Massless Lifshitz Field Theory for Arbitrary z

: By using the notion of fractional derivatives, we introduce a class of massless Lifshitz scalar field theory in (1+1)-dimension with an arbitrary anisotropy index z . The Lifshitz scale invariant ground state of the theory is constructed explicitly and takes the form of Rokhsar-Kivelson (RK). We show that there is a continuous family of ground states with degeneracy parameterized by the choice of solution to the equation of motion of an auxiliary classical system. The quantum mechanical path integral establishes a 2d/1d correspondence with the equal time correlation functions of the Lifshitz scalar field theory. We study the entanglement properties of the Lifshitz theory for arbitrary z using the path integral representation. The entanglement measures are expressed in terms of certain cross ratio functions we specify, and satisfy the c -function monotonicity theorems. We also consider the holographic description of the Lifshitz theory. In order to match with the field theory result for the entanglement entropy, we propose a z -dependent radius scale for the Lifshitz background. This relation


Introduction
Invariance under global scaling transformation t → λt, x i → λx i , λ > 0, (1.1) plays important roles in physics.Apart from describing the fixed point of renormalization group (RG) flow and critical phenomena in field theory, scaling symmetry also finds important applications in particle physics at very high energies and in the microwave background in cosmology.In relativistic field theory, it has been shown that scale symmetry always gets enhanced to the full conformal symmetry.This is not so in non-relativistic field theory.A particularly well-known generalization of the standard scaling symmetry is the Lifshitz scaling symmetry t → λ z t, x i → λx i , λ > 0, which is characterized by the dynamical exponent z ∈ R. Lifshitz symmetry has found important applications in many physical systems.For instance, the z = 2 Lifshitz scalar field theory in (2+1) dimensions [1] is known to describe the critical point of the well-known Rokhsar-Kivelson Quantum dimer model [2,3].The z = 2 Lifshitz field theory in (3+1) dimensions has appeared in the ghost condensation modified gravity scenario [4] and describes a fluid with a non-relativistic dispersion relation with β being some dimensionful energy scale of the theory.Note that for z < 1, the dispersion relation is acausal due to the existence of superluminal modes [5] and the violation of the null energy conditions of the holographic dual [6].Therefore we will focus on z ≥ 1 in this paper.As the Lifshitz scaling (1.2) is defined for general z, it is interesting to ask how the Lifshitz scaling (1.2) symmetry for other values of z can be realized field theoretically.
In the literature, only Lifshitz scalar field theory with integer z has been considered.In this paper, we employ the mathematical notion of fractional derivatives to propose an action for the massless Lifshitz theory for arbitrary values of z in any dimensions.Given a field theory, one of the first properties to understand is its ground state, which is an important first step to the understanding of the physical system.For example, in particle physics, the symmetry of the ground state dedicates the particle spectrum and their interaction in the low energy theory.In statistical mechanics, the ground state of a system provides most of the thermodynamic properties of the system.The entanglement properties of ground state has found intimate relation with topological order [7][8][9] and quantum critical phenomena [10][11][12][13] in quantum many body system.In the literature, it is known that the z = 2 massless Lifshitz theory has a ground state which takes on a special form of Rokhsar-Kivelson (RK) [2], in the sense that the ground state is local and is given by a superposition of quantum states with a quantum mechanical amplitude c [ϕ] specified by the probability distribution of a classical system.Once we have constructed the massless Lifshitz theory for arbitrary z, it is interesting to construct its ground state and examine whether the ground state is still of the form of RK, or is it just for the special value of z = 2.The point z = 2 is in fact somewhat special since it is known that the Lifshitz scaling transformation (1.2) combines with the Galilean boost to the Schrodinger group only for this value.This is analogous to the situation in relativistic theory (z = 1) where the scaling symmetry (1.1) combines with the Lorentz boosts to the conformal group.Therefore, it is meaningful to ask if the RK form of the ground state is a consequence of the Schrodinger symmetry or not.In this paper, we show that the Lifshitz ground state of the (1+1)-dimensional massless theory always take the form of RK, even when there is no presence of Schrodinger symmetry.It turns out that except for z = 2, there is a continuous family of degenerate Lifshitz ground states for z > 1.We show how the correlators in these Lifshitz ground state can be computed in terms of the path integral of a 1-dimensional auxiliary quantum mechanical system.
