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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfc_et/content/2301.00294v1.pdf,len=612
+page_content='Modular Hamiltonian of the scalar field in the semi  infinite line: dimensional reduction for spherically  symmetric regions  Marina Huerta∗ and Guido van der Velde†  Centro Atómico Bariloche, 8400-S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfc_et/content/2301.00294v1.pdf'}
+page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfc_et/content/2301.00294v1.pdf'}
+page_content=' de Bariloche, Río Negro, Argentina  Abstract  We focus our attention on the one dimensional scalar theories that result from dimen-  sionally reducing the free scalar field theory in arbitrary d dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfc_et/content/2301.00294v1.pdf'}
+page_content=' As is well known,  after integrating out the angular coordinates, the free scalar theory can be expressed as  an infinite sum of theories living in the semi-infinite line, labeled by the angular modes  {ℓ, ⃗m}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfc_et/content/2301.00294v1.pdf'}
+page_content=' We show that their modular Hamiltonian in an interval attached to the origin is, in  turn, the one obtained from the dimensional reduction of the modular Hamiltonian of the  conformal parent theory in a sphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfc_et/content/2301.00294v1.pdf'}
+page_content=' Remarkably, this is a local expression in the energy  density, as happens in the conformal case, although the resulting one-dimensional theories  are clearly not conformal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfc_et/content/2301.00294v1.pdf'}
+page_content=' We support this result by analyzing the symmetries of these  theories, which turn out to be a portion of the original conformal group, and proving that  the reduced modular Hamiltonian is in fact the operator generating the modular flow in  the interval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfc_et/content/2301.00294v1.pdf'}
+page_content=' By studying the spectrum of these modular Hamiltonians, we also provide an  analytic expression for the associated entanglement entropy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfc_et/content/2301.00294v1.pdf'}
+page_content=' Finally, extending the radial  regularization scheme originally introduced by Srednicki, we sum over the angular modes  to successfully recover the conformal anomaly in the entropy logarithmic coefficient in even  dimensions, as well as the universal constant F term in d = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfc_et/content/2301.00294v1.pdf'}
+page_content='  1  Introduction: Modular flow and modular Hamiltonian  The successful application of information theory tools to quantum field theory (QFT) along the  last decades, has given place to the solid current consensus that these tools must be definitively  incorporated into the usual QFT machinery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfc_et/content/2301.00294v1.pdf'}
+page_content=' In this context, the study of quantities related  to different information measures for quantum field theories gains relevance and with them, the  study of states reduced to a region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfc_et/content/2301.00294v1.pdf'}
+page_content=' These states are described by reduced (local) density matrices  that live in the core of the definition of all the information measures referenced to spatial regions  R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfc_et/content/2301.00294v1.pdf'}
+page_content=' From the quantum field algebraic perspective [1], each region R is attached to the algebra  ∗e-mail: marina.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfc_et/content/2301.00294v1.pdf'}
+page_content='huerta@cab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfc_et/content/2301.00294v1.pdf'}
+page_content='cnea.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfc_et/content/2301.00294v1.pdf'}
+page_content='gov.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfc_et/content/2301.00294v1.pdf'}
+page_content='ar  †e-mail: guido.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfc_et/content/2301.00294v1.pdf'}
+page_content='vandervelde@ib.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfc_et/content/2301.00294v1.pdf'}
+page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfc_et/content/2301.00294v1.pdf'}
+page_content='ar  1  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfc_et/content/2301.00294v1.pdf'}
+page_content='00294v1  [hep-th]  31 Dec 2022  of the degrees of freedom localized in R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfc_et/content/2301.00294v1.pdf'}
+page_content=' The reduced state to a local algebra of operators in a  region can be expressed, in presence of a cutoff, as a density matrix  ρ = e−K  tre−K ,  (1)  where the exponent K is the modular Hamiltonian operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfc_et/content/2301.00294v1.pdf'}
+page_content=' This convenient way of encoding  the reduced state admits an interesting interpretation of the entanglement entropy as the ther-  modynamic entropy of a system in equilibrium at temperature 1, but with respect to the modular  Hamiltonian K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfc_et/content/2301.00294v1.pdf'}
+page_content=' Moreover, there is a time notion associated to the state through the modular  Hamiltonian, whose evolution is implemented by the unitary operator in the algebra  U(τ) = ρiτ ∼ e−iτK .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfc_et/content/2301.00294v1.pdf'}
+page_content='  (2)  The induced evolution of operators O(τ) = U(τ)OU(−τ) is called the modular flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfc_et/content/2301.00294v1.pdf'}
+page_content=' This is a  purely quantum transformation, which becomes trivial in the classical limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfc_et/content/2301.00294v1.pdf'}
+page_content='  Historically, the earliest recognition of the structural importance of modular flows can be found  in the algebraic formulation of QFT [2, 3] and more recently, in the framework of the study of  different information measures and statistical properties of reduced states in QFT [4, 5, 6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfc_et/content/2301.00294v1.pdf'}
+page_content='  The modular Hamiltonian is a fundamental constitutive part of the relative entropy and plays  an essential role in the entropy bounds formulations and proof of several energy conditions  [7, 8, 9, 10, 11, 12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfc_et/content/2301.00294v1.pdf'}
+page_content=' Besides, profiting that entanglement and relative entropy have well established  geometric duals for holographic QFT [13, 14, 15], modular Hamiltonians have also been used to  clarify localization properties of degrees of freedom in quantum gravity [16, 17, 18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfc_et/content/2301.00294v1.pdf'}
+page_content='  Currently, our knowledge of the explicit form of modular Hamiltonians reduces mostly to  some examples where the modular flow is local, and it is primarily determined by spacetime  symmetries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfc_et/content/2301.00294v1.pdf'}
+page_content='  This is the case for the Rindler wedge x1 > |t| in Minkowski space and any QFT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfc_et/content/2301.00294v1.pdf'}
+page_content=' Choosing the  causal region to be the half spatial plane x1 > 0 and t = 0 then, the rotational symmetry of the  euclidean theory allows us to express the reduced density matrix corresponding to the vacuum  state in terms of the energy density T00  ρ = k e−2π  �  x1>0 dd−1x x1T00(x) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfc_et/content/2301.00294v1.pdf'}
+page_content='  (3)  The above expression manifestly reveals a non trivial connection between entanglement in  vacuum and energy density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfc_et/content/2301.00294v1.pdf'}
+page_content=' Moreover, in equation (3), the exponent corresponds to the modular  Hamiltonian for half space which results to be an integral of a local operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfc_et/content/2301.00294v1.pdf'}
+page_content=' K is in fact 2π  times the generator of boosts restricted to act only on the right Rindler wedge  K = −2π  �  x1>0  dd−1x x1T00(x) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfc_et/content/2301.00294v1.pdf'}
