The thermodynamic formalism of multifractals assigns a mass exponent τ(q) and a singularity spectrum f(α) to any normalised distribution. Applied to the eigenvalues of a reduced density matrix, it turns Rényi entropies into generalised dimensions and the refined Rényi entropy into f(α_q). We use this to separate two notions that are often conflated: the multifractality of single-particle wave functions at localisation transitions, and the multifractal structure of many-body entanglement spectra. For Gaussian fermionic states we prove that the refined Rényi entropy is exactly the entropy of an ideal Fermi gas at temperature 1/q whose single-particle levels are the entanglement energies; an analogous ideal Bose-gas result holds for bosonic Gaussian states. Consequently, the entanglement singularity spectrum is always non-negative and concave. Its dependence on wave-function multifractality enters only through the density of entanglement energies, which is controlled by overlaps between different eigenstates rather than by single-state moments. The identity is verified to 5×10⁻¹⁰ on critical power-law random banded matrices. For tensor-network cascades with correlations between layers, modelled by a Markov chain, the mass exponent is set by a Perron root. Its first-order correction is linear in the subleading eigenvalue λ₂, and it is real-analytic for λ₂ < 1. A first-order (Maxwell) transition appears only in the reducible limit λ₂ = 1. At the quantum Hall transition, Weyl-group invariance makes the anomalous exponents functions of u = q(1−q). The measured non-parabolicity is therefore the coefficient of the u² (higher-Casimir) term. This term appears at four loops in the 2+ε expansion with the observed sign, and its non-vanishing excludes theories in which the multifractal operators are current-algebra (Sugawara) primaries. For Gauss–Bonnet cosmic branes in d = 4, we confirm that the second Taylor coefficient of τ(q) tracks C_T/a, and we find that the refined Rényi entropy turns negative at finite q for positive Gauss–Bonnet coupling. We also explain why no correspondence between holographic couplings and the quantum Hall non-parabolicity should be expected. Companion paper: R. Chen, Holographic Entanglement of Fractal Regions, DOI 10.5281/zenodo.23189575. Code, data and figures: https://github.com/Ruqing1963/multifractal-quantum-entanglement
The thermodynamic formalism of multifractals assigns a mass exponent τ(q) and a singularity spectrum f(α) to any normalised distribution. Applied to the eigenvalues of a reduced density matrix, it turns Rényi entropies into generalised dimensions and the refined Rényi entropy into f(α_q). We use this to separate two notions that are often conflated: the multifractality of single-particle wave functions at localisation transitions, and the multifractal structure of many-body entanglement spectra. For Gaussian fermionic states we prove that the refined Rényi entropy is exactly the entropy of an ideal Fermi gas at temperature 1/q whose single-particle levels are the entanglement energies; an analogous ideal Bose-gas result holds for bosonic Gaussian states. Consequently, the entanglement singularity spectrum is always non-negative and concave. Its dependence on wave-function multifractality enters only through the density of entanglement energies, which is controlled by overlaps between different eigenstates rather than by single-state moments. The identity is verified to 5×10⁻¹⁰ on critical power-law random banded matrices. For tensor-network cascades with correlations between layers, modelled by a Markov chain, the mass exponent is set by a Perron root. Its first-order correction is linear in the subleading eigenvalue λ₂, and it is real-analytic for λ₂ < 1. A first-order (Maxwell) transition appears only in the reducible limit λ₂ = 1. At the quantum Hall transition, Weyl-group invariance makes the anomalous exponents functions of u = q(1−q). The measured non-parabolicity is therefore the coefficient of the u² (higher-Casimir) term. This term appears at four loops in the 2+ε expansion with the observed sign, and its non-vanishing excludes theories in which the multifractal operators are current-algebra (Sugawara) primaries. For Gauss–Bonnet cosmic branes in d = 4, we confirm that the second Taylor coefficient of τ(q) tracks C_T/a, and we find that the refined Rényi entropy turns negative at finite q for positive Gauss–Bonnet coupling. We also explain why no correspondence between holographic couplings and the quantum Hall non-parabolicity should be expected. Companion paper: R. Chen, Holographic Entanglement of Fractal Regions, DOI 10.5281/zenodo.23189575. Code, data and figures: https://github.com/Ruqing1963/multifractal-quantum-entanglement
