arXiv subjects

Erik Tonni

Publications and source records attributed to Erik Tonni.

At least 19 recordsLinked to original sources

arXiv2026

Measurement-induced entanglement Hamiltonian

We study the entanglement Hamiltonian of an infinite hopping chain in its ground state, after partial projective measurements in the occupation basis. For a segment separated by two measurement regions from the rest of the chain, we show that the reduced density matrix can be related to a grand-canonical state via a conformal mapping and a gauge transformation in the underlying field-theory description. The entanglement Hamiltonian is then described by a local inverse temperature that vanishes as a square root around the endpoints and is independent of the particular measurement outcome. In sharp contrast, the local chemical potential is shown to be related to the induced charge density in the segment. Hence the entanglement Hamiltonian of a post-selected state contains much more information on the measurement outcome than the respective entropy.

cond-mat.stat-mech↗

arXiv2026

Jones index from Rényi entropies in the Ising conformal field theory

We study the relation between the Jones index and the Rényi entropies of two disjoint intervals on the line and of the ground state for a generic value of the Rényi index in the two conformal field theory models given by the Ising model and a free Majorana fermion, where Haag duality is satisfied. The analytic expressions of the crossing asymmetry for all the submodels displaying a violation of the Haag duality that are closed under the fusion rules are obtained. In the limiting regime where the two intervals become adjacent, the leading term of the expansion of the crossing asymmetry provides the Jones global index, for any finite value of the Rényi index.

hep-th↗

arXiv2026

A contour for the entanglement negativity of bosonic Gaussian states

We construct a contour function for the logarithmic negativity and the logarithm of the moments of the partial transpose of the reduced density matrix for multimode bosonic Gaussian states of a free lattice model. In one spatial dimension, numerical results are obtained for harmonic chains either in the ground state or at finite temperature, by considering, respectively, either a subsystem made by two adjacent or disjoint blocks on the line or a bipartition of the circle. The contour function of the logarithmic negativity diverges only at the entangling points, while the contour function for the logarithm of the moments of the partial transpose is divergent also at the boundary of the bipartite subsystem, as functions of the position. In a two-dimensional conformal field theory, analytic expressions that describe these divergencies are discussed. In one spatial dimension, we explore the partial derivative of the logarithmic negativity of two adjacent intervals with respect to the logarithm of the harmonic ratio of their lengths while their ratio and the other parameters are kept fixed. Considering the ground state of the harmonic chain on the line and in the massive regime, we report numerical results showing that this quantity displays a monotonically decreasing behaviour.

cond-mat.stat-mech↗

arXiv2026

Entanglement negativity for a free scalar chiral current

We study the entanglement negativity for the free, scalar chiral current in two spacetime dimensions, which is a simple model violating the Haag duality in regions with nontrivial topology. For the ground state of the system, both on the line and on the circle, we consider the setups given by two intervals, either adjacent or disjoint. We find analytic expressions for the moments of the partial transpose of the reduced density matrix and the logarithmic negativity. In the limit of small separation distance, this expression yields the same subleading topological contribution occurring in the mutual information. In the limit of large separation distance between the two intervals, the exponential decay of the logarithmic negativity is obtained from its analytic expression. The analytic formulas are checked against exact numerical results from a bosonic lattice model, finding a perfect agreement. We observe that, since the chiral current generates the neutral subalgebra of the full chiral Dirac fermion theory, this analysis highlights how symmetries produce nontrivial features in the entanglement structure that are analogue to those ones already observed in the mutual information for regions with nontrivial topology.

hep-th↗

arXiv2025

Modular Hamiltonian of the massive scalar field on the half line: A numerical approach

We study the modular Hamiltonian of an interval for the ground state of a massive free scalar field on the half line with Robin boundary conditions, by employing a numerical method. When the interval is adjacent to the boundary, we find numerical evidence that the modular Hamiltonian is non-local, except for the limiting cases of the massless scalar satisfying either Dirichlet or Neumann boundary conditions. When the interval is separated from the boundary, the numerical analysis indicates that the modular Hamiltonian is non-local for all these boundary conditions and any value of the mass.

hep-th↗

arXiv2025

Continuous majorization in quantum phase space for Wigner-positive states and proposals for Wigner-negative states

In quantum resource theory, one is often interested in identifying which states serve as the best resources for particular quantum tasks. If a relative comparison between quantum states can be made, this gives rise to a partial order, where states are ordered according to their suitability to act as a resource. In the literature, various different partial orders for a variety of quantum resources have been proposed. In discrete variable systems, vector majorization of Wigner functions in discrete phase space provides a natural partial order between quantum states. In the continuous variable case, a natural counterpart would be continuous majorization of Wigner functions in quantum phase space. Indeed, this concept was recently proposed and explored (mostly restricting to the single-mode case) in Van Herstraeten, Jabbour, Cerf, Quantum 7, 1021 (2023). In this work, we develop the theory of continuous majorization in the general $N$-mode case. In addition, we propose extensions to include states with finite Wigner negativity. For the special case of the convex hull of $N$-mode Gaussian states, we prove a conjecture made by Van Herstraeten, Jabbour and Cerf. We also prove a phase space counterpart of Uhlmann's theorem of majorization.

