Holographic subregion complexity of a 1+1 dimensional $p$-wave superconductor

We analyze the holographic subregion complexity in a $3d$ black hole with the vector hair. This $3d$ black hole is dual to a $1+1$ dimensional $p$-wave superconductor. We analyze the holographic subregion complexity across the holographic $1+1$ dimensional $p$-wave superconductor phase transition by fixing $q$ or $T$. The behavior of the subregion complexity depends on the gravitational coupling constant divided by the gauge coupling constant. When this ratio is less than the critical value, the subregion complexity increases as temperature becomes low. This behavior is similar to the one of the holographic $1+1$ dimensional $s$-wave superconductor arXiv:1704.00557. When the ratio is larger than the critical value, the subregion complexity has a non-monotonic behavior as a function of $q$ or $T$. We also find a discontinuous jump of the subregion complexity as a function of the size of the interval. The subregion complexity has the maximum when it wraps the almost entire spatial circle. The condensate does not almost vary the subregion complexity, while charge density varies it.


Introduction
Entanglement entropy is an important non-local quantity in quantum information, capturing geometric aspects of field theories (e.g. an area law [1] and the strong subadditivity). The entanglement entropy counts the number of degrees of freedom in the quantum entangled state [2,3], while it turns out to be an order parameter of the phase transition as in the Wilson loop operator in gauge theories (see quantum critical phase transitions [6]). Duality between strongly coupled gauge theories and the weakly coupled gravity called the gauge/gravity correspondence [5] has been a powerful tool to analyze the entanglement entropy. 1 The gravity dual to the entanglement entropy is given by the minimal surface called Ryu-Takayanagi surface [7,8,9]. Ryu-Takayanagi surface is a useful way of analyzing the entanglement entropy of strongly coupled systems. The holographic entanglement entropy has been an order parameter of the confinement/deconfinement phase transition [10]- [14] and superconductor phase transitions [15]- [21].
Besides, the complexity in quantum information describes the minimal number of gates of any quantum circuit to obtain a desired target state from a reference state. The holographic dual of the complexity has recently been remarked. First, the holographic complexity was conjectured by Susskind in dual black holes [22]. The holographic complexity in a black hole is given by the surface of the Einstein-Rosen bridge. The holographic complexity grows linearly in time before QFT approaches the thermal equilibrium as expected. The complexity of a state is in proportion to the volume of the codimension-1 maximal bulk surface V in general (complexity=volume conjecture C ∼ V /κ 2 l). However, the length scale l in the complexity=volume conjecture is unclear for separate backgrounds.
Besides, the complexity=action conjecture improves the ambiguity of the length. Following this conjecture, the Einstein-Hilbert action in the Wheeler-DeWitt patch turns out to be the holographic dual of complexity [23,24]. Choosing a right length scale, it reproduces the same result as in the complexity=volume conjecture. 2 Our purpose is the computation of the holographic complexity of subregions. Namely, we evaluate the holographic complexity of the mixed state by tracing out states of a separate region. The holographic subregion complexity is proportional to the volume surrounded by the minimal surface (Ryu-Takayanagi surface) [34] as follows (see also a generalization [35]): where γ A is an area of the extremal surface and R is the radius of curvatures in the background. 3 The subregion complexity leads to a discontinuous jump at the transition, which is confirmed by computing the integration of the volume form [39] and using the Gauss-Bonnet theorem [36,42]. In the context of the tensor network, numerical results in the Ising model in a squared lattice reproduce an expected linear law behavior of the holographic subregion complexity. 4 In [20], the holographic complexity is computed during the 1+1d s-wave superconductor transition. Like an asymptotic AdS background, the subregion complexity has divergences, while its finite part does not behave as in the entanglement entropy. The subregion complexity decreases with increase of temperature.
