NUPHB14499S0550-3213(18)30328-610.1016/j.nuclphysb.2018.11.012The AuthorsHigh Energy Physics – TheoryEntanglement entropy at higher orders for the states of a = 3 θ = 1 Lifshitz theoryRohitMishraabHarvendraSinghab⁎h.singh@saha.ac.inaTheory Division, Saha Institute of Nuclear Physics, 1/AF Bidhannagar, Kolkata 700064, IndiaTheory DivisionSaha Institute of Nuclear Physics1/AF BidhannagarKolkata700064IndiabHomi Bhabha National Institute, Anushakti Nagar, Mumbai 400094, IndiaHomi Bhabha National InstituteAnushakti NagarMumbai400094India⁎Corresponding author.Editor: Leonardo RastelliAbstractWe evaluate the entanglement entropy of strips for boosted D3-black-branes compactified along the lightcone coordinate. The bulk theory describes 3-dimensional a=3 θ=1 Lifshitz theory on the boundary. The area of small strips is evaluated perturbatively up to second order, where the leading term has a logarithmic dependence on strip width l, whereas entropy of the excitations is found to be proportional to l4. The entanglement temperature falls off as 1/l3 on expected lines. The size of the subsystem has to be bigger than typical ‘Lifshitz scale’ in the theory. At second order, the redefinition of temperature (or strip width) is required so as to meaningfully describe the entropy corrections in the form of a first law of entanglement thermodynamics.1IntroductionThe AdS/CFT correspondence [1] has remained a central idea for holographic studies in string theory. The holography relates conformal field theory living on the boundary of anti-de Sitter spacetime with the gravity theory within the bulk. Along these lines finding the entanglement entropy of strongly coupled quantum systems at criticality has also been a focus of several studies [2,4]. In these calculations entanglement entropy can be obtained [2,3] by estimating the area of codimension two surfaces embedded inside the bulk geometry. The boundary of such extremal surfaces coincides with the boundary of the subsystem in the CFT. Recently it has been observed that the excitations in the CFT follow entanglement laws similar to the black hole thermodynamic laws [5–7]; see also [9], [12], [10], [13]. It is understood now that the entanglement entropy (SE) and the energy of small excitations (E) in AdS spacetime obey a definite relation△E=TE△SE+V△P+μE△N This equation is described as the first law of entanglement thermodynamics. The charge contributions can simply arise for a boosted black-brane vacua [7], where the charged excitations could be either Kaluza–Klein (KK) momentum modes along a compactified brane direction or the dual winding modes of a string.The backgrounds of our interest here are the nonrelativistic Lifshitz spacetimes. We would like to holographically study these solutions and check if similar entanglement law could be written for them. Typically a Lifshitz like geometry [14] has a line elementds2=−dt2z2a+dx12+⋯+dxD2z2+dz2z2 as subspace where time and space scale asymmetrically (z→λz,t→λat,xi→λxi). The Lorentz symmetry is explicitly broken. The parameter a is called the dynamical exponent of time and D is total spatial dimensions. Whereas a hyperscaling violating Lifshitz (hvLif) geometryds2=zθ(−dt2z2a+dx12+⋯+dxD2z2+dz2z2) has an overall conformal factor [4]. So an overall scaling gμν→λθgμν of the metric is involved. As an unique example in ten dimensions, especially Lif4a=2×S1×S5 vacua are recently constructed in [15], as solutions of 10-dimensional massive type IIA supergravity theory [16]. These are understood to describe strongly coupled Lifshitz a=2 theory in three spacetime dimensions at the fixed point. In these bulk solutions ‘massive’ strings are tied up with D2–D8 parallel brane system that exhibit scaling symmetry. These vacua can be related via massive/generalized T-duality [17] to D3–D7 axion ‘flux’ vacua [18]. In the ordinary type IIA/B string theory and M-theory, the boosted black brane solutions compactified along a lightcone direction, can also give rise to Lifshitz