Magnetic Catalysis in Antiferromagnetic Films
aa r X i v : . [ c ond - m a t . s t r- e l ] J a n Magnetic Catalysis in Antiferromagnetic Films
Christoph P. Hofmann a a Facultad de Ciencias, Universidad de ColimaBernal D´ıaz del Castillo 340, Colima C.P. 28045, Mexico
September 12, 2018
Abstract
We study the low-temperature behavior of antiferromagnets in two spa-tial dimensions that are subjected to a magnetic field oriented perpendic-ular to the staggered magnetization order parameter. The evaluation ofthe partition function is carried to two-loop order within the systematiceffective Lagrangian technique. Low-temperature series that are valid inweak magnetic and staggered fields are derived for the pressure, staggeredmagnetization, and magnetization. Remarkably, at T =0, the staggeredmagnetization is enhanced by the magnetic field, implying that the phe-nomenon of magnetic catalysis also emerges in antiferromagnetic films. The thermodynamic properties of antiferromagnets in two spatial dimensions havebeen the topic of numerous studies. Within microscopic, phenomenological, andnumerical approaches, the free energy density, staggered magnetization, and otherobservables have been explored extensively at low temperatures [1–17] – in particu-lar also in magnetic fields [18–52]. Still, a fully systematic analysis of how a weakmagnetic field, in presence of a weak staggered field, affects the low-energy physics ofantiferromagnetic films – both at T =0 and finite temperature – appears to be lacking.Instead of relying on phenomenological or microscopic techniques such as modifiedspin-wave theory, the present analysis is based on the effective Lagrangian methodthat has the virtue of being fully systematic. The crucial point is that the relevantdegrees of freedom in an antiferromagnet at low temperatures – the spin waves or1agnons – are the Goldstone bosons of a spontaneously broken internal symmetry: O (3) → O (2). Goldstone boson effective field theory has been developed in theeighties in the context of quantum chromodynamics [54, 55], but the same universalprinciples can be applied to condensed matter systems [56, 57], where the phenomenonof spontaneous symmetry breaking is ubiquitous.Within effective field theory, the thermodynamic properties of antiferromagnets intwo spatial dimensions have been analyzed in Refs. [58–61]. Some of these studies –apart from the inclusion of a staggered field – also consider the effect of an externalmagnetic field. However a systematic discussion of how the thermodynamic variablesand the physics at T =0 depend on these fields, has not yet been presented. In partic-ular, the situation where the magnetic field is oriented perpendicular to the staggeredfield, has not been discussed on the effective level so far. This motivates the presentwork where we systematically investigate the impact of a perpendicular magnetic fieldonto the low-energy behavior of d =2+1 antiferromagnets. We evaluate the partitionfunction up to two-loop order, derive the low-temperature series for the free energydensity, pressure, staggered magnetization, and magnetization, and also consider thebehavior of the system at zero temperature.In the absence of a magnetic field, the spin-wave interaction does not yet manifestitself: up to two-loop order, the low-temperature series just correspond to the freemagnon gas. In nonzero magnetic fields, however, the spin-wave interaction leads tointeresting effects. In the pressure – irrespective of the strength of the magnetic andstaggered field – the interaction among the magnons is repulsive. Regarding the orderparameter at finite temperature , we also observe subtle effects: if the temperature israised from T =0 to finite T – while keeping the strength of the staggered and magneticfield fixed – the order parameter decreases as a consequence of the spin-wave interac-tion. Remarkably, at zero temperature, the staggered magnetization is enhanced inweak magnetic fields. This phenomenon – magnetic catalysis – has been observed inquantum chromodynamics, graphene, topological insulators, and other systems. Fi-nally, the perpendicular magnetic field – both at zero and finite temperature – causesthe magnetization to take positive values, signaling that the spins get tilted into themagnetic field direction.The article is organized as follows. The incorporation of the perpendicular mag-netic field, along with some essential information on the effective Lagrangian method,is discussed in Sec. 2. The perturbative evaluation of the free energy density up to Strictly speaking, at finite temperature and in two spatial dimensions, spontaneous symmetrybreaking does not occur because of the Mermin-Wagner theorem [53]. However, the low-temperaturephysics is still dominated by the spin waves and the staggered magnetization is different from zeroat low T and weak fields. In this sense the staggered magnetization is still referred to as orderparameter in the present study. See footnote 1. As we comment in subsection 4.2, magnetic catalysis in antiferromagnetic films is different be-cause no charged particles or Landau levels are involved. T =0 and discuss the phenomenon ofmagnetic catalysis. Finally, in Sec. 5 we present our conclusions. Technical details onvertices with an odd number of magnon lines and the evaluation of a specific two-loopdiagram can be found in two separate appendices. Antiferromagnets in two spatial dimensions are described by the quantum Heisenbergmodel, H = − J X n.n. ~S m · ~S n , J = const., (2.1)where the summation extends over all nearest neighbor spins on a bipartite lattice,and the exchange integral J is negative. The Heisenberg Hamiltonian is invariantunder global internal O(3) symmetry. The antiferromagnetic ground state, however,is only invariant under O(2). As a consequence of the spontaneously broken rota-tion symmetry, two spin-wave branches – or two magnon particles – emerge in thelow-energy spectrum. If the O(3) symmetry is exact, the two degenerate spin-wavebranches follow the dispersion law ω ( ~k ) = v | ~k | + O ( ~k ) , ~k = ( k , k ) , (2.2)with v as spin-wave velocity. According to Goldstone’s theorem, both excitationsobey lim ~k → ω ( ~k ) = 0 . (2.3)The symmetric model can be extended by incorporating a