The holographic complexity has also been computed for probing string backgrounds [37] and anisotropic black branes [38]. For the computation of the subregion complexity, we focus on a specific holographic model dual to the 1+1 dimensional p-wave superconductor phase transition. 3d SU (2)Yang-Mills term and the Einstein Hilbert action are dual to a 1+1 dimensional p-wave superconductor as proven in the probe limit [25,26]. In the large N limit, one can evade Coleman-Mermin-Wagner theorem in this lower dimensional system [31]: quantum fluctuations preventing formation of condensates are suppressed in the large N limit. The holographic entanglement entropy is computed in a fully backreacted metric of a 1 + 1 dimensional p-wave superconductor across the phase transition [21]. The backreacted metric turns out to be a black hole with the vector hair in the condensed phase, while it turns out to be the AdS 3 charged black hole in the normal phase. It is shown that the order of the p-wave superconductor phase transition varies depending on the strength of the coupling constant (see appendix A).
In this paper, we compute the holographic subregion complexity in a fully backreacted metric of a 1 + 1 dimensional p-wave superconductor. We make use of the divergent form of the holographic complexity analyzed in [40,41]. After analyzing the coefficient of the divergent term by varying the size of the subregion, we specify the size dependence of this coefficient. Subtracting the divergent term, we analyze the finite part of the subregion complexity. The finite part of the subregion complexity should also depend on the strength of the coupling constant. In main section, we show that the subregion complexity as a function of T or q suddenly jumps near the phase transition for the large ratio of the gravitational coupling constant to the gauge coupling constant.
In section 2, we review 3d Einstein-Hilbert and SU (2) Yang-Mills action, which are dual to the 1 + 1 dimensional p-wave superconductor. To analyze the holographic subregion complexity, we compute the backreaction of the Yang-Mills term into the metric. In section 3, we compute both the holographic entanglement entropy and the holographic subregion complexity in the holographic 1 + 1 dimensional p-wave superconductor phase transition. We compute the holographic subregion complexity by fixing q or T (or both quantities).

Backreactions of the Yang-Mills term
The SU (2) Yang-Mills theory for the AdS 3 black hole has been a holographic model of the p-wave superconductor [25,26]. In this section, we review the holographic p-wave superconductor in 3d Einstein-Hilbert action with SU (2) Yang-Mills term. We consider the action of the Einstein-Hilbert and the SU (2) Yang-Mills term as where the field strength is defined as Note that this normalization of the Yang-Mills term is convenient when it is compared with the one of the Maxwell theory. That is, using tr(T a T b ) = δ ab /2, the kinetic term is as in F a µν F a µν /4g 2 Y M . By performing the coordinate transformation, a general ansatz for the metric is given by where y is compactified with the periodicity y ∼ y + 2πL. The function f (z) is the blackening factor which gives the position of the black hole horizon at z = z h . The ansatz for the background non-Abelian gauge field becomes in the radial gauge (see also [29,30]) where σ a (a = 1, 2, 3) are Pauli matrices. The Einstein equations derived from eq. (2.1) turn out to be following three equations: where the last equation is the z-component corresponding to the constraint equation.
Here, we have introduced a parameterκ = κ/g Y M , which has dimension −1. In addition, EOM in terms of Yang-Mills fields are written as Due to the dependence of these EOM only on the dimensionless combinationκ/L, L is set to be 1 in remaining section.

The normal phase
We then solve the Einstein equation of motion derived from the action eq. (2.1). In the normal phase, the y component of the gauge field is zero, while non-zero A 3 t = φ produces the charge density and breaks SU (2) gauge symmetry into U (1) 3 . The energy momentum tensor turns out to be those without non-linear terms. We then know the charged AdS 3 black hole solution [32,33] with the unit AdS radius as the solution to the Einstein equation of motion as follows: where f (z) = 1 − (z/z 0 ) 2 +κ 2 q 2 z 2 log(z/z 0 ) and the black hole horizon is located at z = z 0 . Here, φ(z) is required to be regular at the position of the horizon z = z 0 . The squared horizon position is inversely proportional to regularized mass M 0 = (L/z 0 ) 2 which satisfies the BPS-like bound M 0 ≥κ 2 q 2 /2 [32]. The BPS-like bound is saturated at the zero temperature. Due to the non-normalizable log term, the gauge field obeys the alternative boundary condition, for which the charge density q is considered as the source. The chemical potential turns out to be µ = −q log(z 0 ).

