Holographic renormalized Entanglement and entropic c − function

We compute holographic entanglement entropy (EE) and the renormalized EE in AdS solitons with gauge potential for various dimensions. The renormalized EE is a cutoff-independent universal component of EE. Via Kaluza-Klein compactification of S 1 and considering the low-energy regime, we deduce the ( d − 1)-dimensional renormalized EE from the odd-dimensional counterpart. This corresponds to the shrinking circle of AdS solitons, probed at large l . The minimal surface transitions from disk to cylinder dominance as l increases. The quantum phase transition occurs at a critical subregion size, with renormalized EE showing non-monotonic behavior around this size. Across dimensions, massive modes decouple at lower energy, while degrees of freedom with Wilson lines contribute at smaller energy scales.


Introduction
Quantum entanglement entropy stands as a pivotal concept in quantum mechanics, offering insight into the level of entanglement among distinct segments of a quantum system.By quantifying the entanglement between different components, this entropy provides a metric for the extent of shared information.Its implications span various facets of quantum mechanics, encompassing quantum information theory, black hole physics, and condensed matter physics.Notably, the entanglement entropy of subsystem A quantifies the entangled degrees of freedom within a given quantum field theory [1,2].Within the context of condensed matter physics, this entropy displays divergence at critical junctures of quantum critical phase transitions, assuming the role of an order parameter [3].This phenomenon encapsulates the geometric essence of field theories, manifested in an area law that draws parallels between subregion entanglement entropy and black hole entropy.
Introducing the Ryu-Takayanagi formula establishes a holographic counterpart for entanglement entropy [5,6,7], emerging as a robust tool for dissecting strongly coupled systems traditionally resistant to conventional analysis.In specific contexts, this formula has served as an order parameter, signaling the onset of confinement/deconfinement phase transitions within confining gauge theories [10,11,12,13,14,15]. 1 The transitions emanate from the interplay between two minimal surfaces, resulting in the post-transition confinement phase entanglement entropy becoming trivial at the infrared limit.Moreover, the holographic entanglement entropy (HEE) emerges as a probing tool for phase transitions in holographic superconductors [17,18,19,20,21,22,23] as well as for unveiling topological phases of matter [24].
The entropic c−function concept provides deeper insights into entanglement entropy in quantum systems [8,9].It represents the logarithmic derivative of entanglement entropy with respect to subsystem size, revealing the intricate interplay between entanglement and subsystem dimensions.The general entropic c−function, proposed by [10], efficiently quantifies degrees of freedom in confining theories and yields the central charge of the corresponding conformal field theory (CFT) [25].In the context of a quantum field theory dual to an AdS soliton, the behavior of the entropic c−function, as it decreases with increasing length, effectively serves as a probe for the deconfinement phase transition.Recently, a study [26] computed the entropic c−function for a striped entangling surface in the same background.Intriguingly, this function displays non-monotonic behavior with increasing background gauge field strength.Importantly, the entropic c−function for the strip incorporates both A-type and B-type anomalies due to the coexistence of these two anomaly types.[28] has derived constraints on anisotropic RG flows from holographic entanglement entropy.
Our focus lies in assessing the degrees of freedom through entanglement entropy using a spherical entangling surface, with a specific emphasis on anomaly effects.The renormalized entanglement entropy, calculated from the entanglement entropy of this spherical surface, offers a solution to the issue of cutoff dependence [27].This renormalized quantity, independent of the cutoff, measures degrees of freedom in quantum entangled states at an energy scale of 1/l, yielding the central charge in conformal field theory (CFT).In the context of four-dimensional (4D) CFT, it manifests as the A-type anomaly, in alignment with the C-theorem: the renormalized entanglement entropy decreases in the infrared (IR) limit as anticipated.[29] derived the renormalized entanglement entropy for a kink region which reduced to a universal positive finite term in the UV limit.
