Chapter 2 Mean-Field Theory and the Gaussian Approximation

Transcription

1 Chapter Mean-Feld Theory and the Gaussan Approxmaton The Wlsonan renormalzaton group (RG) was nvented n order to study the effect of strong fluctuatons and the mutual couplng between dfferent degrees of freedom n the vcnty of contnuous phase transtons. Before embarng on the theory of the RG, let us n ths chapter descrbe two less sophstcated methods of dealng wth ths problem, namely the mean-feld approxmaton and the Gaussan approxmaton. Wthn the mean-feld approxmaton, fluctuatons of the order parameter are completely neglected and the nteractons between dfferent degrees of freedom are taen nto account n some smple average way. The Gaussan approxmaton s n some sense the leadng fluctuaton correcton to the mean-feld approxmaton. Although these methods are very general and can also be used to study quantum mechancal many-body systems, for our purpose t s suffcent to ntroduce these methods usng the nearest-neghbor Isng model n D dmensons as an example. The Isng model s defned n terms of the followng classcal Hamltonan, H = N J j s s j h j= N s. (.) Here, and j label the stes r and r j of a D-dmensonal hypercubc lattce wth N stes, the varables s =±correspond to the two possble states of the z-components of spns localzed at the lattce stes, the J j denote the exchange nteracton between spns localzed at stes r and r j, and h s the Zeeman energy assocated wth an external magnetc feld n the z-drecton. The above model s called classcal because t does not nvolve any noncommutng operators. By smply rotatng the magnetc feld n the x-drecton we obtan a quantum Isng model, as dscussed n Exercse.. The quantum mechancal orgn of magnetsm s hdden n the exchange energes J j. Due to the exponental decay of localzed wave functons, t s often suffcent to assume that the J j are nonzero only f the stes r and r j = For example, n quantum many-body systems the self-consstent Hartree Foc approxmaton can be vewed as a varant of the mean-feld approxmaton, whle the Gaussan approxmaton s usually called random phase approxmaton. Kopetz, P. et al.: Mean-Feld Theory and the Gaussan Approxmaton. Lect. Notes Phys. 798, 3 5 () DOI.7/ c Sprnger-Verlag Berln Hedelberg

2 4 Mean-Feld Theory and the Gaussan Approxmaton are nearest neghbors on the lattce. Denotng the nonzero value of J j for nearest neghbors by J, we obtan from Eq. (.) the nearest-neghbor Isng model H = J j s s j h N s. (.) Here, j denotes the summaton over all dstnct pars of nearest neghbors. In order to obtan thermodynamc observables, we have to calculate the partton functon, Z(T, h) = e β H... exp β J s s j + βh s, {s } s =± s =± s N =± j (.3) where we have ntroduced the notaton β = /T for the nverse temperature. Whle n one dmenson t s qute smple to carry out ths summaton (see Exercse.), the correspondng calculaton n D = s much more dffcult and untl now the exact Z(T, h) forh s not nown. For h = the partton functon of the twodmensonal Isng model was frst calculated by Onsager (944), who also presented an exact expresson for the spontaneous magnetzaton at a conference n 949. A proof for hs result was gven n 95 by C. N. Yang. See the textboos by Wanner (966), by Huang (987), or by Matts (6) for pedagogcal descrptons of the exact soluton for h =. In dmensons D = 3 there are no exact results avalable, so one has to rely on approxmatons. The smplest s the mean-feld approxmaton dscussed n the followng secton. =. Mean-Feld Theory.. Landau Functon and Free Energy Mean-feld theory s based on the assumpton that the fluctuatons around the average value of the order parameter are so small that they can be neglected. Let us therefore assume that the system has a fnte magnetzaton, 3 m = s {s j } e H/T s {s j } e H/T, (.4) The nverse temperature β = /T should not be confused wth the order-parameter exponent β. Because these notatons are both standard we shall adopt them here; usually the meanng of the symbol β s clear from the context. 3 For lattce models t s convenent to dvde the total magnetc moment by the number N of lattce stes and not by the total volume. For smplcty we use here the same symbol as for the magnetzaton n Eq. (.3), whch has unts of nverse volume.

3 . Mean-Feld Theory 5 where we have used the fact that by translatonal nvarance the thermal expectaton values s are ndependent of the ste label. Wrtng s = m + δs wth the fluctuaton δs = s m,wehave s s j = m + m(δs + δs j ) + δs δs j = m + m(s + s j ) + δs δs j. (.5) We now assume that the fluctuatons are small so that the last term δs δs j whch s quadratc n the fluctuatons can be neglected. Wthn ths approxmaton, the Isng Hamltonan (.) s replaced by the mean-feld Hamltonan H MF = m J j j ( h + j ) J j m s = N zj m (h + zjm)s, (.6) where the second lne s vald for the nearest-neghbor nteractons, and z = D s the number of nearest neghbors (coordnaton number) of a gven ste of a D- dmensonal hypercubc lattce. As the spns n the mean-feld Hamltonan (.6) are decoupled, the partton functon factorzes nto a product of N ndependent terms whch are just the partton functons of sngle spns n an effectve magnetc feld h eff = h + zjm, Z MF (T, h) = e β NzJm / {s } e β(h+zjm) s = e β NzJm / e β(h+zjm) + e β(h+zjm)] Wrtng ths as we obtan n mean-feld approxmaton = e β NzJm / coshβ(h + zjm)]] N. (.7) Z MF (T, h) = e β NL MF(T,h; m), (.8) L MF (T, h; m) = zj ] m T ln coshβ(h + zjm)]. (.9) As wll be explaned n more detal below, the functon L MF (T, h; m) sanexample for a Landau functon, whch descrbes the probablty dstrbuton of the order parameter: the probablty densty of observng for the order parameter the value m s proportonal to exp β NL MF (T, h; m)]. The so far unspecfed parameter m s now determned from the condton that the physcal value m of the order parameter