The condensed phase
In this section, we consider the condensed phase, where both φ(z) and w(z) are non-zero. The charged AdS 3 black hole is unstable when q is large. The black hole acquires the vector hair to go to the stable configuration called the condensed phase. A vector operator dual to w(z) condenses in the condensed phase, while it breaks parity symmetry as well as remaining U (1) 3 spontaneously. The critical point for the p-wave superconductor phase transition is determined from the scaling analysis. In the probe limit, critical charge density is q c = 21.7T H [25]. We analyze the order of the phase transition between the normal phase and the condensed phase by varying the coupling constantκ.
At the AdS boundary z → 0, fields are expanded as where µ 0 is the chemical potential, the parameter W 0 is the VEV of a vector order parameter, and v w is the source conjugate to W 0 . z 0 and n 0 are constant parameters. The black hole horizon is expected at z = z h and satisfying f (z h ) = 0. The regular boundary condition φ(z h ) = 0 is imposed at the black hole horizon. The analytic expansion near the black hole horizon z = z h is given by where (α 1 , β 1 , β 2 , , δ 1 , γ 1 , γ 2 ) are constants. The Hawking temperature of this black hole solution turns out to be Substituting the expansion eq. (2.8) into the EOM eq. (2.4) and eq. (2.5), we obtain 4 independent parameters (α 1 , β 1 , γ 1 , z h ). Other parameters are fixed by these 4 parameters as well asκ. We then solve the EOM starting from the black hole horizon. The constraint equation of eq. (2.4) is solved at the black hole horizon. We numerically solve first two of equations (2.4) and equations (2.5), specifying regularity conditions at the horizon z = z h .
At the AdS boundary, we specify the boundary conditions W 0 , v w = 0, n 0 = 1. The vanishing source v w shows the superconductor boundary condition, which describes the spontaneous symmetry breaking of residual U (1) symmetry generated by A 3 µ . The superconductor boundary condition is similar to the one imposed on the charged scalar [27,28] in the holographic s-wave superconductor.
Note that there are scaling symmetry in the EOM as follows: One can use first symmetry to fix z h = 1. Second symmetry can be used to fix the parameter n 0 = 1, which yields the standard asymptotic AdS 3 metric. The behavior of W 0 /q is plotted as the function of q c /q in Fig. 1 at fixed temperature T H = 0.15. In the figure, critical charge density q c is determined from the thermodynamic stability between normal and condensed phases. See appendix A. The critical value q c depends on the coupling constantκ: q c = 189T H , 33.5T H , 21.7T H forκ 2 = 0.5, 0.1, 2 × 10 −6 , respectively. 5 For all coupling constantsκ, W 0 is zero at small charge density. Whenκ 2 < 0.31, W 0 suddenly increases from zero at the critical density q = q c . The condensate W 0 has the scaling behavior ∼ 1.18 1 − q c /q. This implies the second order phase transition. Whenκ 2 > 0.31, W 0 jumps to be non-zero at the critical density q = q c , where W 0 does not follow a scaling behavior.

Holographic complexity of the subregion
In this section, we compute the holographic complexity of the subregion in the holographic d = 1 + 1 p-wave superconductor phase transition. We analyze time independent subregion complexity via holography [34]. We start with the metric of the 3-dimensional black hole eq. (2.2).
