Entanglement Negativity in Galilean Conformal Field Theories

We obtain the entanglement negativity for various bipartite zero and finite temperature pure and mixed state configurations in a class of $(1+1)$-dimensional Galilean conformal field theories. In this context we establish a construction for computing the entanglement negativity for such bipartite states involving a suitable replica technique. Our construction exactly reproduces certain universal features observed for entanglement negativity of corresponding states in relativistic (1+1)-dimensional conformal field theories.


Introduction
Characterization of quantum entanglement has emerged as a central theme in the study of diverse phenomena ranging from condensed matter physics to issues of quantum gravity and black holes. A significant role in these developments involve the measure of entanglement entropy which is defined by the von Neumann entropy of the reduced density matrix for a given subsystem of a bipartite quantum system and characterizes quantum entanglement in a pure state. This is simple to compute for finite systems but often intractable for extended quantum many body systems like quantum field theories. A formal definition of the entanglement entropy for such extended systems may be obtained through the replica technique but is in general difficult to compute. Interestingly as demonstrated in [1][2][3], this quantity may be computed in (1 + 1)dimensional relativistic conformal field theories (CF T 1+1 ) through the above mentioned replica technique.
It is well known however in quantum information theory, that the entanglement entropy fails to be a viable measure for the characterization of mixed state entanglement as it receives contributions from correlations irrelevant to the entanglement of the mixed state under consideration. This renders the characterization of mixed state entanglement to be a subtle and complex issue. This issue was addressed in a classic work by Vidal and Werner [4] in which they introduced a computable measure termed entanglement negativity which characterized an an upper bound on the distillable entanglement of the mixed state under consideration. 1 The entanglement negativity was defined as the logarithm of the trace norm for the partially transposed density matrix for a bipartite system with respect to one of the subsystems. It was demonstrated by Plenio in [5] that unlike usual entanglement measures the entanglement negativity is non convex however it has been shown in quantum information theory that it is an entanglement monotone. Interestingly in a series of communications Calabrese et al. in [6][7][8] advanced a replica technique to compute the entanglement negativity for bipartite systems described by a CF T 1+1 .
In a separate context a class of (1 + 1)-dimensional Galilean invariant conformal field theories (GCF T 1+1 ) has been obtained in [9][10][11][12][13][14][15][16][17][18][19][20][21][22][23] through the İnönü-Wigner contraction of the relativistic conformal algebra for CF T 1+1 s. This contraction involves distinct scalings of the space and the time coordinates and hence breaks the Lorentz symmetry of the CF T 1+1 to a Galilean symmetry. The procedure modifies the original generators of the relativistic conformal algebra but leaves their numbers unchanged. The representation of the resulting Galilean conformal algebra (GCA) may then be utilized to determine the correlation functions of the primary fields in the GCF T 1+1 .
Interestingly the authors in [12,13] utilized a replica technique based on [1,2] to obtain the entanglement entropy of bipartite zero and finite temperature states in a GCF T 1+1 . As mentioned earlier entanglement negativity serves as a suitable measure for characterizing mixed state entanglement. This naturally leads to the extremely interesting issue of determining the entanglement negativity for such bipartite states in a GCF T 1+1 . In this article we address this significant issue and establish a construction involving a replica technique similar to that described [1,2,12,13], to obtain the entanglement negativity for certain zero and finite temperature bipartite states in a GCF T 1+1 . Interestingly our analysis reproduces the universal features of entanglement negativity for corresponding bipartite states of relativistic CF T 1+1 s [6][7][8] in a GCF T 1+1 which strongly validates our construction.
This article is organized as follows. In Section 2, we describe the entanglement negativity measure in quantum information theory and briefly review the replica technique for computing this quantity for various bipartite states in a relativistic CF T 1+1 . Subsequently in Section 3, the replica technique for computing the entanglement entropy of bipartite states in a GCF T 1+1 is reviewed. Following this in Section 4, we compute the entanglement negativity for various bipartite pure and mixed states in a GCF T 1+1 employing the above replica technique. Finally in Section 5 we present a summary of our work and conclusions.