4 6 Mean-Feld Theory and the Gaussan Approxmaton maxmzes ts probablty dstrbuton, correspondng to the mnmum of the Landau functon, L MF (T, h; m) m =. (.) m From Eq. (.9) we then fnd that the magnetzaton m n mean-feld approxmaton satsfes the self-consstency condton m = tanhβ(h + zjm )], (.) whch defnes m = m (T, h) as a functon of T and h. The mean-feld result for the free energy per ste s thus f MF (T, h) = L MF (T, h; m (T, h)). (.) The mean-feld self-consstency equaton (.) can easly be solved graphcally. As shown n Fg.., for h the global mnmum of L MF (T, h; m) occurs always at a fnte m. On the other hand, for h = the exstence of nontrval solutons wth m depends on the temperature. In the low-temperature regme T < zj there are two nontrval solutons wth m, whle at hgh temperatures T > zj our self-consstency equaton (.) has only the trval soluton m =, see Fg... In D dmensons the mean-feld estmate for the crtcal temperature s therefore T c = zj = DJ. (.3) For D = ths s certanly wrong, because we now from the exact soluton (see Exercse.) that T c = n one dmenson. In two dmensons the exact crtcal temperature of the nearest-neghbor Isng model satsfes snh(j/t c ) = (Onsager 944), whch yelds T c.69j and s sgnfcantly lower than the mean-feld predcton of 4J. As a general rule, n lower dmensons fluctuatons are more mportant and tend to dsorder the system or at least reduce the crtcal temperature... Thermodynamc Crtcal Exponents For temperatures close to T c and small β h the value m of the magnetzaton at the mnmum of L MF (T, h; m) s small. We may therefore approxmate the Landau functon (.9) by expandng the rght-hand sde of Eq. (.9) up to fourth order n m and lnear order n h.usng ln cosh x] = ln + x x 4 + O(x 6 ), (.4)

5 . Mean-Feld Theory 7 Fg.. Graphcal soluton of the mean-feld self-consstency equaton (.) for h >. The nset shows the behavor of the correspondng Landau functon L MF (T, h; m) defnedneq.(.9).for T > T c or h suffcently large the Landau functon exhbts only one mnmum at fnte m >. For T < T c and h suffcently small, however, there s another local mnmum at negatve m, but the global mnmum of L MF (T, h; m) s stll at m > we obtan from Eq. (.9), L MF (T, h; m) = f + r m + u 4! m4 hm +..., (.5) wth f = T ln, (.6a) r = zj ( ) T zj T T c, (.6b) T ( zj ) 4 u = T Tc, (.6c) T

6 8 Mean-Feld Theory and the Gaussan Approxmaton Fg.. Graphcal soluton of the mean-feld self-consstency equaton (.) for h =. The upper fgure shows the typcal behavor n the dsordered phase T > T c, whle the lower fgure represents the ordered phase T < T c. The behavor of the Landau functon s shown n the nsets: whle for T > T c t has a global mnmum at m =, t develops for T < T c two degenerate mnma at ± m where the approxmatons are vald close to the crtcal temperature, where T T c T c and zj/t. Obvously, the sgn of the coeffcent r changes at T = T c, so that for h = the global mnmum of L MF (T, h; m) fort > T c evolves nto a local maxmum for T < T c, and two new mnma emerge at fnte values of m, as shown n Fg... The crucal pont s now that for a small reduced temperature t = (T T c )/T c the value of m at the mnma of L MF (T, h; m) s small compared wth unty, so that our expanson (.5) n powers of m s justfed a posteror. Tang the dervatve of Eq. (.5) wth respect to m, t s easy to see that Eq. (.) smplfes to L MF (T, h; m) m = rm + u m 6 m3 h =. (.7)

7 . Mean-Feld Theory 9 The behavor of thermodynamc observables close to T c s now easly obtaned: (a) Spontaneous magnetzaton: Settng h = n Eq. (.7) and solvng for m,we obtan for T T c (wth r ), 6r m = ( t) /. (.8) u Comparng ths wth the defnton (.6) of the crtcal exponent β, we conclude that the mean-feld approxmaton predcts for the Isng unversalty class β = /, ndependently of the dmenson D. (b) Zero-feld susceptblty: To obtan the mean-feld result for the susceptblty exponent γ, we note that for small but fnte h and T T c we may neglect the terms of order m 3 n Eq. (.7), so that m (h) h/r, and hence the zero-feld susceptblty behaves for t as χ = m (h) h h= r. (.9) T T c It s a smple exercse to show that χ T T c also holds for T < T c.the susceptblty exponent s therefore γ = wthn mean-feld approxmaton. (c) Crtcal sotherm: The equaton of state at the crtcal pont can be obtaned by settng r = n Eq. (.7), mplyng m (h) ( ) h /3, (.) u and hence the mean-feld result δ = 3. (d) Specfc heat: The specfc heat C per lattce ste can be obtaned from the thermodynamc relaton C = T f MF (T, h), (.) T where the mean-feld free energy per ste f MF (T, h) s gven n Eq. (.). Settng h =, we fnd from Eq. (.7) for T > T c that f MF (t, ) = f because m = n ths case. On the other hand, for T < T c we may substtute Eq. (.8), so that f MF (t, ) = f 3 r u, T < T c. (.) Settng r T T c and tang two dervatves wth respect to T, we obtan C T c f T, T > T c, (.3a)

8 3 Mean-Feld Theory and the Gaussan Approxmaton T c f T + 3 T c u, T < T c. (.3b) Note that accordng to Eq. (.6c) u T c so that 3T c /u 3/. We conclude that wthn the mean-feld approxmaton the specfc heat s dscontnuous at T c, so that C t, mplyng α =. Note that the mean-feld results are consstent wth the scalng relatons (.3) and (.4), α = β + γ = β(δ + ). In order to obtan the exponents ν and η, we need to calculate the correlaton functon G(r), whch we shall do n Sect..3 wthn the so-called Gaussan approxmaton. However, a comparson of the mean-feld results α =, β = /, γ =, and δ = 3 wth the correct values gven n Table. at the end of Sect.. shows that the mean-feld approxmaton s not sutable to obtan quanttatvely accurate results, n partcular n the physcally accessble dmensons D = and D = 3.. Gnzburg Landau Theory Our ultmate goal wll be to develop a systematc method for tang nto account the fluctuatons neglected n mean-feld theory, even f the fluctuatons are strong and qualtatvely change the mean-feld results. As a frst step to acheve ths ambtous goal, we shall n ths chapter derve from the mcroscopc Isng Hamltonan (.) an effectve (classcal) feld theory whose felds ϕ(r) represent sutably defned spatal averages of the fluctuatng magnetzaton over suffcently large domans such that ϕ(r) vares only slowly on the scale of the lattce spacng. The concept of an effectve feld theory representng coarse-graned fluctuatons averaged over larger and larger length scales les at the heart of the Wlsonan RG dea. It turns out that n the vcnty of the crtcal pont the form of the effectve coarse-graned acton Sϕ] s constraned by symmetry and can be wrtten down on the bass of phenomenologcal consderatons. Such a strategy has been adopted by Gnzburg and Landau (95) to develop a phenomenologcal theory of superconductvty whch s partcularly useful to treat spatal nhomogenetes... Exact Effectve Feld Theory To derve the effectve order-parameter feld theory for the Isng model, let us wrte the partton functon of the Hamltonan (.) n compact matrx form, Z = exp β J j s s j + βh {s } j s = {s } ] exp st Js + h T s, (.4)

