Viktor Eisler, Ingo Peschel
We consider a local quench where two free-fermion half-chains are coupled via a defect. We show that the logarithmic increase of the entanglement entropy is governed by the same effective central charge which appears in the ground-state properties and which is known exactly. For unequal initial filling of the half-chains, we determine the linear increase of the entanglement entropy.
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We consider a section of a half-filled chain of free electrons and its entanglement with the rest of the system in the presence of one or two interface defects. We find a logarithmic behaviour of the entanglement entropy with constants depending continuously on the defect strength.
We consider fermionic and bosonic quantum chains where a defect separates two subsystems and compare the corresponding entanglement spectra. With these, we calculate their Rényi entanglement entropies and obtain analytical formulae for the continuously varying coefficient of the leading logarithmic term. For the bosonic case we also present numerical results.
We study the ground-state entanglement of two halves of a critical transverse Ising chain, separated by an interface defect. From the relation to a two-dimensional Ising model with a defect line we obtain an exact expression for the continuously varying effective central charge. The result is relevant also for other fermionic chains.
We study the out-of-equilibrium time evolution after a local quench connecting two anisotropic spin-1/2 XXZ Heisenberg open chains via an impurity bond. The dynamics is obtained by means of the adaptive time-dependent density-matrix renormalization group. We show that the entanglement entropies (Von Neumann and Rényi), in the presence of a weakened bond depend on the sign of the bulk interaction. For attractive interaction (Δ< 0), the defect turns out to be irrelevant and the evolution is asymptotically equivalent to the one without defect obtained by conformal field theory. For repulsive inte…
We study the entanglement properties of anisotropic open spin one-half Heisenberg chains with a modified central bond. The entanglement entropy between the two half-chains is calculated with the density-matrix renormalization method (DMRG).We find a logarithmic behaviour with an effective central charge c' varying with the length of the system. It flows to one in the ferromagnetic region and to zero in the antiferromagnetic region of the model. In the XX case it has a non-universal limit and we recover previous results.
We study free electrons on an infinite half-filled chain, starting in the ground state with a bond defect. We find a logarithmic increase of the entanglement entropy after the defect is removed, followed by a slow relaxation towards the value of the homogeneous chain. The coefficients depend continuously on the defect strength.
The entanglement of different parts of a quantum system is expected to be proportional to the common interface area. Therefore alterations across the interface will lead to changes on the behavior of entanglement entropy. In this work, the effects of bond defects at the boundaries of Kitaev ladders are considered. We find a logarithmic scaling for the ground state entanglement entropy between the two pieces. The prefactor of the logarithm (effective central charge) varies continuously with the defect strength. The energy dispersion is also obtained and sharp features in the von Neumann entangl…
The entanglement entropy for the ground state of a XY spin chain is related to the corner transfer matrices of the triangular Ising model and expressed in closed form.
We calculate the ground state entanglement entropy between two heterogeneous parts of a free fermion chain. The two parts could be XX chains with different parameters or an XX half chain connected with a quantum Ising half chain. It is shown that logarithmic behavior holds if the two parts are conformally critical. In other cases, area law holds with abundant convergence behaviors. In particular, when XX chain at Lifshitz point is connected with a conformally or Lifshitz critical part, entanglement entropy converges algebraically with a fractional subleading index.
Entanglements and Whitehead Products: Generalizing Kleman's Construction to Higher-Dimensional Defects
本文回顾了同伦论中Whitehead积的物理解释，将其视为有序介质中拓扑缺陷的纠缠，并将Kleman构造推广到高维缺陷情形。
Zero-mode entanglement across a conformal defect
本文研究含扩展零模的共形缺陷自由费米子链混合基态，利用全计数统计分析纠缠谱态密度，并解析得到均匀链及中心缺陷半链的 Rényi 熵变化与不随尺寸衰减的奇偶效应。
Entanglement Through Topological Defects: Reconciling Theory with Numerics
本文通过引入新的减少密度矩阵制备方法，解决了拓扑缺陷与纠缠熵理论预测之间的矛盾，并展示了其在Ising模型中对群状和对偶缺陷的有效性。
The quantum Ising chain of length, L, which is separated into two parts by localized or extended defects is considered at the critical point where scaling of the interface magnetization is non-universal. We measure the entanglement entropy between the two halves of the system in equilibrium, as well as after a quench, when the interaction at the interface is changed for time t>0. For the localized defect the increase of the entropy with log(L) or with log(t) involves the same effective central charge, which is a continuous function of the strength of the defect. On the contrary for the extende…
We study solvable spin chains where either fields or couplings vary linearly in space and create a sandwich-like structure of the ground state. We find that the entanglement entropy between two halves of a chain varies logarithmically with the interface width. After quenching to a homogeneous critical system, the entropy grows logarithmically in time in the XX model, but quadratically in the transverse Ising chain. We explain this behaviour and indicate generalizations to other power laws.