+page_content='  (4)  The modular flow ρiτ moves operators locally following the orbits of the one parameter group  of boost transformations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfc_et/content/2301.00294v1.pdf'}
+page_content=' On the other hand, it is interesting to note that from equation (3),  the vacuum state in half space corresponds to a thermal state of inverse temperature 2π with  respect to the boost operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfc_et/content/2301.00294v1.pdf'}
+page_content=' This is directly connected to the Unruh’s effect [19] according  to which accelerated observers see the vacuum as a thermally excited state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfc_et/content/2301.00294v1.pdf'}
+page_content=' For an observer  following a trajectory given by a boost orbit, the state looks like a thermal state with respect  to the proper time ˜τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfc_et/content/2301.00294v1.pdf'}
+page_content=' For these trajectories, the proper time and the boost parameter s are  2  proportional s = a˜τ with a the proper acceleration of the observer, constant along boost orbits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfc_et/content/2301.00294v1.pdf'}
+page_content='  In turn, this implies there is a relation K = ˜H/a between the boost operator and the proper  time Hamiltonian ˜H of the accelerated observer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfc_et/content/2301.00294v1.pdf'}
+page_content=' For such an observer there is a thermal bath  at (proper time) temperature T = 2π  a .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfc_et/content/2301.00294v1.pdf'}
+page_content='  The other very well known example where symmetries again facilitate the derivation of the  exact modular Hamiltonian is the case of conformal field theories (CFT) for spheres in any  dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfc_et/content/2301.00294v1.pdf'}
+page_content='  For a CFT, Poincare symmetries are enlarged to the conformal group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfc_et/content/2301.00294v1.pdf'}
+page_content='  These  theories are characterized by having a traceless, symmetric and conserved stress tensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfc_et/content/2301.00294v1.pdf'}
+page_content=' This  enlarges the number of conserved currents related to space-time symmetries which in general can  be written as  jµ = aν Tνµ + bαν xα Tνµ + c xν Tνµ + dα (x2gαν − 2 xαxν) Tνµ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfc_et/content/2301.00294v1.pdf'}
+page_content='  (5)  The corresponding conserved charges depend on parameters aµ, determining translations, the  antisymmetric bµν, giving Lorentz transformations, c, related to dilatations, and dµ, for the so  called special conformal transformations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfc_et/content/2301.00294v1.pdf'}
+page_content='  Since there is a conformal transformation that maps the Rindler wedge to causal regions with  spherical boundary, and the same transformation leaves the vacuum invariant for a CFT, then,  the modular Hamiltonian is just the transformed Rindler modular Hamiltonian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfc_et/content/2301.00294v1.pdf'}
+page_content=' It is easy to get  K = 2π  �  |⃗x|<R  dd−1x R2 − r2  2R  T00(⃗x) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfc_et/content/2301.00294v1.pdf'}
+page_content='  (6)  In this example, K is again local and proportional to T00, with a proportionality weight function  β(r) ≡ R2−r2  2R .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfc_et/content/2301.00294v1.pdf'}
+page_content='  Except for the two examples discussed above, the vacuum of a QFT in the Rindler wedge and  the vacuum of a CFT in the sphere, there are only some other few known modular Hamiltonians,  either local or not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfc_et/content/2301.00294v1.pdf'}
+page_content=' The local ones in general are derived profiting from symmetry transformations  that leave the state invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfc_et/content/2301.00294v1.pdf'}
+page_content=' This is for example the case of the modular Hamiltonian for CFTs  in 1 + 1 dimensions in presence of a global or local quench [20, 21, 22, 23, 24, 25, 26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfc_et/content/2301.00294v1.pdf'}
+page_content=' However,  on general grounds, from the point of view of quantum information we do not expect locality to  hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfc_et/content/2301.00294v1.pdf'}
+page_content=' In general, K will be given by a non local and non linear combination of the field operators  at different positions inside the region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfc_et/content/2301.00294v1.pdf'}
+page_content='  An example of a non local modular Hamiltonian which has been explicitly computed is the one  for the vacuum state of the free massless fermion in d = 2 for several disjoint intervals [27, 28, 29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfc_et/content/2301.00294v1.pdf'}
+page_content='  In this case K has a local term proportional to the energy density and an additional non local  part given by a quadratic expression in the fermion field that connects in a very particular way  points located in different intervals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfc_et/content/2301.00294v1.pdf'}
+page_content='  In this paper we calculate the modular Hamiltonian for the vacuum state of non conformal  (1 + 1) dimensional theories in the interval (0, R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfc_et/content/2301.00294v1.pdf'}
+page_content=' These theories are defined in the semi infinite  line, and result from the dimensional reduction of the d dimensional free massless scalar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfc_et/content/2301.00294v1.pdf'}
+page_content=' Our  strategy is to calculate the modular Hamiltonian of the reduced system by profiting of the known  modular Hamiltonian of CFTs in spheres in any dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfc_et/content/2301.00294v1.pdf'}
+page_content='  The free massless scalar in d space time dimensions can be dimensionally reduced to a sum  of one dimensional theories, one for each angular mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfc_et/content/2301.00294v1.pdf'}
+page_content='  Since the reduction is obtained by  integrating over the angular coordinates, these systems live in the semi infinite line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfc_et/content/2301.00294v1.pdf'}
+page_content=' From the  algebraic point of view, this is convenient when studying algebras assigned to spherical regions to  calculate, for example, the entanglement entropy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfc_et/content/2301.00294v1.pdf'}
+page_content=' In these coordinates, the local algebra assigned  3  R  R  Figure 1: The sphere of radius R corresponds to intervals of length R with one edge in the origin in  the radial semi infinite line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfc_et/content/2301.00294v1.pdf'}
+page_content='  to the region can be easily written in terms of fields φ(r, Ω) with nice localization properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfc_et/content/2301.00294v1.pdf'}
+page_content=' For  example, points in the semi infinite line correspond to shells in the original space and intervals  connected to the origin, to d-spheres (see figure 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfc_et/content/2301.00294v1.pdf'}
+page_content=' Concretely, in the radial coordinate, the  canonical Hamiltonian for the massless free scalar decomposes as a sum over angular modes Hℓ⃗m  H =  �  ℓ⃗m  Hℓ⃗m .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfc_et/content/2301.00294v1.pdf'}
+page_content='  (7)  with (ℓ⃗m) the angular mode label.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfc_et/content/2301.00294v1.pdf'}
+page_content=' In fact, there is a family of one dimensional Hamiltonian Hℓ⃗m  for each dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfc_et/content/2301.00294v1.pdf'}
+page_content=' In turn, the same decomposition occurs for the modular Hamiltonian (6)  K =  �  ℓ⃗m  Kℓ⃗m .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfc_et/content/2301.00294v1.pdf'}
+page_content='  (8)  Taking into account that the vacuum state for a system composed by independent subsystems  is a product of density matrices, here ρ = ⊗ρℓ⃗m, then it is immediate to identify the modular  Hamiltonian mode Kℓ⃗m with the modular Hamiltonian of the one dimensional reduced system  Hℓ⃗m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfc_et/content/2301.00294v1.pdf'}