9 . Gnzburg Landau Theory 3 where s and h are N-dmensonal column vectors wth s] = s and h] = βh, and J s an N N-matrx wth matrx elements J] j = J j = β J j.wenowuse the followng mathematcal dentty for N-dmensonal Gaussan ntegrals, vald for any postve symmetrc matrx A, ( N = dx π ) e xt Ax+x T s = det A] / e st A s, (.5) to wrte the Isng nteracton n the followng form (we dentfy J = A ) e st Js = where we have ntroduced the notaton Dx]exp xt J x + x T s ] Dx]exp xt J x ], (.6) Dx] N = dx π. (.7) In fact, the dentty (.6) can be easly proven wthout usng the formula (.5) by redefnng the ntegraton varables n the numerator va the shft x = x + Js and completng the squares, (x + Js) T J (x + Js) + (x + Js) T s = x T J x + st Js. (.8) In a sense, Eq. (.6) amounts to readng the Gaussan ntegraton formula from rght to left, ntroducng an auxlary ntegraton to wrte the rght-hand sde n terms of a Gaussan ntegral. Analogous transformatons turn out to be very useful to ntroduce sutable collectve degrees of freedom n quantum mechancal many-body systems; n ths context transformatons of the type (.6) are called Hubbard Stratonovch transformatons (Hubbard 959, Stratonovch 957, Kopetz 997). The Isng partton functon (.4) can now be wrtten as Z = Dx]exp xt J x ] {s } exp ( h + x) T s ] Dx]exp xt J x ]. (.9) For a gven confguraton of the Hubbard Stratonovch feld x, the spn summaton n the numerator factorzes agan n a product of ndependent terms, each descrbng the partton functon of a sngle spn n a ste-dependent magnetc feld h + x /β. Therefore, ths spn summaton can easly be carred out,

10 3 Mean-Feld Theory and the Gaussan Approxmaton exp ( h + x) T s ] N = {s } = s =± e (βh+x )s = N cosh(βh + x )] = N ] = exp ln cosh(βh + x )]. (.3) = The partton functon (.9) can thus be wrtten as Dx]e Sx] Z = Dx]exp xt J x ] = det J Dx]e Sx], (.3) where Sx] = xt J x N ln cosh(βh + x )]. (.3) = To understand the physcal meanng of the varables x, we calculate the expectaton value of ts -th component x S Dx]e Sx] x Dx]e Sx]. (.33) Introducng an auxlary N-component column vector y = (y,...,y N ) T to wrte x = lm e xt y, (.34) y y and performng the Gaussan ntegraton usng Eq. (.6), we have the followng chan of denttes, x S = lm y = lm y = y Dx]exp xt J x ] {s } exp ( h + x) T s + x T y ] Dx]e Sx] {s } exp (s + y)t J(s + y) + h T s ] {s } exp st Js + h T s ] y {s } e β H Js] {s } e β H = Js], (.35) or n vector notaton, x S = J s. (.36)

11 . Gnzburg Landau Theory 33 In order to ntroduce varables ϕ whose expectaton value can be dentfed wth the average magnetzaton m = s, we smply defne so that the expectaton value of the -th component of ϕ s ϕ = J x, (.37) ϕ S = J x S] = s =m. (.38) Substtutng ϕ = J x n Eqs. (.3) and (.3), we fnally obtan the followng exact representaton of the partton functon, Z = Dϕ]e Sϕ] ] = det J Dϕ]exp ϕt Jϕ Dϕ]e Sϕ], (.39) where the effectve acton Sϕ] Sx Jϕ] s gven by Sϕ] = β J j ϕ ϕ j j N ( ln cosh β h + = N j= ) ] J j ϕ j. (.4) Equaton (.39) expresses the partton functon of the Isng model n terms of an N-dmensonal ntegral over varables ϕ whose expectaton value s smply the magnetzaton per ste. The ntegraton varables ϕ can therefore be nterpreted as the fluctuatng magnetzaton. The nfnte-dmensonal ntegral obtaned from Eq. (.39) n the lmt N s an example of a functonal ntegral. The resultng effectve acton Sϕ] then defnes a (classcal) effectve feld theory for the orderparameter fluctuatons of the Isng model. We shall refer to the ntegraton varables ϕ as the felds of our effectve feld theory. It should be noted that Eq. (.38) does not precsely generalze to hgher order correlaton functons. If we proceed as above and use the dentty x x...x n = lm y y... e xt y y y n (.4) to carry out the same transformatons as n Eq. (.35), we arrve at ϕ ϕ...ϕ n S = s s...s n + { averages nvolvng at most } n ofthespnss,..., s n, (.4) where the lower order correlaton functons nvolvng n and less spn varables arse from the repeated dfferentaton of the y-dependent terms n the exponent of the second lne of Eq. (.35). Fortunately, for short-range J j these addtonal terms

12 34 Mean-Feld Theory and the Gaussan Approxmaton can be neglected as long as the dstances between the external ponts,..., n are large compared wth the range of J j. For example, for n = wehave ϕ ϕ j S = s s j J ] j s s j. (.43) We thus conclude that we may also use the effectve acton Sϕ] gven n Eq. (.4) to calculate the long dstance behavor of correlaton functons nvolvng two and more spns... Truncated Effectve Acton: ϕ 4 -Theory The effectve acton gven n Eqs. (.39) and (.4) s formally exact but very complcated. In order to mae progress, let us assume that the ntegral (.39) s domnated by confguratons where the ntegraton varables ϕ are n some sense small, so that we may expand the second term n the effectve acton (.4) n powers of the ϕ, truncatng the expanson at fourth order. Keepng n mnd that physcally ϕ represents the fluctuatng magnetzaton, we expect that ths truncaton can only be good n the vcnty of the crtcal pont, where the magnetzaton s small. Whether or not ths truncaton s suffcent to obtan quanttatvely correct results for the crtcal exponents s a rather subtle queston whch wll be answered wth the help of the RG. 4 Wth the expanson (.4) we obtan Sϕ] = N ln + β + β4 j h + j J j ϕ ϕ j β h + j J j ϕ j ] J j ϕ j ] 4 + O(ϕ 6 ). (.44) Snce our lattce model has dscrete translatonal nvarance, we may smplfy Eq. (.44) by Fourer transformng the varables ϕ to wave vector space, defnng ϕ = e r ϕ, (.45) N where the wave vector sum s over the frst Brlloun zone of the lattce, whch may be chosen as μ < π/a, where a s the lattce spacng and μ =,...,D labels the components. Imposng for convenence perodc boundary condtons, the 4 It turns out that the quartc truncaton of the expanson of Eq. (.4) can be formally justfed close to four dmensons. In the physcally most nterestng dmenson D = 3 the term nvolvng sx powers of the ϕ cannot be neglected. However the truncated ϕ 4 -theory s n the same unversalty class as the orgnal theory and the effect of all couplngs neglected can be absorbed by a redefnton of the remanng couplngs.

