arXiv subjects

Marius Lemm

Publications and source records attributed to Marius Lemm.

68 records · Page 4Linked to original sources

arXiv2018

Knabe's spectral gap method for open boundary conditions

In 1988, Knabe found a "finite-size criterion" to determine whether a frustration-free quantum spin chain with periodic boundary conditions is uniformly gapped in the thermodynamic limit. The criterion provides a threshold for the spectral gap at a finite system size such that, if the threshold is exceeded for a fixed system size, then the chain with periodic boundary conditions is uniformly gapped. We extend Knabe's result to frustration-free spin chains equipped with open boundary conditions. We now obtain two finite-size criteria: The first one is identical to Knabe's criterion and we interpret it as a bulk criterion. The second one controls the spectral gaps at smaller system sizes and can be interpreted as a new edge criterion. Heuristically, it excludes the presence of thermodynamically gapless excitations living near the edge.

math-ph↗

arXiv2017

On multivariate trace inequalities of Sutter, Berta and Tomamichel

We consider a family of multivariate trace inequalities recently derived by Sutter, Berta and Tomamichel. These inequalities generalize the Golden-Thompson inequality and Lieb's three-matrix inequality to an arbitrary number of matrices in a way that features complex matrix powers. We show that their inequalities can be rewritten as an $n$-matrix generalization of Lieb's original three-matrix inequality. The complex matrix powers are replaced by resolvents and appropriate maximally entangled states. We expect that the technically advantageous properties of resolvents, in particular for perturbation theory, can be of use in applications of the $n$-matrix inequalities, e.g., for analyzing the rotated Petz recovery map in quantum information theory.

math-ph↗

arXiv2017

Actuation of thin nematic elastomer sheets with controlled heterogeneity

Nematic elastomers and glasses deform spontaneously when subjected to temperature changes. This property can be exploited in the design of heterogeneously patterned thin sheets that deform into a non-trivial shape when heated or cooled. In this paper, we start from a variational formulation for the entropic elastic energy of liquid crystal elastomers and we derive an effective two-dimensional metric constraint, which links the deformation and the heterogeneous director field. Our main results show that satisfying the metric constraint is both necessary and sufficient for the deformation to be an approximate minimizer of the energy. We include several examples which show that the class of deformations satisfying the metric constraint is quite rich.

cond-mat.soft↗

arXiv2017

Information-theoretic limitations on approximate quantum cloning and broadcasting

We prove new quantitative limitations on any approximate simultaneous cloning or broadcasting of mixed states. The results are based on information-theoretic (entropic) considerations and generalize the well known no-cloning and no-broadcasting theorems. We also observe and exploit the fact that the universal cloning machine on the symmetric subspace of $n$ qudits and symmetrized partial trace channels are dual to each other. This duality manifests itself both in the algebraic sense of adjointness of quantum channels and in the operational sense that a universal cloning machine can be used as an approximate recovery channel for a symmetrized partial trace channel and vice versa. The duality extends to give control on the performance of generalized UQCMs on subspaces more general than the symmetric subspace. This gives a way to quantify the usefulness of a-priori information in the context of cloning. For example, we can control the performance of an antisymmetric analogue of the UQCM in recovering from the loss of $n-k$ fermionic particles.

quant-ph↗

arXiv2016

Condensation of fermion pairs in a domain

We consider a gas of fermions at zero temperature and low density, interacting via a microscopic two body potential which admits a bound state. The particles are confined to a domain with Dirichlet (i.e. zero) boundary conditions. Starting from the microscopic BCS theory, we derive an effective macroscopic Gross-Pitaevskii (GP) theory describing the condensate of fermion pairs. The GP theory also has Dirichlet boundary conditions. Along the way, we prove that the GP energy, defined with Dirichlet boundary conditions on a bounded Lipschitz domain, is continuous under interior and exterior approximations of that domain.

math-ph↗

arXiv2016

Programming complex shapes in thin nematic elastomer and glass sheets

Nematic elastomers and glasses are solids that display spontaneous distortion under external stimuli. Recent advances in the synthesis of sheets with controlled heterogeneities have enabled their actuation into non-trivial shapes with unprecedented energy density. Thus, these have emerged as powerful candidates for soft actuators. To further this potential, we introduce the key metric constraint which governs shape changing actuation in these sheets. We then highlight the richness of shapes amenable to this constraint through two broad classes of examples which we term nonisometric origami and lifted surfaces. Finally, we comment on the derivation of the metric constraint, which arises from energy minimization in the interplay of stretching, bending and heterogeneity in these sheets.

cond-mat.soft↗

arXiv2016

On Anomalous Lieb-Robinson Bounds for the Fibonacci XY Chain

We rigorously prove a new kind of anomalous (or sub-ballistic) Lieb-Robinson bound for the isotropic XY chain with Fibonacci external magnetic field at arbitrary coupling. It is anomalous in that the usual exponential decay in $x-vt$ is replaced by exponential decay in $x-vt^α$ with $0<α<1$. In fact, we can characterize the values of $α$ for which such a bound holds as those exceeding $α_u^+$, the upper transport exponent of the one-body Fibonacci Hamiltonian. Following the approach of \cite{HSS11}, we relate Lieb-Robinson bounds to dynamical bounds for the one-body Hamiltonian corresponding to the XY chain via the Jordan-Wigner transformation; in our case the one-body Hamiltonian with Fibonacci potential. We can bound its dynamics by adapting techniques developed in \cite{DT07, DT08, D05, DGY} to our purposes. We also explain why our method does not extend to yield anomalous Lieb-Robinson bounds of power-law type for the random dimer model.

math-ph↗

arXiv2016

On the Hölder regularity for the fractional Schrödinger equation and its improvement for radial data