+page_content=' The Hamiltonian Hℓ⃗m does not correspond to a conformal relativistic theory due to an  extra quadratic term proportional to 1/r2, whose proportionality constant depends on the di-  mension of the original problem and the angular mode ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfc_et/content/2301.00294v1.pdf'}
+page_content='  Surprisingly, we find that Kℓ⃗m is  still local and proportional to the energy density T001, with the same weight function β(r) that  characterizes the modular Hamiltonian for CFTs in spheres.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfc_et/content/2301.00294v1.pdf'}
+page_content=' Our analytic results coincide with  the suggested continuum limit of the entanglement Hamiltonian of blocks of consecutive sites in  massless harmonic chains, recently studied in [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfc_et/content/2301.00294v1.pdf'}
+page_content='  This article is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfc_et/content/2301.00294v1.pdf'}
+page_content='  In section 2 we explicitly carry out the dimensional  reduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfc_et/content/2301.00294v1.pdf'}
+page_content=' We write the scalar field in a basis of hyper-spherical harmonics, and after integrating  out the angular coordinates we are left with a Hamiltonian for the reduced systems Hℓ⃗m of the  form  Hℓ⃗m = 1  2  �  dr  �  �π2  ℓ⃗m + (∂r �φℓ⃗m)2 + µd(ℓ)  r2  �φ2  ℓ⃗m  �  ,  (9)  with  µd(ℓ) = (d − 4)(d − 2)  4  + ℓ(ℓ + d − 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfc_et/content/2301.00294v1.pdf'}
+page_content='  (10)  In section 3 the same procedure is followed to find the modular Hamiltonian  Kℓ,⃗m = 2π  �  |⃗x|<R  dr R2 − r2  2R  T ℓ,⃗m  00 (⃗x) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfc_et/content/2301.00294v1.pdf'}
+page_content='  (11)  1Since translational invariance is lost, there is no conserved energy momentum tensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfc_et/content/2301.00294v1.pdf'}
+page_content=' The notation for the  energy density is just a matter of convention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfc_et/content/2301.00294v1.pdf'}
+page_content='  4  In some way, the reduced theory, manifestly invariant under dilatations but non conformal, keeps  the memory of the conformal symmetry of the parent d-dimensional theory [31, 32], with the  same local modular Hamiltonian as that representing the vacuum of a CFT in a sphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfc_et/content/2301.00294v1.pdf'}
+page_content=' We delve  into this in section 4, where we show that the reduced theories preserve an SL(2, R) symmetry,  and that the modular transformation belongs to this subgroup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfc_et/content/2301.00294v1.pdf'}
+page_content=' The modular Hamiltonian (11)  written as a Noether charge can be correctly interpreted as the local operator implementing the  modular flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfc_et/content/2301.00294v1.pdf'}
+page_content='  In section 5 we solve the spectrum of the modular Hamiltonian (11) and compute the entan-  glement entropy in a segment connected to the origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfc_et/content/2301.00294v1.pdf'}
+page_content=' We find the analytic expression  S(ℓ, d) = 1  6 log R  ϵ − iπ  2  � ∞  0  ds  s  sinh2(πs) log  �4isΓ [is] Γ [−1 + d/2 + ℓ − is]  Γ [−is] Γ [−1 + d/2 + ℓ + is]  �  ,  (12)  which is logarithmically divergent, with coefficient 1/6 as expected for (1+1) theories, and has a  constant term that depends both on the mode ℓ and the space time dimensions d of the original  theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfc_et/content/2301.00294v1.pdf'}
+page_content=' Although the above integral cannot in general be solved analytically, we make some  useful approximations to extract relevant information out of it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfc_et/content/2301.00294v1.pdf'}
+page_content=' Moreover, by summing over ℓ we  are able to recover the conformal anomaly in the logarithmic coefficient for the free scalar field  in even dimensions, as well as the constant universal F term in d = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfc_et/content/2301.00294v1.pdf'}
+page_content=' In doing the sum over  the angular modes ℓ, we introduce a novel regularization implemented by a damping exponential  exp[−ℓϵ/R], with the same cutoff ϵ that regularizes the radial coordinate r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfc_et/content/2301.00294v1.pdf'}
+page_content=' This procedure  generalizes the radial regularization scheme introduced by Srednicki in [33], where it is explicitly  stated that for d ⩾ 4 regularization by a radial lattice turns out to be insufficient and the sum  over partial waves does not converge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfc_et/content/2301.00294v1.pdf'}
+page_content=' We end the discussion with some concluding remarks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfc_et/content/2301.00294v1.pdf'}
+page_content='  2  Spherical coordinates  The free scalar action in spherical coordinates reads  S = 1  2  �  dtdrrd−2dΩ  �  −(∂0φ)2 + (∂rφ)2 − φ  r2∆Sd−2φ  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfc_et/content/2301.00294v1.pdf'}
+page_content='  (13)  With the aim of reducing the above to a single integral in the radial direction, we Fourier  transform the scalar field in the angular coordinates, using the real hyper-spherical harmonics  as basis functions,  φ(⃗r) =  �  ℓm1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfc_et/content/2301.00294v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfc_et/content/2301.00294v1.pdf'}
+page_content='md−3  φℓm1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfc_et/content/2301.00294v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfc_et/content/2301.00294v1.pdf'}
+page_content='md−3(r)Y m1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfc_et/content/2301.00294v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfc_et/content/2301.00294v1.pdf'}
+page_content='md−3  ℓ  (ˆr),  (14)  with  ∆Sd−2Y m1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfc_et/content/2301.00294v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfc_et/content/2301.00294v1.pdf'}
+page_content='md−3  ℓ  (ˆr) = −ℓ(ℓ + d − 3)Y m1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfc_et/content/2301.00294v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfc_et/content/2301.00294v1.pdf'}
+page_content='md−3  ℓ  (ˆr),  (15)  �  Sd−2 dΩY m1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfc_et/content/2301.00294v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfc_et/content/2301.00294v1.pdf'}
+page_content='md−3  ℓ  (ˆr)Y  m′  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfc_et/content/2301.00294v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfc_et/content/2301.00294v1.pdf'}
+page_content='m′  d−3  ℓ′  (ˆr) = δℓℓ′δm1m′  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfc_et/content/2301.00294v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfc_et/content/2301.00294v1.pdf'}
+page_content='δmd−3m′  d−3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfc_et/content/2301.00294v1.pdf'}
+page_content='  (16)  After integrating the angular coordinates, we are left with  S = 1  2  �  ℓ⃗m  �  dtdrrd−2  �  −(∂0φℓ⃗m)2 + (∂rφℓ⃗m)2 + ℓ(ℓ + d − 3)  r2  φ2  ℓ⃗m  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfc_et/content/2301.00294v1.pdf'}
+page_content='  (17)  5  However, the theory looks simpler when defined in terms of the rescaled field �φℓ⃗m = r  d−2  2 φℓ⃗m,  whose canonically conjugated momentum is �πℓ⃗m ≡ ∂0�φℓ⃗m,  S = 1  2  �  ℓ⃗m  �  dtdr  �  �−(∂0�φℓ⃗m)2 + rd−2  �  ∂r  � �φℓ⃗m  r  d−2  2  ��2  + ℓ(ℓ + d − 3)  r2  �φ2  ℓ⃗m  �  � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfc_et/content/2301.00294v1.pdf'}
+page_content='  (18)  Functional variation with respect to the field leads to the equation of motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfc_et/content/2301.00294v1.pdf'}
+page_content=' Nevertheless, in  order for the variational problem to be well posed we should impose specific boundary conditions  at r = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfc_et/content/2301.00294v1.pdf'}
+page_content=' In fact,  δS =  �  ℓ⃗m  ��  dtdr  �  ∂2  0 �φℓ⃗m −  1  r  d−2  2 ∂r  �  rd−2∂r  � �φℓ⃗m  r  d−2  2  ��  + ℓ(ℓ + d − 3)  r2  �φℓ⃗m  �  δ�φℓ⃗m  +  �  dt  �  r  d−2  2 ∂r  � �φℓ⃗m  r  d−2  2  �  δ�φℓ⃗m  ������  ∞  0  �  ,  (19)  which requires either δ�φℓ⃗m(r = 0, t) = 0 (Dirichlet boundary conditions) or r  d−2  2 ∂r  � �φℓ ⃗m  r  d−2  2  �  → 0  (analogous to the ordinary Neumann boundary conditions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfc_et/content/2301.00294v1.pdf'}