The computation of renormalized entanglement entropy for a spherical entangling surface within the AdS soliton framework with a gauge potential remains unexplored.The gauge potential's interpretation in this background involves a twisted boundary condition along a circle within the cigar direction.This contributes to the negative Casimir energy of the dual field theory, which can eventually become positive.An intriguing aspect emerges from the interplay of Wilson lines, capable of inducing mass shifts in charged particles [30].
It becomes desirable to capture such alterations through the renormalized entanglement entropy.Conversely, contrasting the analysis presented in [26] for a striped entangling surface in the same background, the renormalized entanglement entropy in R 1,2 × S 1 quantum field theory exhibits solely B-type anomaly characteristics.Generally, this quantity doesn't adhere to the C-theorem.Therefore, an engaging pursuit lies in investigating this aspect, including scenarios in higher-dimensional cases.
This study focuses on introducing a holographic renormalized entanglement entropy (HREE), which encompasses the finite portion of entanglement entropy.We extensively investigate HREE's behavior across diverse scenarios to uncover the universal properties of quantum phase transitions.Our analysis reveals a phase transition between disk and cylinder geometries.The dominance of the disk type is evident for small l, while the cylinder shape prevails for larger l.This critical size marks the occurrence of a quantum phase transition.Notably, this transition is an outcome of the large N limit and is absent in free theory.The anticipated function of the renormalized entanglement entropy is to quantify the degrees of freedom in the dual quantum field theory (QFT).We probe HREE's response to changes in operator mass and gauge potential.Specifically, as the operator mass decreases, we expect HREE to increase significantly for larger l, given the decoupling of massive degrees of freedom.
The rest of this paper is organized as follows.In section II, we analyze the UV structure of the entanglement entropy.We derive the renormalized entanglement entropy of QFT dual to AdS solitons with gauge potential.We discuss properties of the renormalized entanglement entropy in both odd and even dimensions.Section III mainly focuses on holographic stress-energy tensors in AdS solitons with the gauge potential.In sections IV and V, we analyze the quantum phase transition of HREE in higher dimensional backgrounds.
We end with conclusions and prospects in section VI.Some calculation details are presented in the appendices.

The UV structure of the entanglement entropy
The d + 1-dimensional AdS solitons with gauge potential exhibit a geometry akin to a cigar.
In this setup, a compact circle gradually contracts to zero size in the bulk, completing the geometry.This behavior is detailed in [26].The dual theory on R 1,d−2 × S 1 transforms into a confining theory with a discernible energy gap in this context.Notably, this theory incorporates Wilson lines along the S 1 direction, which inherently alters the boundary conditions.
We calculate the entanglement entropy in this theory.The spherical entangling surface is chosen to have the topology S d−3 × S 1 , where the entangling surface wraps another circle S 1 [31].For d = 4, S 1 × S 1 is a cylinder with one identified direction, i.e., a torus.
Because a spherical entangling surface for QFT on R 1,d−1 has a different topology S d−2 , we find that the UV scaling structure is different from those of QFT on R 1,d−1 .According to [31], the UV divergent structure of entanglement entropy S U V is of the form where S U V,0 is the UV structure of entanglement entropy of QFT S (0) EE on R 1,d−1 and L ϕ is the periodicity along a circle S 1 of the cigar.
Two UV scaling structures are related to each other.Using (1) and operating differentiation on S EE , the renormalized entanglement entropy (the UV-independent part of the entanglement entropy) then becomes [31] where f d defines the operation of the renormalized entanglement entropy for R 1,d−1 [27]: EE , d = even. ( Recall that the first line in ( 2) is equivalent to the d − 1 dimensional renormalized entanglement entropy on R 1,d−2 (the second line in (3)) up to coefficients.
Especially we obtain The formula (4) for QFT on R 1,1 × S 1 corresponds to the well-established expression of the entropic c-function on R 1,1 .Through Kaluza-Klein reduction along S 1 , the renormalized entanglement entropy effectively embodies the 2-dimensional entropic c−function in the low-energy limit.In systems respecting Lorentz symmetry, the 2-dimensional entropic cfunction is both non-negative and monotonically increasing.For R ≪ L ϕ (in the UV limit), the renormalized entanglement entropy mirrors the behavior of a 3-dimensional system.