Recall that the holographic entanglement entropy is proportional to the area of a minimal surface γ A . The 1-dimensional strip subregion with the size l is considered. Using the metric eq. (2.2), the embedding scalar of the surface γ A satisfies the EOM where z = z * is the turning point for the surface. Integrating the EOM, the embedding scalar turns out to be x(z) satisfies x( ) = l/2 as well as x(z * ) = 0. Due to symmetry of the curve at the turning point z = z * , the factor of 1/2 appears in front of l. The minimal surface ends on the particular end points. The holographic entanglement entropy is the minimal surface divided by the gravitational constant The divergent part of S EE is of the form S EE ∼ 4π κ 2 log( ). Apart from the divergence, the finite part of the entanglement entropy S EE f in is interesting to analyze. The holographic entanglement entropy S EE f in in the normal phase was analyzed by using charged black holes with hyperbolic horizons [52] and 2nd order excitations [49,53]. The charge q dependence of the finite part S EE f in was analyzed in [21], whenκ 2 < 0.31. The finite part S EE f in has a cusp at the intersecting critical point between normal and condensed phases. The finite part behaves non-monotonically in the regime where the amount of the entanglement in the charge sector competes with the effect of the condensate. Wheñ κ 2 > 0.31, S EE f in is analyzed in appendix B. While the holographic entanglement entropy turns out to be multivalued at a region of q < q i , the holographic entanglement entropy behaves similarly as in the one ofκ 2 < 0.31 at large q > q i .
Unlike the charge density q dependence, we do not find any critical sizes of phase transition varying T /T i . Especially, the finite part always decreases with decrease of T /T i at enough low temperature. The amount of the quantum entanglement decreases due to both decrease of temperature and the formed condensate. Thus, we do not have competition of two effects, namely, the formation of the condensate and decrease of temperature.
By contrast, the holographic complexity of the subregion is proposed to be proportional to the volume surrounded by the minimal surface γ A . This subregion has the size l. Following [34], the holographic complexity is defined as Substituting the metric eq. (2.2) into the formula eq. (3.4), the holographic complexity of the subregion turns out to be where the central charge is defined as c = 12π/κ 2 . By using scaling symmetry of the first line in eq. (2.10), the physics parameters, the entanglement entropy, and the subregion complexity are transformed into Due to the presence of scaling symmetry, it is convenient to use dimensionless parameters such as T l and ql.
One needs to use the numerics to compute the holographic subregion complexity. First, we find z * by following the argument around eq. (3.2) and fixing the parameter l. Secondly, we obtain x(z) from eq. (3.2) to perform the double integration in eq. (3.5). The subregion complexity is divergent itself. It can be shown that the divergent part of C( ) is proportional to only 1/ . The coefficient of the divergent part is given by The parameter l d should be a function of the size of the interval l. 6 One needs to subtract this singular part to pick up the finite contribution κ 2 HC fin .

The holographic complexity as a function of q or T
We consider two separate coupling constantsκ 2 = 0.1 and 0.5 in our numerical computation. We specify the coefficient l d and the finite part κ 2 HC fin for each coupling constant. We find that l d is linearly equal to l in the numerics. This linear behavior of the subregion complexity is also observed in the Ising model on the squared lattice in the context of the tensor network [42]. HC fin Figure 5: The finite part HC fin as a function of T /T c with fixed q, l, and fixedκ 2 = 0.5 (T c ∼ 0.0053q = 0.053). In the normal phase, HC fin decreases with increase of T /T c . By contrast, the holographic complexity in the condensed phase turns out to be multivalued at a specific range of parameters. It increases with decrease of T /T c after the phase transition point, having a peak at low temperature.