Entanglement negativity
We begin by briefly reviewing the definition of entanglement negativity in quantum information theory [4](see [24] for a review). In this context we consider a tripartite system in a pure state constituted by the subsystems A 1 , A 2 and B, where A = A 1 ∪ A 2 and B = A c represent the rest of the system. We assume that the Hilbert space H for the bipartite system A factorizes as H = H 1 ⊗ H 2 , where H 1 , H 2 are the Hilbert spaces for the subsystems A 1 and A 2 respectively. The partial transpose of the reduced density matrix for the bipartite system A with respect to the second subsystem is defined as and |e (2) j are the bases for the Hilbert spaces H 1 and H 2 . The entanglement negativity for the bipartite mixed state configuration A ≡ A 1 ∪ A 2 may then be defined as where the trace norm Tr|ρ T 2 A | is given by the sum of absolute eigenvalues of ρ T 2 A .

Entanglement negativity in a conformal field theory
As mentioned in the Introduction the entanglement negativity for bipartite states in a CF T 1+1 may be computed through a replica technique [6,7]. For this purpose the tripartite system in a pure state is described by the spatial intervals and B as depicted in Fig. 1. The entanglement negativity E for the bipartite mixed state configuration described by A ≡ A 1 ∪ A 2 may then be obtained utilizing a replica technique involving the analytic continuation of the quantity Tr(ρ T 2 A ) ne through even sequences of n = n e to n e → 1 as follows As described in [6,7] the quantity on the right hand side of the above equation may be expressed as a four point twist correlator on the complex plane as

Negativity for a single interval
The entanglement negativity for the pure state configuration of a single interval A 1 of length l = u 2 − v 2 in an infinite system described by a CF T 1+1 may now be obtained by considering the limit u 2 → v 1 , v 2 → u 1 in which the interval A = A 1 ∪ A 2 describes the full system with B → ∅ where ∅ is the null set. In this limit the four point twist correlator in eq. (2.4) reduces to the following two point twist correlator in the replica approach As explained in [6,7] the n e -sheeted Riemann surface decouples into two independent (n e /2)sheeted Riemann surfaces which leads to the following expression, From eq. (2.6), the scaling dimensions of the twist fields T 2 ne andT 2 ne are given as Also the scaling dimension of twist fields T ne andT ne are given by [1] ∆ ne = c 12 n e − 1 n e . (2.8) The entanglement negativity for the pure state configuration of a single interval may then be obtained using eq. (2.3) and (2.6) in the replica limit n e → 1 as [7] where c 1/2 is a non universal constant appearing in the two point twist correlator and a is an UV cut off. Note that the entanglement negativity for the pure state is described by the Renyi entropy of order half as expected from quantum information theory. We observe that for a pure vacuum state, the relationship between the entanglement negativity and entanglement entropy [2] of a single interval in the CF T 1+1 is given as (2.10) As described in [7], the entanglement negativity for the single interval in a finite system of length L with periodic boundary condition may also be obtained from the corresponding two point twist correlator in eq. (2.5) on a cylinder of circumference L. In this case, the conformal map from the complex plane to the cylinder is given by z → w = iL 2π ln z. The entanglement negativity may then be obtained from eq. (2.3) as where once again c 1/2 is a non universal constant and a is an UV cutoff.

Negativity for adjacent intervals
Having described the entanglement negativity for the pure state configuration of a single interval we now focus our attention on a bipartite zero temperature mixed state configuration of adjacent intervals with lengths given as v 1 − u 1 = l 1 , v 2 − u 2 = l 2 . This is described by the limit v 1 → u 2 in which the four point twist correlator in eq. (2.4) reduces to a three point twist correlator [25] on the complex plane as follows where c ne and C Tn eT 2 ne Tn e are constants appearing in the two point and three point twist correlators respectively. Insertion of the conformal weights for the twist operators in eq. (2.12) now leads to the following expression for the entanglement negativity of the mixed state under consideration [7] E = c 4 ln l 1 l 2 (l 1 + l 2 )a + constant, (2.13) where a is a UV cutoff for the CF T 1+1 . The entanglement negativity for the mixed state of adjacent intervals in a finite sized system of length L with a periodic boundary condition may now be obtained from the corresponding three point twist correlator eq. (2.12) on a cylinder of circumference L. The conformal map from the complex plane to a cylinder is given by z → w = iL 2π ln z. The entanglement negativity in this case is then given as [7] where a is a UV cutoff. The corresponding entanglement negativity for the mixed state of adjacent intervals at a finite temperature T = 1/β involves the three point twist correlator on an infinite cylinder of circumference β where the Euclidean time direction is now compactified. This may be determined through the conformal map from the complex plane to the cylinder given as z → w = β 2π ln z. This leads to the following expression for the entanglement negativity of the finite temperature mixed state as follows [7] where a is a UV cutoff.