13 . Gnzburg Landau Theory 35 wave vectors are quantzed as μ = πn μ /L, wth n μ =,,...,N μ, where N μ s the number of lattce stes n drecton μ such that D μ= N μ = N s the total number of lattce stes. Usng the dentty N e ( ) r = δ,, (.46) we obtan for the Fourer transform of the terms on the rght-hand sde of Eq. (.44), β β 4 β J j ϕ ϕ j = β J ϕ ϕ, j (.47) ] β J j ϕ j = J J ϕ ϕ, (.48) j ] 4 β 4 J j ϕ j = j N,, 3, 4 δ , J J J 3 J 4 ϕ ϕ ϕ 3 ϕ 4, (.49) where J s the Fourer transform of the exchange couplngs J j J(r r j ), J = e r J(r ). (.5) Because ϕ and J j are real and J( r) = J(r), we have ϕ = ϕ and J = J.In Fourer space Eq. (.44) thus reduces to Sϕ] = Nln β J = h Nϕ = + β J ( β J )ϕ ϕ + β4 N,, 3, 4 δ , J J J 3 J 4 ϕ ϕ ϕ 3 ϕ 4 +O(ϕ 6, h, hϕ 3 ). (.5) Fnally, we antcpate that suffcently close to the crtcal pont only long-wavelength fluctuatons (correspondng to small wave vectors) are mportant. We therefore expand the functon J appearng n Eq. (.5) n powers of. From Eq. (.5) t s easy to show that for nearest-neghbor nteractons on a D-dmensonal hypercubc lattce wth coordnaton number z = D, J = Jz a ] + O( 4 ) = T c a ] + O( 4 ), (.5) z

14 36 Mean-Feld Theory and the Gaussan Approxmaton where T c = zj s the mean-feld result for the crtcal temperature, see Eq. (.3). The coeffcent of the quadratc term n Eq. (.5) can then be wrtten as β J ( β J ) = a (r + c ) + O( 4 ), (.53) wherewehaveassumedthat T T c T c and the constants r and c are defned by r = T T c, a T c (.54) c = z = D. (.55) In the lmt of nfnte volume V = Na D, the dscrete set of allowed wave vectors merges nto a contnuum, so that the momentum sums can be replaced by ntegratons accordng to the followng prescrpton, V d D (π) D. (.56) It s then convenent to normalze the felds dfferently, ntroducng a new (contnuum) feld ϕ()va 5 Defnng ϕ() = a V ϕ. (.57) f = N V ln = a D ln (.58) and the (dmensonful) couplng constants u = a D 4 (β J = ) 4 a D 4, (.59) h = β J = h βh, a +D/ a+d/ (.6) where the approxmatons are agan vald for T T c T c, we fnd that our lattce acton Sϕ] n Eq. (.5) reduces to 5 The dfferent normalzatons of lattce and contnuum felds are represented by dfferent postons of the momentum labels: whle n the lattce normalzaton the momentum label s attached to the dmensonless feld ϕ as a subscrpt, the label of the contnuum feld ϕ() s wrtten n bracets after the feld symbol.

15 . Gnzburg Landau Theory 37 S Λ ϕ] = Vf h ϕ( = ) + r + c ] ϕ( )ϕ() + u (π) D δ( )ϕ( )ϕ( )ϕ( 3 )ϕ( 4 ), 4! 4 3 (.6) where t s understood that the momentum ntegratons n Eq. (.6) have an mplct ultravolet cutoff < Λ a whch taes nto account that n dervng Eq. (.6) we have expanded J for small wave vectors. The functonal S Λ ϕ] s called the Gnzburg Landau Wlson acton and descrbes the long-wavelength order-parameter fluctuatons of the D-dmensonal Isng model, n the sense that the ntegraton over the felds ϕ() yelds the contrbuton of the assocated longwavelength fluctuatons to the partton functon. The contrbuton of the neglected short-wavelength fluctuatons to the partton functon can be taen nto account mplctly by smply redefnng the feld-ndependent part f and the couplng constants r, c, and u of our effectve acton (.6). Moreover, also the prefactor det J = e Tr ln J of Eq. (.39) can be absorbed nto a redefnton of the feldndependent constant, f V Tr ln J f. Actually, for perodc boundary condtons the egenvalues of the matrx J are smply gven by J /T J = /T c =, so that at long wavelengths and to leadng order n T T c we may approxmate the factor det J by unty. The functonal ntegral (.39) representng the partton functon of the Isng model can thus be wrtten as Z = Dϕ]e S Λ ϕ]. (.6) It s sometmes more convenent to consder the effectve acton S Λ ϕ] n real space. Defnng the real-space Fourer transforms of the contnuum felds ϕ() va ϕ(r) = e r ϕ(), (.63) and usng the dentty e r = δ(r), (.64) we obtan from Eq. (.6), S Λ ϕ] = d D r f + r ϕ (r) + c ] u ] ϕ(r) + 4! ϕ4 (r) h ϕ(r). (.65) For obvous reasons the classcal feld theory defned by ths expresson s called ϕ 4 -theory. In the frst part of ths boo, we shall use ths feld theory to llustrate the

16 38 Mean-Feld Theory and the Gaussan Approxmaton man concepts of the RG. Strctly speang, the dentty (.64) s not qute correct f we eep n mnd that we should mpose an mplct ultravolet cutoff Λ on the -ntegraton. Roughly, the contnuum feld ϕ(r) corresponds to the coarse-graned magnetzaton, whch s averaged over spatal regons wth volume of order Λ D contanng (Λ a) D spns. The quantty e S Λ ϕ] s proportonal to the probablty densty for observng an order-parameter dstrbuton specfed by the feld confguraton ϕ(r). Accordng to Eq. (.6), the partton functon s gven by the average of e S Λ ϕ] over all possble order-parameter confguratons. Wth ths probablstc nterpretaton of the effectve acton (.65), we can now gve a more satsfactory nterpretaton of the mean-feld Landau functon L MF (T, h; m) defned n Eqs. (.8) and (.9). Let us therefore evaluate the functonal ntegral (.6) n saddle pont approxmaton, where the entre ntegral s estmated by the ntegrand at a sngle constant value ϕ of the feld whch mnmzes the acton S Λ ϕ]. Settng ϕ(r) ϕ n Eq. (.6) we obtan Z d ϕ π e S Λ ϕ], (.66) wth S Λ ϕ] = V f + r ϕ + u ] 4! ϕ4 h ϕ. (.67) Note that n Eq. (.66) we stll ntegrate over all confguratons of the homogeneous components ϕ of the order-parameter feld; the quantty e S Λ ϕ] s therefore proportonal to the probablty of observng for the homogeneous component of the order parameter the value ϕ. Requrng consstency of Eq. (.67) wth our prevous defntons (.8) and (.5) of the Landau functon, we should dentfy S Λ ϕ] = β NL MF (T, h; m) = β N f + r m + u ] 4! m4 hm, (.68) whch s ndeed the case f ϕ = a D/ m, (.69) eepng n mnd the defntons of f, r, and u n Eqs. (.6a), (.6b), and (.6c) on the one hand, and of f, r, and u n Eqs. (.58), (.54), and (.59) on the other hand. For V the value of the one-dmensonal ntegral (.66) s essentally determned by the saddle pont of the ntegrand, correspondng to the most probable value of ϕ. The physcal value ϕ of the order-parameter feld s therefore fxed by the saddle pont condton, S Λ ϕ] ϕ = r ϕ + u ϕ 6 ϕ3 h =, (.7)