In a relativistic quantum field theory, the Reeh-Schlieder theorem [14] states that all field variables in any one region are entangled with variables in other regions.This means the entanglement is so strong that it leads to an ultraviolet divergent entanglement entropy between adjoining regions in spacetime.In the article [15], it is explained how this ultraviolet divergence of entanglement is in fact a property of the algebras of observables.As entanglement has not much to do with the speed of light, we expect to have a similar story for the non-relativistic Lifshitz field theory.However, the detailed form of entanglement encoded in different entanglement observables will certainly be different.Previously, entanglement entropy in anisotropic Lifshitz field theories have been analyzed in a series of literature in the context of the quantum Lifshitz model (QLM) [16] which is a special type of Lifshitz field theory in (2 + 1)-dimensions with z = 2 [17][18][19][20][21][22][23][24][25][26][27].Subsequently the QLM was generalized to (d + 1)-dimensions with z = d, d being the number of spatial dimensions.The holographic realization of the Lifshitz symmetry for the arbitrary z was studied in [28] using the RT prescription in higher curvature Lifshitz gravity.Further recent progress has been made in the holographic interpretation of such theory by computing both equal-space and equal-time two-point correlation functions and their exact matching in both field theory and bulk sides [29][30][31].The entanglement entropy has been studied for the z = 2 quantum Lifshitz models [32] followed by the investigation of logarithmic negativity in (1 + 1) and (2 + 1)-dimensions [33,34].Added to this, entanglement entropy of Lifshitz field theory for arbitrary z has also been investigated with cMERA techniques [35][36][37].Very recently, reflected entropy [38] and Markov gap [39] were analyzed for (1+1)-dimensional Lifshitz field theory with z = 2 in [40].
Once we have found the massless Lifshitz field theory and its ground states for arbitrary z, it is an interesting question to study how the entanglement properties of the Lifshitz theory depend on the time anisotropy index z.In this paper, we analyze the entanglement and correlation properties of the (1+1)-dimensional Lifshitz theory for a number of observables.We find that the entanglement entropy increases with respect to z.This is in agreement with the expectations that the increase of z leads to long-range interaction of the theory which typically translates to the growing of the entanglement entropy.Using the kernel of the theory, we extend our computations to the system of two adjacent and disjoint intervals to analytically obtain the Rényi entropy, reflected entropy and the Markov gap.It is worthy to note that our measures can be expressed in terms of cross ratios with components of the form η(l, z) ∼ ( i l z−1 i )/( j l z−1 j ) that depend on the lengths on the intervals l i .Moreover, the standard c-function monotonicity defined via the entanglement entropy along the RG flow, is satisfied and its satisfaction turns out to be equivalent to the satisfaction of the null energy conditions and the presence of causality in the theory [41].
The holographic study of our class of massless Lifshitz scalar theory is also interesting.It has been proposed [42] that the holographic dual of a general Lifshitz field theory in (d + 1)-dimensions is given by the bulk metric The metric is not Lorentz-invariant, but supports the scaling symmetry given by that is consistent with the Lifshitz symmetry (1.2) of the field theory.Such bulk solutions can be understood as arising from the Einstein-Proca type bulk action which consists of a massive vector field [43,44].Using the RT formula, we compute the entanglement entropy for the strongly coupled holographic Lifshitz field theory.As we will show in section 4, the form of the holographic entropy agrees with the result (3.6) of the free theory.This is because the form of the UV divergent part of entanglement entropy is universal.It is possible that, just as in the case of 2d conformal field theory, the coefficient of the entanglement entropy is protected by a non-renormalization theorem.Assuming so, we are leaded to propose a relation between the radius of curvature L of the Lifshitz background and the Newton constant.This is analogous to the Brown-Henneaux relation [45] 3R 2l P = c for the standard AdS 3 /CFT 2 duality.The relation has a novel z-dependence and is consistent with the fact that the Lifshitz vacuum respects a z-dependent scaling symmetry.Furthermore, the anisotropy between the temporal and spatial directions in LFTs makes it interesting to investigate the behavior of entanglement entropy in the time direction.In this context, we compute the time-like entanglement entropy using the holographic prescription of [46,47] (See [48][49][50][51][52][53][54][55][56][57] for recent developments).The holographic result is z-dependent and suggests that there exists a fundamental definition of time-like entanglement entropy other than employing analytic continuation.

