aa r X i v : . [ h e p - t h ] S e p UK/09-04
C P N − Models at a Lifshitz Point
Sumit R. Das and Ganpathy Murthy
Department of Physics and Astronomy,University of Kentucky, Lexington, KY 40506
USA b [email protected], [email protected] Abstract
We consider CP N − models in d + 1 dimensions around Lifshitz fixed points with dynamicalcritical exponent z , in the large-N expansion. It is shown that these models are asymptociallyfree and dynamically generate a mass for the CP N − fields for all d = z . We demonstratethat, for z = d = 2, the initially nondynamical gauge field acquires kinetic terms in a waysimilar to usual CP N − models in 1 + 1 dimensions. Lorentz invariance emerges generically inthe low-energy electrodynamics, with a nontrivial dielectric constant given by the inverse massgap and a magnetic permeability which has a logarithmic dependence on scale. At a specialmulticritical point, the low-energy electrodynamics also has z = 2, and an essentially singulardependence of the effective action on B = ǫ ij ∂ i A j .onlinear sigma models are ubiquitous in a variety of areas in theoretical physics. In thispaper we will deal with the CP N − model [1], whose fields are N component complex vectors ~φ ( t, x ) constrained by ~φ ⋆ · ~φ = 1 g (1)and fields which differ by an overall (space-time dependent) phase are identified, ~φ ( t, x ) ∼ e iθ ( t,x ) ~φ ( t, x ) (2)The identification is incorporated by introducing a non-dynamical U (1) gauge field, A µ . Theconventional, relativistic, action is S = 12 Z dt Z d d x ( D µ ~φ ) ⋆ ( D µ ~φ ) (3)where D µ ≡ ∂ µ + iA µ (4)Integrating out A µ leads to a nonlinear action which involves only the ~φ fields. As is wellknown, in d = 1 the model (3) is asymptotically free and generates a mass m for the fields ~φ by dimensional transmutation, as can be explicitly seen in the ’t Hooft large-N expansion N → ∞ g → g N = λ = fixed [1]. At the same time, the initially nondynamicalgauge field acquires a standard kinetic energy term, with a gauge coupling constant given by m . This is the simplest example of a dynamical emergence of gauge dynamics.The d = 2 model is interesting for condensed matter applications. In fact, the O (3) nonlinearsigma model with three component unit vector ˆ n can be rewritten as the CP model via theidentification ¯ φ~σφ = ˆ n . It is evident that local phase transformations of the φ fields do notaffect the “gauge-invariant” field ˆ n . Now there is a usual order-disorder transition: In themagnetically ordered phase of ˆ n , the φ field is condensed, and gauge field is gapped out by theHiggs mechanism. However, gauge field dynamics appears in the disordered phase when the φ fields become massive. Normally, the gauge fields also become massive and φ fields becomeconfined on the paramagnetic side of the transition due to the compactness of the gauge field[2]. However, suppressing the monopoles [3] of the gauge field (which correspond to “hedgehog”configurations of the original ˆ n fields) leads to a new critical point [4], and a paramagnetic phasewith a gapless photon [5]. There are conjectures that such a model with a noncompact gaugefield also describes a possible non-Landau, deconfined, critical point [6] between the Ne´el andbond-ordered phases of the d = 2 quantum antiferromagnet.In this paper we consider UV modifications of these models, which correspond to Lifshitz-likefixed points with a dynamical critical exponent z , S L = 12 Z dt Z d d x h ( D ~φ ) ⋆ ( D ~φ ) + α ( D i ~φ ) ⋆ ( D i ~φ ) + |D z ~φ | i (5)1here the operator D z is a sum of O ( d ) invariant terms containing z factors of the spatialcovariant derivative D i . For example, in z = 2 |D z φ | ≡ a ( D i D j ~φ ) ⋆ · ( D i D j ~φ ) + b ( D ~φ ) ⋆ · ( D ~φ ) (6)with a, b ≥ z we would have many more terms correspondingto various orderings of the D i .At the fixed point α = 0, one needs to scale the time and space coordinates as t → γ z t x → γt (7)Such fixed points, called Lifshitz fixed points, have a variety of applications in classical con-densed matter systems [7]. They also have a connection to quantum dimer models [8], whichare defined with a “kinetic” term which flips dimers on parallel bonds, and a “potential” termwhich gives an energy to every flippable plaquette. Finally, there is a constraint that every lat-tice site should have one and only one dimer touching it. Recently, it was realized [9, 