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M. A. Rajabpour

Publications and source records attributed to M. A. Rajabpour.

At least 55 records · Page 3Linked to original sources
arXiv2020

Lattice Bisognano-Wichmann modular Hamiltonian in critical quantum spin chains

We carry out a comprehensive comparison between the exact modular Hamiltonian and the lattice version of the Bisognano-Wichmann (BW) one in one-dimensional critical quantum spin chains. As a warm-up, we first illustrate how the trace distance provides a more informative mean of comparison between reduced density matrices when compared to any other Schatten $n$-distance, normalized or not. In particular, as noticed in earlier works, it provides a way to bound other correlation functions in a precise manner, i.e., providing both lower and upper bounds. Additionally, we show that two close reduced density matrices, i.e. with zero trace distance for large sizes, can have very different modular Hamiltonians. This means that, in terms of describing how two states are close to each other, it is more informative to compare their reduced density matrices rather than the corresponding modular Hamiltonians. After setting this framework, we consider the ground states for infinite and periodic XX spin chain and critical Ising chain. We provide robust numerical evidence that the trace distance between the lattice BW reduced density matrix and the exact one goes to zero as $\ell^{-2}$ for large length of the interval $\ell$. This provides strong constraints on the difference between the corresponding entanglement entropies and correlation functions. Our results indicate that discretized BW reduced density matrices reproduce exact entanglement entropies and correlation functions of local operators in the limit of large subsystem sizes. Finally, we show that the BW reduced density matrices fall short of reproducing the exact behavior of the logarithmic emptiness formation probability in the ground state of the XX spin chain.

cond-mat.str-el↗
arXiv2019

Bipartite entanglement entropy of the excited states of the free fermions and the harmonic oscillators

We study general entanglement properties of the excited states of the one dimensional translational invariant free fermions and coupled harmonic oscillators. In particular, using the integrals of motion, we prove that these Hamiltonians independent of the gap (mass) has infinite excited states that can be described by conformal field theories with integer or half-integer central charges. In the case of free fermions, we also show that because of the huge degeneracy in the spectrum, even a gapless Hamiltonian can have excited states with an area-law. Finally, we study the universal average entanglement entropy over eigenstates of the Hamiltonian of the free fermions introduced recently in L. Vidmar, et al., Phys. Rev. Lett. 119, 020601 (2017) https://link.aps.org/doi/10.1103/PhysRevLett.119.020601 . In particular, using a duality relation, we propose a method which can be useful for experimental measurement of the universal average entanglement entropy. Part of our conclusions can be extended to the quantum spin chains that are associated with the free fermions via Jordan-Wigner(JW) transformation.

cond-mat.str-el↗
arXiv2019

Area law and universality in the statistics of subsystem energy

We introduce Rényi entropy of a subsystem energy as a natural quantity which closely mimics the behavior of the entanglement entropy and can be defined for all the quantum many body systems. For this purpose, consider a quantum chain in its ground state and then, take a subdomain of this system with natural truncated Hamiltonian. Since the total Hamiltonian does not commute with the truncated Hamiltonian, the subsystem can be in one of its eigenenergies with different probabilities. Using the fact that the global energy eigenstates are locally close to diagonal in the local energy eigenbasis, we argue that the Rényi entropy of these probabilities follows an area law for the gapped systems. When the system is at the critical point, the Rényi entropy follows a logarithmic behavior with a universal coefficient. Consequently, our quantity not only detects the phase transition but also determines the universality class of the critical point. Moreover we show that the largest defined probabilities are very close to the biggest Schmidt coefficients. We quantify this by defining a {\it{truncated}} Shannon entropy which its value is almost indistinguishable from the {\it{truncated}} von Neumann entanglement entropy. Compare to the entanglement entropy, our quantity has the advantage of being associated to a natural observable (subsystem Hamiltonian) that can be defined (and probably measured) easily for all the short-range interacting systems. We support our arguments by detailed numerical calculations performed on the transverse field XY-chain.