We present the results of magnetic properties and entanglement of the distorted diamond chain model for azurite using pure quantum exchange interactions. The magnetic properties and concurrence as a measure of pairwise thermal entanglement have been studied by means of variational mean-field like treatment based on Gibbs-Bogoliubov inequality. Such a system can be considered as an approximation of the natural material azurite, Cu3(CO3)2(OH)2. For values of exchange parameters, which are taken from experimental results, we study the thermodynamic properties, such as azurite specific heat and ma…
We study zero-temperature XX chains and transverse Ising chains and join an initially separate finite piece on one or on both sides to an infinite remainder. In both critical and non-critical systems we find a typical increase of the entanglement entropy after the quench, followed by a slow decay towards the value of the homogeneous chain. In the critical case, the predictions of conformal field theory are verified for the first phase of the evolution, while at late times a step structure can be observed.
Long-distance genuine multipartite Entanglement between Magnetic Defects in Spin Chains
该研究针对一维横场 XX 自旋-1/2 链中的三个局域磁性缺陷，结合解析与数值方法分析束缚态条件，证明缺陷 RDM 在整个参数空间均可具有长距离 GME，并给出其并发度的量化方法。
Entanglement evolution after a global quench across a conformal defect
本文研究了一维含杂质量子系统在全局淬火后的纠缠演化，通过引入边界扭曲场将纠缠熵线性增长的斜率与有效中心荷关联，并在自由费米子链中通过准粒子假设验证，发现场论与格点结果存在微小差异。
Charge-resolved entanglement in the presence of topological defects
本文研究存在拓扑缺陷时Su-Schrieffer-Heeger模型的电荷分辨纠缠熵，结合解析与数值方法揭示了缺陷引发的纠缠结构变化及纠缠均分破缺的共同机制。
We consider a quench in a free-fermion chain by joining two homogeneous half-chains via a defect. The time evolution of the entanglement negativity is studied between adjacent segments surrounding the defect. In case of equal initial fillings, the negativity grows logarithmically in time and essentially equals one-half of the Rényi mutual information with index $α= 1/2$ in the limit of large segments. In sharp contrast, in the biased case one finds a linear increase followed by the saturation at an extensive value for both quantities, which is due to the backscattering from the defect and can …
Entanglement entropy (EE) in critical quantum spin chains described by 1+1D conformal field theories contains signatures of the universal characteristics of the field theory. Boundaries and defects in the spin chain give rise to universal contributions in the EE. In this work, we analyze these universal contributions for the critical Ising and XXZ spin chains for different conformal boundary conditions and defects. For the spin chains with boundaries, we use the boundary states for the corresponding continuum theories to compute the subleading contribution to the EE analytically and provide su…
We study the half-chain entanglement entropy in the ground state of the spin-1/2 XX chain across an extended random defect, where the strength of disorder decays with the distance from the interface algebraically as $Δ_l\sim l^{-κ}$. In the whole regime $κ\ge 0$, the average entanglement entropy is found to increase logarithmically with the system size $L$ as $S_L\simeq\frac{c_{\rm eff}(κ)}{6}\ln L+const$, where the effective central charge $c_{\rm eff}(κ)$ depends on $κ$. In the regime $κ<1/2$, where the extended defect is a relevant perturbation, the strong-disorder renormalization group met…
We consider the entanglement entropy for the 2D Ising model at the conformal fixed point in the presence of interfaces. More precisely, we investigate the situation where the two subsystems are separated by a defect line that preserves conformal invariance. Using the replica trick, we compute the entanglement entropy between the two subsystems. We observe that the entropy, just like in the case without defects, shows a logarithmic scaling behavior with respect to the size of the system. Here, the prefactor of the logarithm depends on the strength of the defect encoded in the transmission coeff…
The twofold twist defects in the $D(\mathbb{Z}_k)$ quantum double model (abelian topological phase) carry non-abelian fractional Majorana-like characteristics. We align these twist defects in a line and construct a one dimensional Hamiltonian which only includes the pairwise interaction. For the defect chain with even number of twist defects, it is equivalent to the $\mathbb{Z}_k$ clock model with periodic boundary condition (up to some phase factor for boundary term), while for odd number case, it maps to $\mathbb{Z}_k$ clock model with duality twisted boundary condition. At critical point, f…