staggered field ~H s and amagnetic field ~H , H = H − X n ~S n · ~H − X n ( − n ~S n · ~H s , (2.4)that both explicitly break O(3)-invariance. Now the spontaneously broken symmetryis only approximate: the spin-wave branches exhibit an energy gap, i.e., the magnonsare no longer Goldstone bosons as they become massive. In particle physics it iscommon to call such excitations pseudo-Goldstone bosons .Let us turn to the effective description of the d =2+1 antiferromagnet in presence ofstaggered and magnetic fields. This situation has been discussed in detail in sectionsIX-XI of Ref. [62] (see also Ref. [56]). Here we merely list the relevant expressions.3he basic low-energy degrees of freedom – the two antiferromagnetic magnon fields –we denote by U a = ( U , U ), and collect them in a unit vector U i , U i = ( U , U a ) , U = √ − U a U a , a = 1 , , i = 0 , , . (2.5)The ground state of the antiferromagnet is represented by ~U = (1 , , L eff – contains two space-time derivatives, L eff = F D µ U i D µ U i + Σ s H is U i , (2.6)where the covariant derivative is D U i = ∂ U i + ε ijk H j U k , D r U i = ∂ r U i ( r = 1 , . (2.7)The magnetic field H i is incorporated through the time component of the covariantderivative D U i . On the other hand, the staggered field H is couples to Σ s that rep-resents the order parameter: the staggered magnetization at zero temperature, zeroexternal fields, and infinite volume. Apart from Σ s , a second low-energy effectiveconstant appears in L eff : the quantity F that is related to the spin stiffness ρ (orhelicity modulus) by ρ = F . Note that the magnetic field counts as order p like thetime derivative, whereas the staggered field is of order p .The subleading piece in the effective Lagrangian is of order p , L eff = e ( D µ U i D µ U i ) + e ( D µ U i D ν U i ) + k Σ s F ( H is U i )( D µ U k D µ U k )+ k Σ s F ( H is U i ) + k Σ s F H is H is , (2.8)and contains five next-to-leading order (NLO) effective constants whose numericalvalues have to be determined or estimated to make the effective field theory predictive(see below).We now comment on an important issue related to Lorentz-invariance. The leadingand next-to-leading effective Lagrangians are Lorentz-invariant. In view of the factthat the underlying bipartite lattices are not even space-rotation invariant, why is ourapproach legitimate? The first observation is that the leading piece L eff is strictly(pseudo-)Lorentz-invariant, the spin-wave velocity v taking the role of the speed oflight. This accidental symmetry emerges because lattice anisotropies only show up atorder p (and beyond) in the effective Lagrangian [59]. On the other hand, in L eff one should include all additional terms that are permitted by the lattice geometry.However, as we explain below, these effects only start manifesting themselves at next-to-next-to leading order in the low-temperature expansion which is beyond two-loop4ccuracy we pursue in the present evaluation. This perfectly justifies maintaining a(pseudo-)Lorentz-invariant structure also in L eff .In the following we consider the scenario where the magnetic field points into adirection perpendicular to the staggered field, ~H ⊥ = (0 , H, , ~H s = ( H s , , . (2.9)While we have chosen ~H ⊥ to point into the 1-direction, the physics would be thesame had we chosen the 2-direction. Note that the staggered field points into the 0-direction, aligned with the staggered magnetization order parameter or ground state ~U = (1 , , L eff gives rise to the following magnondispersion relations, ω I = r ~k + Σ s H s F + H ,ω II = r ~k + Σ s H s F . (2.10)These results coincide with the expressions derived within microscopic or phenomeno-logical descriptions – see, e.g., Refs. [63, 64]. Remarkably, one of the magnons is notaffected by the magnetic field. The structure of the dispersion relation is relativisticin both cases, and the corresponding ”magnon mass terms” are identified as M I = Σ s H s F + H , M II = Σ s H s F . (2.11)Note that the staggered field emerges linearly while the dependence on the magneticfield is quadratic. If the fields are switched off, we reproduce the linear and ungappeddispersion relation Eq. (2.2). It is convenient to utilize dimensional regularization in the perturbative evaluationof the partition function. The zero-temperature propagators for antiferromagneticmagnons, in presence of ~H s = ( H s , ,
0) and ~H ⊥ = (0 , H, I,II ( x ) = (2 π ) − d Z d d p e ipx ( M I,II + p ) − = Z ∞ d ρ (4 πρ ) − d/ e − ρM I,II − x / ρ , (2.12)where M I and M II are defined in Eq. (2.11). The corresponding thermal propagatorsin Euclidean space are given by G I,II ( x ) = ∞ X n = −∞ ∆ I,II ( ~x, x + nβ ) , β = 1 T . (2.13) We have set the spin-wave velocity v to one. b a Figure 1: Low-temperature expansion of the partition function for the d =2+1 an-tiferromagnet: Feynman diagrams up to two-loop order T . Filled circles refer to L eff , while the vertex associated with the subleading piece L eff is represented by thenumber 4. Each loop is suppressed by one power of T .We emphasize that the magnetic and staggered field are treated as perturbationsthat explicitly break O(3) invariance of the Heisenberg Hamiltonian. As long as thesefields are weak, the O(3) symmetry still is approximate, and our basic setting is valid:it is conceptually consistent to start from the collinear antiferromagnetic ground stateand interpret the two magnons as oscillations of the staggered magnetization order pa-rameter. It is well-known, however, that in presence of magnetic fields perpendicularto the staggered magnetization, the spins get tilted, creating a canted (non-collinear)phase (see, e.g., Refs. [23, 43, 49]). If the canting angle is large, the magnetic fieldcan no longer be considered as a small perturbation. Rather the canted phase shouldbe chosen as the starting configuration underlying the perturbative expansion. Mostimportantly, since the spontaneous symmetry breaking pattern then is O(3) →
1, nottwo, but three Goldstone fields emerge in the low-energy spectrum. This scenario,i.e., the low-temperature physics of canted phases, we postpone for future studies – inthe present investigation we consider weak magnetic fields where only two spin-wavebranches are relevant.