cond-mat.str-el↗
arXiv2019

Entanglement Hamiltonian of quantum critical chains and conformal field theories

We consider a lattice version of the Bisognano-Wichmann (BW) modular Hamiltonian as an ansatz for the bipartite entanglement Hamiltonian of the quantum critical chains. Using numerically unbiased methods, we check the accuracy of the BW-ansatz by both comparing the BW Rényi entropy to the exact results, and by investigating the size scaling of the norm distance between the exact reduced density matrix and the BW one. Our study encompasses a variety of models, scanning different universality classes, including transverse field Ising, Potts and XXZ chains. We show that the Rényi entropies obtained via the BW ansatz properly describe the scaling properties predicted by conformal field theory. Remarkably, the BW Rényi entropies faithfully capture also the corrections to the conformal field theory scaling associated to the energy density operator. In addition, we show that the norm distance between the discretized BW density matrix and the exact one asymptotically goes to zero with the system size: this indicates that the BW-ansatz can be also employed to predict properties of the eigenvectors of the reduced density matrices, and is thus potentially applicable to other entanglement-related quantities such as negativity.

cond-mat.stat-mech↗
arXiv2019

Return amplitude after a quantum quench in the XY chain

We determine an exact formula for the transition amplitude between any two arbitrary eigenstates of the local $z$-magnetization operators in the quantum XY chain. We further use this formula to obtain an analytical expression for the return amplitude of fully polarized states and the Néel state on a ring of length $L$. Then, we investigate finite-size effects in the return amplitude: in particular quasi-particle interference halfway along the ring, a phenomenon that has been dubbed traversal~\cite{FE2016}. We show that the traversal time and the features of the return amplitude at the traversal time depend on the initial state and on the parity of $L$. Finally, we briefly discuss non-analyticities in time of the decay rates in the thermodynamic limit $L\rightarrow\infty$, which are known as dynamical phase transitions.

cond-mat.stat-mech↗
arXiv2018

Light-cone velocities after a global quench in a non-interacting model

We study the light-cone velocity for global quenches in the non-interacting XY chain starting from a class of initial states that are eigenstates of the local $z$-component of the spin. We point out how translation invariance of the initial state can affect the maximal speed at which correlations spread. As a consequence the light-cone velocity can be state-dependent also for non-interacting systems: a new effect of which we provide clear numerical evidence and analytic predictions. Analogous considerations, based on numerical results, are drawn for the evolution of the entanglement entropy.

cond-mat.str-el↗
arXiv2017

Full counting statistics of the subsystem energy in the free fermions and the quantum spin chains

We calculate the full counting statistics (FCS) of a subsystem energy in free fermionic systems by means of the Grassmann variables. We demonstrate that the generating function of these systems can be written as a determinant formula with respect to the Hamiltonian couplings and by employing the Bell's polynomials, we derive exact formulas for the subsystem energy moments. In addition, we discuss the same quantities in the quantum XY spin chain, and we demonstrate that at the critical regimes the fluctuations of the energy moments decay like a power-law as we expect from the conformal field theory arguments, while in non-critical regimes, the decay is exponential. Furthermore, we discuss the full counting statistics of subsystem energy in the quantum XX chain.

cond-mat.str-el↗
arXiv2017

Multipartite entanglement and quantum Fisher information in conformal field theories

Bipartite entanglement entropy of a segment with the length $l$ in $1+1$ dimensional conformal field theories (CFT) follows the formula $S=\frac{c}{3}\ln l+γ$, where $c$ is the central charge of the CFT and $γ$ is a cut-off dependent constant which diverges in the absence of an ultraviolet cutoff. According to this formula, systems with larger central charges have {\it{more}} bipartite entanglement entropy. Using quantum Fisher information (QFI), we argue that systems with bigger central charges not only have larger bipartite entanglement entropy but also have more multipartite entanglement content. In particular, we argue that since systems with smaller {\it{smallest scaling dimension}} have bigger QFI, the multipartite entanglement content of a CFT is dependent on the value of the smallest scaling dimension present in the spectrum of the system. We show that our argument seems to be consistent with some of the existing results regarding the von Neumann entropy, negativity, and localizable entanglement in $1+1$ dimensions. Furthermore, we also argue that the QFI decays under renormalization group (RG) flow between two unitary CFTs. Finally, we also comment on the non-conformal but scale invariant systems.