Negativity for a single interval at a finite temperature
In [8] it was shown that the entanglement negativity for the case of a mixed state of a single interval at a finite temperature was much more subtle. As described there a naive computation of the entanglement negativity through the evaluation of the corresponding two point twist correlator on an infinite cylinder leads to an incorrect result due to subtle geometrical reasons. For this purpose it was necessary to consider the configuration of a single interval complemented by two other large but finite intervals adjacent to it which was described by a four point twist correlator. The entanglement negativity for the single interval could then be obtained through a bipartite limit in which the two other intervals are taken to be infinite, following the replica limit to arrive at (2.16) In the above equation T n (±L) are the twist fields at the branch points located at the extremities of the auxiliary intervals adjacent on either side of the single interval and the subscript β denotes that the four point twist correlator is evaluated on a cylinder of circumference β = 1 T . It is possible to determine the four point twist correlator described in eq. (2.16) on the complex plane upto a non universal function of the cross ratio F ne (x) in both the limits x → 1 and x → 0 as Here ∆ ne , ∆ ne are the scaling dimensions of T ne and T 2 ne respectively. As described in [8] the non universal arbitrary function F ne (x) at the limits x → 1 and x → 0 is given as where C ne is a constant depending upon the full operator content of the theory. It is now possible to evaluate the four point twist correlator on a cylinder of circumference β through the standard conformal map from the complex plane to the cylinder via z → w = β/2π ln z. This leads us to the entanglement negativity of a single interval at a finite temperature as follows [8]

(2.19)
Note that the first two terms in the above expression are universal and the other terms involving the function f (e −2πl/β ) and the constant are non universal. Interestingly the eq. (2.19) may be expressed in the following intriguing form where S A and S th A denote the entanglement entropy and the thermal entropy of the finite temperature mixed state in question respectively. The above expression demonstrates that the universal part of the entanglement negativity is described by the elimination of the thermal entropy from the entanglement entropy for the finite temperature mixed state justifying its characterization as the upper bound on the distillable entanglement in quantum information theory.

Entanglement entropy in a Galilean conformal field theory
In this section we briefly recapitulate the salient features of a class of non relativistic CF T 1+1 with Galilean invariance (GCF T 1+1 ) [9][10][11][12][13] and describe the characterization of the entanglement entropy in these theories. The GCF T 1+1 above involves the Galilean conformal algebra (GCA) which may be obtained from the usual relativistic conformal algebra through a İnönü-Wigner contraction that necessitates a rescaling of the space and time coordinates as follows with ǫ → 0. This is equivalent to considering the velocity v i ∼ ǫ going to zero. The generators of the (1+1)-dimensional GCA in the plane representation [12] are given as 2 which leads to the Lie algebra with a central extension as Here the quantities c L and c M are the central charges for the GCA 3 . A state in the GCF T 1+1 for the highest weight representation of the GCA [10,11], is labelled by the conformal weights h L and h M given as The two point correlator for primary fields V (x, t) in a GCF T 1+1 may be determined from the Galilean conformal symmetry characterized by the GCA as follows [11] are the weights of the primary fields V 1 and V 2 respectively and In an exactly similar fashion it is also possible to determine the three point function of primary fields in a GCF T 1+1 to be as follows [11] (3.6) Here the V i 's are primary fields with weights Similarly, the four point function of primary fields in the GCF T 1+1 may be expressed as which are the non-relativistic analogues of the cross ratio x described in Section 2.4. Note that the function G(t, x t ) is a non universal function of the cross ratios which depend on the specific operator content of the GCF T 1+1 .