solutions [19,21]. The latter class of brane solutions all have conformal scaling (or hyperscaling) properties as listed in [21], see [22] for discussion of θ=1 case. There are other instances also in gauged supergravities where Lifshitz vacua can be obtained, see for example [25].In this work we shall only study boosted black D3-brane system in lightcone coordinates, with one lightcone coordinate compactified on a circle [28]. These compactified bulk solutions describe a thermal state of 3-dimensional Lifa=3 theory with hyperscaling parameter θ=1. The corresponding ground state (at zero temperature) is described by ten-dimensional solutions discussed in [19]. These zero temperature solutions were obtained by performing a double scaling limit in which horizon size is taken to vanishing value associated with an infinite boost. The fact that there is hyperscaling violation in these Lifshitz solutions (when explicit lightcone compactification is performed) was pointed out subsequently in the work [22]. Actually these hvLif theories fall in a special category where θ=D−1 (D=2 is number of spatial dimensions). Precisely for these hvLif states the entanglement entropy has logarithmic violation [4,23,24]. We shall be encountering some of these results in our work as we progress. Both the solutions (finite temperature and the zero temperature one) allow us to embed codimension-2 strip like surfaces (at constant lightcone time) inside the bulk. We evaluate the area of small strips using perturbative method up to second order, by using the procedure introduced in [7]. We find that for small strips (but bigger than some critical size) the leading term in the entanglement entropy has indeed logarithmic dependence on strip width l. Whereas the subleading term which accounts for the entropy of excitations goes as l4. The entanglement temperature falls off as 1l3. These results are on expected lines. Quite importantly, the (KK) charge contribution in the first law is present at the first order itself, unlike in the relativistic cases studied in [7] where the contribution of the charges appears only at the second order. At the second order, once again we find that the first law relation requires the entanglement temperature (and strip width) to be suitably corrected or renormalized.The paper is presented as follows. In section-2 we write down the Lifa=3,θ=1 hyperscaling solutions of our interest both with black hole excitations and the zero temperature counterpart. In section-3 we evaluate the entanglement entropy at first order and present the form of first law. In section-4 we obtain second order corrections and rewrite the new form of first law and determine the corrections to associated thermodynamic quantities. We find that the strip width (so also subsystem volume) has to be renormalized and redefined. The final summary is presented in section-5.2Entanglement entropy for a=3 θ=1 Lifshitz systemIt has been known that the boosted AdS5×S5 black hole background compactified along a light-cone coordinate can describe excitations of hvLifa=3 system [19,21]. These black hole solutions were first explored for their non-relativistic properties in [28]. These type IIB string vacua can be written as(1)ds2=L2(−zl4fz6(dx+)2+z24zl4(dx−−ω)2+dx12+dx22z2+dz2fz2)+L2dΩ52, supported by constant dilaton and a self-dual 5-form field strength. The function f is(2)f(z)=1−z4z04, where z=z0 is the black hole horizon. It will be assumed that x− is compactified on a circle of radius r−. The fiber 1-form is given by(3)ω=zl4z4(2−z4z04)dx+ The radius of curvature L is taken very large in string units (α′=1) so that the stringy excitations are suppressed. The parameter zl is an intermediate (free) UV scale, rather we shall suitably call it as ‘Lifshitz scale’ in the theory. We take a wide parameter range such that z0≫zl. This is so because we wish to study small excitations only. (Also let us note that at any stage the Lifshitz scale zl can be related to z0 through the boost of