We now evaluate the partition function for the d =2+1 antiferromagnet subjectedto the magnetic and staggered fields defined in Eq. (2.9). The relevant Feynmandiagrams up to two-loop order are shown in Fig. 1. The crucial point is that we aredealing with a systematic low-temperature expansion of the partition function whereeach magnon loop is suppressed by one power of temperature. The free Bose gascontribution is given by the one-loop graph 3 (order T ), while the two-loop graph 4 b The perturbative evaluation of the partition function is described in more detail in section 2 ofRef. [60] and in appendix A of Ref. [65]. Regarding the effective Lagrangian technique in general,the interested reader may consult Refs. [66–68]. c d Figure 2: Low-temperature expansion of the partition function for the d =2+1 an-tiferromagnet: additional Feynman diagrams up to two-loop order T emerging inpresence of a perpendicular magnetic field. Filled circles refer to L eff . Each loop issuppressed by one power of T .is of order T .The incorporation of a perpendicular magnetic field generates extra vertices thatinvolve an odd number of magnon lines. With respect to L eff , the explicit terms areproportional to one time derivative and read iF H (cid:16) U ∂ U − U ∂ U (cid:17) . (3.1)These contributions, along with those originating from L eff , create vertices with1 , , , . . . magnon lines: in presence of a perpendicular magnetic field, the set ofFeynman diagrams has to be extended by the graphs depicted in Fig. 2. Note that inthe diagrams depicted in Fig. 1, the magnetic field manifests itself implicitly in thethermal propagator G I ( x ) through M I .The tree graphs 2 and 4 a merely contribute to the energy density at zero temper-ature, z = − Σ s H s − F H ,z a = − ( k + k ) Σ s H s F − k Σ s H s F H − ( e + e ) H . (3.2)The dominant temperature-dependent contribution comes from one-loop graph 3, z = − (4 π ) − d/ Γ( − d ) n M dI + M dII o − n g I + g II o = − n g I + g II o − π ((cid:16) Σ s H s F + H (cid:17) / + (cid:16) Σ s H s F (cid:17) / ) . (3.3)Note that the Gamma function is finite in two spatial dimensions,lim d → (4 π ) − d/ Γ( − d ) = 112 π . (3.4)7he quantities g I and g II are the kinematical functions related to the free magnongas, g I,IIr ( H s , H, T ) = 2 Z ∞ d ρ (4 πρ ) d/ ρ r − exp (cid:16) − ρM I,II (cid:17) ∞ X n =1 exp( − n / ρT ) . (3.5)Next, the two-loop graph 4 b contributes with z b = − Σ s H s F (cid:16) G I − G II (cid:17) − H F (cid:16) G I (cid:17) = − Σ s H s F n ( g I ) − g I g II + ( g II ) o − H F ( g I ) + Σ s H s πF (r Σ s H s F + H − √ Σ s H s F ) g I + H πF r Σ s H s F + H g I − Σ s H s πF (r Σ s H s F + H − √ Σ s H s F ) g II − Σ s H s π F − s H s H π F − H π F + Σ / s H / s π F r Σ s H s F + H , (3.6)where G I,II are the thermal propagators evaluated at the origin, G I,II = G I,II ( x ) | x =0 = g I,II − M I,II π . (3.7)In the absence of the magnetic field, we have G I = G II , such that the entire two-loopcontribution vanishes.Finally, the explicit evaluation of diagram 4 c yields zero, z c = 0 , (3.8)while the sunset diagram amounts to z d = 2 H F Z T d d x G I ( x ) ∂ G I ( x ) ∂ G II ( x ) . (3.9)This integral over the torus T = R d s × S , with circle S defined as − β/ ≤ x ≤ β/
2, is divergent in the ultraviolet. The renormalization of this expression and theevaluation of the thermal sums is described in appendix B. The finite contribution tothe free energy density is given by z d = 2 F s ( σ, σ H ) T . (3.10)The dimensionless function s ( σ, σ H ) is defined in Eq. (B.14), and the dimensionlessparameters σ and σ H are σ = √ Σ s H s πF T = m F T , σ H = H πT = m H F T . (3.11)8 H Σ , Σ H L Σ H Σ- - Figure 3: [Color online] The function s ( σ, σ H ), where σ and σ H are the dimensionlessparameters σ = √ Σ s H s / (2 πF T ) and σ H = H/ (2 πT ).A plot of s ( σ, σ H ) is provided in Fig. 3.Remarkably, up to two-loop order, the NLO effective constants e , e , k , k , k only show up in the tree graph 4 a . These constants – that are a priori unknown –hence only matter in temperature-independent contributions. The low-temperatureexpansion, in particular the impact of the spin-wave interaction, is governed by theleading effective Lagrangian L eff . Up to two-loop order, the thermal properties ofthe d =2+1 antiferromagnet are thus rigorously captured by our effective field theoryapproach that is based on (pseudo-)Lorentz invariance. The specific geometry of theunderlying