Entanglement entropy of a single interval at zero temperature
Having described the two and the three point correlation functions of the primary fields in a GCF T 1+1 , we now briefly review the replica technique for the computation of the entanglement entropy of a bipartite state described by a single interval in a GCF T 1+1 [12,13]. The computation of the entanglement entropy in a GCF T 1+1 is similar to that in a relativistic CF T 1+1 with certain modifications. Unlike a relativistic CF T 1+1 where it was possible to utilize a fixed time slice due to the Lorentz invariance, in a GCF T 1+1 it is required to consider an interval with a Galilean boost. Such a boosted interval A may be described by the end points (x 1 , t 1 ) and (x 2 , t 2 ), as shown in the Fig. 2 where the complement B = A c denotes the rest of the system.  [13] . In this case the partition function Z n (A) is defined on an n-sheeted surface Σ n , consisting of n copies of the GCFT plane (x, t) glued together. The corresponding transformation from the GCFT plane (x, t) to the (x ′ , t ′ ) coordinates on the n-sheeted surface Σ n , is given as [13] The entanglement entropy for the bipartite pure state A of a single interval in the GCF T 1+1 may then be characterized through the product of n two point twist field correlators on the GCFT plane as follows [13] Trρ n (3.10) In the above expression Φ n (x 1 , t 1 ) and Φ −n (x 2 , t 2 ) are the twist and the anti twist fields localized at the end points of the interval A with weights given as h L = c L 24 (1 − 1 n 2 ) and h M = c M 24 (1 − 1 n 2 ) which may be determined from the GF CT 1+1 conformal Ward identities and k n are some normalization constants. Utilziing the above expression the entanglement entropy for the bipartite pure state described by a single interval in the GCF T 1+1 may then be obtained as [13] where a is a UV cutoff.

Entanglement entropy at finite temperature
For the corresponding finite temperature bipartite mixed state of the single interval denoted by A the entanglement entropy is characterized by the two point twist correlator for the GCF T 1+1 on a cylinder with circumference β = 1 T where T is the temperature 4 in a GCF T 1+1 [1]. The transformation law for a primary field φ in relativistic CF T 1+1 from CFT plane to a cylinder (z → w = β 2π ln z) is given asφ An Inönü-Wigner contraction (t → t, x → ǫx) of the above coordinate transformation z → w = β 2π ln z where the coordinates in the Lorentzian signature are described by z = t+x andz = t−x, leads to the conformal map from the coordinates (x, t) on the GCFT plane to the coordinates (ξ, ρ) on the cylinder as follows [14] t A similar contraction of eq. (3.12) provides the appropriate transformation of the GCFT primary fields from the GCF T plane to the cylinder as follows [14] Φ (3.14) 4 Note that unlike in a relativistic CF T1+1 the holomorphic and the antiholomorphic sectors do not decouple in a GCF T1+1. Hence a clear identification of the coordinates as timelike or spacelike on the cylinder are not directly evident. The consistency of the construction is validated by explicit limits.
From the above transformations for the primary fields in the GCF T 1+1 given in eq. (3.14), the two point twist correlator on the cylinder may be expressed as follows [12] Φ n (ξ 1 , The entanglement entropy for the finite temperature mixed state of a single interval described by the end points (ξ 1 , ρ 1 ) and (ξ 2 , ρ 2 ) on the cylinder, in the GCF T 1+1 is then obtained from the eq. (3.15) as follows [12,13] S A = c L 6 ln β πa sinh πξ 12 β + πc M 6β ρ 12 coth πξ 12 β . (3.16) The limit β → ∞ for the above expression leads to the earlier zero temperature result for the entanglement entropy of the pure state described in eq. (3.11) which validates the consistency of the result.

Entanglement entropy for finite size
The entanglement entropy for the zero temperature pure state of a single interval in a finite system of length L with a periodic boundary condition may be computed from the corresponding two point twist correlator for the GCF T 1+1 on a cylinder of circumference L. The relevant conformal map from the GCFT plane described by the coordinates (x, t) to the cylinder in this case is given as follows where (ξ, ρ) are the coordinates on the cylinder. The GCFT primaries under this transformation transforms asΦ Using the above transformation of fields, the two point twist correlator on the cylinder is described as As earlier the entanglement entropy for the zero temperature pure state of a single interval (ξ 1 , ρ 1 ) and (ξ 2 , ρ 2 ) for a finite sized system described by the GCF T 1+1 on a spatial cylinder is then given from eq. (3.19) as follows [12,13] where a is a UV cut-off and ρ 12 =| ρ 1 − ρ 2 |. The limit L → ∞ for the above expression once again leads to the eq. (3.11) which refers to an infinite system validating the result.