We have studied quantum phase transition induced by a quench in different one dimensional spin systems. Our analysis is based on the dynamical mechanism which envisages nonadiabaticity in the vicinity of the critical point. This causes spin fluctuation which leads to the random fluctuation of the Berry phase factor acquired by a spin state when the ground state of the system evolves in a closed path. The two-point correlation of this phase factor is associated with the probability of the formation of defects. In this framework, we have estimated the density of defects produced in several one d…
Entanglement entropy~(EE) contains signatures of many universal properties of conformal field theories~(CFTs), especially in the presence of boundaries or defects. In particular, {\it topological} defects are interesting since they reflect internal symmetries of the CFT, and have been extensively analyzed with field-theoretic techniques with striking predictions. So far, however, very few ab-initio, lattice computations of these predictions have been available. Here, we present an ab-initio analysis of EE for the Ising model in the presence of a topological defect. While the behavior of the EE…
We investigate the presence of multipartite entanglement in macroscopic spin chains. We discuss the Heisenberg and the XY model and derive bounds on the internal energy for systems without multipartite entanglement. Based on this we show that in thermal equilibrium the above mentioned spin systems contain genuine multipartite entanglement, even at finite modest temperatures.
Coupling a many-body system to a thermal environment typically destroys the quantum coherence of its state, leading to an effective classical dynamics at the longest time scales. We show that systems with anyon-like defects can exhibit universal late-time dynamics that is stochastic, but fundamentally non-classical, because some of the quantum information about the state is topologically protected from the environment. Our coarse-grained model describes one-dimensional systems with domain-wall defects carrying Majorana modes. These defects undergo Brownian motion due to coupling with a bath. S…
Entanglement asymmetry in CFT and its relation to non-topological defects
本文利用CFT中全局对称性与非拓扑缺陷的对应关系研究一维临界系统的纠缠不对称性，通过XY自旋链与无质量Majorana费米子映射，得到任意强度n缺陷的精确标度维数及log ℓ/ℓ项前因子。
We simulate the entanglement dynamics in a critical random quantum Ising chain with generic perturbations using the time-evolving block decimation algorithm. Starting from a product state, we observe super-logarithmic growth of entanglement entropy with time. The numerical result is consistent with the analytical prediction of Vosk and Altman using a real-space renormalization group technique.
Kapitza resistance in the chain models with internal defects is considered. For the case of the linear chain, the exact analytic solution for the boundary resistance is derived for arbitrary linear time-independent conservative inclusion or defect. A simple case of isolated isotopic defects is explored in more detail. Contrary to the bulk conductivity in the linear chain, the Kapitza resistance is finite. However, the universal thermodynamic limit does not exist in this case. In other terms, the exact value of the resistance is not uniquely defined and depends on the way of approaching the inf…
The thermal entanglement of a two-qubit anisotropic Heisenberg $XYZ$ chain under an inhomogeneous magnetic field b is studied. It is shown that when inhomogeneity is increased to certain value, the entanglement can exhibit a larger revival than that of less values of b. The property is both true for zero temperature and a finite temperature. The results also show that the entanglement and critical temperature can be increased by increasing inhomogeneous exteral magnetic field.
Entanglement asymmetry and symmetry defects in boundary conformal field theory
本文利用二维边界共形场论，通过纠缠不对称性研究边界条件对全局对称性破缺的影响，计算了领头阶及次领头阶修正，并扩展镜像方法处理含可逆对称性缺陷的动力学淬火情形。
Flow-Based Sampling for Entanglement Entropy and the Machine Learning of Defects
提出利用生成模型结合基于流的方法与复制技巧、围绕格点缺陷采用定制神经网络架构的新技术，用于计算格点量子场论中雷尼纠缠熵，在二维和三维φ⁴标量场论测试中优于现有蒙特卡罗方法且缺陷尺寸扩展性能良好。
We consider two prototypical quantum models, the spin-1/2 XY chain and the quantum Ising chain and study their entanglement entropy, S(l,L), of blocks of l spins in homogeneous or inhomogeneous systems of length L. By using two different approaches, free-fermion techniques and perturbational expansion, an exact relationship between the entropies is revealed. Using this relation we translate known results between the two models and obtain, among others, the additive constant of the entropy of the critical homogeneous quantum Ising chain and the effective central charge of the random XY chain.