10] thatneutral (ungauged) one-component Lifshitz fixed points describe special points (the Rokhsar-Kivelson (RK) points) of quantum dimer models [8] on bipartite lattices, where the field φ is aheight variable dual to a bond which may or may not contain a dimer [11, 12]. The standardRK point describes the transition between the smooth and rough phases of the height and ismulticritical, in the sense that more than one parameter needs to be tuned to attain the fixedpoint [9, 10]. However, it is possible to construct models with enough symmetries such thatthe fixed point can be obtained as a regular crtical point describing, for example, the phasetransition between two different types of bond-ordered states in a bilayer honeycomb lattice[13].While the above are examples of z = d = 2 theories of neutral scalars, examples of z = 2 gaugetheories also occur in condensed matter, in the description of algebraic spin liquids in d = 3[14, 15] and topological critical phases in d = 2[16].Note that the action of Eq. (5) still contains a single time-derivative of the gauge fields A i ,though it has higher spatial derivatives. Thus, even though one cannot easily integrate out the A µ to obtain a pure spin model, the gauge field is non-dynamical to begin with. In the spiritof the renormalization group, one expects that the model defined by Eq. (5) for N = 2 is inthe universality class of a z = d = 2 O(3) nonlinear sigma model.Recently, Lifshitz-type theories have been suggested as UV completions of low energyLorentz invariant theories of gravity and gauge dynamics [17]. This is because theories whichare non-renormalizable at the usual Lorentz-invariant UV fixed points with z = 1 can becomerenormalizable for non-trivial z . The idea that Lorentz symmetry violation can be regardedas UV regulators of field theories has been around for a while. See [18] for a recent discus-sion and references. In the same spirit, recently such Lifshitz fixed points have been proposed2s UV completions of four fermion theories similar to the Nambu-Jona Lasinio model in 3+1dimensions and discussed as possible candidates for physics at the weak scale [19], [20].We will find that for any choice of D z the theory defined by (6) is asymptotically free forall d = z and generates a mass gap, pretty much as the models in [20] - so that the theory isalways in a disordered phase. As a result, the gauge field acquire a kinetic term.This means that the possible emergence of Lorentz invariance at low energies is a little morenon-trivial since this has to happen in the gauge as well in the scalar sector. For z = d = 2we explicitly show that this indeed happens generically, and discuss the possibilities for higher d = z .For special choices of D z (the a = 0 multicritical point in our case) something even moreinteresting happens: one obtains a z = 2 electrodynamics with a standard ~E term, but theleading term in B = ǫ ij ∂ i A j is ( ∇ B ) . This naively suggests that a constant B costs noenergy: however a more careful calculation reveals that there is a nonanalytic dependence onconstant B of the form B / exp ( − πm/B ). Note that the analytic terms in our z = 2 , d = 2electrodynamics have the same form as the gauge theory descriptions of algebraic spin liquidsin d = 3 [14, 16], but appear to be dual to the gauge description of the transition between twobond-ordered phases [13] or the topological critical phase[16] in d = 2 in which it is the ~E term which is replaced by ( ǫ ij ∂ i E j ) . We will study the large-N limit of the model (5) with the constraint (1), using standard tech-niques [21]. The coupling g in the model (5) becomes dimensionless under Lifshitz scaling (7)when d = z . This may be seen by dimension counting : t has length dimension z , so that thelength dimensions of ~φ and g are [ ~φ ] ∼ [ L ] z − d [ g ] ∼ [ L ] d − z (8)Whether the coupling is marginally relevant or marginally irrelevant at z = d depends on thedynamics. To investigate this we use standard large-N techniques. Imposing the constraint (1)by a largrange multiplier field χ ( t, x ) we get the action S L = 12 Z dt Z d d x " ( D ~φ ) ⋆ ( D ~φ ) + α ( D i ~φ ) ⋆ ( D i ~φ ) + |D z ~φ | + χ ( t, x ) | ~φ | ( t, x ) − g ! (9)Integrating out the field ~φ we get the effective action S eff = N { Tr log h − D + ( − z ( D z ) + χ ( t, x ) i − g Z dtd d x χ ( t, x ) } (10)3t N = ∞ the functional integral over A µ ( t, x ) and