17 .3 The Gaussan Approxmaton 39 whch s equvalent to the mean-feld self-consstency equaton (.7). In summary, the mean-feld Landau functon defned n Eq. (.8) can be recovered from the functonal ntegral representaton (.6) of the partton functon by gnorng spatal fluctuatons of the order parameter; the correspondng saddle pont equaton s equvalent to the mean-feld self-consstency equaton (.). Because of ts probablstc nterpretaton the Landau functon cannot be defned wthn the framewor of thermodynamcs, whch only maes statements about averages. The concept of a Landau functon s further llustrated n Exercse., where the Landau functons of nonnteractng bosons are defned and calculated..3 The Gaussan Approxmaton Whle the mean-feld approxmaton amounts to evaluatng the functonal ntegral (.6) n saddle pont approxmaton, the Gaussan approxmaton retans quadratc fluctuatons around the saddle pont and thus ncludes the lowest-order correcton to the mean-feld approxmaton n an expanson n fluctuatons around the saddle pont. In the feld-theory language, the Gaussan approxmaton corresponds to descrbng fluctuatons n terms of a free feld theory, where the fluctuatons wth dfferent momenta and frequences are ndependent. In condensed matter physcs, the Gaussan approxmaton s closely related to the so-called random phase approxmaton. It turns out that qute generally the Gaussan approxmaton s only vald f the dmensonalty D of the system s larger than a certan upper crtcal dmenson D up, whch we have already encountered n the context of the scalng hypothess n Sect..3, see the dscusson after Eq. (.3). Because for the Isng unversalty class D up = 4, the Gaussan approxmaton s not suffcent to descrbe the crtcal behavor of Isng magnets n expermentally accessble dmensons. Nevertheless, n order to motvate the RG t s nstructve to see how and why the Gaussan approxmaton breas down for D < D up. For smplcty, we set h = n ths secton..3. Gaussan Effectve Acton To derve the Gaussan approxmaton, we go bac to our Gnzburg Landau Wlson acton S Λ ϕ] defned n Eq. (.65). Let us decompose the feld ϕ(r) descrbng the coarse-graned fluctuatng magnetzaton as follows, or n wave vector space ϕ(r) = ϕ + δϕ(r), (.7) ϕ() = (π) D δ() ϕ + δϕ(). (.7) Here, ϕ s the mean-feld value of the order parameter satsfyng the saddle pont equaton (.7) and δϕ(r) descrbes nhomogeneous fluctuatons around the saddle

18 4 Mean-Feld Theory and the Gaussan Approxmaton pont. Substtutng Eq. (.7) nto the acton (.65) and retanng all terms up to quadratc order n the fluctuatons, we obtan n momentum space S Λ ϕ + δϕ] V f + r ϕ + u ] 4! ϕ4 + r ϕ + u ] 6 ϕ3 δϕ( = ) + r + u ϕ + c ] δϕ( )δϕ(). (.73) Ths s the Gaussan approxmaton for the Gnzburg Landau Wlson acton. To further smplfy Eq. (.73), we note that the second lne on the rght-hand sde of Eq. (.73) vanshes because the coeffcent of δϕ( = ) satsfes the saddle pont condton (.7). The frst lne on the rght-hand sde of Eq. (.73) can be dentfed wth the mean-feld free energy. Explctly substtutng for ϕ n Eq. (.73) the saddle pont value, ϕ = { for r >, 6r /u for r <, (.74) we obtan for the effectve acton n Gaussan approxmaton for T > T c, where ϕ = and δϕ = ϕ, S Λ ϕ] = Vf + On the other hand, for T < T c, where r <, we have r + c ] ϕ( )ϕ(), T > T c. (.75) and hence S Λ ϕ] = V f 3 r u r r ϕ + u 4! ϕ4 = 3, u (.76) r + u ϕ = r, (.77) ] + r + c ] δϕ( )δϕ(), T < T c. (.78) Wth the help of the Gaussan effectve acton gven n Eqs. (.75) and (.78), we may now estmate the effect of order-parameter fluctuatons on the mean-feld results for the thermodynamc crtcal exponents. Moreover, because the Gaussan approxmaton taes the spatal nhomogenety of the order-parameter fluctuatons nto account, we may also calculate the crtcal exponents ν and η assocated wth the order-parameter correlaton functon.

19 .3 The Gaussan Approxmaton 4.3. Gaussan Correctons to the Specfc Heat Exponent Because n the thermodynamc lmt both the Gaussan approxmaton and the meanfeld approxmaton predct the same contrbuton from the homogeneous fluctuatons represented by ϕ to the free energy 6, Gaussan fluctuatons do not modfy the mean-feld predctons for the exponents β, γ, and δ, whch are related to the homogeneous order-parameter fluctuatons. Wthn the Gaussan approxmaton, we therefore stll obtan β = /, γ =, and δ = 3. On the other hand, the Gaussan approxmaton for the specfc heat exponent α s dfferent from the mean-feld predcton α =, because the fluctuatons wth fnte wave vectors gve a nontrval contrbuton Δf to the free energy per lattce ste, whch accordng to Eqs. (.75) and (.78) can be wrtten as e β NΔf = Dϕ]exp c ] (ξ + )δϕ( )δϕ(). (.79) Here, we have ntroduced the length ξ va { c ξ = r for T > T c, r for T < T c. (.8) In Sect..3.3 we shall dentfy the length ξ wth the order-parameter correlaton length ntroduced n Sect... Recall that for T > T c we may wrte δϕ() = ϕ() because n ths case ϕ =. To evaluate the Gaussan ntegral n Eq. (.79), t s convenent to dscretze the ntegral n the exponent wth the help of the underlyng lattce and use the assocated lattce normalzaton of the felds, whch accordng to Eq. (.57) amounts to the substtuton δϕ() = a V δϕ. Then we obtan from Eq. (.79), e β NΔf dδϕ = dreδϕ dimδϕ π, ˆn> π π ] exp c a (ξ + ) δϕ, (.8) where the product n the square braces s restrcted to those wave vectors whose projecton ˆn onto an arbtrary drecton ˆn s postve. Ths restrcton s necessary n order to avod double countng of the fnte -fluctuatons, whch are represented by a real feld ϕ whose Fourer components satsfy ϕ = ϕ. Snce the ntegratons 6 Recall that the mean-feld approxmaton for the free energy s f MF (t, ) = f for T > T c,and f MF (t, ) = f 3 r u for T < T c, see Eq. (.). Tang the dfferent normalzaton of S Λ ϕ] nto account, these expressons correspond to the feld-ndependent terms on the rght-hand sdes of Eqs. (.75) and (.78).