quant-ph↗

arXiv2025

Entanglement Hamiltonian for the massless Dirac field on a segment with an inhomogeneous background

We study the entanglement Hamiltonian of an interval for the massless Dirac field in an inhomogeneous background on a segment where the same boundary condition at both its endpoints is imposed, and in its ground state. We focus on a class of metrics that are Weyl equivalent to the flat metric through a Weyl factor that depends only on the spatial coordinate. The explicit form of the entanglement Hamiltonian is written as the sum of a local and a bilocal term. The weight function of the local term allows us to study a contour function for the entanglement entropies. For the model obtained from the continuum limit of the rainbow chain, the analytic expressions are compared with exact numerical results from the lattice, showing an excellent agreement.

hep-th↗

arXiv2025

Entanglement Hamiltonian of two disjoint blocks in the harmonic chain

We study the entanglement Hamiltonian of two disjoint blocks in the harmonic chain on the line and in its ground state. In the regime of large mass, the non vanishing terms are only the on-site and the nearest-neighbour ones. Analytic expressions are obtained for their profiles, which are written in terms of piecewise linear functions that can be discontinuous and display sharp transitions as the separation between the blocks changes. In the regime of vanishing mass, where the matrices characterising the entanglement Hamiltonian contain couplings at all distances, we explore the location of the subdominant terms and some combinations of matrix elements that are useful for the continuum limit, comparing the results with the corresponding ones for the free chiral current. The single-particle entanglement spectra of these entanglement Hamiltonians are also investigated.

cond-mat.stat-mech↗

arXiv2025

Modular transport in two-dimensional conformal field theory

We study the quantum transport generated by the bipartite entanglement in two-dimensional conformal field theory at finite density with the $U(1) \times U(1)$ symmetry associated to the conservation of the electric charge and of the helicity. The bipartition given by an interval is considered, either on the line or on the circle. The continuity equations and the corresponding conserved quantities for the modular flows of the currents and of the energy-momentum tensor are derived. We investigate the mean values of the associated currents and their quantum fluctuations in the finite density representation, which describe the properties of the modular quantum transport. The modular analogues of the Johnson-Nyquist law and of the fluctuation-dissipation relation are found, which encode the thermal nature of the modular evolution.

hep-th↗

arXiv2025

Modular evolutions and causality in two-dimensional conformal field theory

In two-dimensional conformal field theories (CFT) in Minkowski spacetime, we study the spacetime distance between two events along two distinct modular trajectories. When the spatial line is bipartite by a single interval, we consider both the ground state and the state at finite different temperatures for the left and right moving excitations. For the free massless Dirac field in the ground state, the bipartition of the line given by the union of two disjoint intervals is also investigated. The modular flows corresponding to connected subsystems preserve relativistic causality. Locality along the modular flows of some fields is explored by evaluating their (anti-)commutators. In particular, the bilocal nature of the modular Hamiltonian of two disjoint intervals for the massless Dirac field provide multiple trajectories leading to Dirac delta contributions in the (anti-)commutators even when the initial points belong to different intervals, thus being spacelike separated.

hep-th↗

arXiv2024

Holographic thermal entropy from geodesic bit threads

The holographic bit threads are an insightful tool to investigate the holographic entanglement entropy and other quantities related to the bipartite entanglement in AdS/CFT. We mainly explore the geodesic bit threads in various static backgrounds, for the bipartitions characterized by either a sphere or an infinite strip. In pure AdS and for the sphere, the geodesic bit threads provide a gravitational dual of the map implementing the geometric action of the modular conjugation in the dual CFT. In Schwarzschild AdS black brane and for the sphere, our numerical analysis shows that the flux of the geodesic bit threads through the horizon gives the holographic thermal entropy of the sphere. This feature is not observed when the subsystem is an infinite strip, whenever we can construct the corresponding bit threads. The bit threads are also determined by the global structure of the gravitational background; indeed, for instance, we show that the geodesic bit threads of an arc in the BTZ black hole cannot be constructed.

hep-th↗

arXiv2023

Entanglement entropies of an interval for the massless scalar field in the presence of a boundary

We study the entanglement entropies of an interval for the massless compact boson either on the half line or on a finite segment, when either Dirichlet or Neumann boundary conditions are imposed. In these boundary conformal field theory models, the method of the branch point twist fields is employed to obtain analytic expressions for the two-point functions of twist operators. In the decompactification regime, these analytic predictions in the continuum are compared with the lattice numerical results in massless harmonic chains for the corresponding entanglement entropies, finding good agreement. The application of these analytic results in the context of quantum quenches is also discussed.

hep-th↗

arXiv2023

Krylov complexity of modular Hamiltonian evolution

We investigate the complexity of states and operators evolved with the modular Hamiltonian by using the Krylov basis. In the first part, we formulate the problem for states and analyse different examples, including quantum mechanics, two-dimensional conformal field theories and random modular Hamiltonians, focusing on relations with the entanglement spectrum. We find that the modular Lanczos spectrum provides a different approach to quantum entanglement, opening new avenues in many-body systems and holography. In the second part, we focus on the modular evolution of operators and states excited by local operators in two-dimensional conformal field theories. We find that, at late modular time, the spread complexity is universally governed by the modular Lyapunov exponent $λ^{mod}_L=2π$ and is proportional to the local temperature of the modular Hamiltonian. Our analysis provides explicit examples where entanglement entropy is indeed not enough, however the entanglement spectrum is, and encodes the same information as complexity.

