我们开发了一种系统框架，基于改进的热核方法计算带电量子系统中的对称分解纠缠熵（SREE）。尽管传统Sommerfeld公式在中性系统中有效，但在引入规范场或化学势时因不完整的残差预设和渐近边界条件的违反而受限。通过使用相位因子重构热核的解析结构，我们推导出一个全局收敛的展开式，协调了离散残差求和与连续谱分解。我们进一步将此框架应用于Gaussian连续多尺度纠缠重整化近似（cMERA）态，并展示纠缠熵（EE）可以表示为cMERA流函数。特别是，在存在化学势的情况下，我们获得了对称分解纠缠熵的流方程。此公式自然扩展到任意时空维数，并在μ→0极限下恢复中性系统的已知结果。我们通过两种设置验证了我们的框架：（1）使用扭操作技术与（1+1）维共形场论（CFT）预测的精确一致；（2）与S1 x H^(d-1)几何的双生熵计算的一致性。我们的结果既统一了带电和中性纠缠熵的处理，又将其扩展到实空间重整化框架，为探测共形场论、其双生对偶和cMERA表示中的对称分解纠缠提供了稳健工具。
英文摘要：
We develop a systematic framework for computing symmetry-resolved entanglement entropies (SREE) in charged quantum systems based on an improved heat kernel approach. Although the conventional Sommerfeld formula proves effective for neutral systems, it encounters limitations when gauge fields or chemical potentials are introduced due to incomplete residue prescriptions and violations of asymptotic boundary conditions. By reconstructing the analytic structure of the heat kernel using a phase factor, we derive a globally convergent expansion that reconciles discrete residue summations with continuous spectral decompositions. We further apply this framework to Gaussian continuous multi-scale entanglement renormalization ansatz (cMERA) states and show that the entanglement entropy (EE) can be expressed in terms of the cMERA flow functions. In particular, we obtain a symmetry-resolved entanglement entropy flow equation in the presence of a chemical potential. This formulation extends naturally to arbitrary spacetime dimensions and recovers established results for neutral systems in the mu -> 0 limit. We validate our framework through two settings: (1) exact agreement with (1+1)-dimensional conformal field theory (CFT) predictions using twist-operator techniques, and (2) consistency with holographic entropy calculations on S1 x H^(d-1) geometries. Our results both unify the treatment of charged and neutral entanglement entropy and extend this treatment to real-space renormalization frameworks, providing a robust tool for probing symmetry-resolved entanglement in conformal field theories, their holographic duals, and cMERA representations.
We consider the form factor bootstrap approach of integrable field theories to derive matrix elements of composite branch-point twist fields associated with symmetry resolved entanglement entropies. The bootstrap equations are determined in an intuitive way and their solution is presented for the massive Ising field theory and for the genuinely interacting sinh-Gordon model, both possessing a $\mathbb{Z}_{2}$ symmetry. The solutions are carefully cross-checked by performing various limits and by the application of the $Δ$-theorem. The issue of symmetry resolution for discrete symmetries is als…
In this paper, we apply the form factor bootstrap approach to branch point twist fields in the $q$-state Potts model for $q\leq 3$. For $q=3$ this is an integrable interacting quantum field theory with an internal discrete $\mathbb{Z}_3$ symmetry and therefore provides an ideal starting point for the investigation of the symmetry resolved entanglement entropies. However, more generally, for $q\leq 3$ the standard Rényi and entanglement entropies are also accessible through the bootstrap programme. In our work we present form factor solutions both for the standard branch point twist field with …
Symmetry Resolved Entanglement Entropy in a Non-Abelian Fractional Quantum Hall State
SISSA — International School for Advanced Studies（国际高等研究院）；INFN（意大利国家核物理研究所）；Center for Computational Quantum Physics, Flatiron Institute（平顿研究所计算量子物理中心）；Department of Physics, University of Toronto（多伦多大学物理系）；Laboratoire de Physique de l’Ecole normale supérieure, ENS, Université PSL, CNRS, Sorbonne Université, Université Paris-Diderot, Sorbonne Paris Cité（巴黎高等师范学院物理实验室，巴黎文理研究大学，法国国家科学研究中心，索邦大学，巴黎第七大学，索邦巴黎城）；Department of Physics, Princeton University（普林斯顿大学物理系）；Sorbonne Université, CNRS, Laboratoire de Physique Théorique et Hautes Energies, LPTHE（索邦大学，法国国家科学研究中心，理论与高能物理实验室）
We study the symmetry resolved entanglement entropies in gapped integrable lattice models. We use the corner transfer matrix to investigate two prototypical gapped systems with a U(1) symmetry: the complex harmonic chain and the XXZ spin-chain. While the former is a free bosonic system, the latter is genuinely interacting. We focus on a subsystem being half of an infinitely long chain. In both models, we obtain exact expressions for the charged moments and for the symmetry resolved entropies. While for the spin chain we found exact equipartition of entanglement (i.e. all the symmetry resolved …
We consider the symmetry resolved Rényi entropies in the one dimensional tight binding model, equivalent to the spin-1/2 XX chain in a magnetic field. We exploit the generalised Fisher-Hartwig conjecture to obtain the asymptotic behaviour of the entanglement entropies with a flux charge insertion at leading and subleading orders. The o(1) contributions are found to exhibit a rich structure of oscillatory behaviour. We then use these results to extract the symmetry resolved entanglement, determining exactly all the non-universal constants and logarithmic corrections to the scaling that are not …