bipartite lattice is irrelevant as far as the structure of the low-temperatureexpansion is concerned. Alternatively, this can be seen as follows. Lattice anisotropiesmodify the dispersion relation ω ( ~k ) = v | ~k | + O ( ~k ) (3.12)at order ~k – the specific terms and coefficients indeed depend on the lattice geometry.While the linear term in the dispersion relation yields the dominant contribution oforder T in the free energy density, the corrections ∝ ~k contribute at order T which is beyond our scope. Therefore our (pseudo-)Lorentz-invariant framework isperfectly legitimate: we make no mistake by merely considering the leading term inthe dispersion relation.The lattice structure only reflects itself in the numerical values of the leading-order effective constants F and Σ s that have been determined with high-precision9oop-cluster algorithms. For the square lattice [69] they read ρ = 0 . J , Σ s = 0 . /a , v = 1 . J a ( S = ) , (3.13)for the honeycomb lattice [70] they are ρ = 0 . J , ˜Σ s = 0 . , v = 1 . J a ( S = ) , (3.14)with ˜Σ s = 3 √
34 Σ s a . (3.15)Note that the spin stiffness ρ as well as Σ s , much like the spin-wave velocity v , aregiven in units of J (exchange integral) and a (lattice size). The low-temperature physics of the system can be captured by various dimensionlessratios. As independent quantities we define the parameters m, m H and t as m ≡ √ Σ s H s πF , m H ≡ H πF , t ≡ T πF . (4.1)For the effective low-energy expansion to be consistent, the temperature as well as thestaggered and magnetic field must be small compared to the scale Λ that characterizesthe microscopic system. The natural scale in the Heisenberg antiferromagnet is theexchange integral J . In the present study, we define low temperatures and weak fieldsby T, H, M II ( ∝ p H s ) . . J . (4.2)The factors 2 π in Eq. (4.1) were introduced in analogy to the relevant scale in quantumchromodynamics (see Ref. [60]). The point is that for the antiferromagnet – both onthe square and honeycomb lattice – the denominator 2 πF is of the order of J . Theparameters m, m H , t hence measure temperature and field strength relative to theunderlying microscopic scale.Whereas temperature and magnetic field can be arbitrarily small, it should benoted that the staggered field can not be switched off. This is a consequence of theMermin-Wagner theorem [53] and the fact that the staggered magnetization – unlikethe magnetization – represents the order parameter. As we have discussed on previousoccasions, the domain where the effective expansion fails due to the smallness of thestaggered field, is tiny. The interested reader is referred to Figs. 2 and 3 of Ref. [61].10 .1 Pressure We first discuss the pressure, defined by P = z − z . (4.3)The quantity z includes all terms in the energy density that do not depend on tem-perature. Introducing dimensionless functions h i ( m, m H , t ) as g ( m, m H , t ) = T h ( m, m H , t ) , g ( m, m H , t ) = T h ( m, m H , t ) ,g ( m, m H , t ) = h ( m, m H , t ) T , g ( m, m H , t ) = h ( m, m H , t ) T , (4.4)the structure of the low-temperature series becomes more transparent because powersof temperature are explicit. For the pressure we get P ( T, H s , H ) = ˜ p T + ˜ p T + O ( T ) , ˜ p ( T, H s , H ) = n h I + h II o , ˜ p ( T, H s , H ) = m F t ( h I − h II ) + m H F t ( h I ) − m πF t (q m + m H − m )(cid:16) h I − h II (cid:17) − m H πF t q m + m H h I − F s ( σ, σ H ) . (4.5)The dominant term of order T (graph 3) corresponds to the free Bose gas. Theterm of order T (graphs 4 b and 4 d ) represents the leading interaction contribution.In the absence of a perpendicular magnetic field, the spin-wave interaction does notmanifest itself at two-loop order: the corresponding coefficient ˜ p is zero. On theother hand, if a perpendicular magnetic field is present, the behavior of the system isquite interesting: in Fig. 4 we depict the ratio ξ P ( T, H s , H ) = P int ( T, H s , H ) P Bose ( T, H s , H ) = ˜ p T ˜ p T (4.6)that measures strength and sign of the spin-wave interaction in the pressure relativeto the free Bose gas contribution. The plots refer to the temperatures T / πF = 0 . T / πF = 0 . It has been argued previously that the limit H s → P m H m Ξ P m H m Figure 4: [Color online] Spin-wave interaction manifesting itself in the pressure –measured by ξ P ( T, H s , H ) – of d =2+1 antiferromagnets as a function of magneticand staggered field at the temperatures T / πF = 0 .