The subregion's topology is not a disk D but rather L × S 1 with an interval L, while the entangling surface forms S 1 .Moving to formula (5) for QFT on R 1,3 × S 1 , it captures one variant of the 4-dimensional renormalized entanglement entropy on R 1,3 .Kaluza-Klein reduction along S 1 approximates the renormalized entanglement entropy on R 1,3 in the low-energy regime.However, the behavior of the 4-dimensional renormalized entanglement entropy can be either negative or positive, displaying non-monotonic tendencies.In this scenario, the subregion's topology does not correspond to a ball B 4 but rather B 3 × S 1 , while the entangling surface takes the form of S 2 × S 1 .

Renormalized entanglement entropy of 4d QFT
This section examines the entanglement entropy and renormalized entanglement entropy for various cases: a free scalar, Dirac fermion, and a 4-dimensional conformal field theory (CFT) on R 1,2 × S 1 .Additionally, we provide an overview of the trace anomaly in general CFT, which is intricately linked to the logarithmic term present in the entanglement entropy.
The entanglement entropy can be derived from the effective action w = − log Z in a d(= 4)-dimensional manifold featuring conical singularities.By taking the limit n → 1, the entanglement entropy assumes an analytical expression involving w on a manifold with such singularities.Notably, the effective action w generally exhibits logarithmic divergence, which is connected to the concept of conformal anomaly.We consider the infinitesimal rescaling g µν → (1 − 2δλ)g µν .We then have The equation stands as the trace anomaly, indicating the deviation from the traceless condition ⟨T µ µ ⟩ = 0 within Quantum Field Theory (QFT) for Conformal Field Theory (CFT).The trace anomaly is characterized by polynomials of the curvature tensor, a formulation contingent on the dimension d.Notably, in odd dimensions, the trace anomaly must satisfy the condition of vanishing.
When we define the length scale R 1 of the subregion, this scale is related to rescaling the metric (6).Thus, one obtains the following formula: Here, the entanglement entropy has been replaced with a function of the partition function in the manifold with conical singularities.The above-mentioned formula relates the entanglement entropy to the trace anomaly.
To evaluate the entanglement entropy, we examine a subsystem σ with a cylindrical configuration where one direction is identified as ϕ ∼ ϕ + L ϕ .Interestingly, this subsystem aligns with the one in QFT corresponding to d + 1-dimensional AdS solitons.Due to this, a conical singularity arises, characterized by a curvature tensor proportional to a delta function.The resulting logarithmic contribution to the entanglement entropy is expressed as S EE = s log(ϵ/R 1 ) + . . ., where ϵ represents the ultraviolet cut-off.Remarkably, this logarithmic term can also be derived by integrating the entanglement entropy of a 3d free theory [32,33].s is expressed in terms of extrinsic curvatures [34].According to [32,33], where E 2 is the Euler density and I 2 is a Weyl invariant.Compared with the normalization of [27], we have a = 360a 4 and c = 120c 4 .c = 1 for a real scalar and c = 6 for Dirac fermion.Coefficients are consistent with trace anomalies.
For CFT on a cylinder of length L ϕ and radius l, s becomes By using the renormalized entanglement entropy for cylinder type topology in (2), we obtain This formula shows that the renormalized entanglement entropy agrees with the coefficient s.Furthermore, according to [27], s agrees with the renormalized entanglement entropy for a spherical entangling surface S 2 as follows: 3 The AdS soliton with the gauge potential The AdS soliton is achieved through a double Wick rotation of the AdS black hole, following the Einstein equation.It corresponds to the QFT system with anti-periodic boundary conditions [35].In our investigation, we apply this approach to the Reissner Nordstrom AdS black hole [36], performing an analytical continuation of the metric in both temporal and spatial dimensions.The metric of the AdS soliton with the gauge potential becomes [26] ds where .