Entanglement negativity in a GCF T 1+1
Having reviewed the computation of the entanglement entropy for various zero and finite temperature bipartite pure and mixed state configurations in a GCF T 1+1 , in this section we now turn our attention to the definition of the entanglement negativity for such bipartite states.
To this end we consider the configuration described by the intervals and B = A c describes the rest of the system as depicted in the are the end points of the intervals A 1 , A 2 respectively. Following the procedure described in [7] it is possible to define the partial transpose of the reduced density matrix ρ A with respect to the second interval A 2 which corresponds to the exchange of the row and the column indices for the interval A 2 . The replica technique to compute the trace norm of the partially transposed reduced density matrix follows from that described in Section 2. Accordingly in the present case the twist fields at the end points of the second interval are reversed in the four point correlator to arrive at where Φ ±ne are the twist fields in the GCF T 1+1 and the replica index n e is even. As described in eq. (2.3) for the relativistic CF T 1+1 , the entanglement negativity for the bipartite state depicted by the configuration of disjoint intervals in the GCF T 1+1 may also be defined through the replica limit n e → 1 as follows

Negativity for a single interval
Utilizing the expression in eq. (4.2) given above we may now obtain the entanglement negativity for the pure vacuum state configuration of a single interval in the GCF T 1+1 through the bipartite limit of B → ∅ described by u 2 → v 1 and v 2 → u 1 where the interval A 2 now describes the rest of the system. In this limit the four point twist correlator in eq. (4.1) reduces to a two point twist correlator as follows As described in [7] the n e -sheeted surface Σ ne decouples into two independent (n e /2)-sheeted surfaces in this case and hence in this limit the two point twist correlator in eq. (4.3) reduces to the following The corresponding weights of the twist fields Φ ±ne may then be determined as described in Section 3 to be ∆ ne = c L 24 n e − 1 ne and χ ne = c M 24 n e − 1 ne . 5 Comparing (4.3) and (4.4), we find the weights of twist fields Φ 2 ±ne as (4.5) Upon substitution of the weights for the twist fields from eq. (4.5) in eq. (4.3) we may express the left hand side of eq. (4.3) as follows The analytic continuation n e → 1 leads us to the trace norm of the reduced density matrix describing the pure state of a single interval in a GCF T 1+1 as where x 12 , t 12 are as defined earlier. Finally, the entanglement negativity for the bipartite pure state configuration of a single interval A in a GCF T 1+1 may be obtained from the above expression as follows Note that the first two terms on the right hand side of the above equation are universal whereas the last term is a non universal constant that depends on the specific GCF T 1+1 being considered. Interestingly from eq. (4.8) we observe that the entanglement negativity for the bipartite pure state of a single interval in a GCF T 1+1 is given by the Renyi entropy of order half similar to that of a relativistic CF T 1+1 . From (3.11) for the entanglement entropy of the pure state of a single interval in a GCF T 1+1 the above equation may be expressed in the following form Note that the form of the universal part of the entanglement negativity is identical to the corresponding form in a relativistic CF T 1+1 as given in eq. (2.10). This is extremely significant as our construction reproduces the universal features of entanglement negativity in relativistic CF T 1+1 s illustrating that these are purely determined by the conformal symmetry. Naturally this also serves as a strong consistency check for the validity of our construction.

Entanglement negativity of a single interval for finite size systems
It is now possible to extend the above results for the entanglement negativity for the bipartite pure state of a single interval in a GCF T 1+1 to a finite size system of length L with a periodic boundary condition. For this purpose we employ the transformation of the primary fields given in eq. (3.18) to express the two point twist correlator on the cylinder as follows where (∆ (2) ne , χ (2) ne ) are the weights of Φ 2 ±ne . Utilizing the transformations described in eq. (3.17) to the above eq. (4.10), the two point twist correlator on the cylinder is given as Substituting the weights of twist fields from eq. (4.5) in above expression and using eq. (4.2), we arrive at the entanglement negativity for the bipartite pure state of a single interval in a finite sized system described by a GCF T 1+1 on a cylinder as follows Once again we observe that the entanglement negativity for the pure state of a single interval in a finite sized system described by the GCF T 1+1 is given by the Renyi entropy of order half and the universal part is proportional to the entanglement entropy given in (3.20) through the following expression