02 (left) and
T / πF = 0 . The staggered magnetization order parameter can be extracted from the free energydensity by Σ s ( T, H s , H ) = − ∂z ( T, H s , H ) ∂H s . (4.7)The low-temperature series takes the structure Σ s ( T, H s , H ) = Σ s (0 , H s , H ) + ˜ σ T + ˜ σ T + O ( T ) , ˜ σ ( T, H s , H ) = − Σ s F (cid:16) h I + h II (cid:17) . (4.8)The spin-wave interaction comes into play at order T . Again, in zero magnetic field,there is no interaction term at two-loop order: ˜ σ ( T, H s ,
0) = 0.To explore the impact of the spin-wave interaction in the order parameter, weconsider the ratio ξ Σ s ( T, H s , H ) = Σ s,int ( T, H s , H ) | Σ s,Bose ( T, H s , H ) | = ˜ σ T | ˜ σ | T , (4.9)that we depict in Fig. 5 for the temperatures
T / πF = { . , . } . The quantity ξ Σ s ( T, H s , H ) is negative in the parameter region we consider. Negative ξ Σ s means We do not display the coefficient ˜ σ since the expression is rather lengthy – it can trivially beobtained from z b given in Sec. 3. S s m H m - - Ξ S s m H m - - - - Figure 5: [Color online] Spin-wave interaction manifesting itself in the staggeredmagnetization – measured by ξ Σ s ( T, H s , H ) – of d =2+1 antiferromagnets as a func-tion of magnetic and staggered field at the temperatures T / πF = 0 .
02 (left) and
T / πF = 0 . T =0 to finite T – while keeping H s and H fixed– the order parameter decreases due to the spin-wave interaction.Recall that it makes no sense to address the two-dimensional system in very weakstaggered fields within our framework, because the effective expansion breaks downwhen one approaches the limit H s → In our plots we have chosen the staggeredfield strength as 0 . ≤ m . . , m = √ Σ s H s πF . (4.10)This guarantees that the effects we observe are indeed physical and not just artifactsof our effective calculation extrapolated to a forbidden parameter region.At zero temperature, the order parameter is given byΣ s (0 , H s , H )Σ s = 1 + m p m + m H m m H − m p m + m H − m m H p m + m H + 8 π F ( k + k ) m + 4 π F k m H ,m = √ Σ s H s πF , m H = H πF , Σ s = Σ s (0 , , . (4.11)In contrast to finite temperature, at T =0, next-to-leading order effective constantsarise in the low-energy expansion of the staggered magnetization. The actual values of A detailed discussion of how this relates to the Mermin-Wagner theorem, can be found at theend of section 4 in Ref. [71]. d =2+1 anti-ferromagnet, none of these options seems to be available. Still, their magnitude canbe estimated. According to Ref. [72] they are very small, of order | k | ≈ | k | ≈ | k | ≈ π F ≈ . F , (4.12)much like the other NLO effective constants e and e . It should be noted that theabove estimate concerns their magnitude, but leaves open their signs. However thesecorrections are small – moreover, the dominant contributions in the series (4.11) donot involve NLO effective constants.At T =0 and in zero magnetic field, the series is characterized by powers of √ H s ,Σ s (0 , H s ,
0) = Σ s + Σ s / πF p H s + 2Σ s F ( k + k ) H s + O ( H / s ) , (4.13)and in zero staggered field by powers of H ,Σ s (0 , , H ) = Σ s + Σ s πF H + Σ s F n k + 5128 π F o H + O ( H ) . (4.14)While the order parameter is indeed expected to increase when the staggered fieldbecomes stronger, the behavior with respect to the magnetic field comes rather un-expectedly: in the series (4.14), the term linear in H is small, but positive. Theorder parameter thus increases when a weak perpendicular magnetic field is applied.Notice that the subleading correction (order H ) involves the NLO effective constant k whose sign remains open. Still, the behavior of the order parameter in weak mag-netic fields is dominated by the leading term that is strictly positive. We emphasizethat this result is universal in the sense that the term of order H is the same for anybipartite lattice: the only difference between, e.g., the square and honeycomb lattice,concerns the actual values of the effective constants Σ s and F .The phenomenon that the order parameter is enhanced by an external magneticfield when the order parameter is already present in zero magnetic field, is called magnetic catalysis according to Ref. [73]. It has been observed in quantum chro-modynamics, where the quark condensate – the order parameter of the spontaneouslybroken chiral symmetry – increases at T =0 in presence of a magnetic field [73–76].Magnetic catalysis has also been reported in condensed matter systems like graphene[77] and three-dimensional topological insulators [78]. The fact that the staggeredmagnetization is enhanced at T =0 in square lattice antiferromagnets subjected to amagnetic field perpendicular to the order parameter, has been reported in Ref. [46]. The exception is Ref. [69] where the combination k + k of NLO effective constants was deter-mined using a loop-cluster algorithm. It is perfectly legitimate at T =0 to consider the limit H s →
0. Only at finite T it is inconsistentto switch off the staggered field in our effective field theory approach. T m H m - - - S T m H m - - - Figure 6: [Color online] Temperature-dependent part of the magnetization – measuredby Σ T ( T, H s , H ) – of d =2+1 antiferromagnets as a function of magnetic and staggeredfield at T / πF = 0 .