Here, we set ε = −1 2 , and define transformation.As the Wilson line vanishes at the tip of the soliton (z = z + ), the gauge connection remains regular there.The radial coordinate z in ( 12) is confined to z ≤ z + , while the ϕ direction follows the periodicity ϕ → ϕ + 1/M 0 to prevent conical singularities.
The Kaluza-Klein mass M 0 of the ϕ direction is given by The formula (14) can also be rewritten in terms of M 0 and a ϕ as follows: There is also a minus branch.However, z + is divergent at small a ϕ in that case, and the background does not approach the AdS d+1 soliton.It can be shown that the solution with the plus sign in ( 15) is always more stable than the one in the minus branch.
The boundary stress tensor µν for field theory dual to the above background was computed in our previous work [26].Here, we quote the results of the boundary stress tensor for later use.For more details, please refer to [26] with The boundary energy then is [37 The boundary energy can change the sign when we change Wilson lines (gauge potential).
In other words, M is negative for a ϕ < 2πM 0 /(d − 1), while it can become positive when . For a ϕ = 0, it realizes Casimir energy of 4d SYM theory on R 3 ×S 1 [35].
This behavior is analogous to Casimir energy of fields with twisted boundary conditions in 2d CFT [30].Casimir energy is different between the periodic and anti-periodic boundary conditions.

The holographic entanglement entropy
In this section, we compute the entanglement entropy [5,6].The entangling surface is specified by z = 0 at R = l, and 0 ≤ ϕ ≤ L ϕ at a constant time slice in the background (12).Its topology becomes S 1 × S d−3 , where S 1 and S d−3 be of radius L ϕ and l respectively.Note that the topology of the entangling surface differs from the theory without the compactification of the ϕ direction.The surface action becomes where L = √ det g ind and g ind is the induced metric.The holographic entanglement entropy is given by with A minimized.Recall that L d−1 /κ 2 is dimensionless.We omit the AdS radius (L = 1) for the convenience.We solve EOM derived from ( 21) to obtain the minimal surface.The One should specify the IR boundary condition.In fact, there are two kinds of RT surfaces, as drawn in Fig.
(1) schematically.The turning point of the disk type RT surface is R(z t ) = 0.
Moreover, the surface of the disk type is smooth at the bulk.The embedding scalar must satisfy R ′ (z t ) = ∞.The surface ends at the tip of the soliton z t = z + for a cylinder case.
Varying A in terms of l and fixing z = ϵ, the Hamiltonian-Jacobi method (please refer to appendix A for a brief review of this method) becomes [27] where Figure 1: The RT surfaces corresponding to the small (left) and large (right) subsystem in the AdS-Soliton background.If the subsystem is a spherical region, then the RT surface has a disk (cylinder) topology for the small (large) subsystem.
The first term of ( 25) drops out due to the following IR boundary conditions Equation ( 25) only depends on the solution near the AdS boundary, and an asymptotic expansion is useful.
We compute the asymptotic expansion of the embedding scalar near z = 0.The UV behavior of R(z) has the following ansatz: where the log term arises in (27) similar to the Fefferman-Graham expansion of fields in the AdS spacetime [38,39].We can determine coefficients a n and b n after substituting the ansatz mentioned above into (24).Below we will analyze the cases with d = 4, 5, 3 in detail.

d = 4
Let us begin with the d = 4 case, where the boundary QFT lives on R 1,2 × S 1 , and the topology of the entangling surface becomes S 1 × S 1 .By substituting expansion of R(z) near the boundary z = 0 (27) into equation of motion, one can obtain Here the higher order terms can be determined by parameters l and a 4 (l).The coefficient a 4 (l) can not be determined from the UV expansion of the EOM.Instead, a 4 (l) has information determined by the IR boundary condition.
Substituting (28) into (25), the l derivative of the surface becomes Divergent pieces are defined up to a logarithmic term.Because the inside of the log term should be dimensionless, there are no unique ways to remove it.However, the renormalized entanglement entropy is finite and does not depend on the cut-off.Thus, it has unique descriptions.