hep-th↗

arXiv2023

Entanglement and negativity Hamiltonians for the massless Dirac field on the half line

We study the ground-state entanglement Hamiltonian of several disjoint intervals for the massless Dirac fermion on the half-line. Its structure consists of a local part and a bi-local term that couples each point to another one in each other interval. The bi-local operator can be either diagonal or mixed in the fermionic chiralities and it is sensitive to the boundary conditions. The knowledge of such entanglement Hamiltonian is the starting point to evaluate the negativity Hamiltonian, i.e. the logarithm of the partially transposed reduced density matrix, which is an operatorial characterisation of entanglement of subsystems in a mixed states. We find that the negativity Hamiltonian inherits the structure of the corresponding entanglement Hamiltonian. We finally show how the continuum expressions for both these operators can be recovered from exact numerical computations in free-fermion chains.

cond-mat.stat-mech↗

arXiv2023

Probing RG flows, symmetry resolution and quench dynamics through the capacity of entanglement

We compare the capacity of entanglement with the entanglement entropy by considering various aspects of these quantities for free bosonic and fermionic models in one spatial dimension, both in the continuum and on the lattice. Substantial differences are observed in the subleading terms of these entanglement quantifiers when the subsystem is made by two disjoint intervals, in the massive scalar field and in the fermionic chain. We define $c$-functions based on the capacity of entanglement similar to the one based on the entanglement entropy, showing through a numerical analysis that they display a monotonic behaviour under the renormalisation group flow generated by the mass. The capacity of entanglement and its related quantities are employed to explore the symmetry resolution. The temporal evolutions of the capacity of entanglement and of the corresponding contour function after a global quench are also discussed.

cond-mat.stat-mech↗

arXiv2023

Sequences of resource monotones from modular Hamiltonian polynomials

We introduce two infinite sequences of entanglement monotones, which are constructed from expectation values of polynomials in the modular Hamiltonian. These monotones yield infinite sequences of inequalities that must be satisfied in majorizing state transitions. We demonstrate this for information erasure, deriving an infinite sequence of "Landauer inequalities" for the work cost, bounded by linear combinations of expectation values of powers of the modular Hamiltonian. These inequalities give improved lower bounds for the work cost in finite dimensional systems, and depend on more details of the erased state than just on its entropy and variance of modular Hamiltonian. Similarly one can derive lower bounds for marginal entropy production for a system coupled to an environment. These infinite sequences of entanglement monotones also give rise to relative quantifiers that are monotonic in more general processes, namely those involving so-called $σ$-majorization with respect to a fixed point full rank state $σ$; such quantifiers are called resource monotones. As an application to thermodynamics, one can use them to derive finite-dimension corrections to the Clausius inequality. Finally, in order to gain some intuition for what (if anything) plays the role of majorization in field theory, we compare pairs of states in discretized theories at criticality and study how majorization depends on the size of the bipartition with respect to the size of the entire chain.

quant-ph↗

arXiv2022

Modular conjugations in 2D conformal field theory and holographic bit threads

We study the geometric action of some modular conjugations in two dimensional (2D) conformal field theories. We investigate the bipartition given by an interval when the system is in the ground state, either on the line or on the circle, and in the thermal Gibbs state on the line. We find that the restriction of the corresponding inversion maps to a spatial slice is obtained also in the gauge/gravity correspondence through the geodesic bit threads in a constant time slice of the dual static asymptotically AdS background. For a CFT in the thermal state on the line, the modular conjugation suggests the occurrence of a second world which can be related through the geodesic bit threads to the horizon of the BTZ black brane background. An inversion map is constructed also for the massless Dirac fermion in the ground state and on the line bipartite by the union of two disjoint intervals.

hep-th↗

arXiv2022

Entanglement entropies of an interval in the free Schrödinger field theory at finite density

We study the entanglement entropies of an interval on the infinite line in the free fermionic spinless Schrödinger field theory at finite density and zero temperature, which is a non-relativistic model with Lifshitz exponent $z=2$. We prove that the entanglement entropies are finite functions of one dimensionless parameter proportional to the area of a rectangular region in the phase space determined by the Fermi momentum and the length of the interval. The entanglement entropy is a monotonically increasing function. By employing the properties of the prolate spheroidal wave functions of order zero or the asymptotic expansions of the tau function of the sine kernel, we find analytic expressions for the expansions of the entanglement entropies in the asymptotic regimes of small and large area of the rectangular region in the phase space. These expansions lead to prove that the analogue of the relativistic entropic $C$ function is not monotonous. Extending our analyses to a class of free fermionic Lifshitz models labelled by their integer dynamical exponent $z$, we find that the parity of this exponent determines the properties of the bipartite entanglement for an interval on the line.

hep-th↗