02 (left) and
T / πF = 0 . d =2+1 antiferromagnets whereno charged particles are involved in its low-energy description. The fact that thestaggered magnetization grows in presence of a weak perpendicular magnetic field, issimply due to the suppression of quantum fluctuations of the order parameter vectorby the magnetic field. Still, according to the definition given in Ref. [73], we aredealing with magnetic catalysis. The low-temperature expansion of the magnetization,Σ(
T, H s , H ) = − ∂z ( T, H s , H ) ∂H , (4.15)takes the form Σ( T, H s , H ) = Σ(0 , H s , H ) + ˆ σ T + ˆ σ T + O ( T ) , ˆ σ ( T, H s , H ) = − Hh I . (4.16)The free Bose gas contribution is proportional to one power of temperature, while thespin-wave interaction is contained in the T -term. The coefficient ˆ σ can trivially be obtained from z b given in Sec. 3. int T m H m S int T m H m Figure 7: [Color online] Spin-wave interaction manifesting itself in the magnetization– measured by Σ intT ( T, H s , H ) – of d =2+1 antiferromagnets as a function of magneticand staggered field at the temperatures T / πF = 0 .
03 (left) and
T / πF = 0 . T / πF = { . , . } , we plot the total temperature-dependent part of the magnetizationΣ T ( T, H s , H ) = ˆ σ T + ˆ σ T F . (4.17)The quantity Σ T is negative in the entire parameter domain we consider. NegativeΣ T means that the magnetization decreases when we go from from T =0 to finite T while keeping H s and H fixed. This is what one would expect.Remarkably, the quantity Σ intT ( T, H s , H ) = ˆ σ T F , (4.18)that only takes into account the spin-wave interaction part, is positive as we illus-trate in Fig. 7 that refers to the temperatures T / πF = { . , . } . PositiveΣ intT ( T, H s , H ) means that if the temperature is raised from T =0 to finite T – whilekeeping H s and H fixed – the magnetization grows due to the spin-wave interaction.This result appears to be rather counterintuitive. But it is important to point out thatwe are dealing with weak effects originating from the spin-wave interaction. The dom-inant behavior at finite temperature is given by the free Bose gas term. Indeed, thetotal temperature-dependent magnetization (not just the interaction part), is strictlynegative according to Fig. 6. 16inally, at zero temperature, the magnetization amounts toΣ(0 , H s , H ) F = 2 π m H + π m H q m + m H + πm H + 5 π m m H − π m m H p m + m H + 32 π F ( e + e ) m H + 16 π F k m m H ,m = √ Σ s H s πF , m H = H πF . (4.19)Again, NLO effective constants – e , e , k – show up in subleading corrections. If themagnetic field is switched off, the magnetization tends to zero as it should,lim H → Σ(0 , H s , H ) = 0 . (4.20)In the limit H s →
0, the expansion in the magnetic field involves integer powers of H ,Σ(0 , , H ) = F H + H π + n e + e ) + 18 π F o H + O ( H ) . (4.21)The leading contributions are positive, whereas the sign of H -term remains open.The leading terms, however, do not involve NLO effective constants, such that themagnetization takes positive values in presence of the magnetic field. As one wouldexpect, the magnetization in the direction of the magnetic field no longer is zero, sincethe spins get tilted. We have considered the low-energy properties of antiferromagnetic films subjected tomagnetic fields perpendicular to the staggered magnetization order parameter. Withineffective field theory we have systematically derived the low-temperature expansionsfor the free energy density, pressure, order parameter, and magnetization.In presence of a weak magnetic field, the spin-wave interaction in the pressure isrepulsive, irrespective of the strength of the magnetic and staggered field. The orderparameter decreases due to the spin-wave interaction, when the temperature is raisedfrom T =0 to finite T – while keeping H s and H fixed. Finally, the magnetization –both at zero and finite temperature – takes positive values: the spins get tilted intothe direction of the external perpendicular magnetic field.At zero temperature, both the magnetization and staggered magnetization growwhen a perpendicular magnetic field is applied. While this behavior is expected for themagnetization, the enhancement of the order parameter in presence of the magneticfield comes rather unexpectedly. It implies that the phenomenon of magnetic catalysis– well-known in quantum chromodynamics, graphene and other condensed mattersystems – also emerges in antiferromagnetic films.17 cknowledgments The author thanks J. O. Andersen, A. Auerbach, T. Brauner, H. Leutwyler, I. A.Shovkovy and R. R. P. Singh for correspondence.