lightcone coordinates, i.e. one can write zl2=z02/λ, with λ≥1 being the lightcone boost parameter.)Further we shall take the a=3 θ=1 Lifshitz solutions [19] as the ground state. Let us explain it here. Recalling [19], one can take simultaneous double limits λ→∞,z0→∞, while keeping the ratio λz02=1zl2 (say) fixed. These limits take us to (hyperscaling) Lifshitz a=3 vacua, namely(4)dsLif2=L2(−zl4(dx+)2z6+z24zl4(dx−−2zl4z4dx+)2+dx12+dx22+dz2z2+dΩ52)≡L2(−dx+dx−z2+z24zl4(dx−)2+dx12+dx22+dz2z2+dΩ52) This zero temperature background is characterized by the scale zl, which also defines the charge (number) density of the states of this system at zero temperature [19]. It only suggests that a=3 hyperscaling Lifshitz ground state system exists for any given zl. The zl is treated as Lifshitz (intermediate) scale in the black hole solution (1), when we switch on the temperature. That is we are interested in the excitations around the hyperscaling Lifshitz vacua (4).The entanglement entropy has also been studied for these BH systems in the work [26]. It was pointed out there that due to the boost there is an asymmetry in the entanglement along various directions of the boundary theory. This asymmetry should show up in the entanglement entropy calculations and the first law as well. Up to first order in perturbative expansion (for small subsytem) it has been explicitly shown that it indeed is the case [8]. The observation was that the entanglement asymmetry is entirely due to pressure asymmetry in the theory. But in the current paper we shall be studying only those strip systems which lie in the transverse to the boost direction, it is because the solutions are compactified along the boosted lightcone coordinate, so that we can view it as a hyperscaling Lifshitz theory. The entanglement entropy along the boost direction can also be studied at higher orders, but this will require some careful considerations, because the constant time slices will not be existing. Technically one has to resort to a covariant slice analysis [3]. Up to first order the calculations are all easy and there are no such hurdles. But beyond first order we have to only consider covariant approach such as [3]. We leave it for a separate investigation.2.1Small strip systemsThe entanglement entropy for a subsystem on the boundary of the background (1) can be studied by using Ryu–Takayanagi proposal [2]. Here x1 and x2 are two flat directions along the brane, while spatial lightcone coordinate x− is compactified. We choose a strip along x1 direction with an interval −l/2≤x1≤l/2. We wish to embed co-dimension two strip (a constant x+ surface) inside the bulk geometry. The two straight boundaries of the extremal strip surface coincide with the two ends of the interval △x1=l. The size of other coordinates are taken as; x−≃x−+2πr−, 0≤x2≤l2, where l2 is taken very large, l2≫l.Following Ryu–Takayanagi prescription the entanglement entropy of a strip subsystem is given in terms of the geometrical area of co-dimension two surface (with light-cone time x+ taken constant everywhere on the surface). We thus get(5)SE≡AStrip4G5=L3πr−l22G5zl2∫ϵz⁎dz1zf1+f(∂zx1)2 where G5 is 5-dimensional Newton's constant. Here ϵ∼0, is the cut-off scale in UV. (We need to pay special attention to z=zl scale in the theory. The bulk geometry (1) is not well defined beyond z=zl, as the size of the x− circle becomes sub-stringy in z<zl near boundary region. A way to overcome this problem is that beyond z=zl one can switch over to T-dual type-IIA background, where the circle size instead will increase. Doing this however does not affect the entropy functional given in (5). Hence so far as the area functional is concerned it appears immune to z=zl. Nevertheless zl is an important scale in the Lifshitz theory and we can add suitable counter terms as we shall discuss next.) The z⁎ is the turning point of the strip. Next the area functional is extremized through the equation of motion(6)dx1dz=zz⁎f11−(zz⁎)2 It implies that