20 4 Mean-Feld Theory and the Gaussan Approxmaton n Eq. (.8) treat the real and the magnary parts of δϕ as ndependent varables, we have to correct for ths double countng by ntegratng only over half of the possble values of. The functonal ntegral s now reduced to a product of onedmensonal Gaussan ntegrals, whch are of course a specal case of our general Gaussan ntegraton formula (.5) for N =, dx π e A x = A / = e ln A. (.8) We thus obtan from Eq. (.8) for the correcton from Gaussan fluctuatons to the free energy per lattce ste, Δf = T N ] ln c a (ξ + ), (.83) where t s understood that the sum s regularzed va an ultravolet cutoff <Λ. The correspondng correcton to the specfc heat per lattce ste s ΔC = T Δf T T = T N = T T = T N { ξ ] N { T T ] ln c a (ξ + ) + T ] } ln c a (ξ + ) T ξ + ] T T ξ ] ξ ] N } ξ + ξ +. (.84) In the thermodynamc lmt, the lattce sums can be converted nto ntegrals usng Eq. (.56) so that for n =, lm N N ξ + ] = a D n ξ + ] n Λ = K D a D d ξ + ] = K Da D ξ n D n D Here, the numercal constant K D s defned by K D = Λ ξ x D dx + x ]. (.85) n Ω D (π) = D D π D/ Γ (D/), (.86) where Ω D s the surface area of the D-dmensonal unt sphere. Usng the fact that accordng to Eqs. (.54) and (.8) the dervatve of the square of the nverse correlaton length wth respect to temperature s gven by

21 .3 The Gaussan Approxmaton 43 T ξ = c r T, for T > T c, for T < T = c c r T A t c a T c, (.87) where, for T > T c, A t =, for T < T c, (.88) we obtan from Eq. (.84) for t T T c /T c, ΔC = K D A t c K D A t c ( ) ξ 4 D Λ ξ x D dx a + x ] ( ) a D Λ ξ dx x D ξ + x ]. (.89) Keepng n mnd that close to the crtcal pont the dmensonless parameter Λ ξ s large compared wth unty, t s easy to see that the second term on the rghthand sde of Eq. (.89) can be neglected n comparson wth the frst one, because the dmensonless ntegral s for D > proportonal to (Λ ξ) D, and has for D < a fnte lmt for Λ ξ. On the other hand, the behavor of the frst term on the rght-hand sde of Eq. (.89) depends on whether D s larger or smaller than the upper crtcal dmenson D up = 4. For D > D up the ntegral depends on the ultravolet cutoff and s proportonal to (Λ ξ) D 4, so that ΔC (ξ/a) 4 D (Λ ξ) D 4 = (Λ a) D 4 whch gves rse to a fnte correcton to the specfc heat. Hence, for D > 4 long-wavelength Gaussan fluctuatons merely modfy the sze of the jump dscontnuty n the specfc heat, but do not qualtatvely modfy the mean-feld result α =. On the other hand, for D < 4 the frst ntegral on the rght-hand sde of Eq. (.89) remans fnte for Λ ξ, I D = x D (D )π dx = + x ] 4sn( (D )π ), (.9) so that to leadng order for large ξ r / t / the contrbuton from Gaussan fluctuatons to the specfc heat s ΔC = K D I A t D c ( ) ξ 4 D t ( D/). (.9) a Ths s more sngular than the jump dscontnuty predcted by mean-feld theory and mples α = D/. We thus conclude that n Gaussan approxmaton the specfc heat exponent s gven by

22 44 Mean-Feld Theory and the Gaussan Approxmaton α = { D for D < D up = 4, for D D up. (.9) It should be noted that wthn the Gaussan model consdered here the scalng relatons α = β + γ = β(δ + ) see Eqs. (.3) and (.4)] are volated for D < 4. Ths falure of the Gaussan approxmaton wll be further dscussed n Sect Correlaton Functon Next, we calculate the order-parameter correlaton functon G(r r j ) of the Isng model and the assocated crtcal exponents ν and η n Gaussan approxmaton. We defne G(r r j ) va the followng thermal average, G(r r j ) = a D δs δs j a D {s } e β H δs δs j {s } e β H, (.93) where δs = s s =s m s the devaton of the mcroscopc spn from ts average. Usng the dentty (.43) we can express G(r r j ) for dstances large compared wth the range of the J j as a functonal average over all confguratons of the fluctuatng order-parameter feld, G(r r j ) = a D δϕ δϕ j S a D Dϕ]e Sϕ] δϕ δϕ j Dϕ]e Sϕ]. (.94) The prefactor a D s ntroduced such that G(r r j ) can also be wrtten as G(r r j ) = δϕ(r )δϕ(r j ) S, (.95) where the contnuum feld ϕ(r) s related to the real-space lattce feld ϕ va ϕ(r ) = a D/ ϕ, (.96) whch follows from the defntons (.63), (.45), and (.57), see also Eq. (.69). Expandng δϕ and δϕ j n terms of ther Fourer components δϕ defned analogously to Eq. (.45), we obtan G(r r j ) = a V = V, e ( r + r j ) δϕ δϕ S e (r r j ) G( ), (.97)