Massless Lifshitz scalar theory and ground state
In this section, we employ the definition of fractional calculus to introduce the Lifshitz scalar field theory for an arbitrary real value of z.For the case of (1 + 1)-dimensions, we show that the ground state of the massless Lifshitz field theory can be constructed explicitly and takes on the form of RK.We call these Lifshitz ground states as they are Lifshitz scaling invariant.We show that the theory possess a 2d/1d duality in that the equal time correlation functions of the 2-dimensional Lifshitz theory can be determined in terms of the path integral of a 1-dimensional quantum mechanical system.We also show that the trace involving powers of the reduced density matrix can be computed explicitly without introducing the twist operators.

Massless Lifshitz scalar field theory in (d + 1)-dimensions
We start by considering the following action for the massless Lifshitz scalar field theory in (d + 1)-dimensions with arbitrary critical exponent z, where κ > 0 is a constant and the operator ) z e ikx = k 2z e ikx on the plane wave basis.This form of action has also been considered independently in [30] for integer z and in [58] for arbitrary z.The above action is invariant under the Lifshitz scaling symmetry (1.2) and the transformation Here ∆ ϕ is the scaling dimension of the scalar field ϕ.Note that a self-interacting potential ϕ d+z ∆ ϕ with fractional power can be added to the theory (2.2) and still preserve the Lifshitz scaling symmetry.For z = 1, the massless action (2.2) is Lorentzian invariant, which is extended to the full conformal symmetry due to the presence of scaling symmetry.For z = 2, the theory has the Galilean boost, which is extended to the Schrodinger symmetry due to the presence of the Lifshitz scaling symmetry.For other values of z, the theory has the Lifshitz symmetry group which does not admit the Galilean boost.
The Green's function of the theory which is given by where For the case of equal-time and equal-space separations, G can be evaluated using the method of residues.
One can obtain the equal-time correlators as 1 Another scalar field theory realization of the Lifshitz scaling symmetry is given by the action This is an interesting theory as well.However unlike (2.2), the action (2.1) does not admit a canonical formulation and the quantization of the model becomes obscure.
and the equal-space correlators given by For general configuration of positions, the expression for G is quite complicated and is given by integrals of hypergeometric functions.Previously, the Green's function of the theory (2.2) was determined for general point configurations for the case z = d [30].In this case the scalar field has vanishing scaling dimensions and the two point functions display a logarithmic singularity at short distance.Recently, the results (2.6) and (2.7) were also obtained by [31].Here, we consider the massless Lifshitz theory (2.2) defined for arbitrary z and obtain these results directly as special cases.