Subtracting the singular part with a coefficient in eq. (3.7), we compute the finite part of the subregion complexity κ 2 HC fin . We plot the finite part κ 2 HC fin fixing the size of the interval l, temperature T H , andκ 2 in Fig. 2 and 3. We plot the finite part κ 2 HC fin fixing l, q, andκ 2 in Fig. 4 and 5. In both cases, when the size l is smaller than 1/T H (the extremal limit) or 1/q, the finite part κ 2 HC fin in the condensed phase (solid curves) is almost equal to the one of the normal phase (dashed curves). The two behaves differently when lT H 1 or lq 1. Note that the intersecting point arises from q = q i (T = T i ). While the intersecting point coincides with the critical point q i = q c for κ 2 = 0.1(< 0.31), q i = q c forκ 2 = 0.5(> 0.31). Whenκ 2 = 0.5(> 0.31), the intersecting point does not seem to have physical meanings.
In the normal phase of both coupling constants, the finite part κ 2 HC fin decreases with increase of T /T c or q c /q. Whenκ 2 = 0.1, the finite part κ 2 HC fin in the condensed phase behaves similarly. This implies that the ordered phase at high density is a more complicated system. Whenκ 2 = 0.5, the finite part κ 2 HC fin turns out to be multivalued at a specific range of the parameter q c /q or T /T c . As opposed to smallκ < 0.31, the finite part HC fin in the condensed phase has a peak at the intermediate regime after increasing at low temperature in Fig. 5. These behaviors are separate from the holographic entanglement entropy, which decreases at low temperature or high density.
One can define opening angles θ o around the intersecting point q i or T I between two curves of two phases. The opening angles θ o increase whenκ 2 increases as observed in the holographic s-wave superconductor [20]. Whenκ 2 < 0.31, θ o is small, being similar to the probe limit. Whenκ 2 > 0.31, θ o can turn out to be larger than π/2. The extremal limit T H l 1 of opening angles is interesting. In the extremal limit and forκ 2 < 0.31, the holographic complexity of the condensed phase coincides with the one of the normal phase. The opening angles between the normal and condensed phases are small enough in the extremal limit. By contrast, while the holographic complexity coincides between two phases forκ 2 > 0.31 in the extremal limit, the open angles between two phases are large enough. See the right hand side of Fig. 3. The open angles decrease with increase of temperature.

The holographic complexity as a function of lq
In this section, we compute κ 2 HC fin as well as S EE f in as a function of lq. After increasing the size l, the minimal surface is attached to the black hole horizon. A part of the surface attached to the black hole horizon explains the thermal entropy. Following [45], moreover, the minimal surface wraps the black hole horizon of a BTZ black hole for the enough large size of the interval, which gives the contribution of the thermal entropy. Thus, we introduce the following surface where S ent is thermal entropy. That is, the difference S EE (2π − δ) − S EE (δ) (δ 1) is equal to the thermal entropy S ent . This equality is also satisfied by the entanglement entropy of 2d free massless Dirac Fermions on the 2-torus [45].
Actually, the surface wrapping the black hole horizon minimizes the holographic entanglement entropy when the size l is larger than a critical size l c . There is a phase transition at a critical size l c as a function of l. In Fig. 6  the complexity also has the following form: where C entire is the subregion complexity of the entire spatial boundary (3.10) C entire does not give any finite part in an AdS 3 black hole. In eq. (3.9), the singular part is proportional to the size l due to the cancellation between two terms. The finite part of the holographic complexity κ 2 HC fin is plotted as a function of a dimensionless size lq at fixed temperature in Fig. 7. The finite part κ 2 HC fin increases with increase of the size of the interval. Due to the topological phase transition of the minimal surface surrounding the volume of the subregion complexity at critical sizes, the finite part κ 2 HC fin suddenly jumps at a critical size l c q = 25.7, 20.5 (q = 5) for T /T c = 0.48, 0.17, respectively. The critical size l c is not dependent on the magnitude of the subregion complexity but the magnitude of the holographic entanglement entropy. The critical size decreases with decrease of temperature. lq = 10π corresponds to the entire spatial boundary. In the figure, the finite part turns out to be maximum at lq = 10π. This finite part at zero charge density approaches π (smaller than the one at finite density) after the jump from −π to the tolopogically different configuration. That is the opposite sign of the finite part. In the right hand side of Fig. 7, the blue curve is almost the same as in the charged AdS 3 black hole with same parameters.