χ ( t, x ) is dominated by the saddle pointof (10). We will assume that the saddle point is translationally invariant and rotationallysymmetric in the d spatial dimensions with a vanishing gauge field strength. Thus we may set χ ( t, x ) = χ in the saddle point equation . This also means that so far as the saddle pointequation is concerned, all possible terms in D z contribute equally2 N Z dk d d k (2 π ) d +1 k + α~k + ( ~k ) z + χ = 1 g (11)For any finite α , this integral is logarithmically divergent for d = z and behaves as log Λ z m ,where Λ is a cutoff on the spatial momentum ~k . This immediately implies that a solution tothe gap equation is m ≡ χ ∼ Λ z exp [ − Ag N ] (12)where A is a positive real number . Since m is (to leading order in 1 /N ) the physical massof the ~φ field (i.e. in a lorentzian signature this is the lowest value of the energy of a singleparticle state), it is clear from (12) that the coupling g has to be asymptotically free, with abeta function Λ dd Λ g = − g NA (13)It is useful to evaluate the integral in (11) for our primary case of interest, z = d = 2. For α, m ≪ Λ we get m = 2Λ e − πg N − α α ≫ Λ and leads to a linearly divergentanswer in this case.Dynamical mass generation for this model is thus almost exactly identical to that in thefour-fermion model of Ref. [20]. The effective action for the gauge field A µ and the fluctuationsof χ ( t, x ) has to be now obatined by substituting χ ( t, x ) = χ + 1 √ N δχ A µ ( t, x ) → √ N A µ ( t, x ) (15)in (10). Clearly, this will generate kinetic terms for δχ and A µ . The effect of these will be toprovide corrections to the leading order propagator of the ~φ fields which is simply the integrandof (11). Accordingly, the parameter α will be renormalized. If we go off the critical surfacecontaining the Lifshitz fixed point, the renormalized value of α will be nonzero. Clearly, when In principle there could a condensation of the field strength. However we will soon see in Section 2.1 thatfor d = z = 2 the effective action for a constant B = ǫ ij ∂ i A j is always larger than the action with B = 0.This rules out condensation of B . Our assumption that the field strength vanishes at the saddle point is thusjustified only a posteriori . We do not have a proof that this continues to hold for all d = z , but this appears tobe plausible. √ α , the propagator of the ~φ will be dominated bythe α ren k term. Therefore at low energies when α = 0, Lorentz invariance is recovered with aspeed of light given by √ α . d = z = 2 , α = 0 Emergence of Lorentz symmetry at low energies in the gauge field sector is more non-trivial,especially when α = 0, which is the case we will concentrate on. By gauge invariance, theinduced action for the gauge fields must be functionals of the field strengths F i and F ij andtheir derivatives. In addition, it must be symmetric under spatial rotations. For a Lorentzsymmetry to emerge, this effective action must contain combinations like ǫ F i + 1 µ F ij (16)with constant ǫ , µ . In that one can now rescale t, x, A , A i to get a standard Lorentz invariantformThe length dimensions of the dielectric constant ǫ and magnetic permeability µ may beeasily seen to be [ ǫ ] ∼ [ L ] z − d +2 [ µ ] ∼ [ L ] d + z − (17)so that the speed of light c = 1 / √ µ ǫ has length dimensions[ c ] ∼ [ L ] − z (18)as it should.It is not at all obvious that terms like (16) have to emerge at α = 0, since the parent theoryhas z = 2. In fact we will show that for special choices of the operator D z this will not happen.However, we will find that for generic choices of D z , terms like (16) do appear.Let us first address this question for z = d = 2, using the form (6). For this purpose, it issufficient to consider the effective action (10) with δχ = 0, so that we essentially have S eff = N { Tr log h − D + ( a + b )( D i D i ) + a ( B ( t, x )) − iaǫ ij ( ∂ i B ) D j + m i − ( B = 0 term ) } (19)where we have used the commutation relation[ D i , D j ] = iF ij (20)and for d = 2 renamed F = B ( t, x ). 