We compute the renormalized entanglement entropy, corresponding to DOF at an energy scale E ∼ 1/l.According to (2), the 4d renormalized entanglement entropy becomes where we have used the commutation relation [∂ l , l∂ l ] = ∂ l .Recall that the central charge , where κ 2 10 = π 3 L 5 κ 2 .The renormalized entanglement entropy depends on the entangling surface and the trace anomaly [27] as follows: where ∂A is the entangling surface (see also [6,34]).In 4 dimensions, we have an A-type anomaly a 4 and a B-type anomaly c 4 on the entangling surface.E 2 is the Euler density and I 2 is a Weyl invariant.For the spherical entangling surface and the Weyl invariant is zero.Thus, S d=4 ren = 4a 4 .The renormalized entanglement entropy will satisfy the C-theorem in that case.Because the entangling surface is S 1 × S 1 for QFT on R 1,2 × S 1 , however, the Euler number is zero.Only the B-type anomaly remains.
The renormalized entanglement entropy can be non-monotonic since there is no universal C-theorem for B-type anomalies.
To compute the renormalized entanglement entropy, we need a 4 (l), l appearing in (28) and S (not confused with anomaly a 4 in (32)).A numerical result of a 4 (l) is obtained in Fig. 2. For pure imaginary a ϕ , it leads to results of geometric entropy [40].The geometric entropy is related to entanglement entropy via the double Wick rotation [41,42].The disk shape is dominant for small l, and the cylinder shape is dominant for large l. a 4 (l) becomes multi-valued near the phase transition at a critical length between disk and cylinder surfaces.
Multi-valued behavior is also observed for other a ϕ .
Substituting the boundary expansion (28) into the action (21) and expanding at a small z, we obtain the following divergent part of A = A f in + A div : The final states will be product states.
Thus, the divergent structure of entanglement entropy A is where the log dependence is included in the finite part S f in .S f in = A f in − 2πL ϕ 1 8l log l.
To find minimal surfaces, one must compute the on-shell action of (21).Minimal surfaces between the disk and the cylinder dominate the phase in Fig. 3.The quantum phase transition occurs at a critical length l c .Yellow and Blue curves show that the confinement occurs and decreases DOF [43,10,11].Recall that a ϕ increases Casimir energy of dual QFT, and then the entanglement entropy increases with the increase of a ϕ .
S f in is different from S ren because the cut-off dependence is removed at S ren .Actually, S f in is related to S ren via Considering ( 29), S f in (l) satisfies the following relation: Due to (36), Recall that (37) is the finite part of the minimal surface A. The renormalized entanglement entropy S ren is finite.Substituting ( 36) and ( 37) into (35), S ren is rewritten as follows: The coefficient of 1/l comes from only the logarithmic term of S, which is brought from the Weyl anomaly.The formula (38) also depends on a function of a 4 (l) unlike the holographic entanglement entropy of the spherical entangling surface in 4 dimensions [6].For the spherical entangling surface, here, HREE describes the A type anomaly in CFT: S d=4 ren = 4a 4 .Recall that QFT dual to the AdS soliton with gauge potential breaks conformal invariance.
The first three terms in (38) represent terms breaking conformal invariance and the last term is to realize the small l limit of HREE (CFT behavior) [34].Fig. 4 shows the renormalized entanglement entropy S ren for several a ϕ .It becomes non-monotonic behavior, which is similar to a behavior of GPPZ flow [27].Intuitively, the renormalized entanglement entropy is also a detector of the effective DOF of entangling states at the energy scale El ∼ 1.For large energy E/M 0 ∼ 1/(M 0 l) > 1, S ren decreases as a function of lM 0 .Because the degrees of freedom with Wilson lines contribute to large a ϕ and energy, however, S ren slowly decreases until the critical length (see green and red curves).For small E/M 0 < 1, the renormalized entanglement entropy can not detect effective DOF and can almost become a constant.Even if the renormalized entanglement entropy increases after the quantum phase transition, it satisfies a kind of C-theorem:

d = 5
We proceed with d = 5 case, where the AdS boundary expansion for d = 5 has the following form Similar to the case a 4 (l) discussed in the previous section, the parameter a 5 (l) is not determined from the AdS boundary expansion but determined from the IR boundary condition.