hep-th↗
arXiv2017

On the possibility of complete revivals after quantum quenches to a critical point

In a recent letter, J. Cardy, Phys. Rev. Lett. \textbf{112}, 220401 (2014), the author made a very interesting observation that complete revivals of quantum states after quantum quench can happen in a period which is a fraction of the system size. This is possible for critical systems that can be described by minimal conformal field theories (CFT) with central charge $c<1$. In this article, we show that these complete revivals are impossible in microscopic realizations of those minimal models. We will prove the absence of the mentioned complete revivals for the critical transverse field Ising chain analytically, and present numerical results for the critical line of the XY chain. In particular, for the considered initial states, we will show that criticality has no significant effect in partial revivals. We also comment on the applicability of quasi-particle picture to determine the period of the partial revivals qualitatively. In particular, we detect a regime in the phase diagram of the XY chain which one can not determine the period of the partial revivals using the quasi-particle picture.

cond-mat.str-el↗
arXiv2016

Entanglement entropy after selective measurements in quantum chains

We study bipartite post measurement entanglement entropy after selective measurements in quantum chains. We first study the quantity for the critical systems that can be described by conformal field theories. We find a connection between post measurement entanglement entropy and the Casimir energy of floating objects. Then we provide formulas for the post measurement entanglement entropy for open and finite temperature systems. We also comment on the Affleck-Ludwig boundary entropy in the context of the post measurement entanglement entropy. Finally, we also provide some formulas regarding modular hamiltonians and entanglement spectrum in the after measurement systems. After through discussion regarding CFT systems we also provide some predictions regarding massive field theories. We then discuss a generic method to calculate the post measurement entanglement entropy in the free fermion systems. Using the method we study the post measurement entanglement entropy in the XY spin chain. We check numerically the CFT and the massive field theory results in the transverse field Ising chain and the XX model. In particular, we study the post meaurement entanglement entropy in the infinite, periodic and open critical transverse field Ising chain and the critical XX model. The effect of the temperature and the gap is also discussed in these models.

cond-mat.str-el↗
arXiv2016

Finite size corrections to scaling of the formation probabilities and the Casimir effect in the conformal field theories

We calculate formation probabilities of the ground state of the finite size quantum critical chains using conformal field theory (CFT) techniques. In particular, we calculate the formation probability of one interval in the finite open chain and also formation probability of two disjoint intervals in a finite periodic system. The presented formulas can be also interpreted as the Casimir energy of needles in particular geometries. We numerically check the validity of the exact CFT results in the case of the transverse field Ising chain.

cond-mat.stat-mech↗
arXiv2016

Classification of the sign of the critical Casimir force in two dimensional systems at asymptotically large separations

We classify the sign of the critical Casimir force between two finite objects separated by a large distance in the two dimensional systems that can be described by conformal field theory (CFT). In particular, we show that as far as the smallest scaling dimension present in the spectrum of the system is smaller than one, the sign of the force is independent of the shape of the objects and can be determined by the elements of the modular $S$-matrix of the CFT. The provided formula for the sign of the force indicates that the force is always attractive for equal boundary conditions independent of the shape of the objects. However, different boundary conditions can lead to attractive or repulsive forces. Using the derived formula, we prove the known results regarding the Ising model and the free bosons. As new examples, we give detailed results regarding the Q=3-states Potts model and the compactified bosons. In particular, for the latter model we show that Dirichlet boundary condition does not always lead to an attractive force.

cond-mat.stat-mech↗
arXiv2016

Entanglement entropy after a partial projective measurement in $1+1$ dimensional conformal field theories: exact results