5 .1 Constant Magnetic Field It is useful to first evaluate this for a constant B . Then the problem in evaluating the effectiveaction reduces to the problem of determining the eigenvalues of the operator H ( B ) = − D + ( a + b )( − D − D ) + aB (21)which is closely related to the problem of Landau diamagnetism. Let us choose a Landau gauge A = A = 0 A = B x (22)Consider the system to be in a large box with size in the time direction T and spatial sizes L , L . For large enough T, L , L the eigenvalue of ∂ can be taken to be continuous, whichwe will call p . It is straightforward to see that the eigenvalues of H ( B ) are κ ( p , n ) = p + ( a + b ) B (2 n + 1) + aB (23)with a degeneracy of the level n given by d ( n ) = BL L π (24)To evaluate the effective action (19) we use the Nambu-Schwinger-de Witt representation, − S eff = N Z ∞ dss e − m s Tr e − sH ( B ) − ( B = 0 term ) (25)Using (24) and (23) we haveTr e − sH ( B ) = e − saB ∞ X n =0 T Z ∞−∞ dp π BL L π e − [ sp +4( a + b ) B s ( n + ) ] = V T B π r πs e − saB ϑ [0 | iB s ( a + b ) /π ] (26)where V = L L denotes the spatial volume and ϑ [ w | τ ] is a Jacobi theta function . Toexamine the small B behavior it is useful to use standard theta function identities to writeTr e − sH ( B ) = V T πs √ a + b e − saB ϑ [0 | iπ/ (4 B s ( a + b ))]= V T πs √ a + b e − saB ∞ X k = −∞ ( − k e − π k sB a + b ) (27)The theta function in (27) may be written in a product represenation as ϑ [0 | iπ/ (4 B s ( a + b ))] = ∞ Y n =1 (1 − e − (2 n − π B s ( a + b ) ) (1 − e − nπ B s ( a + b ) ) (28) We are grateful to Al Shapere for pointing out that an efficient way to manipulate this sum is to recognizethis as a theta function. a, b ≥
0, this immediately shows thatTr e − sH ( B ) < Tr e − sH (0) (29)so that S eff ( B ) > S eff (0) (30)for any m . This provides a justification for setting B = 0 in the saddle point equation whichdetermines m . The result (30) in fact holds for all d with z = 2.The integral over s in (27) can be performed, leading to the effective action S eff ( B ) V T = − X k =0 ( − k Bm π k s aB m K πkmB vuut aB m a + b − π √ a + b Z ∞ dss e − m s (cid:16) e − saB − (cid:17) (31)where K denotes a modified Bessel function. In deriving (31) we have noted that the k = 0term in the sum in (27) is the sole contribution when B = 0 and subtracted that.The first term on the right hand side of (31) has a non-analytic dependence on B for small B . This follows from the asymptotic behavior of the modified Bessel function. For B ≪ m thesum in (31) is dominated by the k = 1 term, which leads to S eff ( B ) − S eff (0) V T ≃ B m b π √ e − πmB √ b (32)The second term contains various powers of B .Therefore, at the multicritical point a = 0, the effective action vanishes for B = 0 in a non-analytic fashion .For any a = 0 the final form of the effective action begins with a term proportional to B ,It follows from (25) that the coefficient of this term is divergent (proportional to Γ(0)). Tounderstand this, we use a version of dimensional regularization by adding ( d −
2) extra spatialdirections. Now the first line of (26) will be modified toTr e − sH ( B ) = e − saB ∞ X n =0 T Z ∞−∞ dp π L d − Z d d − p (2 π ) d − BL L π e − sp − sp − a + b ) B s ( n + ) (33)This leads to the following coefficient of B in the effective action (25) a Z ∞ dss e − m s s − d − = a Γ( − d ) m − d (34) Note that there is an overall factor of N in the effective action (39). Since we are performing a 1 /N expansion, so that the fields have to be rescaled as in (15). The factor of N cancels for the terms which arequadratic in the fields. We have performed the sum in (27) using a Euler-McLaurin expansion and verified that there are nopolynomial terms in B in this case to very high orders. − d ) as required. In a dimensional regularization schemewe introduce a scale κ with length dimensions −
2, and write this coefficient as1 µ κ − d (35)where µ is dimensionless. Then as ǫ →
0, the finite part of 1 /µ becomes1 µ ∼ log (cid:18) κm (cid:19) (36) In this section we will evaluate the effective action using a standard heat kernel method forgeneral fields F i ( t, x ) and F ij ( t, x ). The calculation presented above for constant magnetic fieldshows that at the special point a = 0 the effective action has a non-analytic dependence on B .We would like to determine whether the action contains terms which are analytic in derivativesof B . As shown in the Appendix , the small s expansion of the heat kernel is of the formTr e − sH ( E,B ) = Z d x dt ∞ X n =1 b n ( x ) s n − (37)which leads to the effective action S eff = N ∞ X n =1 Z d x dt b n ( x ) Γ( n − m n − (38)As in the previous subsection the term with n = 4 can be handled via dimensional regulariza-tion. As explained in the appendix, we should therefore