Numerically, a 5 (l) is plotted in Fig. 6 for fixed values of M 0 and a ϕ .Once we have the expansion (39), by making use of ( 25), we can obtain the following derivative where Ω 2 = 4π.The first term is the cut-off dependent term.There are no log divergences.
The finite part of dA/dl is determined by the second term, which is similar to d = 3 cases: a term corresponding to trace anomaly is not present in odd dimensions.
Substituting the expansion (39) into the action (21) and expanding in terms of small z, we can obtain the divergent part of entanglement entropy which has the following divergent structure Since the second term in the parentheses does not depend on l, O(ϵ −1 ) part is absent in (40).The l dependence of A(l) differs from the CFT one.We plotted the finite part of the entanglement entropy in Fig. 7, where it is shown that the disk-shaped RT surface dominates the behavior for small l and cylinder shape dominates for large l.The finite part  A f in has the quantum phase transition at the critical length.Making use of ( 40) and ( 41), the finite part satisfies the following relation Next consider the 5-dimensional renormalized entanglement entropy defined in (2), which is Substituting (41), the renormalized entanglement entropy can be rewritten as The renormalized entanglement entropy is plotted in Fig. 8 with different a ϕ .The renormalized entanglement entropy counts the DOF of the entangling states at an energy scale E ∼ 1/l.It has non-monotonic behaviors.The quantum phase transition happens at critical limits.And S ren approaches to zero at large distances.Fig. 8 (right) shows that massive DOF decouples other modes at low energy soon.This behavior is easily seen when a ϕ = 0, where scaling symmetry M 0 → λM 0 , l → λ −1 l arises.This is the symmetry of the action and EOM.Scaling symmetry simultaneously rotates a ϕ when it is nonzero.The scaling symmetry implies that the critical length is inversely proportional to M 0 : l c = c 5 /M 0 .
The quantum phase transition quickly occurs for large masses, while it slowly occurs for small masses.After the phase transition, the renormalized entanglement entropy gradually becomes constant.
5 d = 3 (a striped shape) In this section, we analyze the holographic entanglement entropy for d = 3.The configuration for d = 3 is equal to a striped boundary shape.We start with d dimensional striped shapes, and d = 3 is a special case.To consider a striped shape, we replace R 2 dΩ d−3 with dx 2 ⊥ in d + 1 dimensional AdS soliton with a gauge field (12) as follows: where R is along (−∞, ∞) unlike the radial direction of polar coordinate systems.Right: S ren for d = 3.The curve is for a ϕ = 0, 1, 2/ √ 3 from the left to the right.S ren monotonically decreases as a function of lM 0 .S ren becomes 0 at a large distance.It implies that theory is a product state there [44].
We consider a strip with a length of l along the R direction (−l/2 ≤ R ≤ l/2) and choose R = R(z) as an embedding scalar.The surface action becomes where V d−3 was used to replace the volume of d−3 dimensional space spanned by x ⊥ .Factor 2 comes from two contributions of the minimal surface.Note that the lagrangian density of (45) does not depend explicitly on R.This simplifies the analysis for a rectangular region more than for a circular region.The momentum doesn't explicitly depend on R for a striped It implies that massive modes M > 1/l quickly decouple others and remain product states.
boundary shape.Solving the condition Π =const and imposing the IR boundary condition dz/dR| z=zt = 0, we have This formula demonstrates that z = z t is the turning point dz/dR| z=zt = 0.The equation (46) gives the profile of the minimal surface satisfying R(ϵ) = −l/2 and R(z t ) = 0.
Using the AdS boundary expansion where The Hamilton-Jacobi equation for a striped shape has the same form as ( 25) where and H(z t ) = 0 due to the IR boundary condition.When (46) is substituted into (25), the l derivative of the surface becomes Note that the minimal surface must satisfy strong subadditivity.It requires that the minimal surface must be concave [45] as follows: Actually, one can show the following inequality: because h(z) satisfies the following condition: According to [26], in addition, ∂z t /∂l has two values.It is positive for a minimal surface and negative for an unphysical curve.