We study the symmetry resolved entanglement entropies in one-dimensional systems with boundaries. We provide some general results for conformal invariant theories and then move to a semi-infinite chain of free fermions. We consider both an interval starting from the boundary and away from it. We derive exact formulas for the charged and symmetry resolved entropies based on theorems and conjectures about the spectra of Toeplitz+Hankel matrices. En route to characterise the interval away from the boundary, we prove a general relation between the eigenvalues of Toeplitz+Hankel matrices and block …
The excess entanglement resulting from exciting a finite number of quasiparticles above the ground state of a free integrable quantum field theory has been investigated quite extensively in the literature. It has been found that it takes a very simple form, depending only on the number of excitations and their statistics. There is now mounting evidence that such formulae also apply to interacting and even higher-dimensional quantum theories. In this paper we study the entanglement content of such zero-density excited states focusing on the symmetry resolved entanglement, that is on 1+1D quantu…
In a quantum many-body system that possesses an additive conserved quantity, the entanglement entropy of a subsystem can be resolved into a sum of contributions from different sectors of the subsystem's reduced density matrix, each sector corresponding to a possible value of the conserved quantity. Recent studies have discussed the basic properties of these symmetry-resolved contributions, and calculated them using conformal field theory and numerical methods. In this work we employ the generalized Fisher-Hartwig conjecture to obtain exact results for the characteristic function of the symmetr…
We generalise the form factor bootstrap approach to integrable field theories with U(1) symmetry to derive matrix elements of composite branch-point twist fields associated with symmetry resolved entanglement entropies. The bootstrap equations are solved for the free massive Dirac and complex boson theories, which are the simplest theories with U(1) symmetry. We present the exact and complete solution for the bootstrap, including vacuum expectation values and form factors involving any type and arbitrarily number of particles. The non-trivial solutions are carefully cross-checked by performing…
The presence of a global internal symmetry in a quantum many-body system is reflected in the fact that the entanglement between its subparts is endowed with an internal structure, namely it can be decomposed as sum of contributions associated to each symmetry sector. The symmetry resolution of entanglement measures provides a formidable tool to probe the out-of-equilibrium dynamics of quantum systems. Here, we study the time evolution of charge-imbalance-resolved negativity after a global quench in the context of free-fermion systems, complementing former works for the symmetry-resolved entang…
We investigate the symmetry resolution of entanglement in the presence of long-range couplings. To this end, we study the symmetry-resolved entanglement entropy in the ground state of a fermionic chain that has dimerised long-range hoppings with power-like decaying amplitude -- a long-range generalisation of the Su-Schrieffer-Heeger model. This is a system that preserves the number of particles. The entropy of each symmetry sector is calculated via the charged moments of the reduced density matrix. We exploit some recent results on block Toeplitz determinants generated by a discontinuous symbo…
Symmetry Resolved Entanglement Entropy: Equipartition under Driven and Non-unitary Evolution in a Compact Boson CFT
SISSA and INFN Sezione di Trieste（的里雅斯特国际高等研究学院和意大利国家核物理研究所的里雅斯特分部）；Theory Division, Saha Institute of Nuclear Physics, A CI of Homi Bhabha National Institute（萨哈核物理研究所理论部，霍米·巴哈国家研究所附属机构）
In this paper we continue the programme initiated in Part I, that is the study of entanglement measures in the sine-Gordon model. In both parts, we have focussed on one specific technique, that is the well-known connection between branch point twist field correlators and measures of entanglement in 1+1D integrable quantum field theory. Our papers apply this technique for the first time to a non-diagonal theory with an involved particle spectrum, the sine-Gordon model. In this Part II we focus on a different entanglement measure, the symmetry resolved entanglement, and develop its associated tw…
Similarly to the system Hamiltonian, a subsystem's reduced density matrix is composed of blocks characterized by symmetry quantum numbers (charge sectors). We present a geometric approach for extracting the contribution of individual charge sectors to the subsystem's entanglement measures within the replica trick method, via threading appropriate conjugate Aharonov-Bohm fluxes through a multi-sheet Riemann surface. Specializing to the case of 1+1D conformal field theory, we obtain general exact results for the entanglement entropies and spectrum, and apply them to a variety of systems, ranging…
The study of the entanglement dynamics plays a fundamental role in understanding the behaviour of many-body quantum systems out of equilibrium. In the presence of a globally conserved charge, further insights are provided by the knowledge of the resolution of entanglement in the various symmetry sectors. Here, we carry on the program we initiated in [Phys. Rev. B 103, L041104 (2021)], for the study of the time evolution of the symmetry resolved entanglement in free fermion systems. We complete and extend our derivations also by defining and quantifying a symmetry resolved mutual information. T…