Massless Lifshitz scalar theory and Lifshitz ground state in (1 + 1) dimensions
In this paper, we will consider the Lifshitz theory in (1+1)-dimensions for arbitrary z > 1.
As shown in the appendix A of [30] where the analysis actually applies for arbitrary value of z, the theory (2.2) admits a RK vacuum.However, in addition to the RK form of the vacuum, one is also interested in the properties of the vacuum, e.g.expectation value (2.20) of operators or entanglement properties.As we will see below, many interesting properties of the RK vacuum requires the knowledge of the solution to the equation of motion of an associated 1-dimensional quantum system.For the theory (2.2), the equation of motion takes the form is a fractional derivative with the modulus |k| z as its eigenvalue.This is a different fractional derivative from the one (2.9)we are going to define below, and the construction of solution appears to be more nontrivial.Instead let us consider the following definition of the Lifshitz theory where the fractional derivative ∇ z x is defined as and a choice of the branch of the multi-valued function w z , w ∈ C is chosen.For example, Logw = Log|w| + iθ, −3π/2 < θ ≤ π/2 with a cut at the positive imaginary axis.Here, the fractional derivative is defined through (2.9) via some generalized notion of the Fourier analysis which involves some appropriate momentum contour integral, see, e.g., example in appendix A. We note that while the action (2.8) is equivalent to (2.2) when written in momentum space, (2.8) and (2.2) are different in the presence of boundary.The discussion of boundary Lifshitz theory is interesting, and will be left for future consideration.The construction of RK vacuum for the theory (2.8) resembles to that of [30].The Hamiltonian of the theory can be written in the factorized form where ) are generalized annihilation and creation operators.They satisfy the following commutation relation where ) is used for the canonical momentum Π.Now since dxA † (x)A(x) is positive semi-definite, a ground state of the theory may be defined by using the position space annihilation operator under the Lifshitz scaling (1.2).Therefore |Ψ 0 ⟩ is Lifshitz scaling invariant and we will call such |Ψ 0 ⟩ a Lifshitz ground state.
In the Schrodinger representation Π(x) = −i ∂ ∂ϕ(x) , (2.15) turns into a differential equation for This can be solved easily and the ground state of the Lifshitz theory is given by where is a normalization factor.It is interesting to note that the ground state wavefunctional (2.18) takes the form of RK [2], it is given by a superposition of quantum states with a quantum mechanical amplitude c[ϕ] given by the Boltzmann weight of a classical system, c[ϕ] ∝ e −S cl [ϕ]/2 .In this case, the classical system is a 1d particle theory with action given by S cl [ϕ]/2, and as such, Z get the interpretation as the corresponding partition function.
With the vacuum (2.18), the expectation value of operators at equal time is given by the path integral Note that path integral (2.20) is not to be confused with the ordinary path integral expression of the original Lifshitz theory (2.2), which involves the 2d Lifshitz action instead of the action S cl of the auxiliary quantum mechanical system.This equivalence of the 2dimensional field theory with a 1-dimensional system here applies only for the equal time correlation functions and is due to the specific RK form of the Lifshitz vacuum.While this relation is interesting, the path integral (2.20) requires further specification before it can be computed explicitly.To explain this, let us consider a path integral over ϕ satisfying an arbitrary specified condition of the form ϕ( where is the ground state wavefunction and n is an arbitrary positive integer.As usual, the integral can be evaluated by integrating out the fluctuations around the classical solution ϕ c to the equation of motion of S cl .This gives [59] where F (x i , x f ) := Dδϕ e −nS cl (δϕ)/2 is obtained by integrating out the fluctuation that satisfies the conditions δϕ(x i ) = δϕ(x f ) = 0 and is a function of x i , x f only.Except for the case of z = 2, the condition alone is not enough to fix an unique solution to (2.22) for general z > 2. This is clear for integer z > 2 as the equation has derivative higher than order 2. To construct the solution to (2.22) for general z, we can make use of the definition (2.9) for the fractional derivative and use the Fourier analysis with appropriate choice of integral contour as described in appendix A. Now using (A.1), the equation of motion (2.22) is immediately solved by functions for arbitrary a.The general solution to (2.22) and (2.24) is then given by where l := x f − x i and there c n are arbitrary constants.Note that an upper bound N f is needed in (2.26) in order for ϕ to be finite at the endpoint x = x i .For non-integer z, N z = [z] and there are [z] − 1 free parameters.For integer z, N z = z − 1 as the term n = z is a constant and this has been taken into account already in (2.26).Therefore in this case there are z − 2 free parameters.This analysis is consistent with the case of z = 2.We can make a change of variable to t := (x − x i )/l, then with g(t) given by the "polynomial" (2.28) Note that g(t) is independent of the boundary condition (2.24).Therefore, the evaluation of the path integral (2.20) requires a specification of the function g only once and for all.We may interpret this as saying that the Lifshitz theory has a family of ground states whose degeneracy is parameterized by the function g.Back to the vacuum expectation value (2.20), we can decompose the functional integration in regions and rewrite it as an integral of the insertion points x i over the product of the O i 's with the path integral propagator (2.29) With a choice of g made for the theory, the kernel K is well defined and one obtains where γ := κc and c := For example, for the solution which satisfies the Neumann boundary conditions at x = x i : we have (2.34) Note that for any given g, the kernel K satisfies the same composition property, where l 12 is given by (2.36) In the following, we will consider the entanglement properties for a general Lifshitz ground state.It turns out that the choice of the ground state only affects the constant term in the entanglement quantities we computed for the Lifshitz theory.
Finally, we remark that in addition to the Lifshitz ground state |Ψ 0 ⟩ which is defined in terms of the position space annihilation operator A(x), one may also consider the Fock vacuum |0⟩ defined by the momentum space annihilation operator a k obtained from the canonical quantization of the theory.Although also a ground state, |0⟩ is different from |Ψ 0 ⟩ since A(x) is given by a nontrivial Bogoliubov transformation of the momentum space operators a k , a † k .Our analysis for the Lifshitz ground state holds valid only for z > 1.For example, the RK vacuum of Lifshitz theory is not defined for z = 1 while the Fock vacuum is defined for z = 1 and is continuous there.We will see below that they display completely different entanglement properties.