Next, we consider the case in which the minimal surface is disconnected.We focus where it is easy to see that one of the solutions is R = const, which is for the disconnected minimal surface.Then the area of the surface is and the corresponding entanglement entropy 4 When a ϕ is pure imaginary a ϕ = ia ′ ϕ , the second condition is replaced with where z t = z + .The finite part of S EE is −4πL ϕ /(κz + ).For disconnected surfaces, s 3 (l) = 0.
One can compute the renormalized entanglement entropy from the entanglement entropy S EE .We apply the renormalized entanglement entropy formula defined in (2): S ren = l∂ l S substituting d = 3.Because dual 3-dimensional QFT theory is defined on R 1,1 × S 1 , the renormalized entanglement entropy will give the result of 2d QFT on R 1,1 in the low energy limit.
The renormalized entanglement entropy can be written in terms of l and s 3 (l) as follows: where where k is the Chern-Simons level.For the disconnected surface, we see that the entanglement entropy does not depend on the size l of the entangling surface at the boundary, which means We plotted dA s /(3L ϕ dl) = s 3 (l) in Fig. 9 (left).Because the concavity of the entanglement entropy [46][47] means A ′′ s ≤ 0, A ′ s monotonically decreases as a function of l.We plotted the renormalized entanglement entropy in Fig. 9 (right).The renormalized entanglement entropy detects the DOF of the entangling states at the energy scale El ∼ 1 again.Because massive degrees of freedom decouple in the low energy limit, S ren decreases as a function of l (with energy E ∼ 1/l). 5This monotonic behavior (S ′ ren (l) ≤ 0) will also be the consequence of Lorentz symmetry and strong subadditivity [45].The quantum phase transition happens at critical lengths l c M 0 = 0.19, 0.24, 0.34 for a ϕ = 0, 1, 2/ √ 3, respectively.S ren becomes 0 at large distances.Fig. 10 changes M 0 after fixing a ϕ .It shows that massive modes M 0 > 1/l quickly decouple others at low energy, remaining product states.When a ϕ = 0, one can recover scaling symmetry M 0 → λM 0 , l → λ −1 l, in addition to bulk parameters.This is the symmetry of the action and EOM.Scaling symmetry simultaneously changes a ϕ if it is nonzero.Thus, the critical length is the same as those for different M 0 and is given by M 0 = c/l c , where c is a constant.The phase transition happens soon for large masses and slowly for small masses.
Our results agree with the fact that the renormalized entanglement entropy is equal to the formula of the entropic c-function l∂ l S EE on R 1,1 , which becomes observable in renormalizable theory.The entropic c-function on R 1,1 counts DOF and depends on the radius l(∼ 1/E) including the information in S EE (l 1 ) − S EE (l 2 ) (l 1 > l 2 ).It is known that the c function for a massive scalar field exponentially decreases for large r [8].Finally, there remains no degree of freedom (zero entropic c function) at a large distance.Recall that the trace anomaly exists in 2 dimensions.The monotonic quantity is the Euler term in the trace anomaly there.

Small subregions
By employing the our previous results, where z t can be expressed in terms of l for small l we obtain where we have defined āϕ = 1 − (z + a ϕ ) 2 .For the notation e −1 , k −1 please refer to [26] for more details.The expansion (63) agrees with the holographic entanglement entropy in [26], and the formula (51) is also valid for other dimensions realizing the holographic entanglement entropy.The result is compared with our previous paper [26] in the appendix B.

Summary and discussion
We computed holographic EE and the renormalized EE in the AdS soliton with the gauge potential for several dimensions.The disk shape of the minimal surface was dominant for small l, and the cylinder shape was dominant for large l as similar to [31].The quantum phase transition occurs at a critical size of the subregion.