The following thesis focuses on the scaling of entanglement entropy in lower dimensions and is divided into three main parts. Chapter 2 studies the thermal reduced density matrices in fermion and spin systems on ladders. Chapter 3 studies the many-body localization phase transition in a Rokhsar-Kivelson type wave function. Chapter 4 studies the subleading correction term of entanglement entropy in 2+1 dimensional scale invariant systems. Chapter 5 studies the bulk-boundary correspondence in 3 + 1 dimensional topological phases and its entanglement entropy. In chapter 2, we investigate the reduced density matrices for a model of free fermions on a two-leg ladder (gapped by the inter-chain tunneling operator) and in 1/2 spin systems on a ladder with a gapped ground state using exact solutions and several controlled approximations. We calculate the reduced density matrix and the entanglement entropy for a leg of the ladder (i.e. cut made between the chains). In the fermionic system we find the exact form of the reduced density matrix for one of the chains and determine the entanglement spectrum explicitly. Here we find that in the weak tunneling limit of the ladder the entanglement entropy of one chain of the gapped ladder has a simple and universal form dictated by conformal invariance. In the case of the spin system, we consider the strong coupling limit by using perturbation theory and get the reduced density matrix by the Schmidt decomposition. The entanglement entropies of a general gapped system of two coupled conformal field theories (in 1+1 dimensions) is discussed using the replica trick and scaling arguments. We show that 1) for a system with a bulk gap, the reduced density matrix has the form of a thermal density matrix, 2) the long-wavelength modes of one subsystem (a chain) of a gapped coupled system are always thermal, 3) the von Neumann entropy equals to the thermodynamic entropy of one chain, and 4) the bulk gap plays the role of effective temperature. In chapter 3, we construct a family of many-body wave functions to study the many-body localization phase transition. The wave functions have a Rokhsar-Kivelson form, in which the weight for the configurations are chosen from the Gibbs weights of a classical spin glass model, known as the Random Energy Model, multiplied by a random sign structure to represent a highly excited state. These wave functions show a phase transition into an MBL phase. In addition, we see three regimes of entanglement scaling with subsystem size: scaling with entanglement corresponding to an infinite temperature thermal phase, constant scaling, and a sub-extensive scaling between these limits. Near the phase transition point, the fluctuations of the Renyi entropies are non-Gaussian. We find that Renyi entropies with different Renyi index transition into the MBL phase at different points and have different scaling behavior, suggesting a multifractal behavior. In chapter 4, we study the universal scaling behavior of the entanglement entropy of critical theories in 2+1 dimensions. We specially consider two fermionic scale-invariant models, free massless Dirac fermions and a model of fermions with quadratic band touching, and numerically study the two-cylinder entanglement entropy of the models on the torus. We find that in both cases the entanglement entropy satisfies the area law and has the subleading term which is a scaling function of the aspect ratios of the cylindrical regions. We test the scaling of entanglement in both the free fermion models using three possible scaling functions for the subleading term derived from a) the quasi-one-dimensional conformal field theory, b) the bosonic quantum Lifshitz model, and c) the holographic AdS/CFT correspondence. For the later case we construct an analytic scaling function using holography, appropriate for critical theories with a gravitational dual description. We find that the subleading term in the fermionic models is well described, for a range of aspect ratios, by the scaling form derived from the quantum Lifshitz model as well as that derived using the AdS/CFT correspondence (in this case only for the Dirac model). For the case where the fermionic models are placed on a square torus we find the fit to the different scaling forms is in agreement to surprisingly high precision. In chapter 5, we discuss (2+1)-dimensional gapless surface theories of bulk (3+1)-dimensional topological phases, such as the BF theory at level K, and its generalization. In particular, we put these theories on a flat (2+1) dimensional torus T3 parameterized by its modular parameters, and compute the partition functions obeying various twisted boundary conditions. We show the partition functions are transformed into each other under SL(3, Z) modular transformations, and furthermore establish the bulk-boundary correspondence in (3+1) dimensions by matching the modular S and T matrices computed from the boundary field theories with those computed in the bulk. We propose the three-loop braiding statistics can be studied by constructing the modular S and T matrices from an appropriate boundary field theory. We also study the EE for 3 + 1- dimensional topological phase with or without three-loop braiding phase.
We show that the R\'enyi entropies of single particle, critical wave functions for disordered systems contain information about the multifractal spectrum. It is shown for moments of the R\'enyi entropy, Sn, where ∣n∣<1, it is possible to extract universal information about the multifractility of such systems. This is shown through a generic calculation and then illustrated through two example models. We find good agreement between our analytic formula and numerical simulations of the two test models. Our formalism is easily extendable to generic noninteracting fermion models. It is also suggested that recent experimental advances in measuring the multifractal spectrum might allow some moments of the R\'enyi entropy to be measured.