We study the non-equilibrium dynamics of two coupled oscillators interacting with their own heat baths of quantum scalar fields at different temperature $T_1$ and $T_2$ with bilinear couplings between them. We particularly focus on the entanglement or inseparability property of their quantum states. The critical temperatures of two respective oscillators, $T_{1c}$ and $T_{2c}$, higher than which the entanglement disappears, can be determined. It is found that when two damping parameters are largely different, say $γ_1 \ll γ_2$, the critical temperature $T_{1c}$ with respect to the frequency $Ω…
Detecting Symmetry-Resolved Entanglement: A Quantum Monte Carlo Approach
Fudan University（复旦大学）；Shanghai Qizhi Institute and Shanghai Artificial Intelligence Laboratory（上海琦智研究院和上海人工智能实验室）；The Chinese University of Hong Kong（香港中文大学）；Shanghai Research Center for Quantum Sciences（上海量子科学研究中心）；Hefei National Laboratory（合肥国家实验室）；Julius-Maximilians-Universität Würzburg（维尔茨堡大学）；Shanghai Institute for Mathematics and Interdisciplinary Sciences (SIMIS)（上海数学与交叉学科研究院）
Quantum entanglement and its main quantitative measures, the entanglement entropy and entanglement negativity, play a central role in many body physics. An interesting twist arises when the system considered has symmetries leading to conserved quantities: Recent studies introduced a way to define, represent in field theory, calculate for 1+1D conformal systems, and measure, the contribution of individual charge sectors to the entanglement measures between different parts of a system in its ground state. In this paper, we apply these ideas to the time evolution of the charge-resolved contributi…
We consider symmetry-resolved entanglement entropy in AdS${}_3$/CFT${}_2$ coupled to $U(1)$ Chern-Simons theory. We identify the holographic dual of the charged moments in the two-dimensional conformal field theory as a charged Wilson line in the bulk of AdS${}_3$, namely the Ryu-Takayanagi geodesic minimally coupled to the $U(1)$ Chern-Simons gauge field. We identify the holonomy around the Wilson line as the Aharonov-Bohm phases which, in the two-dimensional field theory, are generated by charged $U(1)$ vertex operators inserted at the endpoints of the entangling interval. Furthermore, we de…
The time evolution of the entanglement entropy is a key concept to understand the structure of a non-equilibrium quantum state. In a large class of models, such evolution can be understood in terms of a semiclassical picture of moving quasiparticles spreading the entanglement throughout the system. However, it is not yet known how the entanglement splits between the sectors of an internal local symmetry of a quantum many-body system. Here, guided by the examples of conformal field theories and free-fermion chains, we show that the quasiparticle picture can be adapted to this goal, leading to a…
We report on the calculation of the symmetry resolved entanglement entropies in two-dimensional many-body systems of free bosons and fermions by \emph{dimensional reduction}. When the subsystem is translational invariant in a transverse direction, this strategy allows us to reduce the initial two-dimensional problem into decoupled one-dimensional ones in a mixed space-momentum representation. While the idea straightforwardly applies to any dimension $d$, here we focus on the case $d=2$ and derive explicit expressions for two lattice models possessing a $U(1)$ symmetry, i.e., free non-relativis…
We consider the problem of the decomposition of the Rényi entanglement entropies in theories with a non-abelian symmetry by doing a thorough analysis of Wess-Zumino-Witten (WZW) models. We first consider $SU(2)_k$ as a case study and then generalise to an arbitrary non-abelian Lie group. We find that at leading order in the subsystem size $L$ the entanglement is equally distributed among the different sectors labelled by the irreducible representation of the associated algebra. We also identify the leading term that breaks this equipartition: it does not depend on $L$ but only on the dimension…
In the presence of a conserved quantity, symmetry-resolved entanglement entropies are a refinement of the usual notion of entanglement entropy of a subsystem. For critical 1d quantum systems, it was recently shown in various contexts that these quantities generally obey entropy equipartition in the scaling limit, i.e. they become independent of the symmetry sector. In this paper, we examine the finite-size corrections to the entropy equipartition phenomenon, and show that the nature of the symmetry group plays a crucial role. In the case of a discrete symmetry group, the corrections decay alge…
Holographic entanglement entropy quantities for a three-party system are considered, namely tripartite information, total correlation and secrecy monotone. A holographic approach with Vaidya-AdS is used to calculate time evolution of the entanglement entropy during non-equilibrium heating. We study time dependence of these three entropy quantities.