+page_content=' In the following we will adopt the  former.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfc_et/content/2301.00294v1.pdf'}
+page_content='  The second term in (19) can be further simplified, which leads to the saddle point  ∂2  0 �φℓ⃗m − ∂2  r �φℓ⃗m + µd(ℓ)  r2  �φℓ⃗m = 0,  (20)  with  µd(ℓ) = (d − 4)(d − 2)  4  + ℓ(ℓ + d − 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfc_et/content/2301.00294v1.pdf'}
+page_content='  (21)  This partial differential equation can be solved by separation of variables, and expressed in terms  of the original field φℓ⃗m, the radial eigenfunction problem is a Bessel equation, with solution  jℓ(r) ≡  1  r(d−3)/2Jℓ+ d−3  2 (kr).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfc_et/content/2301.00294v1.pdf'}
+page_content=' Therefore, the solution is  �φℓ⃗m(t, r) = e±ikt√  krJℓ+ d−3  2 (kr),  (22)  which means that �φℓ⃗m ∼ rℓ+d/2−1 near kr ∼ 0, in agreement with the boundary conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfc_et/content/2301.00294v1.pdf'}
+page_content='  Having stated that, it is also possible to rewrite the second term in (18) by getting rid of a  boundary term2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfc_et/content/2301.00294v1.pdf'}
+page_content=' More explicitly,  S =  �  ℓ⃗m  Sℓ⃗m  (23)  where  Sℓ⃗m = 1  2  �  dtdr  �  −(∂0�φℓ⃗m)2 + (∂r �φℓ⃗m)2 + µd(ℓ)  r2  �φ2  ℓ⃗m  �  (24)  can be thought of as the action for a free scalar living in the half line, satisfying Dirichlet  boundary conditions at the origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfc_et/content/2301.00294v1.pdf'}
+page_content=' Note that, unlike the theory we started with, this is not a  CFT because of the last term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfc_et/content/2301.00294v1.pdf'}
+page_content='  2We would be able to ignore the boundary term provided �φ2  ℓ⃗m went to zero faster than r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfc_et/content/2301.00294v1.pdf'}
+page_content=' This is at least  satisfied by the classical configuration (22).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfc_et/content/2301.00294v1.pdf'}
+page_content='  6  The dimensional reduction of the free scalar Hamiltonian can be made following the same  steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfc_et/content/2301.00294v1.pdf'}
+page_content=' But we can alternatively calculate the conserved charge due to time translations associated  directly to the 1 + 1 dimensional action (24), yielding  H = 1  2  �  ℓ⃗m  �  dr  �  �π2  ℓ⃗m + (∂r �φℓ⃗m)2 + µd(ℓ)  r2  �φ2  ℓ⃗m  �  (25)  Once again we stress that �πℓ⃗m and �φℓ⃗m satisfy canonical commutation relations  �  �φℓ⃗m(r), �πℓ′ ⃗m′(r′)  �  = iδℓ,ℓ′δ⃗m,⃗m′δ(r − r′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfc_et/content/2301.00294v1.pdf'}
+page_content='  (26)  3  The sphere modular Hamiltonian  On the other hand, since the free scalar field theory in d spacetime dimensions is conformally  invariant, when the whole system is in its ground state the modular Hamiltonian of a sphere is  K = 1  2  �  |x|<R  dxd−1  �R2 − r2  2R  �  T00.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfc_et/content/2301.00294v1.pdf'}
+page_content='  (27)  However, although the stress tensor involved in this expression must be traceless, the canonical  stress tensor of the free scalar field is  T (c)  µν = ∂µφ∂νφ − 1  2ηµν(∂φ)2,  (28)  which has non vanishing trace T µ  µ = (1 − d/2) (∂φ)2 = (1−d/2)  2  ∂2(φ2)3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfc_et/content/2301.00294v1.pdf'}
+page_content=' Hence, it must be improved  by adding a conserved symmetric tensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfc_et/content/2301.00294v1.pdf'}
+page_content=' A possible choice is  T ′  µν = T (c)  µν − (1 − d/2)  2(1 − d) (∂µ∂ν − ηµν∂2)φ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfc_et/content/2301.00294v1.pdf'}
+page_content='  (29)  Therefore,  K = 1  2  �  |x|<R  dxd−1  �R2 − r2  2R  � �  (∂0φ)2 + (∂iφ)2 − (1 − d/2)  (1 − d) ∂2  i φ2  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfc_et/content/2301.00294v1.pdf'}
+page_content='  (30)  Using the following identities:  (∂iφ)2 = (∂rφ)2 − φ  r2∆Sd−2φ,  (31)  where we have partially integrated the angular piece, and  ∂2  i φ2 =  1  rd−2∂r  �  rd−2∂rφ2�  + 1  r2∆Sd−2φ2,  (32)  we arrive at  K = 1  2  �  ℓ⃗m  �  drrd−2  �R2 − r2  2R  � �  π2  ℓ⃗m + (∂rφℓ⃗m)2 + ℓ(ℓ + d − 3)  r2  φ2  ℓ⃗m−  −(1 − d/2)  (1 − d)  �  ∂2  rφ2  ℓ⃗m + d − 2  r  ∂rφ2  ℓ⃗m  ��  ,  (33)  3This identity holds on-shell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfc_et/content/2301.00294v1.pdf'}
+page_content='  7  In terms of the canonically conjugated operators,  K = 1  2  �  ℓ⃗m  �  dr  �R2 − r2  2R  � �  �π2  ℓ⃗m + (∂r �φℓ⃗m)2 + µd(ℓ)  r2  �φ2  ℓ⃗m−  − d − 2  2(d − 1)  �  3∂r  � �φ2  ℓ⃗m  r  �  + r∂2  r  � �φ2  ℓ⃗m  r  ���  ,  (34)  Note that the second line of (34), together with the prefactor (R2 − r2), is a total derivative in  disguise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfc_et/content/2301.00294v1.pdf'}
+page_content=' Hence,  K = 1  2  �  ℓ⃗m  �� R  0  dr  �R2 − r2  2R  � �  �π2  ℓ⃗m + (∂r �φℓ⃗m)2 + µd(ℓ)  r2  �φ2  ℓ⃗m  �  −  − d − 2  2(d − 1)  �  (R2 − r2)  2R  r∂r  � �φ2  ℓ⃗m  r  �  + R  � �φ2  ℓ⃗m  r  ������  R  0  �  ,  (35)  The boundary terms (coming from the improving) can be interpreted in general, as an ambiguity  in the definition of modular Hamiltonian in a region, and safely ignored as explained in [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfc_et/content/2301.00294v1.pdf'}
+page_content='  Consequently, the modular Hamiltonian of the d dimensional free scalar is  K =  �  ℓ⃗m  Kℓ⃗m,  (36)  where  Kℓ⃗m = 1  2  � R  0  dr  �R2 − r2  2R  � �  �π2  ℓ⃗m + (∂r �φℓ⃗m)2 + µd(ℓ)  r2  �φ2  ℓ⃗m  �  (37)  can be interpreted as the modular Hamiltonian for the vacuum of (24) in a segment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfc_et/content/2301.00294v1.pdf'}
+page_content='  This  identification rests on the fact that the theory decomposes into independent sectors, labeled by  the angular modes, so the state must write as the direct product of the states pertaining to each  sector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfc_et/content/2301.00294v1.pdf'}
+page_content=' But, most remarkably, this modular Hamiltonian is still local in the energy density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfc_et/content/2301.00294v1.pdf'}
+page_content=' In  other words, (37) agrees with the general expression (27) in spite of the reduced one dimensional  theories being non conformal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfc_et/content/2301.00294v1.pdf'}
+page_content=' In the next section we analyse this in detail, paying attention to  the symmetries which survive the dimensional reduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfc_et/content/2301.00294v1.pdf'}
+page_content='  Provided that (37) defines the reduced state of a free field theory, Wick’s theorem guarantees  it can be expressed in terms of the two-point correlators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfc_et/content/2301.00294v1.pdf'}
+page_content=' In fact, for a Gaussian state with  modular Hamiltonian  K =  �  V  dd−1x1dd−1x2 [φ(x1)M(x1, x2)φ(x2) + π(x1)N(x1, x2)π(x2)] ,  (38)  and correlators  X = ⟨φ(x1)φ(x2)⟩ ,  P = ⟨π(x1)π(x2)⟩ ,  (39)  the following relation must be satisfied [35]4  M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfc_et/content/2301.00294v1.pdf'}
+page_content='X = P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfc_et/content/2301.00294v1.pdf'}
+page_content='N  (40)  4Here the product is a bi-local function constructed as  [M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfc_et/content/2301.00294v1.pdf'}
+page_content='X] (x1, x2) ≡  �  V  dyM(x1, y)X(y, x2)  8  In the case at hand,  M(r, r′) = −2πδ(r − r′)  �  β(r)∂2  r + ∂rβ(r)∂r − β(r) µ  r2  �  (41)  and  N(r, r′) = 2πδ(r − r′)β(r) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfc_et/content/2301.00294v1.pdf'}
+page_content='  (42)  Meanwhile, the explicit form of the correlators for the one dimensional theory (24) is [36]  X(r1, r2) =  Γ [ℓ + d/2 − 1]  2Γ  � 1  2  �  Γ  �  ℓ + d−1  2  �  �r1  r2  �ℓ+ d  2 −1  2F1  �  1  2, ℓ + d  2 − 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfc_et/content/2301.00294v1.pdf'}
+page_content=' ℓ + d − 1  2  ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfc_et/content/2301.00294v1.pdf'}