the boundary value x1(0)=l/2 is given by the following integral relation(7)l2=∫0z⁎dzzz⁎1f1−(zz⁎)2 which relates the width l with the turning point z⁎. The turning-point of the strip lies at the mid-point x1(z⁎)=0 of the boundary interval due to symmetry. The entropy for extremal strip system can now be described as(8)SE=L3πr−l22G5zl2∫ϵz⁎dzz1f11−(zz⁎)2 The Lifshitz scale zl is an important fixed parameter in these vacua, but it only appears as a constant multiplier outside the integrand.2.2A hierarchy of scales and perturbative expansionWhen strip width l is small the turning point generically lies in the proximity of asymptotic region. Therefore one can safely assume z⁎≪z0. However, our main focus here will be on those subsystems (or critical surfaces) for which following hierarchy of scales is obeyed(9)zl<z⁎≪z0. This will specially require us to take zl (UV) and z0 (IR) to be widely separated scales. This [z0,zl] interval is known as the ‘Lifshitz window’ region in [20,21]. A large Lifshitz window is desirable here for perturbative expansion to work out properly, as we are seeking to evaluate the entanglement entropy (8) by expanding it around a zero temperature vacua (4) (i.e. treating a=3 θ=1 Lifshitz vacua [19] as the ground state). Under these conditions we can estimate area entropy perturbatively by expanding the integrand around its central value.We first proceed to obtain the perturbative expansion of the l-integral (7) up to first order, assuming z⁎4z04≪1,(10)l=2z⁎∫01dξξ1−ξ2(1+z⁎42z04ξ4+⋯)=2z⁎b0+z⁎5z04b1+⋯ where for simplicity we introduced ξ≡zz⁎ and R≡(1−ξ2).11The value of expansion coefficients b0,b1,b2 can be evaluated, b0=∫01dξξR=12B(1,1/2)=1, b1=∫01dξξ5R=12B(3,1/2)=815, b2=∫01dξξ9R=12B(5,1/2), where B(m,n)=Γ(m)Γ(n)Γ(m+n) are the Beta-functions. The ellipses stand for second and higher order terms which we neglect in this section. By inverting the above series we get a turning point expansion(11)z⁎=z¯⁎(1−z¯⁎4z04b12b0)+⋯ where z¯⁎≡l2b0 is the turning point for pure hvLif ground state (4) (i.e. in the absence of excitations or black holes). This relationship is an important first step before we proceed to the area calculation.Next we consider the area of the strip (8). We evaluate the integral quantity (which is independent of zl)(12)A≡2zl2∫ϵz⁎dzz1f1−(zz⁎)2 by expanding the integrand perturbatively as(13)A≡2zl2(∫ϵ/z⁎1dξξ1R+z⁎42z04∫ϵ/z⁎1dξξ3R+⋯). The contribution of first term is singular when ϵ→0 (near the boundary). Also as mentioned before, going beyond z=zl, the x− circle in (1) becomes sub-stringy, so near boundary region z<zl needs to be carefully considered. We thus note that, in the corresponding dual geometry the size of T-dual x− circle will anyway expand for z<zl. While the functional form of integral in eq. (12) remains unchanged under this duality. Thus there appears to be no pathological problem in the near boundary region 0<z<zl. Nevertheless, to be on the safe side we subtract the following contribution (as a counter term)(14)ACT=2zl2∫ϵzldzz=2zl2lnzlϵ from the area integral A given above. This precisely amounts to subtracting the contribution of two disconnected (no turning point) strips hanging between z=zl and the z=ϵ inside the hvLif geometry (4). Note that ACT has no dependence on z0, which is a parameter controlling the excitations. So it is totally a harmless subtraction from point of view of the excitations (our goal is to know the entropy of the excitations and it will not be affected). So we extract the finite area contribution as(15)Afinite=A−ACT=2zl2(∫ϵ/z⁎1dξξ1R+z⁎42z04∫ϵ/z⁎1dξξ3R+⋯)−2zl2∫ϵ/z⁎zl/z⁎dξξ=2zl2ln2z⁎zl+1zl2z⁎4z04∫01dξξ3R+⋯ and the limit ϵ→0 is understood to have been implemented in the second equality. In the next step by substituting the expansion of z⁎ in the eq. (15), we get up to first order(16)Afinite=2zl2ln2z¯⁎zl−2zl2z¯⁎4z04b12+2zl2z¯⁎4z04a12=2zl2lnlzl+1zl2z¯⁎4z04(a1−b1)=A0+1zl2z¯⁎4z04a15 where a1,b1,⋯ are finite coefficients.22The expansion coefficients are