Negativity for a single interval at a finite temperature
We now turn to the issue of obtaining the entanglement negativity for the finite temperature bipartite mixed state of a single interval following the analysis described for a relativistic CF T 1+1 in [8] and reviewed in Section 2. As earlier the entanglement negativity for the finite temperature mixed state of a single interval in a GCF T 1+1 involves a four point twist correlator on an infinite cylinder arising from the configuration of a single interval sandwiched between two adjacent large but finite intervals. The entanglement negativity for the mixed state in question may then be obtained as earlier, through a bipartite limit subsequent to the replica limit leading to the following expression (4.14) In the above expression the fields Φ ∓ne (±L, ±y) are the twist fields located at the extremities of the auxiliary intervals adjacent to the single interval and the fields Φ 2 −ne (−ξ, −ρ), Φ 2 ne (0, 0) are the twist fields located at the ends of the single interval such that L > ξ > 0 and y > ρ > 0. The coordinates (ξ, ρ) are then defined on the infinite cylinder of circumference β = 1/T where T is the temperature. Now employing eq. (3.7) we may describe the four point twist correlator on the GCFT plane as follows x t ).
(4.15) In the above equation G ne (t, x t ) is an arbitrary non universal function of the cross ratios. This may be fixed in certain limits of the cross ratios t and x t which are related to each other. Hence it suffices to consider the function G ne (t, x t ) in the limits t → 1 and t → 0 and express the four point twist correlator in eq. (4.15) in these limits as follows (4.16) Here the non universal function F ne (t, x/t) of the cross ratios which is related to the function G ne (t, x t ), in the limits t → 1 and t → 0 are given as where C ne is a constant that depends on the full operator content of the theory. We now utilize the conformal map from the GCF T 1+1 plane to the cylinder given by equation (3.13) and the corresponding field transformations from equation (3.14) to express the four point function on the cylinder in the following way x t ).
(4.18) Utilizing the transformations described in eq. (3.13), the cross ratios in eq. (3.8) in the bipartite limit ( L → ∞) are given as follows Note that as described in [8] the replica limit must precede the bipartite limit in the evaluation of the four point twist correlator on the cylinder. Upon implementing the correct order of the of the replica and the bipartite limits the four point twist correlator in eq. (4.18) may finally be expressed as Here a is a UV cut-off and the arbitrary function f (t, x/t) describing the non universal part is given as 21) and the last term is a non universal constant. Note that using eq. (3.16) the above eq. (4.20) may be expressed in the following fashion where S A and S th A denote the entanglement entropy and the thermal entropy corresponding to the mixed state described by the single interval in the GCF T 1+1 and the other terms are non universal and depend on the full operator content of the theory. Note as earlier, the elimination of the thermal entropy from the entanglement entropy in the universal part for the entanglement negativity conforms to its characterization as the distillable entanglement of the mixed state under consideration. Remarkably we observe that the universal part of the entanglement negativity is identical in form to that of a relativistic CF T 1+1 given in eq. (2.20). This illustrates the universality of the result and its sole dependence on the identical conformal structures of the two theories. This completes our analysis for the entanglement negativity for the case of the zero and finite temperature bipartite pure and mixed state of a single interval in a GCF T 1+1 .

Entanglement negativity for adjacent intervals
Having described the various scenarios associated with a single interval in a GCF T 1+1 we now focus on computing the entanglement negativity for the zero temperature mixed state configuration of two adjacent intervals in a GCF T 1+1 . To this end we consider the adjacent limit v 1 → u 2 for the two disjoint intervals in Fig. 3 to arrive at the configuration described in Fig. 4. For this configuration the right hand side of the eq. (4.1) reduces to a three point twist correlator on the GCFT plane as follows .