The renormalized EE, a universal part of EE independent of the cutoff, was computed by operating differentiation on EE [27].By containing modes with KK mass and considering the low energy limit, we continuously derived from odd dimensional renormalized EE to the formula of d − 1 dimensional renormalized EE. 6 Actually, the ϕ circle shrinks to zero at the tip of the AdS soliton (z = z + ), which is probed for large l.The logarithmic term is absent since we don't have a Weyl anomaly in odd dimensions.This is a sort of topology change in the entanglement entropy.In any dimension, massive modes M 0 l > 1 decouple others as a decrease of energy as shown in Fig. 10 and then product states are retained.For high energy limit (l ≪ L ϕ ), the renormalized EE recovers behaviors of the original dimensions because the renormalized EE measures the degrees of freedom in a state with energy E ∼ 1/l.
Because the degrees of freedom with Wilson lines contribute to large a ϕ and high energy, the renormalized EE slowly changes until the critical length (see Fig. 9).The paper [49] also tracked the entanglement entropy across dimensions and found transitions.
In section 5, we analyzed striped surfaces for d = 3.Our results demonstrated that the renormalized EE is positive (non-negative) and satisfies the C theorem.After dimensional reduction, the renormalized entanglement entropy of 2d QFT with Kaluza-Klein modes will also be consistent with the C theorem of the 2-dimensional entropic c−function.
We showed that when d = 3, the renormalized EE for the entangling boundary with a striped shape decreases monotonically and jumps to zero when the size of the entangling surface becomes large, which means that it probes a first-order phase transition and the corresponding boundary field theory runs into a product state at the low energy scale.Unlike the generalized entropic c−function studied in [26], the renormalized EE here behaves monotonically even for the large value of gauge potential a ϕ , and since a ϕ can increase the degrees of freedom and the increase of a ϕ leads to the increase of the renormalized EE, which implies that the renormalized EE is counting the degrees of freedom (DOF) of the boundary field theory.
We analyzed the striped entangling surface in the previous analysis and computed an entropic c−function.This entropic c−function is always positive and non-monotonically behaves.The phase transition happens at a critical length.The non-monotonic behavior is caused by the effective DOF of Wilson lines along the S 1 direction.That is, effective DOF increases the renormalized entanglement entropy.It implies that this Wilson line will decrease the mass of particles such as glueballs [48] because particles of small mass contribute to the entropic c−function at large l.On the other hand, in higher dimensions, c−function.This will be considered as an artifact in the large N and strongly coupled limit.The two are similar in the quantum phase transition's presence and the Wilson lines' effect.The gauge potential will decrease the mass of particles such as glueballs.Thus, more DOF will contribute to HREE and let it increase at a large distance.
It shows that Thus, the motion in which (66) becomes the total derivative is possible.
B Hamilton-Jacobi equations for small l We derive the Hamilton-Jacobi equation for a striped shape for d = 5, 6 in this appendix.
When we restrict to the small size l of the subregion, we can use the analytic expression.
We consider d = 5 first.The turning point, z t is expanded in the small l limit (see [26]), (69) For d = 6, z t is expanded in terms of small l as follows: a dimensionless parameter.The gauge field a (0) ϕ acts as the source for the conserved current and induces a non-zero VEV for the current, ⟨J ϕ ⟩ ̸ = 0.Alternatively, this gauge field can be interpreted as a Wilson line, altering the boundary condition (twisted boundary condition) due to a gauge 2 ε = 1 for the Reissner Nordstrom AdS black hole.

5 Figure 6 :
Figure 6: a 5 as function of l with T = 1 π , a ϕ = 1 2 in the case of d=5.The red line corresponds to disk shaped RT surface and the blue line corresponds to a cylinder shape.The critical length of the phase transition is l c = 1.57.

Figure 9 :
Figure 9: Left: s 3 (l) = dA s /(3L ϕ dl) as a function of lM 0 .The curve is for a ϕ = i/2, 0, 1, 2/ √ 3 from the left to the right.The phase transition happens at critical lengths l c = 0.15, 0.19, 0.24, 0.34 in units of 1/M 0 from the left to the right, respectively.s 3 (l) has two values.The physical curve (an upper one) is consistent with the strong subadditivity.