a1=∫01dξξ31−ξ2=12B(2,1/2)=23, a2=∫01dξξ71−ξ2=12B(4,1/2)=67b1. The leading finite term is simply given by(17)A0=1zl2lnl2zl2 Thus the entanglement entropy for strip can be written as(18)SE=S(0)+L3πr−l24G5a15zl2z¯⁎4z04 where the leading term is(19)S(0)=L3πr−l24G51zl2lnl2zl2 It is clear that S(0)>0 only when l>zl and that is why the hierarchy of the scales (9) was adopted. It also does not look like an AdS5 ground state entropy which instead goes as −1l2 [2]. Therefore the logarithmic dependence on l ought to be recognized as a contribution of a=3, θ=1 Lifshitz ground state [4,23]. This also happens because we have chosen to study x+=constant strip subsystems. Had we chosen to evaluate entanglement entropy for usual x0=constant (fixed Lorentzian time) strips, we instead would get the leading contribution precisely that for AdS5; see [7] for a second order perturbative calculation in relativistic theory.The leading logarithmic term depends on zl (UV scale) and the width l(>zl), and not on z0 (the scale describing the excitations). But both of these quantities are fixed for a given subsystem. Thus S(0) is essentially a fixed quantity and it cannot be viewed as part of the excitations. Subtracting the leading term leaves us with the net vacuum-subtracted entropy of the excitations around Lifshitz theory as(20)△SE(1)=L3πr−l24G5a15zl2l4(2z0)4+higher order corrections This result is true up to first order in the ratio z¯⁎4z04. At higher order there will be further corrections on the right hand side to add. It can be immediately observed that the entanglement entropy of excitations is proportional to l4 and depends on z0 also, the parameter describing excitations. In contrast, for Lorentz covariant AdSd+1 ground state the entropy of excitations rather increases quadratically as l2 [5].3The entanglement first lawThe boundary theory is a 3-dimensional Lifshitz theory, since the lightcone direction, namely x−, is compactified. The excitation energy and the pressure can be obtained by expanding the geometry (1) in Fefferman–Graham coordinates near the boundary [27]. The energy density of the excitations is given by [28](21)E=L3r−16G51z04, whereas the charge (number) density is(22)ρ=Nvolume=L3r−28G51zl4 The charge density in the Lifshitz theory at zero temperature is usually very large whereas other quantities can be vanishingly small [19]. It is obvious here too as ρ∝1zl4 and given our hierarchy of the scales zl<z⁎≪z0. The 'entanglement' chemical potential, obtained by measuring the value of KK field ω+ at the turning point, is(23)μE=1r−(2zl4z⁎4−zl4z04)=1r−(2zl4z¯⁎4+(4b1b0−1)zl4z04)=μELif+1r−(4b1b0−1)zl4z04 where to obtain second equality the turning point expansion (11) has been used. The leading term μELif=1r−2zl4z¯⁎4 is chemical potential corresponding to the hvLif ground state (4). The subleading term is however of universal nature, because it is independent of l. Thus the net change in chemical potential due to excitations is(24)△μE(1)=μE−μELif≃1r−(4b1b0−1)zl4z04 It is remarkable that, using the quantities defined so far, from (20) we can construct the following first law-like relation(25)△SE(1)=1TE(△E+12N△μE(1)) where the net charge contained in the subsystem is simply N=ρl2l and energy of excitations △E=l2lE. The entanglement temperature is given by(26)TE=26zl2π1l3. Importantly the temperature is inversely proportional to the cubic power of the strip width l. This conveys the fact that the dynamical exponent of time for the Lifshitz theory is indeed three, and it collaborates with early work [19].We add some remarks here. In the first law (25) the charge and chemical potential contribution is present at the first order itself, unlike in the relativistic case where no charge appears at the first order. In the relativistic case the charges appeared only at the second order in perturbation, see [7]. The reason for this major difference may be the fact that the charge density is very high in the Lifshitz theory, i.e.