We explore a fruitful connection between the physics of conformal field theories (CFTs) in d-dimensional Minkowski spacetime and the extended gravitational thermodynamics of hyperbolic black holes in (d+1)-dimensional anti-de Sitter spacetime. The CFTs are reduced on a region bounded by a sphere. We show that Renormalization Group (RG) flows between CFTs are specific thermodynamic processes in the (p,V) plane, where the irreversibility of coarse-graining flows from the ultraviolet to the infrared corresponds to the Second Law of thermodynamics, preventing heat from flowing from low temperature…
In a recent paper we studied the entanglement content of zero-density excited states in complex free quantum field theories, focusing on the symmetry resolved entanglement entropy (SREE). By zero-density states we mean states consisting of a fixed, finite number of excitations above the ground state in an infinite-volume system. The SREE is defined for theories that possess an internal symmetry and provides a measure of the contribution to the total entanglement of each symmetry sector. In our work, we showed that the ratio of Fourier-transforms of the SREEs (i.e. the ratio of charged moments)…
We study the scaling of entanglement in low-energy states of quantum many-body models on lattices of arbitrary dimensions. We allow for unbounded Hamiltonians such that systems with bosonic degrees of freedom are included. We show that if at low enough temperatures the specific heat capacity of the model decays exponentially with inverse temperature, the entanglement in every low-energy state satisfies an area law (with a logarithmic correction). This behaviour of the heat capacity is typically observed in gapped systems. Assuming merely that the low-temperature specific heat decays polynomial…
The firewall was introduced into black hole evaporation scenarios as a deus ex machina designed to break entanglements and preserve unitarity (Almheiri et.al., 2013). Here we show that the firewall actually exists and does break entanglements, but only in the context of a virtual reality for observers stationed near the horizon, who are following the long-term evolution of the hole. These observers are heated by acceleration radiation at the Unruh temperature and see pair creation at the horizon as a high-energy phenomenon. The objective reality is very different. We argue that Hawking pair cr…
Entanglement entropy (EE) of a spatial region quantifies correlations between the region and its surroundings. For a free scalar in the adiabatic vacuum in de Sitter space the EE is known to remain low, scaling as the surface area of the region. Here, we study the evolution of entanglement after the universe transitions from de Sitter to flat space. We concentrate on the case of a massless minimally coupled scalar. We find numerically that, after the de Sitter stage ends, the EE and the Rényi entropy rapidly grow and saturate at values obeying the volume law. The final state of the subsystem (…
We study the energy and entanglement dynamics of $(1+1)$D conformal field theories (CFTs) under a Floquet drive with the sine-square deformed (SSD) Hamiltonian. Previous work has shown this model supports both a non-heating and a heating phase. Here we analytically establish several robust and `super-universal' features of the heating phase which rely on conformal invariance but not on the details of the CFT involved. First, we show the energy density is concentrated in two peaks in real space, a chiral and anti-chiral peak, which leads to an exponential growth in the total energy. The peak lo…
We test the proposal of arXiv:2012.11274 for the holographic computation of the charged moments and the resulting symmetry-resolved entanglement entropy in different excited states, as well as for two entangling intervals. Our holographic computations are performed in $U(1)$ Chern-Simons-Einstein-Hilbert gravity, and are confirmed by independent results in a conformal field theory at large central charge. In particular, we consider two classes of excited states, corresponding to charged and uncharged conical defects in AdS${}_3$. In the conformal field theory, these states are generated by the…
We argue that Landau-Fermi liquids do not have any gravity duals in the purely classical limit. We employ the logarithmic behavior of entanglement entropy to characterize the existence of Fermi surfaces. By imposing the null energy condition, we show that the specific heat always behaves anomalously. We also present a classical gravity dual which has the expected behavior of the entanglement entropy and specific heat for non-Fermi liquids.
We consider a holographic model of two 1+1-dimensional heat baths at different temperatures joined at time $t=0$, such that a steady state heat-current region forms and expands in space for times $t>0$. After commenting on the causal structure of the dual 2+1-dimensional spacetime, we present how to calculate the time-dependent entanglement entropy of the boundary system holographically. We observe that the increase rate of the entanglement entropy satisfies certain bounds known from the literature on entanglement tsunamis. Furthermore, we check the validity of several non-trivial entanglement…