+page_content='  �r1  r2  �2�  ,  (43)  P(r1, r2) =  2Γ(ℓ + d/2)  Γ  � 1  2  �  Γ  �  ℓ + d−1  2  �  (r2  2 − r2  1)  �r1  r2  �ℓ+ d  2 −1 �  A 2F1  �  1  2, ℓ + d  2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfc_et/content/2301.00294v1.pdf'}
+page_content=' ℓ + d − 1  2  ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfc_et/content/2301.00294v1.pdf'}
+page_content='  �r1  r2  �2�  +B 2F1  �  −1  2, ℓ + d  2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfc_et/content/2301.00294v1.pdf'}
+page_content=' ℓ + d − 1  2  ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfc_et/content/2301.00294v1.pdf'}
+page_content='  �r1  r2  �2��  ,  (44)  where A =  �  ℓ + d−1  2  �  (1 − r2  1/r2  2) − 1, B = 1 − ℓ − d/2, and r1 < r2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfc_et/content/2301.00294v1.pdf'}
+page_content='  Using these concrete  expressions it is possible to check that (40) indeed holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfc_et/content/2301.00294v1.pdf'}
+page_content='  4  Symmetries  The locality of (37) suggests the existence of a symmetry with a conserved current such that the  modular Hamiltonian is the corresponding Noether charge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfc_et/content/2301.00294v1.pdf'}
+page_content=' This has to be an endomorphism in  the causal wedge of the region, and must point in the time direction at t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfc_et/content/2301.00294v1.pdf'}
+page_content=' For CFTs in spheres  in any dimensions, this is the conformal transformation that maps the spherical boundary in  itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfc_et/content/2301.00294v1.pdf'}
+page_content=' For an interval (0, R) in the half line, whose causal wedge is a half diamond, the symmetry  transformation leaves the boundary point r = R fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfc_et/content/2301.00294v1.pdf'}
+page_content=' The identification of this symmetry in  the present case is the natural path to justify the locality of (37).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfc_et/content/2301.00294v1.pdf'}
+page_content=' With this aim, we first discuss  the symmetries of the reduced theories with action (24).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfc_et/content/2301.00294v1.pdf'}
+page_content='  The symmetries of (24) are a subgroup of the conformal transformations inherited from higher  dimensions, in particular those which involve only the time and radial coordinates, and that map  the line r = 0 into itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfc_et/content/2301.00294v1.pdf'}
+page_content=' These are  Time translations:  t → t + t0  (45)  Dilatations:  (t, r) → (λt, λr)  (46)  Special conformal transformations with parameter bµ = α  Rˆeµ  t :  (t, r) →  �  tR2 + αR(t2 − r2)  R2 + 2αRt + α2(t2 − r2),  rR2  R2 + 2αRt + α2(t2 − r2)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfc_et/content/2301.00294v1.pdf'}
+page_content='  (47)  Infinitesimally, that is, if we set α = ϵ << 1, then  (t, r) →  �  t − ϵ(t2 + r2)/R, r − 2ϵtr/R  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfc_et/content/2301.00294v1.pdf'}
+page_content='  (48)  9  The generators of the transformations listed above are P0 = i∂t, D = i(t∂t + r∂r), and K0 =  i ((t2 + r2)∂t + 2tr∂r) respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfc_et/content/2301.00294v1.pdf'}
+page_content=' These close an sl(2, R) algebra, which can be expressed in a  more suggestive way identifying L−1 ≡ P0, L0 ≡ D, L1 ≡ K0, so that  i [Lm, Ln]LB = (m − n)Lm+n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfc_et/content/2301.00294v1.pdf'}
+page_content='  (49)  Just for completeness, we note that one would have expected the original conformal group  SO(d, 2) to break into SO(2, 2) ∼ SL(2, R) ⊗ SL(2, R) [31, 32], with six generators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfc_et/content/2301.00294v1.pdf'}
+page_content=' In fact,  besides the three generators already mentioned, there are three more that do not mix the angular  coordinates with (t, r), associated to  Translations in the radial direction:  r → r + r0  (50)  Boosts:  (t, r) → (t + ϵr, r + ϵt)  (51)  Special conformal transformations with parameter bµ = α  Rˆeµ  r  (t, r) →  �  tR2  R2 − 2αRr − α2(t2 − r2),  rR2 + αR(t2 − r2)  R2 − 2αRr − α2(t2 − r2)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfc_et/content/2301.00294v1.pdf'}
+page_content='  (52)  These are ˆeµ  rPµ, ˆeµ  rM0µ and ˆeµ  rKµ, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfc_et/content/2301.00294v1.pdf'}
+page_content=' However, it is easy to see that they fail to become  symmetries of the dimensionally reduced theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfc_et/content/2301.00294v1.pdf'}
+page_content='  Then, the modular symmetry of the reduced theories we are looking for must be a particular  composition of the identified symmetry transformations (45) - (47).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfc_et/content/2301.00294v1.pdf'}
+page_content='  On the other hand, we know that the modular symmetry for the parent conformal theory is  associated to the generator of the boosts as seen from the domain of dependence of the ball [37],  ζ = π  R  �  (R2 − t2 − |⃗x|2)∂t − 2txi∂i  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfc_et/content/2301.00294v1.pdf'}
+page_content='  (53)  In fact, comparing with (45) and (47), we notice that this transformation in the semi infinite line  is the composition of a time translation of parameter ϵπR and a special conformal transformation  of parameter ϵ π  R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfc_et/content/2301.00294v1.pdf'}
+page_content=' Let us check this explicitly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfc_et/content/2301.00294v1.pdf'}
+page_content='  In spherical coordinates, the infinitesimal transformation reads  t −→ t′ = t + ϵ π  R(R2 − t2 − r2)  r −→ r′ = r + ϵ π  R(−2tr)  Ω −→ Ω′ = Ω  (54)  Since the invariance of the kinetic term is guaranteed, we need only to check the invariance of  the quadratic term dtdr/r2, which is less evident.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfc_et/content/2301.00294v1.pdf'}
+page_content=' On the one hand, we have  dtdr  =  dt′dr′  ����  ∂t  ∂t′  ∂t  ∂r′  ∂r  ∂t′  ∂r  ∂r′  ���� = dt′dr′  ����  1 + 2πϵt′/R + O(ϵ2)  2πϵr′/R + O(ϵ2)  2πϵr′/R + O(ϵ2)  1 + 2πϵt′/R + O(ϵ2)  ����  ∼  dt′dr′(1 + 4πϵt′/R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfc_et/content/2301.00294v1.pdf'}
+page_content='  (55)  10  On the other hand,  1  r2 ∼  1  (r′ + 2πϵt′r′/R)2 = 1  r′2(1 − 4πϵt′/R + O(ϵ2)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfc_et/content/2301.00294v1.pdf'}
+page_content='  (56)  Hence,  dtdr  r2  = dt′dr′  r′2  + O(ϵ2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfc_et/content/2301.00294v1.pdf'}
+page_content='  (57)  By Noether’s theorem, there must exist a conserved current associated to (54), which is of the  form5  jµ =  �  δL  δ(∂µφ)∂νφ − Lδµ  ν  �  ζν,  (58)  or, in components,  jt = 1  2  �  (∂tφ)2 + (∂rφ)2 + µ  r2φ2� (R2 − t2 − r2)  R  − 2tr  R∂rφ∂tφ,  (59)  jr = 1  2  �  (∂tφ)2 + (∂rφ)2 − µ  r2φ2�  2tr  R − ∂rφ∂tφ(R2 − t2 − r2)  R  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfc_et/content/2301.00294v1.pdf'}
+page_content='  (60)  Finally, the current above corresponds to a modular Hamiltonian  Kℓ⃗m =  � R  0  drj0(t = 0, r)  = 1  2  � R  0  dr  �R2 − r2  2R  � �  �π2  ℓ⃗m + (∂r �φℓ⃗m)2 + µd(ℓ)  r2  �φ2  ℓ⃗m  �  ,  (61)  the same as (37) deduced in the previous section from different arguments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfc_et/content/2301.00294v1.pdf'}
+page_content='  5  Modular Hamiltonian and entropy  In this section we study the spectrum of the modular Hamiltonian (37).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfc_et/content/2301.00294v1.pdf'}
+page_content=' Solving the eigenfunction  problem allows us to compute the entanglement entropy for an interval attached to the origin,  as a function of the angular mode ℓ and the original spacetime dimension d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfc_et/content/2301.00294v1.pdf'}
+page_content=' Then we sum over  the modes and compare the result with the entanglement entropy of the d-sphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfc_et/content/2301.00294v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfc_et/content/2301.00294v1.pdf'}
+page_content='1  Eigenfunctions  In general, given a quadratic modular Hamiltonian of a region V , of the form  K =  �  V  dd−1x dd−1x′ (φ(x)M(x, x′)φ(x′) + π(x)N(x, x′)π(x′)) ,  (62)  with M and N real symmetric operators, the eigenfunctions are those of the right and left action  of M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfc_et/content/2301.00294v1.pdf'}