ρρc=z04zl4≫1, where ρc=L3r−28G51z02 is some critical (reference) charge density.The von Neumann entanglement entropy SE=−TrσAlnσA of a quantum subsystem A requires the knowledge of a reduced density matrix (obtained by tracing out the states over the complimentary system),(27)σA=e−HAZ where partition function Z=TrAe−HA. The HA is the reduced Hamiltonian describing the subsystem. In this approach, at first order we expect that the modular (entanglement) Hamiltonian HE of the subsystem to be related as, [11,12],(28)△SE(1)=1TE(△E+12N△μE)=<△HE(1)>.A variational form of first law The small fluctuations of bulk parameters (z0,zl) determine the variations of the thermodynamic quantities of boundary nonrelativistic theory. In the present hvLifa=3 case we are interested in the study of the ensembles with fixed KK charges which can only be done by keeping zl fixed, so we will only allow z0 to have a spread. The small variation of chemical potential becomes (at first order)(29)δμE=(4b1−1)zl4r−δ(1z04) as given b0=1. One can see that the product ρ.δμE is of the same order as δE and thus it will eventually be related to it. Hence the states of the system describe a canonical ensemble and therefore knowing the fluctuations of a single quantity, such as δE, is sufficient to describe the state of the system. Under these restrictions (since δS(0)=0 as zl is fixed) we find from eq. (18) the variational form of first law is(30)δSE=1TEδE′ where new energy E′≡E+12μEN has been defined so that the entanglement temperature is the same as (26). For a comparison with the black hole first law, we wish to recall the thermal first law [28] for boosted BH background, which for fixed charge density (zl=fixed), gets reduced to(31)δSth=1TthδE where(32)Tth=zl2πz03 is thermal temperature. It is worthwhile to note that not only the zl dependence in entanglement temperature is exactly the same as that in thermal temperature but the dynamical exponent of time also comes out as 3. Usually for smaller subsystems the entanglement temperature is higher than compared to the thermal one (if any). It is appropriate to compare the two in the present Lifshitz case. The ratio comes out to be(33)TthTE=18(l2z0)3≪1 Since l≪2z0, there will exist a big hierarchy in two temperature scales where the degree is determined by the dynamical exponent of time. Though the ratio remains independent of zl (or charge density of Lifshitz states), but it crucially depends on the value of dynamical exponent, which obviously enhances this hierarchy.4The entropy at second orderWe wish to evaluate the area of the lightcone strip up to next higher order and evaluate corresponding entropy corrections. The higher order results provide us with better precision and improved estimate of the entanglement entropy since exact analytical calculations cannot be done. First the expansion of the turning point has to be obtained. We expand the integrand in eq. (7) up to second order in z⁎4z04≪1, which gives us the series(34)l=2z⁎∫01dξξ1−ξ2(1+z⁎42z04ξ4+3z⁎88z08ξ8+higherorders)=2z⁎(1+z⁎4z04b12+z⁎8z083b28)+⋯ where the ellipses stand for third and higher order terms. By inverting the above expansion one can obtain(35)z⁎=z¯⁎(1+z¯⁎4z04b12+z¯⁎8z08(38b2−b12))−1 where z¯⁎=l/2 is the turning point value for the ground state. The A expansion up to second order is, keeping the counter term same as in the previous section,(36)A−ACT=2zl2(∫ϵ/z⁎1dξξ1R+z⁎42z04∫ϵ/z⁎1dξξ3R+3z⁎88z08∫ϵ/z⁎1dξξ7R+⋯)−2zl2∫ϵ/z⁎zl/z⁎dξξ=2zl2(ln2z⁎zl+z⁎42z04∫01dξξ3R+3z⁎88z08∫01dξξ7R+⋯)=2zl2ln2z⁎zl+1zl2z⁎4z04a1+34zl2z⁎8z08a2 where ϵ→0 limit has been implemented. The coefficients a1,a2 are defined earlier. Substituting the z⁎-expansion (35) in the A expansion (36), we finally get(37)Afinite=A0+A1+A2 where A0 and A1 are the leading order and first order terms, respectively. These are the same as obtained in the previous section. The new contribution at second order is(38)A2≡14zl2(9b12−8a1b1−3(b2−a2))z¯⁎8z08 With this the entanglement entropy calculated up to second order becomes(39)SE=S(0)+L3πr−l24G5(A1+A2). So overall