replace (38) by S eff = N ∞ X n =1 Z d x dt b n ( x ) Γ( n − ǫ ) m n − ǫ ) (39)Let us first evaluate this for a = 0 and b = 1 For this case, explicit calculations yield b ( t, x ) = b ( t, x ) = b ( t, x ) = b ( t, x ) = b ( t, x ) = 0. The leading contribution comes from b ( t, x ). Aftera fairly long calculation we find that this leads to the effective action S eff,a =0 = 112 m Z dtd x (cid:20) F i + 110 ( ∂ i B )( ∂ i B ) (cid:21) (40)Higher powers of field strength are suppressed by powers of N . Note that the term we ob-tained in the previous section for constant B is nonanalytic in B , and formally irrelevant bypower counting since its Taylor series is 0. However, in the absence of this nonanalytic term,any constant B costs no energy, leading to a huge ground state degeneracy. Thus, the term B / e − πm/B is a dangerously irrelevant operator for this special case a = 0 , b = 1. ClearlyLorentz invariance is not regained at low energies in this case.8way from this multicritical point a = 0. As expected from our constant B calculation wenow find that b = 0. It is straightforward to see that b ( t, x ) ∼ F ij F ij (41)Since we are interested in the action at low energies, we should retain only the lowest non-trivialterms with the least number of derivatives. This leads to the low energy effective action forgauge fields, upto numerical factors S eff ∼ Z d x dt [ 1 m F i F i + Γ( ǫ ) m ǫ F ij F ij + · · · ] (42)where the ellipses now stand for terms containing more derivatives and/or more powers of thefield strength. As in the previous subsection, in the spirit of dimensional regularization this isreally S eff ∼ Z d x dt [ 1 m F i F i + log (cid:18) κm (cid:19) F ij F ij + · · · ] (43)so that we have µ given in (35) and ǫ ∼ m (44)Since ǫ , µ are constants independent of ( t, x ) one can now rescale t, x, A , A i to get the form S eff ∼ Z d x dt [ F µν F µν + · · · ] (45)This demonstrates the emergence of approximate low energy Lorentz symmetry with a scaledependent speed of light. Note that this happens even when the (renormalized) parameter α ren in (5) is tuned to zero. At this Lifshitz point there is no lorentz symmetry in the scalar sector.For α ren = 0 the speed of light in the scalar sector is different from that in the gauge sector. It is clear from section (1) that the coupling g in the sigma model is asymtptically free for all d = z . Does this also mean that gauge dynamics emerges in all dimensions? From equation(17) we see that the length dimensions of ǫ is always 2 for all z = d . If such a term appears,one would expect that ǫ ∼ m /z , since from (12) the length dimension of the mass m is z . For z ≥ ~φ . In fact if such an effective action is an expansion of powers of 1 /m (apart from logs) one would expect that the lowest dimension operator which would appearmust have length dimensions [ L ] z + d . In a similar vein, the magnetic permeability would have positive length dimensions. Such a term is also unlikely to come from an effective action. Itwould be interesting to investigate this issue further.9he model studied in this paper gauges the overall U (1) of the symmetry group. One could,instead, consider gauging the entire U ( N ) group to obtain a non-abelian gauged sigma model.This model would generate a mass gap in exactly the same fashion - in fact the gap equation isidentical. It would be interesting to see the effective action for the non-abelian gauge fields inthis case. It appears to us that the heat kernel expansion calculation is quite similar to ours.Another interesting direction is to revisit the linear rather than the non-linear sigma modelaround a Lifshitz fixed point as has been originally considered in classical statistical mechanics.The length dimension of a ( ~φ · ~φ ) coupling is given by ( d − z ), so that this is a relevantoperator for z > d/
3. It would be interesting to explore if there are IR fixed points for d > z = 1 , d = 2. This could have interestingapplications to particle physics. These vector models can be also interesting from the point ofview of AdS/CFT correspondence. Lifshitz fixed points have been argued to have dual gravitydescriptions [23]. On the other hand, the dual of usual vector models are higher spin gaugefields in usual AdS [24], [25]. It would be interesting to see the nature of the gravity duals forthese Lifshitz sigma models.These issues are currently under investigation.