We consider the ground state of the XX chain that is constrained to carry a current of energy. The von Neumann entropy of a block of $L$ neighboring spins, describing entanglement of the block with the rest of the chain, is computed. Recent calculations have revealed that the entropy in the XX model diverges logarithmically with the size of the subsystem. We show that the presence of the energy current increases the prefactor of the logarithmic growth. This result indicates that the emergence of the energy current gives rise to an increase of entanglement.
We study the continuum limit of the entanglement hamiltonians of a block of consecutive sites in massless harmonic chains. This block is either in the chain on the infinite line or at the beginning of a chain on the semi-infinite line with Dirichlet boundary conditions imposed at its origin. The entanglement hamiltonians of the interval predicted by Conformal Field Theory for the massless scalar field are obtained in the continuum limit. We also study the corresponding entanglement spectra and the numerical results for the ratios of the gaps are compatible with the operator content of the Boun…
Entanglement Propagation in Integrable Heisenberg Chains from a New Lens
该研究利用量子关联传递函数（QCTF）公式得到海森堡链单磁振子纠缠演化的精确结果，揭示早期准粒子超最大群速度传播对纠缠的贡献，结果可扩展至多磁振子体系以解释平衡动力学和热力学。
We examine the entanglement of cyclic spin 1/2 chains with anisotropic XY Z Heisenberg couplings of arbitrary range at transverse factorizing magnetic fields. At these fields the system exhibits a degenerate symmetry-breaking separable ground state (GS). It is shown, however, that the side limits of the GS pairwise entanglement at these fields are actually non-zero in finite chains, corresponding such fields to a GS spin-parity transition. These limits exhibit universal properties like being independent of the pair separation and interaction range, and are directly related to the magnetization…
We investigate the entanglement entropy (EE) of gapped S=1 and $S=1/2$ spin chains with dimerization. We find that the effective boundary degrees of freedom as edge states contribute significantly to the EE. For the $S=1/2$ dimerized Heisenberg chain, the EE of the sufficiently long chain is essentially explained by the localized $S=1/2$ effective spins on the boundaries. As for S=1, the effective spins are also $S=1/2$ causing a Kennedy triplet that yields a lower bound for the EE. In this case, the residual entanglement reduces substantially by a continuous deformation of the Heisenberg mode…
A spectral viewpoint on the single defect tight-binding chain
本文从谱视角推导单缺陷紧束缚链的占据概率、位移等解析结果，揭示均方位移非单调性的谱起源，且数值计算验证了临界缺陷强度与最小化均方位移的缺陷强度高度吻合。
Full counting statistics and symmetry resolved entanglement for free conformal theories with interface defects
本文研究通过共形界面耦合的两类一维临界自由理论，利用缺陷几何与均匀几何的关系，解析给出电荷涨落的全计数统计、带电矩和对称分辨Rényi熵，并通过数值验证。
We compute the quantum Renyi relative entropies in an infinite spinless fermionic chain with a defect. Doing a numerical analysis we will show that the resulting quantity depends non trivially on the effective central charge of the theory. Moreover, we will see that an explicit analytic expression can be written for all of them and from that, one can read the quantum fidelity and the relative entropy.