Reduced density matrix and replica
We are interested in the entanglement properties of the Lifshitz ground state |Ψ 0 ⟩ of the theory.The density matrix is given by Consider a subsystem as depicted in figure 1, which consists of N intervals A i = (u i , v i ), where u i < v i are the endpoints of the interval A i and i = 1, • • • , N .Let us denote the complementary intervals as ) denotes the coordinate of the left (resp.right) boundary of the total system.The reduced density matrix ρ A = tr B ρ is obtained by tracing over B := N i=0 B i , the complement of A. Explicitly, it has the matrix elements Here ϕ ′ A , ϕ ′′ A are specified by their values over A: (2.40) Evaluating the path integral, we obtain where K(v i , u i+1 ) is the propagator (2.29) with the corresponding boundary values of ϕ specified by β i := ϕ i (v i ) and α i+1 := ϕ i+1 (u i+1 ).We have suppressed the appearance of the boundary values of ϕ in the notation here.As a result, the trace Z 1 := Dϕ A (ρ A ) ϕ A ,ϕ A is given by Similarly the n-th power of the reduced density can be computed and the trace Z n := Dϕ A (ρ n A ) ϕ A ,ϕ A is given by (2.44) Note that we have assumed Dirichlet boundary condition in the above and so there is no integration over the fields β 0 and α N +1 at the boundary of the theory.Such an integration would be needed if free boundary is considered.We remark that the trace Z n of the density matrix has an interpretation as a partition function Z n = Mn Dϕe −S cl over a n-branched covered manifold M n .In CFT, the partition function can be readily computed with the use of twisted operators.However, it is not clear how to introduce the twist operator in non-conformal field theory.Instead, we find that Z n can be determined directly in the Lifshitz theory without using the formalism of twist operators due to the specific RK form of the Lifshitz vacuum.

Entanglement
Quantum information theory has found many applications in our understanding of QFT, condensed matter physics, quantum gravity and so on with a central aim to measure the degree of correlations between different subsystems of a quantum system.Amongst various quantum entanglement measures, entanglement entropy is the most properly understood entanglement measure in different aspects from QFTs to holography.The entanglement entropy associated with a subsystem A when the full bipartite system A ∪ B is specified by a density matrix ρ with total Hilbert space H = H A ⊗ H B is given by S A = −Tr (ρ A log ρ A ) with the reduced density matrix defined as ρ A = Tr B ρ.However, entanglement entropy does not serve as a good entanglement measure for systems with mixed quantum states as it includes contributions from both classical and quantum correlations.Such discrepancy leads us to study other entanglement measures, for instance, mutual information, entanglement negativity [60,61] and the reflected entropy [38].We compute in this section the entanglement entropy, mutual information and the reflected entropy for the massless Lifshitz theory (2.8) for arbitrary z.

Entanglement entropy and mutual information
In this subsection, we apply the formula (2.44) for the trace of the reduced density matrix to determine the entanglement entropy and mutual information for bipartite subsystems in the Lifshitz ground state of the theory.By definition, for a subsystem A in the system, the Rényi entropy is given by where ρ A is the reduced density matrix; and the entanglement entropy can be obtained as The mutual information between two subsystems A, B is defined by By construction, mutual information provides a measure of the correlation between the two subsystems.If the system is pure, then the mutual information is twice the entanglement entropy of the state.
1.A finite subsystem in an infinite system: For a finite subsystem A of length l in an infinite system, the trace Z n is given by Using (2.30), we obtain the Rényi entropy and the entanglement entropy where ϵ is the UV cut-off and c n = log π γ + log n n−1 and c 1 = log π γ +1 are z-dependent constants.Now since A and its complement B = A c form the total system which is in a pure state, the mutual information is equal to twice of S(A).
A few remarks follow: a) We note that the choice of vacuum does not affect the universal UV part, but only the finite part of the entanglement entropy through the constant γ. b) We note that the Rényi entropy and the entanglement entropy share the same universal UV part and differs only in their constant terms.This is different from the usual case of a conformal vacuum where their UV parts are proportional with a nontrivial n-dependent coefficient: UV .This confirm that the Lifshitz vacuum is different from the conformal vacuum.
2. Two adjacent intervals in a finite system: Next, we consider the case of a finite length divided into two adjacent intervals A and B with length l A and l B respectively.Here A further question that naturally arises is whether our theory in the Lifshitz ground state respects the monotonicity theorems along the RG flow, when the c-function is defined via the entanglement entropy.The monotonicity of the c-function along the RG flow is guaranteed by the strong subadditivity in theories with Lorentz symmetry that are unitary.An appropriate c-function candidate for d-dimensional anisotropic theories has been suggested in [41] based on previous developments [62][63][64][65].For the 2-dimensional theory it takes the simple usual form where β is a constant of normalization.Applying this definition of the c-function to any of the entanglement formulas in this section, for example to (3.6), we find that the right monotonicity along the RG flow, which is for c ′ (l) < 0, is guaranteed when z ≥ 1.This constraint matches nicely the condition of having causality in the theory and satisfies maximally the null energy conditions of the holographic dual theory.