+page_content='N, namely  (N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfc_et/content/2301.00294v1.pdf'}
+page_content='M)us = s2us  (63)  (M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfc_et/content/2301.00294v1.pdf'}
+page_content='N)vs = s2vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfc_et/content/2301.00294v1.pdf'}
+page_content='  (64)  5We remove the tildes and the angular mode labels to avoid cluttering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfc_et/content/2301.00294v1.pdf'}
+page_content='  11  This leads to the alternative way of writing K  K =  �  V  dd−1x  � ∞  0  ds us(x) s v∗  s(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfc_et/content/2301.00294v1.pdf'}
+page_content='  (65)  More concretely,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfc_et/content/2301.00294v1.pdf'}
+page_content=' the problem we are interested in is defined by (41) and (42),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfc_et/content/2301.00294v1.pdf'}
+page_content=' so the eigenfunctions  u and v satisfy the following hypergeometric equations6  �  β2∂2  r + β∂rβ∂r − β2 µ  r2  �  us = −s2us  (66)  �  β2∂2  r + 3β∂rβ∂r +  �  β∂2  rβ + (∂rβ)2 − β2 µ  r2  ��  vs = −s2vs  (67)  The solutions of these equations are7  us(r) = Nu  � r  R  �−1+ d  2 +ℓ �R2 − r2  R2  �−is  2F1  �1  2 − is,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfc_et/content/2301.00294v1.pdf'}
+page_content=' −1 + d  2 + ℓ − is,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfc_et/content/2301.00294v1.pdf'}
+page_content=' d  2 − 1  2 + ℓ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfc_et/content/2301.00294v1.pdf'}
+page_content=' r2  R2  �  vs(r) = Nu  R  β(r)us(r),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfc_et/content/2301.00294v1.pdf'}
+page_content='  (68)  where Nu is a normalization constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfc_et/content/2301.00294v1.pdf'}
+page_content='  Near r ∼ 0 the solutions behave as,  us(r) ∼ vs(r) ∝ r−1+ d  2 +ℓ  (69)  in agreement with the classical profile (22), whereas near r ∼ R they behave as  us(r) ∼ Nu  ��R − r  R  �−is  α(s) + c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfc_et/content/2301.00294v1.pdf'}
+page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfc_et/content/2301.00294v1.pdf'}
+page_content='  �  vs(r) ∼ Nu  ��R − r  R  �−1−is  α(s) + c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfc_et/content/2301.00294v1.pdf'}
+page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfc_et/content/2301.00294v1.pdf'}
+page_content='  �  ,  (70)  with  α(s) =  2−isΓ  � d−1  2 + ℓ  �  Γ [2is]  Γ  �  is + 1  2  �  Γ  �  is + ℓ + d  2 − 1  �.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfc_et/content/2301.00294v1.pdf'}
+page_content='  (71)  It is very important to keep in mind that there is a branch point at r = R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfc_et/content/2301.00294v1.pdf'}
+page_content=' In fact, since the  eigenfunctions must satisfy the orthogonality relation  � R  0  drus(r)v∗  s′(r) = δ(s − s′) ,  (72)  in order to find out the normalization factor Nu we substitute in (72) the leading terms in their  Taylor series expansion (70), because only the region near r ∼ R can contribute with a Dirac  delta function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfc_et/content/2301.00294v1.pdf'}
+page_content=' That results in  � R  0  drus(r)v∗  s′(r) ∼ 2|Nu|2Re [I(s − s′)α(s)α∗(s′) + I(s + s′)α(s)α(s′)] ,  (73)  6For later convenience we renormalize the eigenvalues to absorb a factor 1/(2π)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfc_et/content/2301.00294v1.pdf'}
+page_content='  7There is an additional independent solution, but we dismiss it because it does not go to zero at r = 0, as  mandated by the boundary conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfc_et/content/2301.00294v1.pdf'}
+page_content='  12  where  I(s) ≡  � R  0  dr  R  R − r exp  �  −is log  �R − r  R  ��  = R  � i  s + πδ(s)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfc_et/content/2301.00294v1.pdf'}
+page_content='  (74)  Hence, neglecting the finite terms8, we have that (72) holds provided that  Nu =  1  √  2πR|α(s)|  ,  (75)  save an overall phase that we set to one for convenience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfc_et/content/2301.00294v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfc_et/content/2301.00294v1.pdf'}
+page_content='2  The entropy  As explained in [38] in the context of the free chiral scalar,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfc_et/content/2301.00294v1.pdf'}
+page_content=' we can take advantage of the orthogo-  nality relation to simplify the computation of the entanglement entropy,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfc_et/content/2301.00294v1.pdf'}
+page_content=' which can be expressed  as a regularized integral over a small region behind the end point r = R,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfc_et/content/2301.00294v1.pdf'}
+page_content=' of the form  S(ℓ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfc_et/content/2301.00294v1.pdf'}
+page_content=' d) =  � R−ϵ  0  dr  � ∞  0  ds us(r)g(s)v∗  s(r)  = − lim  δs→0  � R  R−ϵ  dr  � ∞  0  ds us(r)g(s)v∗  s+δs(r) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfc_et/content/2301.00294v1.pdf'}
+page_content='  (76)  with  g(s) = 1 + coth(πs)  2  log  �1 + coth(πs)  2  �  + 1 − coth(πs)  2  log  �1 − coth(πs)  2  �  (77)  Note that since we expect the entanglement entropy of a QFT to diverge due to the short range  correlations between modes at both sides of the boundary,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfc_et/content/2301.00294v1.pdf'}
+page_content=' we regularized it by introducing a  small UV cutoff ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfc_et/content/2301.00294v1.pdf'}
+page_content=' Furthermore, in going from the first to the second line of (76) we shifted the  v sub index, summing over slightly off diagonal elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfc_et/content/2301.00294v1.pdf'}
+page_content=' For fixed δs ̸= 0 the integral defined  on the whole interval vanishes because of (72), leading to an integral just behind the boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfc_et/content/2301.00294v1.pdf'}
+page_content='  This trick allows us to substitute the expansion (70), which is much easier to integrate than the  original solutions (68).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfc_et/content/2301.00294v1.pdf'}
+page_content=' Finally, we get  S(ℓ, d) = 1  6 log R  ϵ − 1  π  � ∞  0  ds g′(s)Arg(α(s)),  (78)  or, more explicitly,  S(ℓ, d) = 1  6 log R  ϵ − iπ  2  � ∞  0  ds  s  sinh2(πs) log  �4isΓ [is] Γ [−1 + d/2 + ℓ − is]  Γ [−is] Γ [−1 + d/2 + ℓ + is]  �  (79)  The logarithmic coefficient 1/6 is the expected result for a (1+1) dimensional theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfc_et/content/2301.00294v1.pdf'}
+page_content=' Meanwhile,  the constant term is expressed in terms of an integral that cannot be solved explicitly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfc_et/content/2301.00294v1.pdf'}
+page_content=' For later  8We also neglect a contribution of the form δ(s + s′), coming from the second term in (73), because it is  non-zero only at s = s′ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfc_et/content/2301.00294v1.pdf'}
+page_content='  13  0  20  40  60  80  100  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfc_et/content/2301.00294v1.pdf'}
+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfc_et/content/2301.00294v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfc_et/content/2301.00294v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfc_et/content/2301.00294v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfc_et/content/2301.00294v1.pdf'}
+page_content='0  Figure 2: Constant term of the entropy at d = 3, as a function of the angular mode ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfc_et/content/2301.00294v1.pdf'}
+page_content=' The red dots represent  the exact numerical value of (81), for ℓ = {1, 2, 5, 10, 15, 20, 30, 40, 100}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfc_et/content/2301.00294v1.pdf'}
+page_content=' The blue curve corresponds to the fit  f(ℓ, d = 3) = c0 + c1 log ℓ, with c0 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfc_et/content/2301.00294v1.pdf'}
+page_content='345 × 10−5 and c1 = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfc_et/content/2301.00294v1.pdf'}