We calculate analytically the Rényi bipartite entanglement entropy $S_α$ of the ground state of $1+1$ dimensional conformal field theories (CFT) after performing a projective measurement in a part of the system. We show that the entanglement entropy in this setup is dependent on the central charge and the operator content of the system. When the measurement region $A$ separates the two parts $B$ and $\bar{B}$, the entanglement entropy between $B$ and $\bar{B}$ decreases like a power-law with respect to the characteristic distance between the two regions with an exponent which is dependent on the rank $α$ of the Rényi entanglement entropy and the smallest scaling dimension present in the system. We check our findings by making numerical calculations on the Klein-Gordon field theory (coupled harmonic oscillators) after fixing the position (partial measurement) of some of the oscillators. We also comment on the post-measurement entanglement entropy in the massive quantum field theories.

hep-th↗
arXiv2016

Fate of the area-law after partial measurement in quantum field theories

We calculate numerically the Rényi bipartite entanglement entropy of the ground state of Klein-Gordon field theory (coupled harmonic oscillators) after fixing the position (partial measurement) of some of the oscillators in $d=1,2$ and $3$ dimensions. We show that after partial measurement as far as the two parts are connected the area-law is valid in generic dimensions except in $d=1$ conformal field theories. When due to the measured region the two parts are disconnected, the entanglement entropy decreases exponentially with respect to the minimum distance between the regions. In the massless case the decay is power-law with a universal dimension-dependent exponent.

hep-th↗
arXiv2016

Formation probabilities and Shannon information and their time evolution after quantum quench in transverse-field XY-chain

We first provide a formula to calculate the probability of occurrence of different configurations (formation probabilities) in a generic free fermion system. We then study the scaling of these probabilities with respect to the size in the case of critical transverse-field XY-chain in the $σ^z$ bases. In the case of the transverse field Ising model, we show that all the "crystal" configurations follow the formulas expected from conformal field theory (CFT). In the case of critical $XX$ chain, we show that the only configurations that follow the formulas of the CFT are the ones which respect the filling factor of the system. By repeating all the calculations in the presence of open and periodic boundary conditions we find further support to our classification of different configurations. Using the developed technique, we also study Shannon information of a subregion in our system. In this respect we distinguish particular configurations that are more important in the study of the scaling limit of the Shannon information of the subsystem. Finally, we study the evolution of formation probabilities, Shannon information and Shannon mutual information after a quantum quench in free fermion system. In particular, for the intial state considered in this paper, we demonstrate that the Shannon information after quantum quench first increases with the time and then saturates at time $t^*=\frac{l}{2}$, where $l$ is the size of the subsystem.

cond-mat.str-el↗
arXiv2016

Formation probabilities in quantum critical chains and Casimir effect

We find a connection between logarithmic formation probabilities of two disjoint intervals of quantum critical spin chains and the Casimir energy of two aligned needles in two dimensional classical critical systems. Using this connection we provide a formula for the logarithmic formation probability of two disjoint intervals in generic $1+1$ dimensional critical systems. The quantity is depenedent on the full structure of the underlying conformal field theory and so useful to find the universality class of the critical system. The connection that we find also provides a very efficient numerical method to calculate the Casimir energy between needles using quantum critical chains. The agreement between numerical results performed on critical transverse field Ising model and XX chain with our exact results is very good. We also comment on the mutual Rényi information of two disjoint intervals.

cond-mat.str-el↗
arXiv2015

Post measurement bipartite entanglement entropy in conformal field theories

We derive exact formulas for bipartite von Neumann entanglement entropy after partial projective local measurement in $1+1$ dimensional conformal field theories with periodic and open boundary conditions. After defining the set up we will check numerically the validity of our results in the case of Klein-Gordon field theory (coupled harmonic oscillators) and spin-$1/2$ XX chain in a magnetic field. The agreement between analytical results and the numerical calculations is very good. We also find a lower bound for localizable entanglement in coupled harmonic oscillators.

cond-mat.stat-mech↗