Reflected entropy and Markov gap
In The reflected entropy is defines as the von Neumann entropy of the reduced density matrix Markov gap has been proposed as a measure of tripartite entanglement [66].It is defined as the difference between the reflected entropy S R (A : B) and the mutual information As the reflected entropy is lower bounded by the mutual information [38], Markov gap is non-negative.The replica method for reflected entropy considers two replica indices m and n, where the former is related to the purification |ρ m/2 AB ⟩ of ρ m AB for positive even integer m, and the latter denotes the usual Rényi index.The reflected density matrix is then given by ρ (m) and the (m, n)-Rényi reflected entropy is defined as, The reflected entropy is obtained by setting m → 1 followed by another limit n → 1.

System of two adjacent intervals:
Let us first consider a bipartite configuration of a finite system with Dirichlet boundary conditions where the two adjacent subsystems A and B have lengths l A and l B respectively as shown in fig. 2. It is not difficult to obtain the trace Z m,n := Tr ρ ) where the propagator is given by (2.30) and ϕ 1 , ϕ 2 are the fields present at the interface between A and B. Using the expression for the kernel, we obtain the (m, n)-reflected entropy for two adjacent intervals and the reflected entropy Note that the (m, n)-reflected entropies for the Lifshitz ground states are independent of m, and the dependence on n only appears in the finite constant piece.Note also that in general for a bipartite pure state, the reflected entropy becomes twice the entanglement entropy.This is exactly what we find here for (3.25) and (3.8).
As for the Markov gap, it is exactly zero for this specific configuration since the two adjacent subsystems constituting the whole system is a pure state.As a result no tripartite entanglement should be detected from the study of this bipartite state.Markov gap being zero correctly serves as a consistency check of our results.

System with two disjoint intervals:
Next let us consider a bipartite mixed state configuration of two disjoint intervals on a finite system with Dirichlet boundary conditions in the boundary.Here we will consider the disjoint subsystems which include the boundaries of the whole system on an interval although it can be generalized for any two intervals at arbitrary positions.This configuration is illustrated in fig. 3. We obtain the trace similar to [40] as where ϕ i and ϕ i+1 are the fields present at the junction point of B 1 ∪ B 2 and A and the kernel follows the same form presented in the calculation for adjacent intervals.Using the kernel (2.30), we obtain where M m,n is the 2n × 2n matrix which takes the same form as [40] M and a = 2 1 + . The determinant is given by where and η(z) is the cross ratio (3.16).As a result, the Rényi reflected entropy is Taking the two consecutive limits m → 1 and n → 1, we obtain the expression for the reflected entropy between two disjoint intervals as follows, As for the Markov gap, it is where η is the cross-ratio.Here we observe a peculiar characteristics of η and the h(B 1 : B 2 ) with the variation of z depending on the sizes of the intervals.For l A < min{l B 1 , l B 2 } and l A = min{l B 1 , l B 2 }, η increases and saturates at the values 1 and 1 2 respectively.Whereas for l A > min{l B 1 , l B 2 }, η shows a decreasing characteristics followed by an brief initial increasing phase and finally converges to 0. Similar to the cross-ratio, the Markov gap also show distinctive behavior depending on the sizes of the subsystem.In fig.4, we have plotted the Markov gaps with increasing z.For l A ≤ min{l B 1 , l B 2 }, h(B 1 : B 2 ) increases up to a constant value whereas for l A > min{l B 1 , l B 2 }, h(B 1 : B 2 ) decays to zero.From this observation, we conclude that with increasing degrees of anisotropy of the Lifshitz field theory, the tripartite entanglement can be enhanced or completely destroyed depending on the sizes of the partitions.