We describe an algorithm for studying the entanglement entropy and spectrum of 2D systems, as a coupled array of $N$ one dimensional chains in their continuum limit. Using the algorithm to study the quantum Ising model in 2D, (both in its disordered phase and near criticality) we confirm the existence of an area law for the entanglement entropy and show that near criticality there is an additive piece scaling as $c_{eff}\log (N)/6$ with $c_{eff} \approx 1$. \textcolor{black}{Studying the entanglement spectrum, we show that entanglement gap scaling can be used to detect the critical point of th…
Spatial Entanglement Sudden Death in Spin Chains at All Temperatures
该研究证明了在所有温度下，旋链的吉布斯态具有有限的纠缠长度，移除特定区间后剩余部分处于分离态。
Defects in two-dimensional conformal field theories (CFTs) contain signatures of their characteristics. In this work, we compute the entanglement entropy (EE) and the entanglement negativity (EN) of subsystems in the presence of energy and duality defects in the Ising CFT using the density matrix renormalization group (DMRG) technique. We show that the EE for the duality defect exhibits fundamentally different characteristics compared to the energy defect due to the existence of localized and delocalized zero energy modes. Of special interest is the nontrivial `finite-size correction' in the E…
We calculate exactly the entanglement content of magnon excited states in the integrable spin-1/2 XXX and XXZ chains in the scaling limit. In particular, we show that as far as the number of excited magnons with respect to the size of the system is small one can decompose the entanglement content, in the scaling limit, to the sum of the entanglement of particular excited states of free fermionic or bosonic theories. In addition we conjecture that the entanglement content of the generic translational invariant free fermionic and bosonic Hamiltonians can be also classified, in the scaling limit,…
We study the entanglement entropy of the quantum trajectories of a free fermion chain under continuous monitoring of local occupation numbers. We propose a simple theory for entanglement entropy evolution from disentangled and highly excited initial states. It is based on generalized hydrodynamics and the quasi-particle pair approach to entanglement in integrable systems. We test several quantitative predictions of the theory against extensive numerics and find good agreement. In particular, the volume law entanglement is destroyed by the presence of arbitrarily weak measurement.
Quantum excitations in lattice systems always propagate at a finite maximum velocity. We probe this mechanism by considering a defect travelling at a constant velocity in the quantum Ising spin chain in transverse field. Independently of the microscopic details of the defect, we characterize the expectation value of local observables at large times and large distances from the impurity, where a Local Quasi Stationary State (LQSS) emerges. The LQSS is strongly affected by the defect velocity: for superluminal defects, it exhibits a growing region where translational invariance is spontaneously …
Entanglement entropy obeys area law scaling for typical physical quantum systems. This may naively be argued to follow from locality of interactions. We show that this is not the case by constructing an explicit simple spin chain Hamiltonian with nearest neighbor interactions that presents an entanglement volume scaling law. This non-translational model is contrived to have couplings that force the accumulation of singlet bonds across the half chain. Our result is complementary to the known relation between non-translational invariant, nearest neighbor interacting Hamiltonians and QMA complete…
We introduce a connection between entanglement induced by interaction and geometric phases acquired by a composite quantum spin system. We begin by analyzing the evaluation of cyclic (Aharonov-Anandan) and non-cyclic (Mukunda-Simon) geometric phases for general spin chains evolving in the presence of time-independent magnetic fields. Then, by considering Heisenberg chains, we show that the interaction geometric phase, namely, the total geometric phase with subtraction of free spin contributions, is directly related to the global (Meyer-Wallach) entanglement exhibited by an initially separable …
In this paper we propose an expression for the entanglement entropy of several intervals in a stationary state of a free, translational invariant Hamiltonian in a fermionic chain. We check numerically the accuracy of our proposal and conjecture a new formula for the asymptotic behaviour of principal sub-matrices of a Toeplitz matrix.
We carry out a systematic study of the exact block entanglement in XXZ spin-chain at Delta=-1/2. We present, the first analytic expressions for reduced density matrices of n spins in a chain of length L (for n<=6 and arbitrary but odd L) of a truly interacting model. The entanglement entropy, the moments of the reduced density matrix, and its spectrum are then easily derived. We explicitely construct the "entanglement Hamiltonian" as the logarithm of this matrix. Exploiting the degeneracy of the ground-state, we find the scaling behavior of entanglement of the zero-temperature mixed state.
It has been recently proven that new types of bulk, local order can ensue due to frustrated boundary condition, that is, periodic boundary conditions with an odd number of lattice sites and anti-ferromagnetic interactions. For the quantum XY chain in zero external fields, the usual antiferromagnetic order has been found to be replaced either by a mesoscopic ferromagnet or by an incommensurate AFM order. In this work we examine the resilience of these new types of orders against a defect that breaks the translational symmetry of the model. We find that, while a ferromagnetic defect restores the…
We introduce a new spin chain which is a deformation of the Fredkin spin chain and has a phase transition between bounded and extensive entanglement entropy scaling. In this chain, spins have a local interaction of three nearest neighbors. The Hamiltonian is frustration-free and its ground state can be described analytically as a weighted superposition of Dyck paths. In the purely spin $1/2$ case, the entanglement entropy obeys an area law: it is bounded from above by a constant, when the size of the block $n$ increases (and $t>1$). When a local color degree of freedom is introduced the entang…
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