Negativity for two adjacent intervals in vacuum for finite size
It is now possible to extend the above analysis to compute the entanglement negativity for the zero temperature mixed state of adjacent intervals for a finite size system described by a GCF T 1+1 on a cylinder of circumference L. For this purpose it is required to evaluate the corresponding three point twist correlator on the cylinder of circumference L utilizing the transformation of a primary field defined on the GCF T plane to the cylinder described in eq. (3.18) to arrive at the following expression ne ) x 13 t 13 , ne , χ (2) ne ) are the weights of the GCF T twist fields Φ ne and Φ 2 −ne respectively. Note that the right hand side of the above equation involves the three point twist correlator on the GCF T plane apart from the leading exponential factor with its coefficient. We may now employ the coordinate transformations described in eq. (3.17) on the right hand side of the above eq. (4.27) and use eq. (4.2), to obtain the entanglement negativity of the zero temperature mixed state under consideration as follows where a is a UV cut off. As earlier we observe from eq. (3.20) that the universal part of the entanglement negativity in the above eq. (4.28) may be expressed in terms of the universal part of the mutual information I(A 1 , A 2 ) between the subsystems A 1 and A 2 as follows

Negativity for two adjacent intervals at a finite temperature
In an exactly similar fashion as above we may now compute the entanglement negativity for the mixed state configuration of adjacent intervals in a GCF T 1+1 at a finite temperature T . To this end it is required to evaluate the three point twist correlator in eq. (4.23) on a cylinder of circumference β = 1/T . Utilizing the transformation for the GCF T primary fields described in eq. (3.14) the three point twist correlator on the cylinder may be expressed as follows ne ) 13 exp − (χ (2) ne ) 30) where (∆ ne , χ ne ), (∆ (2) ne , χ (2) ne ) are the weights of the GCF T twist fields Φ ne and Φ 2 −ne respectively. Note that as earlier the right hand side of the above equation involves the three point twist correlator on the GCF T plane apart from the leading exponential factor with its coefficient. It is now possible to employ the coordinate transformation described in eq. (3.13) on the right hand side of the above eq. (4.30) and use eq. (4.2), to obtain the entanglement negativity of the finite temperature mixed state under consideration as follows where a is a UV cutoff. As earlier we observe from eq. (3.16) that the universal part of the entanglement negativity in eq. (4.31) may be expressed in terms of the mutual information I(A 1 , A 2 ) as follows (4.32)

Summary and Conclusions
To summarize, in this article we have obtained the entanglement negativity for various bipartite pure and mixed state configurations in a class of (1 + 1)-dimensional Galilean conformal field theory. In this context utilizing a replica technique we have computed the entanglement negativity for the pure state configurations of a single interval in infinite and finite size systems described by a GCF T 1+1 . It is observed that, as expected from quantum information theory the pure state entanglement negativity is given by the Renyi entropy of order half. Interestingly our analysis reproduces the universal feature of pure state entanglement negativity in a relativistic CF T 1+1 where it is proportional to the corresponding entanglement entropy. This serves as a strong consistency check of our construction for characterizing the entanglement negativity of such bipartite pure states in a GCF T 1+1 . Subsequently we have investigated the interesting issue of determining the entanglement negativity for mixed state configurations of adjacent intervals for zero, finite temperature and finite size in a GCF T 1+1 . As earlier our results reproduce the universal feature of mixed state entanglement negativity to be proportional to the mutual information for this configuration in a relativistic CF T 1+1 . Following this we have obtained a construction for the non trivial issue of the entanglement negativity for the finite temperature mixed state configuration of a single interval in an infinite system described by a GCF T 1+1 . The replica technique for this scenario involved the determination of the partial transpose of the density matrix over an infinite cylinder which has been effected through a non trivial bipartite limit following the replica limit. Interestingly it was observed that the universal part of the entanglement negativity obtained through the above procedure was characterized by the elimination of the thermal contribution from the entanglement entropy for the finite temperature mixed state under consideration. As earlier the above result exactly reproduces the universal feature of entanglement negativity for the corresponding finite temperature mixed state in a relativistic CF T 1+1 . Naturally this also constitutes a strong consistency check of our construction for the entanglement negativity for both zero and finite temperature mixed states in a GCF T 1+1 .
The systematic procedure to compute the entanglement negativity for mixed states prescribed by our construction should find interesting applications to entanglement issues in (1+1)dimensional non relativistic systems with conformal symmetry in condensed matter physics. It would be extremely significant to characterize a holographic description of the entanglement negativity for GCF T 1+1 in the context of flat holography. We hope to return to these exciting issues and applications in the near future.