We consider the linear, time-independent fractional Schrödinger equation $$ (-Δ)^s ψ+Vψ=f. $$ We are interested in the local Hölder exponents of distributional solutions $ψ$, assuming local $L^p$ integrability of the functions $V$ and $f$. By standard arguments, we obtain the formula $2s-N/p$ for the local Hölder exponent of $ψ$ where we take some extra care regarding endpoint cases. For our main result, we assume that $V$ and $f$ (but not necessarily $ψ$) are radial functions, a situation which is commonplace in applications. We find that the regularity theory "becomes one-dimensional" in the sense that the Hölder exponent improves from $2s-N/p$ to $2s-1/p$ away from the origin. Similar results hold for $\nablaψ$ as well.

math.AP↗

arXiv2016

Multi-component Ginzburg-Landau theory: microscopic derivation and examples

This paper consists of three parts. In part I, we microscopically derive Ginzburg--Landau (GL) theory from BCS theory for translation-invariant systems in which multiple types of superconductivity may coexist. Our motivation are unconventional superconductors. We allow the ground state of the effective gap operator $K_{T_c}+V$ to be $n$-fold degenerate and the resulting GL theory then couples $n$ order parameters. In part II, we study examples of multi-component GL theories which arise from an isotropic BCS theory. We study the cases of (a) pure $d$-wave order parameters and (b) mixed $(s+d)$-wave order parameters, in two and three dimensions. In part III, we present explicit choices of spherically symmetric interactions $V$ which produce the examples in part II. In fact, we find interactions $V$ which produce ground state sectors of $K_{T_c}+V$ of arbitrary angular momentum, for open sets of of parameter values. This is in stark contrast with Schrödinger operators $-\nabla^2+V$, for which the ground state is always non-degenerate. Along the way, we prove the following fact about Bessel functions: At its first maximum, a half-integer Bessel function is strictly larger than all other half-integer Bessel functions.

math-ph↗

arXiv2016

On polynomial Lieb-Robinson bounds for the XY chain in a decaying random field

We consider the isotropic XY quantum spin chain in a random external field in the $z$ direction, with single site distributions given by i.i.d. random variables times the critical decaying envelope $j^{-1/2}$. Our motivation is the study of many-body localization. We investigate transport properties in terms of polynomial Lieb-Robinson (PLR) bounds. We prove a zero-velocity PLR bound for large disorder strength $λ$ and for small $λ$ we show a partial converse, which suggests the existence of non-trivial transport in the model.

math-ph↗

arXiv2015

Monotonicity of quantum relative entropy and recoverability

The relative entropy is a principal measure of distinguishability in quantum information theory, with its most important property being that it is non-increasing with respect to noisy quantum operations. Here, we establish a remainder term for this inequality that quantifies how well one can recover from a loss of information by employing a rotated Petz recovery map. The main approach for proving this refinement is to combine the methods of [Fawzi and Renner, arXiv:1410.0664] with the notion of a relative typical subspace from [Bjelakovic and Siegmund-Schultze, arXiv:quant-ph/0307170]. Our paper constitutes partial progress towards a remainder term which features just the Petz recovery map (not a rotated Petz map), a conjecture which would have many consequences in quantum information theory. A well known result states that the monotonicity of relative entropy with respect to quantum operations is equivalent to each of the following inequalities: strong subadditivity of entropy, concavity of conditional entropy, joint convexity of relative entropy, and monotonicity of relative entropy with respect to partial trace. We show that this equivalence holds true for refinements of all these inequalities in terms of the Petz recovery map. So either all of these refinements are true or all are false.

quant-ph↗

arXiv2015

Heat flows on hyperbolic spaces

In this paper we develop new methods for studying the convergence problem for the heat flow on negatively curved spaces and prove that any quasiconformal map of the sphere $\mathbb{S}^{n-1}$, $n\geq 3$, can be extended to the $n$-dimensional hyperbolic space such that the heat flow starting with this extension converges to a quasi-isometric harmonic map. This implies the Schoen-Li-Wang conjecture that every quasiconformal map of $\mathbb{S}^{n-1}$, $n\geq 3$, can be extended to a harmonic quasi-isometry of the $n$-dimensional hyperbolic space.

math.DG↗

arXiv2014

New Counterexamples for Sums-Differences

We present new counterexamples, which provide stronger limitations to sums-differences statements than were previously known. The main idea is to consider non-uniform probability measures.

math.CO↗

arXiv2014

New Anomalous Lieb-Robinson Bounds in Quasi-Periodic XY Chains

We announce and sketch the rigorous proof of a new kind of anomalous (or sub-ballistic) Lieb-Robinson bound for an isotropic XY chain in a quasi-periodic transversal magnetic field. By "anomalous", we mean that the usual effective light cone defined by $|x|\leq v|t|$ is replaced by the region $|x|\leq v|t|^α$ for some $0<α<1$. In fact, we can characterize exactly the values of $α$ for which this holds as those exceeding the upper transport exponent $α_u^+$ of an appropriate one-body discrete Schrödinger operator. Previous study has produced a good amount of quantitative information on $α_u^+$. The result is obtained by mapping to free fermions, obtaining good dynamical bounds on the one-body level by adapting techniques developed by Damanik, Gorodetski, Tcheremchantsev, and Yessen and then "pulling back" these bounds through the non-local Jordan-Wigner transformation, following an idea of Hamza, Sims, and Stolz. To our knowledge, this is the first rigorous derivation of anomalous many-body transport. We also explain why our method does not extend to yield anomalous LR bounds of power-law type if one replaces the quasi-periodic field by a random dimer field.

math-ph↗