The discontinuous jump of the subregion complexity was originally found in [39,42] by computing topologically different configurations of 2d volume surfaces (i.e. the disc as well as the annulus). The magnitude of the jump is ∆C = 2π in the AdS 3 black hole being independent of temperature. After switching on the charge density, the difference ∆HC fin (l = l c ) is not independent of the charge density but almost 2π.
In summary, the singular part of the subregion complexity gives an expected linear behavior, which is divergent like l/ for those in either eq. (3.5) or eq. (3.9). The finite part of the holographic complexity does not vary a lot by varying the condensate, while the finite part of the entanglement entropy decreases as the condensate increases. These behaviors are also observed in [20]. Interestingly, the maximum of the subregion complexity is not a constant depending on charge density q unlike the one with zero density.

Discussion
We computed the holographic subregion complexity in a fully backreacted metric of the 1 + 1 dimensional p-wave superconductor phase transition. We computed the subregion complexity by fixing q or T (or both quantities). The leading divergence of the subregion complexity was shown to be linear to the size of the interval C ∝ l/ in either eq. (3.5) or eq. (3.9). Even if Ryu-Takayanagi surface wraps the black hole horizon at the large size of the interval, cancellations occur between two terms in eq. (3.9). The same linear behavior was observed in the Ising model on the square lattice.
The finite part of the subregion complexity κ 2 HC fin was plotted by fixing the size of the interval l. In the extremal limit lT H 1 or lq 1, κ 2 HC fin almost agreed between normal and condensed phases except for the region around the intersecting point, where the curve in the condensed phase ended. The curve in the condensed phase behaved differently when lT H 1 (or lq 1). Moreover, it depended on the coupling constant. Whenκ 2 = 0.1 < 0.31, κ 2 HC fin decreased with increase of T /T c or q c /q. This implies that the system at high charge density and low temperature is complicated. This result of the subregion complexity agreed with those of the holographic 1 + 1 dimensional s-wave superconductor [20].
The order of phase transition is varied in the holographic 1 + 1 dimensional p-wave superconductor with the large amount of the backreactionκ 2 = 0.5 > 0.31, while it is not varied in the holographic 1 + 1 dimensional s-wave superconductor [50,51,54] with the backreaction. Due to the large amount of the backreaction, the condensate does not behave as in mean field theories (2nd order phase transition) but suddenly jumps to a finite value at the critical point. This large amount of the backreaction also causes the non-monotonic behavior of the finite part κ 2 HC fin in the condensed phase, while κ 2 HC fin in the normal phase behaves monotonically. Moreover, the finite part turns out to be multi-valued as a function of T /T c or q c /q.