A Vertices with an Odd Number of Magnon Lines
Magnetic fields perpendicular to the staggered magnetization order parameter giverise to vertices that involve an odd number of magnon lines. Explicitly, vertices withone magnon line originate from iF H∂ U + 2 ik Σ s H s F H∂ U , (A.1)while vertices with three magnon lines are generated by iF H n U U ∂ U − ∂ U U U + ∂ U U U o +2 ik Σ s H s F H n U ∂ U U − ∂ U U U o − i ( e + e ) H∂ U ∂ U a ∂ U a + i ( e + e ) H n U ∂ U U + 2 U ∂ U U − ∂ U U U o − ie H∂ U ∂ r U a ∂ r U a − ie H∂ r U ∂ U a ∂ r U a . (A.2)Note that we only consider contributions from L eff and L eff – higher-order piecesof the effective Lagrangian also yield such vertices, but they do not contribute upto order p in the partition function, as we argue below. The additional Feynmandiagrams that can be constructed from the expressions (A.1) and (A.2) are depictedin Fig. 2. According to (A.1), the line emitted (or absorbed) by a one-magnon vertexalways corresponds to U . In case of a three-magnon vertex, according to (A.2), weeither have U U U or U U U – in particular, three magnons of the same type U are never emitted or absorbed simultaneously.An important observation that drastically reduces the number of additional Feyn-man graphs, is that the one-magnon vertices from L eff and L eff are irrelevant. Inthe evaluation of the partition function they lead to integrals of the form Z d x d y d z . . . ( ∂ ) G II ( x − y ) F ( y, z, . . . ) , x = ( x , x , x ) , (A.3)where ∂ is the Euclidean time derivative corresponding to the coordinate x . Thefunction F ( y, z, . . . ), depending on the topology of the diagram, may contain anarbitrary number of propagators that involve additional time and space derivatives.But the point is that – irrespective of the complexity of the diagram – the integrationover the coordinates of the first vertex, i.e., integration over the coordinates x , x , x
18f the one-magnon vertex, is identically zero. One concludes that the relevant newdiagrams must involve vertices with at least three magnon lines.This then leads to the two-loop diagrams 4 c and 4 d of Fig. 2. Any other diagramthat involves vertices with an odd number of magnon lines is at least of order p , i.e.,beyond the scope of the present study. Remarkably, the explicit evaluation of diagram4 c yields zero, z c = 0 , (A.4)while the sunset diagram contributes with z d = 2 H F Z T d d x G I ( x ) ∂ G I ( x ) ∂ G II ( x ) . (A.5)This integral over the torus T = R d s × S , with circle S defined as − β/ ≤ x ≤ β/ B Evaluation of the Sunset Diagram
In order to process the integral (A.5), we decompose the thermal propagators G I,II ( x )as G I,II ( x ) = ∆ I,II ( x ) + G I,II ( x ) , (B.1)where the ∆ I,II ( x ) are the zero-temperature propagators defined in Eq. (2.12). Theintegral then takes the form Z T d d x (cid:16) G I ∂ G I ∂ G II + ∆ I ∂ G I ∂ G II + G I ∂ ∆ I ∂ G II + G I ∂ G I ∂ ∆ II +∆ I ∂ G I ∂ ∆ II + ∆ I ∂ ∆ I ∂ G II + G I ∂ ∆ I ∂ ∆ II + ∆ I ∂ ∆ I ∂ ∆ II (cid:17) . (B.2)The first four integrals over the torus are convergent in d →
3. The four remainingintegrals that involve two or three zero-temperature propagators, however, are singularin the limit d →