change up to second order is(40)△SE(2)=SE−S(0)=L3πr−l24G5⋅a1Q5zl2⋅l424z04 where Q factor is defined as(41)Q=(1−26105z¯⁎4z04)<1 which involves first order term only. It is always smaller than unity. This in fact implies that overall entanglement entropy after the inclusion of second order corrections has indeed decreased,(42)△SE(2)<△SE(1). This is a common observation in many CFTs including the relativistic ones. This calculation ends our perturbative results up to second order. In the next step we would like to see if the second order corrections can be absorbed in the redefinitions of various entanglement quantities like TE and μE.4.1Renormalization of thermodynamic observablesAs we have seen that entanglement entropy of the strip system gets corrected at higher orders in perturbative calculation. It is reasonable to expect that other thermodynamic variables also receive similar corrections at higher orders. We already saw that the chemical potential μE indeed gets corrected. Using second order turning point expansion (35), one can determine(43)μE=1r−(2zl4z⁎4−zl4z04)≃1r−(2zl4z¯⁎4+zl4z04[(4b1b0−1)+z¯⁎4z04(3b2−5b12)])=μELif+1r−(4b1Z−1)zl4z04 where Z=1+z¯⁎4z04(3b2−5b12)4b1. Thus the net difference in chemical potential due to excitations(44)△μE(2)=1r−(4b1Z−1)zl4z04 has second order terms also. It can be seen that entropy expression (40) can be reexpressed as a first law(45)△SE(2)=1TE(△ER+12NR△μE(2)) The corresponding entanglement temperature is given by(46)TE=64zl2π(lR)3≃64zl2πl31(1−135l4(2z0)4) which involves cubic power of renormalized length. Similarly the ‘renormalized’ energy and charge within subsystem are also given in terms of lR(47)△ER=l2lRE,NR=l2lRρ. In the above ‘renormalized’ width of the strip is defined as(48)lR≡l(QZ)14≃lQ˜ where Q˜=1−1105z¯⁎4z04. The new subsystem width lR includes terms only up to first order z¯⁎4z04. The result also suggests lR<l, which is consistent with the fact that at the second order overall entanglement entropy decreases.In summary, in this universal approach all thermodynamic (extensive) quantities describing subsystem are assumed to be dependent on the renormalized width through the volume factor. It is also hypothesized that in terms of the corrected quantities the entanglement first law should hold true at each order, like our approach in [7]. We also speculate that at the second order a new modular Hamiltonian can be inferred as(49)△SE(2)=<△HE(2)>.5SummaryWe calculated the entanglement entropy of a strip like subsystem on the boundary of 10-dimensional boosted black 3-brane solutions. These solutions when compactified along a lightcone coordinate describe excitations of 3-dimensional a=3 θ=1 hyperscaling Lifshitz theory, at fixed charge (momentum) density. The theory has a natural scale zl, determined by the charge density. The area of the strip geometry for constant ‘lightcone time’ is evaluated perturbatively up to second order. The ‘finite’ contribution of the a=3 θ=1 Lifshitz ground state is found to beS(0)=L3πr−l24G5zl2lnl2zl2 where an allowed range is l>2zl. Due to the zl dependence, this entropy is qualitatively different from the 2D CFT entropy which behaves as ∼ln(l/ϵ) (ϵ being UV cutoff) [2].The entanglement entropy of the excitations is however found to be proportional to l4, whereas the entanglement temperature falls off as 1l3. These results are essentially along expected lines, indicating that the dynamical exponent of time for the hyperscaling Lifshitz background is three. Notably these results are distinct when compared with the relativistic counterpart where the entanglement entropy of excitations instead grows as l2, and temperature goes as 1l, at first order [5]. A renormalization of entanglement width is proposed at second order when we try to write down first law of thermodynamics△SE(2)=1TE(△ER+12NR△μE(2)) This conclusion falls along the lines with the hypothesis invoked in an earlier work [7] for the relativistic case. 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