced Density Matrix Structure

We generalize the topological entanglement entropy to a family of topological Renyi entropies parametrized by a parameter alpha, in an attempt to find new invariants for distinguishing topologically ordered phases. We show that, surprisingly, all topological Renyi entropies are the same, independent of alpha for all non-chiral topological phases. This independence shows that topologically ordered ground-state wavefunctions have reduced density matrices with a certain simple structure, and no additional universal information can be extracted from the entanglement spectrum.

cond-mat.str-el↗
arXiv2009

Diffusion on an Ising chain with kinks

We count the number of histories between the two degenerate minimum energy configurations of the Ising model on a chain, as a function of the length n and the number d of kinks that appear above the critical temperature. This is equivalent to count permutations of length n avoiding certain subsequences depending on d. We give explicit generating functions and compute the asymptotics. The setting considered has a role when describing dynamics induced by quantum Hamiltonians with deconfined quasi-particles.

math.CO↗
arXiv2009

Toric-boson model: Toward a topological quantum memory at finite temperature

We discuss the existence of stable topological quantum memory at finite temperature. At stake here is the fundamental question of whether it is, in principle, possible to store quantum information for macroscopic times without the intervention from the external world, that is, without error correction. We study the toric code in two dimensions with an additional bosonic field that couples to the defects, in the presence of a generic environment at finite temperature: the toric-boson model. Although the coupling constants for the bare model are not finite in the thermodynamic limit, the model has a finite spectrum. We show that in the topological phase, there is a finite temperature below which open strings are confined and therefore the lifetime of the memory can be made arbitrarily (polynomially) long in system size. The interaction with the bosonic field yields a long-range attractive force between the end points of open strings but leaves closed strings and topological order intact.

quant-ph↗