3, and need to be considered in detail. Following Ref. [79], we cutout a sphere of radius | S | ≤ β/ Z T d d x ∆ I ∂ G I ∂ ∆ II = Z S d d x ∆ I ∂ G I ∂ ∆ II + Z T \S d d x ∆ I ∂ G I ∂ ∆ II , Z T d d x ∆ I ∂ ∆ I ∂ G II = Z S d d x ∆ I ∂ ∆ I ∂ G II + Z T \S d d x ∆ I ∂ ∆ I ∂ G II , Z T d d x G I ∂ ∆ I ∂ ∆ II = Z S d d x G I ∂ ∆ I ∂ ∆ II + Z T \S d d x G I ∂ ∆ I ∂ ∆ II , Z T d d x ∆ I ∂ ∆ I ∂ ∆ II = Z S d d x ∆ I ∂ ∆ I ∂ ∆ II + Z T \S d d x ∆ I ∂ ∆ I ∂ ∆ II . (B.3)19he evaluation of the integrals over the complement of the torus T \ S poses noproblems in d =3. In the integral over the sphere in line three, we subtract the piece g I = G I | x =0 , G I → G I − g I , (B.4)while in the integrals over the sphere in lines one and two, we perform the subtractions ∂ G I,II → ∂ G I,II − ∂ G I,II | x =0 × x . (B.5)Making use of ∂ G I,II ( x ) | x =0 = g I,II + M I,II g I,II ( d = 3) , (B.6)we end up with Z S d d x ∆ I ∂ G I ∂ ∆ II = Z S d d x ∆ I (cid:16) ∂ G I − x ( g I + M I g I ) (cid:17) ∂ ∆ II + Z S d d x ∆ I x ( g I + M I g I ) ∂ ∆ II , Z S d d x ∆ I ∂ ∆ I ∂ G II = Z S d d x ∆ I ∂ ∆ I (cid:16) ∂ G II − x ( g II + M II g II ) (cid:17) + Z S d d x ∆ I ∂ ∆ I x ( g II + M II g II ) , Z S d d x G I ∂ ∆ I ∂ ∆ II = Z S d d x (cid:16) G I − g I (cid:17) ∂ ∆ I ∂ ∆ II + Z S d d x g I ∂ ∆ I ∂ ∆ II . (B.7)The subtracted integrals over the sphere on the RHS are convergent in d →
3. Thesecond integrals on the RHS we decompose further as Z S d d x ∆ I x ( g I + M I g I ) ∂ ∆ II = Z R d d x ∆ I x ( g I + M I g I ) ∂ ∆ II − Z R\S d d x ∆ I x ( g I + M I g I ) ∂ ∆ II , Z S d d x ∆ I ∂ ∆ I x ( g II + M II g II ) = Z R d d x ∆ I ∂ ∆ I x ( g II + M II g II ) − Z R\S d d x ∆ I ∂ ∆ I x ( g II + M II g II ) , Z S d d x g I ∂ ∆ I ∂ ∆ II = Z R d d x g I ∂ ∆ I ∂ ∆ II − Z R\S d d x g I ∂ ∆ I ∂ ∆ II . (B.8)20he integrals over the complement R \ S are well-defined. The integrals over allEuclidean space are finite in dimensional regularization in the limit d → d → Z R d d x ∆ I x ( g I + M I g I ) ∂ ∆ II = − M I + 2 M II π ( M I + M II ) (cid:16) g I + M I g I (cid:17) , lim d → Z R d d x ∆ I ∂ ∆ I x ( g II + M II g II ) = − πM I (cid:16) g II + M II g II (cid:17) , lim d → Z R d d x g I ∂ ∆ I ∂ ∆ II = − M I + M I M II + M II π ( M I + M II ) g I . (B.9)Finally, the last integral in Eq. (B.2) that contains three zero-temperature propa-gators, is decomposed as Z T d d x ∆ I ∂ ∆ I ∂ ∆ II = Z T \S d d x ∆ I ∂ ∆ I ∂ ∆ II + Z R d d x ∆ I ∂ ∆ I ∂ ∆ II − Z R\S d d x ∆ I ∂ ∆ I ∂ ∆ II . (B.10)The integrals over T \ S and
R \ S are finite, but the integral over all Euclidean spaceis singular in d →
3. The corresponding counterterm C , C = Z R d d x ∆ I ∂ ∆ I ∂ ∆ II , (B.11)can be absorbed by NLO effective constants in z a , Eq. (3.2).In conclusion, the first four integrals in the sunset contribution, Eq. (B.2), arewell-defined and can be evaluated numerically in a straightforward manner, using thefact that the integrals are two-dimensional,d x = 2 πrdrdt . (B.12)The evaluation of the remaining four integrals in Eq. (B.2) is more subtle, but can behandled within dimensional regularization using the method established in Ref. [79].In the limit d →
3, the final – and finite – representation for the free energy density21riginating from the sunset diagram 4 d reads z d = 2 H F Z T d x T + Z T \S d x U + Z S d x V − Z R\S d x W + R ! ,T = G I ∂ G I ∂ G II + ∆ I ∂ G I ∂ G II + G I ∂ ∆ I ∂ G II + G I ∂ G I ∂ ∆ II ,U = ∆ I ∂ G I ∂ ∆ II + ∆ I ∂ ∆ I ∂ G II + G I ∂ ∆ I ∂ ∆ II + ∆ I ∂ ∆ I ∂ ∆ II ,V = ∆ I (cid:16) ∂ G I − x ( g I + M I g I ) (cid:17) ∂ ∆ II + ∆ I ∂ ∆ I (cid:16) ∂ G II − x ( g II + M II g II ) (cid:17) + (cid:16) G I − g I (cid:17) ∂ ∆ I ∂ ∆ II ,W = ∆ I x ( g I + M I g I ) ∂ ∆ II + ∆ I ∂ ∆ I x ( g II + M II g II ) + g I ∂ ∆ I ∂ ∆ II + ∆ I ∂ ∆ I ∂ ∆ II ,R = − M I + 2 M II π ( M I + M II ) (cid:16) g I + M I g I (cid:17) − πM I (cid:16) g II + M II g II (cid:17) − M I + M I M II + M II π ( M I + M II ) g I . 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