We would like to thank Al Shapere for enlightening discussions, and the referee for commentsabout a previous version of the manuscript. The work of S.R.D. was supported in part by aNational Science Foundation Grant NSF-PHY-0555444, and that of GM by NSF-DMR-0703992.GM also thanks the Aspen Center for Physics, where some of this work was carried out.
A Expansion of the Heat Kernel
In this appendix calculate the heat kernel using the technique of [22]. This uses the represen-tation Tr e − s O = Z dtd x Z dωd k (2 π ) e − i ( ωt + k · x ) e − s O e i ( ωt + k · x ) (46)Using the basic identity e − i ( ωt + k · x ) D µ e i ( ωt + k · x ) = ik µ + D µ (47)we have for our case e − i ( ωt + k · x ) H ( E, B ) e i ( ωt + k · x ) = ω + ~k + a + a + a + a (48)where a = − ik ( k i D i ) 10 = − iωD − k i D i ) − k ( D i D i ) a = i [( D i D i )( k j D j ) + ( k j D j )( D i D i ) + 2 D i ( k j D j ) D i ] a = − D + D i D j D i D j (49)Using (48) in (46) and rescaling ω → s / ω ~k → s / ~k (50)we get Tr e − sH ( E,B ) = 1 s Z dtd x Z dωd k (2 π ) e − ( ω + k ) e − G ( E,B ) (51)where G ≡ s a + s a + s a + s a (52)The integrals over ω and ~k can be now evaluated, leading to small-s expansion of the heatkernel of the form (37), leading to the form of the effective action (38).The term with n = 4 has to be treated in dimensional regularization. This means that in(46) we replace d k → d d k , so that after the rescalings (50), the equation (51) becomesTr e − sH ( E,B ) = 1 s d +24 Z dtd x Z dωd d k (2 π ) e − ( ω + k ) e − G ( E,B ) (53)For d = 2 − ǫ we will still evaluate the integrals over ~k by replacing d d k → d k in (53). Thisleads to the small s expansionTr e − sH ( E,B ) = Z d x dt ∞ X n =1 b n ( x ) s n − ǫ (54)This leads to the expression (39).The integrals over ω and ~k can be performed using basic symmetry properties. Thus, e.g. Z ∞−∞ dω e − ω ω n +1 = 0 ( n integer) Z ∞−∞ dω e − ω ω = 12 Z ∞−∞ dω e − ω Z ∞−∞ dω e − ω ω = 34 Z ∞−∞ dω e − ω · · · · · · (55)while Z d k k i k i k n +1 e − k = 0 ( n integer) Z d k k i k j e − k = 12 √ π δ ij Z d k e − k d k ( ~k · ~k ) e − k = 12 Z d k e − k Z d k ( ~k · ~k ) e − k = 1 √ π Z d k e − k Z d k ( ~k · ~k ) k i k j e − k = 12 √ π δ ij Z d k e − k · · · · · · (56)Using these integrals, it is straightforward to see that terms with n = 1 , , ω and/or ~k . A short calculation using the explicitexpressions in (55) and (56) shows that the various terms cancel, leading to b ( x, t ) = 0. Thefirst non-trivial term is therefore for n = 4. Here, after several cancellations one is left with b ( t, x ) ∼ F ij F ij (57)which basically comes from rewriting the second term in a as D i D j D i D j = D i D D i + D i D j [ D i , D j ] = D i D D i + 12 [ D i , D j ][ D i , D j ] = D i D D i + 12 F ij F ij (58)The next nonzero term comes at n = 6. This leads to an electric field term, which arises from Z dωd d k (2 π ) e − ( ω + k ) ( a a a a ) (59)which clearly includes a term D i D D i D (60)and hence to F i F i . The n = 6 term contains other contributions as well. These contain higherderivative terms in the field strength B . Specifically, for the case a = 0 , b = 1, we obtain theterm 1120 m ( ∇ B ) (61) References [1] A. D’Adda, M. Luscher and P. Di Vecchia, Nucl. Phys. B , 63 (1978).[2] A. M. Polyakov, Phys. Lett. B , 82 (1975); A. M. Polyakov, Nucl. Phys. B , 429 (1977).[3] M. Lau and C. Dasgupta, Phys. Rev. B , 7212 (1989).[4] M. Kamal and G. Murthy, Phys. Rev. Lett. , 1911 (1993).[5] O. Motrunich and A. Vishwanath, Phys. Rev. B , 075104 (2005).[6] T. Senthil, L. Balents, S. Sachdev, A. Vishwanath, and M. P. A. Fisher, Science , 1490 (2004);Phys. Rev. B , 14407 (2004).
7] R. M. Hornreich, M. Luban and S. Shtrikman, Phys. Rev. Lett. , 1678 (1975); G. Grinstein,Phys. Rev. B , 4615 (1981).[8] D. S. Rokhsar and S. A. Kivelson, Phys. Rev. Lett. , 2376 (1988); E. Fradkin and S. A.Kivelson, Mod. Phys. Lett. B , 225 (1990).[9] E. Frakdin, D. A. Huse, R. Moessner, V. Oganesyan, and S. L. Sondhi, Phys. Rev. B , 224415(2004).[10] E. Ardonne, P. Fendley and E. Fradkin, Annals Phys. , 493 (2004) [arXiv:cond-mat/0311466];S. Papanikolaou, E. Luijten and E. Fradkin, Phys. Rev. 134514 (2007) [arXiv:cond-mat/0607316][11] C. L. Henley, Jour. Stat. Phys. , 483 (1997).[12] L. S. Levitov, Phys. Rev. Lett. , 92 (1990).[13] A. Vishwanath, L. Balents, and T. Senthil, Phys. Rev. B , 224416 (2004).[14] R. Moessner and S. L. Sondhi, Phys. Rev. B , 184512 (2003).[15] M. Hermele, M. P. A. Fisher, and L. Balents, Phys. Rev. B , 064404 (2004).[16] M. Freedman, C. Nayak, and K. Shtengel, Phys. Rev. Lett. , 147205 (2005).[17] P. Horava, arXiv:0811.2217 [hep-th]. P. Horava, Phys. Rev. D , 084008 (2009) [arXiv:0901.3775[hep-th]].[18] M. Visser, arXiv:0902.0590 [hep-th].[19] D. Anselmi, arXiv:0904.1849 [hep-ph].[20] A. Dhar, G. Mandal and S. R. Wadia, arXiv:0905.2928 [hep-th].[21] See e.g. Erice lectures of S. R. Coleman, “1/N” in Aspects of Symmetry , Press Syndicate ofUniversity of Cambridge, 1985; A.M. Polyakov “Gauge Fields and Strings”, Harwood Publishers,1987.[22] R. I. Nepomechie, Phys. Rev. D , 3291 (1985).[23] S. Kachru, X. Liu and M. Mulligan, Phys. Rev. D , 106005 (2008) [arXiv:0808.1725 [hep-th]].[24] I. R. Klebanov and A. M. Polyakov, Phys. Lett. B , 213 (2002) [arXiv:hep-th/0210114].[25] S. R. Das and A. Jevicki, Phys. Rev. D , 044011 (2003) [arXiv:hep-th/0304093]., 044011 (2003) [arXiv:hep-th/0304093].