We plotted the finite part of the subregion complexity κ 2 HC fin as a function of lq fixing q and T . The formation of the condensate did not almost vary the finite part κ 2 HC fin , while the charge density varies it. Wrapping the almost entire space circle maximized the subregion complexity. We found the discontinuous jump of the finite part κ 2 HC fin . The magnitude of the discontinuous jump depended on the charge density unlike the one of the AdS 3 black hole. Note that the magnitude of the jump is larger than the jump in the Ising model on the squared lattice ∆HC fin ∼ 4 ± 0.3 < 2π [42]. This discrepancy will come from broken rotational symmetry of the squared lattice as well as difference between two models.  To compute the free energy using the AdS/CFT correspondence, we analyze a finite on-shell action in the presence of the Gibbons-Hawking term and a term of the Legendre transformation. These terms reflect a well-defined variation principle. Due to the divergent on-shell action, we also must use counter-terms to cancel divergence [46,47,48] where α 1 = −1/L and α 2 = L. The log term in the second line addes a scale L in the lagrangian. Summing three contributions eq. (2.1) and eq. (A.1) up, one can obtain the finite renormalized action and the free energy as follows: where a dictionary of the AdS/CFT correspondence has been used in the last line. By using the analytic solution of the charged AdS 3 black hole eq. (2.6), we can integrate the free energy in the normal phase to give where V 1 is the volume of the entire spatial circle. In the above free energy, the presence of the log term varies the scaling transformation: the scaling transformation gives an additional term in the free energy [33]. The variation of the free energy in terms of T, q gives thermodynamic quantities such as the entropy S = −dF/dT and the chemical potential µ = dF/dq, respectively. In contrast, the numerical computation is required to compute the free energy in the condensed phase. The difference of the normalized free energy 2κ 2 ∆I G ≡ 2κ 2 (F SF − F n ) is plotted as a function of the normalized temperature T /T c in Fig. 8. For smallκ (κ 2 < 0.31) and fixed charge density, the solution of the condensed phase always has lower free energy. Whenκ is large (κ 2 > 0.31), ∆I G is multi-valued at temperature larger than T c . There occurs the swallow tail of the first order phase transition. In contrast, it is plotted as a function of the normalized charge density in Fig. 9. While the phase transition to the condensed phase occurs at low temperature T < T c , it occurs at large charge density q > q c . In this section, we compute the finite part of the holographic entanglement entropy. Due to the subtraction of the divergent part from S EE , we define I m,f in ≡ κ 2 2π S EE − 2 log( ). Whenκ 2 = 0.5, the finite part of the holographic entanglement entropy is plotted as a function of q/q i in Fig. 10  behaves non-monotonically. Compared withκ 2 < 0.31, the holographic entanglement entropy turns out to be multivalued at a region of small q < q i . For large q > q i , by contrast, the behavior of the holographic entanglement entropy is qualitatively similar to the one ofκ 2 < 0.31. By increasing l, the increasing behavior of I m,f in (l, q) with increase of q is varied into the decreasing behavior at large q. This phase transition occurs because S EE probes the formation of the condensate at large interval l. The decreased DOF due to the formation of the condensate overcomes increasing entanglement of charged states. In figures 11 and 12, the finite part of the holographic entanglement entropy is plotted as a function of T /T i . There is a cusp between two curves of normal and condensed phases. At low temperature, the finite part of the condensed phase is always lower than the one of the normal phase. Whenκ 2 = 0.5 > 0.31, the finite part turns out to be multi-valued around the critical point T c .
Unlike the entanglement entropy as a function of q, we do not find any critical sizes where the finite part of the entanglement entropy has the phase transition. The finite part I m,f in decreases with decrease of T /T i at enough low temperature, while the finite part increases with increase of q/q i at high charge density for the small size. This implies that by decreasing temperature, the amount of the quantum entanglement decreases. Simultaneously, the condensate is formed and degrees of freedom decreases [21].

C Holographic subregion complexity in asymptotically AdS backgrounds
In this section, we analytically compute the holographic subregion complexity in asymptotically AdS backgrounds for comparison. The subregion complexity has a divergent part like 1/ and a finite part.
where z * is the position of the turning point. The boundary condition x( ) = l/2 is imposed and then l = 2z * .
Substituting (C.1) into the holographic subregion complexity eq. (3.5), we can analytically integrate it to give 7 where c = 12π/κ 2 is the central charge. The subregion complexity is divergent like 1/ . By considering a rectangular shape of the surface γ A , the constant term of eq. (C.3) vanishes as follows: The subregion complexity of the whole spatial circle is easily evaluated. It does not have a finite term as follows: . (C.7) At the large size l, Ryu-Takayanagi surface wraps the black hole horizon. Accordingly, the subregion complexity at a large interval eq. (3.9) has a finite term of the opposite sign as follows: l + π . (C.8) Thus, the subregion complexity has its maximum similar to the holographic superconductor in main section when it wraps the almost whole spatial circle.