+page_content='1666.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfc_et/content/2301.00294v1.pdf'}
+page_content='  convenience,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfc_et/content/2301.00294v1.pdf'}
+page_content=' we write it as a sum of two contributions,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfc_et/content/2301.00294v1.pdf'}
+page_content=' one that does not depend neither on the  dimension nor on the angular mode  c ≡ −iπ  2  � ∞  0  ds  s  sinh2(πs) log  �4isΓ [is]  Γ [−is]  �  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfc_et/content/2301.00294v1.pdf'}
+page_content='  (80)  and another which does depend on both parameters  f(ℓ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfc_et/content/2301.00294v1.pdf'}
+page_content=' d) ≡ −iπ  2  � ∞  0  ds  s  sinh2(πs) log  �Γ [−1 + d/2 + ℓ − is]  Γ [−1 + d/2 + ℓ + is]  �  (81)  Although it is unfortunately impossible to find an analytic expression for the integral,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfc_et/content/2301.00294v1.pdf'}
+page_content=' for  sufficiently large modes we can make use of the Stirling’s approximation  log Γ(z) ∼ z log z − z + 1  2 log 2π  z +  N−1  �  n=1  B2n  2n(2n − 1)z2n−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfc_et/content/2301.00294v1.pdf'}
+page_content='  |z| → ∞  (82)  to write  log  �Γ [−1 + d/2 + ℓ − is]  Γ [−1 + d/2 + ℓ + is]  �  ∼ −2is log ℓ +  ∞  �  k=2  ⌊ k+1  2 ⌋  �  m=1  (k − 2)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfc_et/content/2301.00294v1.pdf'}
+page_content='  ℓk−1  ak,m(s)  + 1  2  ∞  �  k=1  ⌊ k+1  2 ⌋  �  m=1  (k − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfc_et/content/2301.00294v1.pdf'}
+page_content='  ℓk  ak,m(s) +  ∞  �  n=1  ∞  �  k=1  ⌊ k+1  2 ⌋  �  m=1  B2n(2n + k − 2)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfc_et/content/2301.00294v1.pdf'}
+page_content='  (2n)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfc_et/content/2301.00294v1.pdf'}
+page_content='ℓ2n+k−1  ak,m(s),  ℓ >> 1,  (83)  where  ak,m(s) =  2i(−1)k+m  (2m − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfc_et/content/2301.00294v1.pdf'}
+page_content=' (k + 1 − 2m)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfc_et/content/2301.00294v1.pdf'}
+page_content='  �  −1 + d  2  �k+1−2m  s2m−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfc_et/content/2301.00294v1.pdf'}
+page_content='  (84)  This means that the constant term grows logarithmically with the mode ℓ, with corrections that  decay as positive powers of 1/ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfc_et/content/2301.00294v1.pdf'}
+page_content=' In fact, performing the integration over the variable s order by  order in the expansion, we can straightforwardly check that the first few leading terms read  f(ℓ, d) ∼ −1  6 log ℓ + a1  ℓ + a2  ℓ2 + a3  ℓ3 + a4  ℓ4 + O  � 1  ℓ5  �  ,  (85)  14  0  5  10  15  20  25  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfc_et/content/2301.00294v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfc_et/content/2301.00294v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfc_et/content/2301.00294v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfc_et/content/2301.00294v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfc_et/content/2301.00294v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfc_et/content/2301.00294v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfc_et/content/2301.00294v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfc_et/content/2301.00294v1.pdf'}
+page_content='1  Figure 3: ∆f(ℓ, 3) ≡ f(ℓ, 3) − f(ℓ = 1, 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfc_et/content/2301.00294v1.pdf'}
+page_content=' Blue: direct numerical integration of (81).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfc_et/content/2301.00294v1.pdf'}
+page_content=' Orange: calculation with a  radial lattice regularization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfc_et/content/2301.00294v1.pdf'}
+page_content=' ℓ = {1, 2, 5, 10, 20}  with  a1  =  1  4 − d  12  (86)  a2  =  7  40 − d  8 + d2  48  (87)  a3  =  3  20 − 7d  40 + d2  16 − d3  144  (88)  a4  =  73  560 − 9d  40 + 21d2  160 − d3  32 + d4  384  (89)  Quite surprisingly, the logarithmic term already approximates f(ℓ, d) at ℓ ∼ O(1) very accurately,  as shown in figure (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfc_et/content/2301.00294v1.pdf'}
+page_content='  In figure (3) we compare the numerical value of (81) with the one obtained from direct calcu-  lation in a radial lattice, again at d = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfc_et/content/2301.00294v1.pdf'}
+page_content=' Since the constant term depends on the regularization  scheme, we subtract the one corresponding to ℓ = 1 and compare ∆f(ℓ, 3) ≡ f(ℓ, 3)−f(ℓ = 1, 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfc_et/content/2301.00294v1.pdf'}
+page_content='  Although it is very hard to achieve good precision in the lattice9, we find reasonable agreement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfc_et/content/2301.00294v1.pdf'}
+page_content='  For example, for ℓ = 10, numerical integration yields ∆f = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfc_et/content/2301.00294v1.pdf'}
+page_content='3737, while the lattice computa-  tion gives ∆f = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfc_et/content/2301.00294v1.pdf'}
+page_content='3795.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfc_et/content/2301.00294v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfc_et/content/2301.00294v1.pdf'}
+page_content='3  Recovering the scalar entropy  As discussed in section 3, the modular Hamiltonian of the free scalar in the sphere is equal to  the sum over ℓ of the modular Hamiltonian pertaining to each one dimensional theory in the  segment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfc_et/content/2301.00294v1.pdf'}
+page_content=' Consequently, we expect that summing (79) must necessarily reproduce the general  structure for the entanglement entropy,  S =  �  #  � R  ϵ  �d−2 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfc_et/content/2301.00294v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfc_et/content/2301.00294v1.pdf'}
+page_content=' + clog log R  ϵ ,  d  even  #  � R  ϵ  �d−2 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfc_et/content/2301.00294v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfc_et/content/2301.00294v1.pdf'}
+page_content=' + F,  d  odd  (90)  that is, an infinite contribution controlled by the area term, and a universal piece, either in  the form of a logarithmic coefficient in even dimensions, which is precisely the trace anomaly  9Roughly speaking, the value of ℓ gives a lower bound for the meaningful radios R/ϵ >> ℓ  15  coefficient associated to the Euler density [39, 40, 41], or a constant term in odd dimensions  [42, 43, 44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfc_et/content/2301.00294v1.pdf'}
+page_content='  To show that this indeed holds, we need to introduce a cutoff in ℓ to regularize the sum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfc_et/content/2301.00294v1.pdf'}
+page_content=' More  concretely, we introduce a damping exponential so that  S =  ∞  �  ℓ=0  λ(ℓ, d)S(ℓ, d)e−ℓϵ/R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfc_et/content/2301.00294v1.pdf'}
+page_content='  (91)  where  λ(ℓ, d) = (2ℓ + d − 3)(ℓ + d − 4)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfc_et/content/2301.00294v1.pdf'}
+page_content='  ℓ!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfc_et/content/2301.00294v1.pdf'}
+page_content=' (d − 3)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfc_et/content/2301.00294v1.pdf'}
+page_content='  (92)  is the density of states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfc_et/content/2301.00294v1.pdf'}
+page_content=' Note that this grows as ℓd−3 for ℓ >> 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfc_et/content/2301.00294v1.pdf'}
+page_content='  Given the complicated expression of the constant term f(ℓ, d), we approximate it by its large  ℓ expansion, leading to  S =  ∞  �  ℓ=1  λ(ℓ, d)  �  1  6 log R  ϵ + c − 1  6 log ℓ +  jmax  �  j=1  aj  ℓj  �  e−ℓϵ/R + λ(0, d)S(0, d) + correction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfc_et/content/2301.00294v1.pdf'}
+page_content='  (93)  The correction above accounts for the error made when approximating f(ℓ, d) by its series ex-  pansion, truncated at O(ℓ−jmax).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfc_et/content/2301.00294v1.pdf'}
+page_content='  It is straightforward to verify that (93) reproduces (90).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfc_et/content/2301.00294v1.pdf'}