Holography
In this section, we study the holographic dual representation of the Lifshitz field theory.
Lifshitz holography [42] has been discussed extensively in the literature, see for example [44] for review.We will use our field theory result for the entanglement entropy to fix the z-dependence of the radius scale of the bulk Lifshitz background.And using that, we make a prediction for the time-like entanglement entropy in Lifshitz field theory.

z-dependent Lifshitz radius
The standard form of the (2+1)-dimensions Lifshitz metric with one-direction anisotropy is given by The above metric is not Lorentz-invariant and supports non-relativistic Lifshitz translational invariance given by t → λ z t, x → λx, r → λr, that is consistent with the Lifshitz symmetry (1.2) for the field theory.This metric appears as solution of the equations of motion of the bulk action given by [43] where the usual Einstein's gravity theory is deformed by the inclusion of a massive vector field given as computation entirely in the Poincaré patch.This can indeed be done straight forwardly as we will show now (with the AdS case covered by z = 1 in our computation below).Consider a time interval A in the two dimensional Lifshitz field theory described by A ≡ [−T, T ].The geodesics needed for computing the holographic TEE of A consists of two space-like geodesics connecting the endpoints of A and infinities plus a time-like geodesic which connects the endpoints of two space-like geodesics.With the Lifshitz geometry described in the Poincaré form (4.1), the equation for the geodesic on t − r plane is given by where c is an integration constant.Now, depending on value of c 2 , we get two classes of geodesics: where T and R are constants.Note that the curve (4.9a) is space-like and describes a geodesics extending from the end point t = T (or t = −T ) of the time-like interval to infinity.The curve (4.9b) is time-like and, for any value of R > 0, can be used to smoothly join the former geodesics at infinity.The geometry is shown in figure 5. Let us denote by L 1 the length of the space-like geodesic from the endpoint t = T to infinity, and by L 2 half the length of the above mentioned connecting time-like geodesic.Then and Utilizing the RT formula, the time-like entanglement entropy may be obtained as where we have used the relation given in (4.5).We remark that in Lorentz invariant field theory, it has been proposed to define timelike entanglement entropy in terms of an analytic continuation based on the Wick rotation.In the present case, the result (4.12) can be obtained by formally replacing in (3.6) the time interval l with the temporal interval 2T as l → (i2T ) 1/z . (4.13) The replacement (4.13) is consistent with the Lifshitz symmetry (1.2).However, due to the presence of fractional derivative, it is not clear how to implement it on the Lifshitz action as a Wick rotation.Nevertheless, the existence of a holographic time-like entanglement entropy in the Lifshitz case suggests that there should be a general definition of time-like entanglement entropy in field theory.It is interesting to understand better how time like entanglement can be defined from first principle and to study its physical meaning.