Entanglement (Rényi) entropies of spatial regions are a useful tool for characterizing the ground states of quantum field theories. In this paper we investigate the extent to which these are universal quantities for a given theory, and to which they distinguish different theories, by comparing the entanglement spectra of the massless Dirac fermion and the compact free boson in two dimensions. We show that the calculation of Rényi entropies via the replica trick for any orbifold theory includes a sum over orbifold twists on all cycles. In a modular-invariant theory of fermions, this amounts to a sum over spin structures. The result is that the Rényi entropies respect the standard Bose–Fermi duality. Next, we investigate the entanglement spectrum for the Dirac fermion without a sum over spin structures, and for the compact boson at the self-dual radius. These are not equivalent theories; nonetheless, we find that (1) their second Rényi entropies agree for any number of intervals, (2) their full entanglement spectra agree for two intervals, and (3) the spectrum generically disagrees otherwise. These results follow from the equality of the partition functions of the two theories on any Riemann surface with imaginary period matrix. We also exhibit a map between the operators of the theories that preserves scaling dimensions (but not spins), as well as OPEs and correlators of operators placed on the real line. All of these coincidences can be traced to the fact that the momentum lattice for the bosonized fermion is related to that of the self-dual boson by a 45° rotation that mixes left- and right-movers.
Using the density matrix renormalization group, we calculated the finite-size corrections of the entanglement $\ensuremath{\alpha}$-R\'enyi entropy of a single interval for several critical quantum chains. We considered models with U(1) symmetry such as the spin-1/2 XXZ and spin-1 Fateev-Zamolodchikov models, as well as models with discrete symmetries such as the Ising, the Blume-Capel, and the three-state Potts models. These corrections contain physically relevant information. Their amplitudes, which depend on the value of $\ensuremath{\alpha}$, are related to the dimensions of operators in the conformal field theory governing the long-distance correlations of the critical quantum chains. The obtained results together with earlier exact and numerical ones allow us to formulate some general conjectures about the operator responsible for the leading finite-size correction of the $\ensuremath{\alpha}$-R\'enyi entropies. We conjecture that the exponent of the leading finite-size correction of the $\ensuremath{\alpha}$-R\'enyi entropies is ${p}_{\ensuremath{\alpha}}=2{X}_{\ensuremath{\epsilon}}/\ensuremath{\alpha}$ for $\ensuremath{\alpha}>1$ and ${p}_{1}=\ensuremath{\nu}$, where ${X}_{\ensuremath{\epsilon}}$ denotes the dimensions of the energy operator of the model and $\ensuremath{\nu}=2$ for all the models.
Rényi entropies, S_{n}, admit a natural generalization in the presence of global symmetries. These "charged Rényi entropies" are functions of the chemical potential μ conjugate to the charge contained in the entangling region and reduce to the usual notions as μ→0. For n=1, this provides a notion of charged entanglement entropy. In this Letter, we prove that for a general d(≥3)-dimensional conformal field theory, the leading correction to the uncharged entanglement entropy across a spherical entangling surface is quadratic in the chemical potential, positive definite, and universally controlled (up to fixed d-dependent constants) by the coefficients C_{J} and a_{2}. These fully characterize, for a given theory, the current correlators ⟨JJ⟩ and ⟨TJJ⟩, as well as the energy flux measured at infinity produced by the insertion of the current operator. Our result is motivated by analytic holographic calculations for a special class of higher-curvature gravities coupled to a (d-2) form in general dimensions as well as for free fields in d=4. A proof for general theories and dimensions follows from previously known universal identities involving the magnetic response of twist operators introduced in A. Belin et al. [J. High Energy Phys. 12 (2013) 059.JHEPFG1029-847910.1007/JHEP12(2013)059] and basic thermodynamic relations.
We study the time evolution of single interval Rényi and entanglement entropies following local quantum quenches in two dimensional conformal field theories at finite temperature for which the locally excited states have a finite temporal width ϵ. We show that, for local quenches produced by the action of a conformal primary field, the time dependence of Rényi and entanglement entropies at order ϵ2 is universal. It is determined by the expectation value of the stress tensor in the replica geometry and proportional to the conformal dimension of the primary field generating the local excitation. We also show that in CFTs with a gravity dual, the ϵ2 correction to the holographic entanglement entropy following a local quench precisely agrees with the CFT prediction. We then consider CFTs admitting a higher spin symmetry and turn on a higher spin chemical potential μ. We calculate the time dependence of the order ϵ2 correction to the entanglement entropy for small μ, and show that the contribution at order μ 2 is universal. We verify our arguments against exact results for minimal models and the free fermion theory.