23 .3 The Gaussan Approxmaton 45 where G( ) = a δϕ δϕ S = a δϕ S (.98) s the Fourer transform of G(r r j ). In the second lne of Eq. (.97) we have used the fact that by translatonal nvarance the functonal average s only nonzero f the total wave vector + vanshes, δϕ δϕ S = δ, δϕ δϕ S. (.99) In the nfnte volume lmt, t s more convenent to wor wth the contnuum felds ϕ() = a V ϕ ntroduced n Eq. (.57); then the relatons (.98) and (.99) can be wrtten n the compact form, 7 δϕ( )δϕ( ) S = (π) D δ( + )G( ). (.) Let us now evaluate the functonal average n Eq. (.98) wthn the Gaussan approxmaton where the long-wavelength effectve acton S Λ ϕ] sgvenby Eqs. (.75) and (.78). Smlar to Eq. (.8), we wrte the Gaussan effectve acton for fnte V as S Λ ϕ] S Λ ϕ ] + c a (ξ + ) δϕ = c a ( ξ + ) δϕ + terms ndependent of δϕ, (.) where n the second lne we have used the fact that the terms = and = gve the same contrbuton to the sum. Usng the fact that wthn the Gaussan approxmaton fluctuatons wth dfferent wave vectors are decoupled, the evaluaton of the functonal average n Eq. (.98) s now reduced to smple one-dmensonal Gaussan ntegratons. Renamng agan, we see that the Gaussan approxmaton G () for the Fourer transform of the correlaton functon G()sfor, G () = a = = dreδϕ π dreδϕ π dρρ 3 e c (ξ + )ρ dρρe c (ξ + )ρ dimδϕ π δϕ exp c a (ξ + ) δϕ ] dimδϕ π exp c a (ξ + ) δϕ ] c (ξ + ) = A t r + c, (.) 7 In a fnte volume V the sngular expresson (π) D δ( = ) should be regularzed wth the volume of the system, (π) D δ( = ) V.

24 46 Mean-Feld Theory and the Gaussan Approxmaton where A t s defned n Eq. (.88) and n the second lne we have ntroduced ρ = a δϕ as a new ntegraton varable. In partcular, at the crtcal pont, where r =, we obtan G () = c, T = T c. (.3) Comparng ths wth the defnton of the correlaton functon exponent η n Eq. (.4), we conclude that n Gaussan approxmaton η =. (.4) Next, let us calculate the correlaton length exponent ν and justfy our dentfcaton of the length ξ defned n Eq. (.8) wth the correlaton length. Therefore, we calculate the correlaton functon G(r) n real space and compare the result wth Eq. (.). For V the Fourer transformaton of the correlaton functon n Gaussan approxmaton can be wrtten as G (r) = e r G () = c e r ξ +. (.5) Although there s an mplct ultravolet cutoff Λ hdden n our notaton, for mn{ r,ξ} Λ the ntegral s domnated by wave vectors much smaller than Λ, so that we may formally move the cutoff to nfnty. To evaluate the D-dmensonal Fourer transform n Eq. (.5), let us denote by the component of parallel to the drecton of r, and collect the other components of nto a (D )-dmensonal vector. Then the ntegraton over can be performed usng the theorem of resdues. Introducng (D )-dmensonal sphercal coordnates n -space we obtan ] G (r) = K exp ξ + r D d D. (.6) c ξ + Although for general D the ntegraton cannot be performed analytcally, the leadng asymptotc behavor for r ξ and for r ξ s easly extracted. In the regme r ξ, the exponental factor cuts off the -ntegraton at / r. Snce for r ξ ths cutoff s large compared wth /ξ, we may neglect ξ as compared wth n the ntegrand.8 The ntegral can then be expressed n terms of the Gammafuncton Γ (D ) and we obtan 8 For D ths approxmaton s not justfed because t generates an artfcal nfrared dvergence. In ths case a more careful evaluaton of the ntegral (.6) s necessary. Because for D our smple Gaussan approxmaton s not justfed anyway, we focus here on D >.

25 .3 The Gaussan Approxmaton 47 G (r) K D c Γ (D ) r D, for r ξ. (.7) In partcular, at the crtcal pont where ξ =, ths expresson s vald for all r. A comparson wth Eq. (.3) confrms agan η =. In the opposte lmt r ξ the ntegraton n Eq. (.5) s domnated by the regme (ξ r ) / ξ. Then we may gnore the -term as compared wth ξ n the denomnator of Eq. (.6) and approxmate n the exponent, so that ξ + r r ξ + ξ r, (.8) G (r) K D ξe r /ξ c d D ξ r e. (.9) The Gaussan ntegraton s easly carred out and we fnally obtan G (r) K ( D D ) e r /ξ D 3 Γ, for r ξ. (.) c ξ D 3 r D Ths has the same form as postulated n Eq. (.), justfyng the dentfcaton of ξ n Eq. (.8) wth the order-parameter correlaton length. Hence, wthn Gaussan approxmaton we obtan c ξ = A t r t /, (.) mplyng for the correlaton length exponent n Gaussan approxmaton ν = /. (.) Note that for D < D up = 4, where the Gaussan approxmaton yelds α = D/ for the specfc heat exponent see Eq. (.9)], the Gaussan results η = and ν = / are also consstent wth the hyper-scalng relatons (.9) and (.3) connectng ν and η wth the thermodynamc crtcal exponents, α = Dν = D/ and γ = ( η)ν = ( ) =..3.4 Falure of the Gaussan Approxmaton n D < 4 The Gaussan approxmaton amounts to the quadratc truncaton (.73) of the Gnzburg Landau Wlson acton Sϕ] defned n Eq. (.4). The mportant queston s now whether the Gaussan approxmaton s suffcent to calculate the crtcal

26 48 Mean-Feld Theory and the Gaussan Approxmaton exponents or not. As already mentoned, the answer depends crucally on the dmensonalty of the system: only for D > D up = 4 the crtcal exponents obtaned wthn the Gaussan approxmaton are correct, whle for D < D up the Gaussan approxmaton s not suffcent. To understand the specal role of the upper crtcal dmenson D up = 4 for the Isng model, let us attempt to go beyond the Gaussan approxmaton by retanng the quartc term n the expanson of Sϕ] npowers of the feld. As shown n Sect..., for small wave vectors our effectve acton then reduces to the ϕ 4 -theory S Λ ϕ] defned va Eq. (.6) or (.65). At the frst sght t seems that f we assume that the couplng constant u assocated wth the quartc term n Eqs. (.6) and (.65) s arbtrarly small, then we may calculate the correctons to the Gaussan approxmaton perturbatvely n powers of u. However, ths strategy fals n D < 4 n the vcnty of the crtcal pont, because the effectve dmensonless parameter whch s relevant for the perturbatve expanson s ū = u ξ 4 D c = (D) ( ξ a ) 4 D, (.3) where we have used Eq. (.59) to approxmate u a D 4. The fact that the perturbatve expanson of the nteracton n the effectve acton (.6) s controlled by the dmensonless couplng ū can be made manfest by ntroducng dmensonless wave vectors = ξ, (.4) and dmensonless felds ϕ( ) = c ξ +D ϕ( /ξ). (.5) Assumng for smplcty T > T c so that r = c /ξ, t s easy to show that n terms of these dmensonless varables our effectve Gnzburg Landau Wlson acton (.6) taes the form S Λ ϕ] = Vf + + ] ϕ( ) ϕ( ) +ū (π) D δ( ) ϕ( ) ϕ( ) ϕ( 3 ) ϕ( 4 ). 4! 3 4 (.6) Because the coeffcent of the quadratc fluctuatons n ths expresson s fxed to unty for small wave vectors, the coeffcent ū n front of the quartc part drectly gves the relatve strength of the quartc nteracton between the fluctuatons as compared wth the free feld theory descrbed by the Gaussan part. The crucal pont s now that close to the crtcal pont the correlaton length ξ t ν dverges,