Summary and discussion
In this paper, we have employed fractional derivative to propose a definition of the massless Lifshitz theory with arbitrary z in general (d + 1)-dimensions.In (1+1)-dimensions, the massless Lifshitz theory admits a Lifshitz scaling invariant ground state which takes on the form of RK.Unlike the usual vacuum in conformal field theory which is defined by the annihilation operators in the momentum space, the Lifshitz ground state is defined by annihilation operators in the coordinate space.As a result, we showed that there is a 2d/1d correspondence between the (1+1)-dimensional Lifshitz field theory and a dual quantum mechanical system defined with a fractional derivative.In order for the path integral of the dual quantum system to be well-defined, a choice of classical solution is needed to be made.This can be interpreted as that the Lifshitz theory actually admit a family of ground states whose degeneracy is parameterized by the choice of the classical solution.
We then computed various bipartite and tripartite entanglement measures in the Lifshitz theory and determined their z-dependence respectively.The entanglement measures can be expressed in terms of simple cross ratios that depend on the lengths of the intervals and the exponent z.Moreover, the standard c-function monotonicity defined via the entanglement entropy along the RG flow, is satisfied for the range of z that is compatible to causality and maximally satisfies the null energy conditions.Finally, we considered a gravity dual corresponding to the Lifshitz vacuum of the Lifshitz field theory.We showed that in order to reproduce the field theory result for the entanglement entropy, the previously considered Lifshitz bulk geometry has to be supplemented by a Lifshitz radius scale that is dependent on z.Using the dual geometry, we studied and computed the time-like entanglement entropy for a time-like subsystem.In Lorentzian theory, the time-like entanglement entropy can be obtained from the space-like one via an analytic continuation.In Lifshitz theory, space and time are different due to their different scaling, and a standard form of Wick rotation that mixes space and time cannot be introduced.Our holographic result suggests that there should be a field theoretic definition of time-like entanglement entropy that does not employ analytic continuation.It is an interesting question for further investigation.
The entanglement observables in the massless Lifshitz scalar theory have been computed in this paper using field theory methods.For the entanglement entropy of an interval in an infinite system, the result can be readily reproduced from holography provided that the mapping relation (4.5) is implemented.The situation is more complicated for entanglement observables defined in a finite system.For CFT, the dual geometry involves a global AdS, suggesting that the global completion of the Poincaré patch of the Lifshitz geometry (4.1) for z ̸ = 1 is needed.This is however quite a non-trivial open problem.We will leave the holographic entanglement computation for massless Lifshitz theory on finite system for future analysis.
There are many interesting directions to follow from our work in this paper.Here, we were confined to computing various entanglement measures in the zero temperature case.It will be interesting to extend our analysis to non-zero temperature in the context of Lifshitz holography which still remains a non-trivial issue.The study of time evolution of entanglement measures in Lifshitz field theory as a function of z is also interesting.Our work is expected to bring new insights in the studies of quantum critical phenomenon in condensed matter systems exhibiting Lifshitz scaling.It is interesting to consider boundary Lifshitz field theory (BLFT) and its holography by generalizing previous AdS/BCFT formulations with Neumann boundary condition [67,68], conformal boundary condition and Dirichlet boundary conditions [69][70][71][72].It is also interesting to analyze the phase structure of timelike entanglement entropy in the aforementioned BLFT similar to [54].We hope to return to these exciting issues in the near future.ϕ(x) = C dq a(q)e iqx for some contour C and some coefficient a(q).It is important to note that for arbitrary β and z, the integrand of ∇ z x ϕ has a branch point singularity at the origin q = 0 and so one has to be careful in finding the right choice of C so that a solution ϕ can be obtained.A naive choice of C over the real axis would not work.With an appropriate choice of contour, one can show that where C is the countor −∞ < q < −ϵ, ϵ < q < ∞ with ϵ → 0. It follows immediately from the definition (A.2) that Now, it is easy to perform a contour deformation and evaluate the line integral in (A.2), see figure 6. Taking into account of the branch cut, we obtain immediately that for β > 0, x β Γ(β+1) , x ̸ = 0 0, x < 0.

Figure 1 :
Figure 1: Schematic for the configuration consisting of intervals A and B.

30 h (B 1 : B 2 )Figure 4 :
Figure4: The Markov gap as function of z for two disjoint intervals B1 and B2.It depends on the relative size of the separation interval A in comparison with the minimum length of B1 and B2.When lA is not the minimum length of the three, the Markov gap, approaches to zero for large z (green curve), otherwise it saturates to a finite value.

Figure 5 :
Figure 5: Geodesics for holographic time-like entanglement entropy computed in the Poincaré patch.The two mirroring curves that reach the boundary provide the real contribution, while the other bulk curve provides the imaginary part of the time-like entanglement entropy.
− z + 1) x β−z for x > 0 and for any real β, z.(A.1)To see this, let us consider the function defined byH β (x)

(A. 4 )Figure 6 :
Figure 6: Contour deformation for x > 0 (left) and x < 0 (right).The line integral on CR vanishes for β > 0 as R → ∞.The dotted green line denotes the branch cut for w z , w ∈ C.
general, given a mixed state density matrix ρ on a Hilbert space H R of finite dimension.It is always possible to purify ρ in the sense that one can always construct a second Hilbert space H S and a pure state |ψ⟩ such that ρ is given by the partial trace of |ψ⟩ ⟨ψ| with respect to H S .Now, for a bipartite system in an arbitrary mixed state ρ AB on a finite Hilbert space H A ⊗ H B , there is a canonical purification defined by a pure state | √ ρ AB ⟩ in a doubled Hilbert space H A ⊗ H B ⊗ H A * ⊗ H B * where A * and B * are dual copies of A and B respectively such that ρ AB = Tr A * B * (| √ ρ AB ⟩⟨ √ ρ AB |).(3.18)