+page_content=' For example, the divergent pieces  come from terms with the general structure  ∞  �  ℓ=1  ℓp  �  log R  ϵ − log ℓ  �  e−ℓϵ/R = −Γ′(p + 1)  �R  ϵ  �p+1  + ζ(−p) log R  ϵ + ζ′(−p),  (94)  ∞  �  ℓ=1  ℓpe−ℓϵ/R = p!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfc_et/content/2301.00294v1.pdf'}
+page_content='  �R  ϵ  �p+1  + ζ(−p),  (95)  with {p | p ∈ N0 ∧ p ≤ d − 3}, and  ∞  �  ℓ=1  1  ℓe−ℓϵ/R = log R  ϵ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfc_et/content/2301.00294v1.pdf'}
+page_content='  (96)  Note that the logarithmic term, only present in even dimensions, stems from (94) and (96).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfc_et/content/2301.00294v1.pdf'}
+page_content='  Based on this observation, it is worth pointing out that in order to compute the logarithmic  coefficient we only need to take into account the first d − 1 terms in the expansion of f(ℓ, d),  that is, jmax = d − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfc_et/content/2301.00294v1.pdf'}
+page_content=' Subleading corrections give finite contributions at most.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfc_et/content/2301.00294v1.pdf'}
+page_content='  Just to explicitly address some relevant specific cases, at d = 6 we get  clog(d = 6) = 29  540 + a1 + 13  6 a2 + 3  2a3 + 1  3a4,  (97)  and, substituting (86), (87), (88), (89),  clog(d = 6) =  1  756,  (98)  16  10  20  30  40  50  60  70  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfc_et/content/2301.00294v1.pdf'}
+page_content='19  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfc_et/content/2301.00294v1.pdf'}
+page_content='18  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfc_et/content/2301.00294v1.pdf'}
+page_content='185  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfc_et/content/2301.00294v1.pdf'}
+page_content='195  corr  Figure 4: Error made in the computation of the constant term F when approximating f(ℓ, 3) by − 1  6 log ℓ + a2  ℓ2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfc_et/content/2301.00294v1.pdf'}
+page_content='  ℓmax is the greatest angular momentum that is summed over.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfc_et/content/2301.00294v1.pdf'}
+page_content=' We see that the correction converges very fast to  ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfc_et/content/2301.00294v1.pdf'}
+page_content='00519641  in agreement with the expected anomaly value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfc_et/content/2301.00294v1.pdf'}
+page_content=' On the other hand, at d = 4 we get  clog(d = 4) = 1  18 + a1 + 2a2,  (99)  which leads to the expected anomaly coefficient  clog(d = 4) = − 1  90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfc_et/content/2301.00294v1.pdf'}
+page_content='  (100)  The case of d = 3 is different from the ones discussed above in that it has no logarithmic term  and the universal piece in the entanglement entropy is associated to the constant term F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfc_et/content/2301.00294v1.pdf'}
+page_content=' In  fact, direct calculation yields, using (86)  clog(d = 3) = 2a1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfc_et/content/2301.00294v1.pdf'}
+page_content='  (101)  Regarding the constant F, the infinite tail in the 1/ℓ expansion must in principle be taken into  account.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfc_et/content/2301.00294v1.pdf'}
+page_content=' For that reason, we regularize the sum taking up to jmax = 2 and then add a finite  contribution which corrects the approximation, giving the exact value for the constant term in  the series of f(ℓ, d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfc_et/content/2301.00294v1.pdf'}
+page_content=' That is,  correction = 2  lim  ℓmax→∞  ℓmax  �  ℓ=1  �  f(ℓ, d = 3) + 1  6 log ℓ − a2  ℓ2  �  (102)  In figure (4) we plot the above correction as a function of ℓmax and show that it converges very  fast to  correction ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfc_et/content/2301.00294v1.pdf'}
+page_content='00519641  (103)  According to (93), another term which contributes is  f(0, 3) = −iπ  2  � ∞  0  ds  s  sinh2(πs) log  �Γ [1/2 − is]  Γ [1/2 + is]  �  ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfc_et/content/2301.00294v1.pdf'}
+page_content='278435.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfc_et/content/2301.00294v1.pdf'}
+page_content='  (104)  Gathering all the pieces together, we finally get  F = π2  3 a2 + ζ′(0)  3  + f(0, 3) + correction ∼ −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfc_et/content/2301.00294v1.pdf'}
+page_content='0638049,  (105)  which is within 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfc_et/content/2301.00294v1.pdf'}
+page_content='003% of the exact value [42, 44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfc_et/content/2301.00294v1.pdf'}
+page_content=' Note that the constant c does not contribute  at all to F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfc_et/content/2301.00294v1.pdf'}
+page_content='  17  6  Final remarks  In this article, we focused on theories in the semi infinite line constructed from the dimensional  reduction of a free scalar in d dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfc_et/content/2301.00294v1.pdf'}
+page_content=' Given that the decomposition of the parent theory H  into independent sectors Hℓ⃗m, labeled by the angular modes, also holds for the vacuum modular  Hamiltonian in spheres K = �  ℓ⃗m Kℓ⃗m, and provided that the vacuum state of the system is  the product ρ = ⊗ρℓ⃗m, then, it is immediate to identify the modular Hamiltonian mode Kℓ⃗m  with the modular Hamiltonian of the one-dimensional reduced system Hℓ⃗m in the interval (0, R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfc_et/content/2301.00294v1.pdf'}
+page_content='  Remarkably, the resulting modular Hamiltonian is local and proportional to the energy density,  with the same weight function β(r) = R2−r2  2R  as the one characteristic of a CFT in a sphere R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfc_et/content/2301.00294v1.pdf'}
+page_content='  We complemented the previous analysis with the study of the symmetries inherited from the  d dimensional conformal theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfc_et/content/2301.00294v1.pdf'}
+page_content=' This approach evidences the fact that the symmetry behind  the locality of the reduced modular Hamiltonian is just the restriction to the semi infinite line of  the original modular symmetry in d dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfc_et/content/2301.00294v1.pdf'}
+page_content=' We identified the conserved current associated  to this symmetry transformation and checked that the Kℓ⃗m found by dimensional reduction  coincides with the Noether charge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfc_et/content/2301.00294v1.pdf'}
+page_content='  On the other hand, the spectral decomposition of the modular Hamiltonian leads to an analytic  expression for the corresponding entanglement entropy (EE) which in turn, after summing over  the angular modes, allowed us to recover the EE of the original d dimensional theory in the sphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfc_et/content/2301.00294v1.pdf'}
+page_content='  To make sense of the sum, we used a novel regularization implemented by a damping exponential  parametrized by the same cutoff ϵ that regularizes the radial coordinate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfc_et/content/2301.00294v1.pdf'}
+page_content=' As we mentioned in  the introduction, in a way, this procedure generalizes the one introduced by Srednicki in [33]  and provides an additional tool to calculate analytically the EE logarithmic coefficient in even  dimensions, the universal constant term in d = 3, among others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfc_et/content/2301.00294v1.pdf'}
+page_content='  It would certainly be interesting to explore in the future the modular Hamiltonian of non-  conformal theories constructed from the dimensional reduction of other free theories, fermions  for example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfc_et/content/2301.00294v1.pdf'}
+page_content=' We expect that the decomposition of the parent theory into independent sectors  must carry over unaltered, as well as the symmetry arguments which justify the resulting modular  Hamiltonian is a conserved charge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfc_et/content/2301.00294v1.pdf'}
+page_content='  Acknowledgments  We thank H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfc_et/content/2301.00294v1.pdf'}
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