27 .3 The Gaussan Approxmaton 49 so that below four dmensons the effectve dmensonless couplng ū defned n Eq. (.3) becomes arbtrarly large for T T c, no matter how small the bare couplng constant u s chosen. As a consequence, for D < 4 any attempt to calculate perturbatvely correctons to the Gaussan approxmaton s bound to fal for temperatures suffcently close to T c. Because the crtcal exponents are defned n terms of the asymptotc behavor of physcal observables for T T c, they cannot be obtaned perturbatvely for D < D up = 4. Obvously, ths problem can only be solved wth the help of nonperturbatve methods, where all orders n the relevant dmensonless couplng ū are taen nto account. The most powerful and general method avalable to deal wth ths problem s the RG. In the ordered phase where ϕ(r) = ϕ s fnte the dmensonless nteracton ū n Eq. (.3) has a nce nterpretaton n terms of the relatve mportance of orderparameter fluctuatons, whch s measured by the followng dmensonless rato, Q = ξ d D r δϕ(r)δϕ(r = ) S ξ d D r ϕ ξ d D rg (r) = ξ D ϕ. (.7) Here, the ntegrals ξ d D r extend over a cube wth volume ξ D enclosng the orgn, the mean-feld value ϕ of the order parameter s gven n Eq. (.74), and the correlaton functon n the numerator s evaluated n Gaussan approxmaton. Intutvely, the quantty Q sets the strength of order-parameter fluctuatons n relaton to the average order parameter, tang nto account that only wthn a volume of lnear extenson ξ the fluctuatons are correlated. If Q s small compared wth unty, then fluctuatons are relatvely wea and t s reasonable to expect that a mean-feld descrbton of the thermodynamcs and the Gaussan approxmaton for the correlaton functon are suffcent. Tang nto account that for r ξ the functon G (r) decays exponentally, the value of the ntegral n the numerator of Eq. (.7) s not changed f we extend the ntegraton regme over the entre volume of the system, so that we arrve at the estmate, ξ d D rg (r) d D rg (r) = G ( = ) = ξ c. (.8) But accordng to Eqs. (.74) and (.8) for T < T c we may wrte ϕ = 6r u = 3c ξ u, (.9) so that we fnally obtan Q = ξ 4 D u 3c = ū 3. (.)

28 5 Mean-Feld Theory and the Gaussan Approxmaton Hence, our dmensonless couplng constant ū defned n Eq. (.3) s proportonal to the rato Q whch measures the relatve mportance of fluctuatons. Obvously, for Q fluctuatons are wea and t s allowed to treat the nteracton term n Eq. (.6) perturbatvely. On the other hand, n the regme Q we are dealng wth a strongly nteractng system, so that mean-feld theory and the Gaussan approxmaton are not suffcent. The condton Q s called the Gnzburg crteron (Gnzburg 96, Amt 974). Exercses. Mean-feld Analyss of the Isng Model n a Transverse Feld Below a few Kelvn, the magnetc propertes of the rare-earth nsulator lthum holmum fluorde (LHoF 4 ) are well descrbed by the Isng model n a transverse feld (also nown as the quantum Isng model). Its Hamltonan reads Ĥ = J, j σ z σ z j Γ σ x. Here, J > s the exchange couplng for nearest-neghbor spns, Γ denotes the strength of the transverse feld and σ z and σ x are Paul matrces whch measure the x- and z-components of the spns resdng on the lattce stes of a hypercubc lattce. Let us denote the egenstates of the σ z by and (wth egenvalues s =±). (a) Show that for Γ = and n the above bass spanned by the egenstates of the σ z, the Hamltonan Ĥ reduces to the Hamltonan of the famlar classcal Isng model. (b) Introduce g Γ/zJ (where z = D s the coordnaton number) to tune a quantum phase transton at T =. Show that for g = (Γ = ) and for g ( J ) you get two qualtatvely dfferent ground states. These states are called the ferromagnetc and the paramagnetc ground states. Why? (c) To derve the mean-feld Hamltonan Ĥ MF of the quantum Isng model, wrte σ z = m z + δσ z wth the fluctuaton matrx δσ z = σ z m z and m z = σ z z σ =Tr e β Ĥ ]. Tr e β Ĥ ] Expand the term σ z σ z j n Ĥ to lnear order n δσ z and neglect terms quadratc n the fluctuaton matrces. (d) Calculate the egenvalues of the sngle ste mean-feld Hamltonan and use these egenvalues to evaluate the partton functon Z MF and the Landau func-

29 Exercses 5 ton L MF. Mnmze the Landau functon wth respect to the magnetzaton m z to obtan a self-consstency condton for m z = mz (T,Γ). (e) Followng the procedure outlned n Sect..., solve the mean-feld consstency equaton graphcally. Analyze n partcular regons n parameter space where m z and draw the phase boundary (n the T -g-plane) separatng ths ferromagnetc phase from the paramagnetc phase wth m z =. What nd of phase transtons can you dentfy? (f) For T = calculate m z explctly and setch m z as a functon of g. Determne the mean- feld value for the crtcal exponent β g defned by m z (g c g) β g. (g) Calculate also m x = σ x and for T = setchmx as a functon of g.. Landau Functons for the Free Bose Gas Consder N nonnteractng bosons n a D-dmensonal harmonc trap wth Hamltonan Ĥ = m E mb m + b m, where E m = ω(m + +m D ), and m =,,,... The partton functon n the canoncal ensemble s gven by Z N = Tr N e β Ĥ, where the trace runs over the Hlbert space wth fxed partcle number N. It can be expressed as Z N = n= Z N (n), where Z N (n) = Tr N (δ n,b + b e β Ĥ ) has a fxed occupaton of the sngle partcle ground state. We defne the dmensonless Landau functon for Bose Ensten condensaton va Z N (n) = e NLBEC N (n/n). Provded L BEC N (q) s smooth and has an absolute mnmum at q n the thermodynamc lmt, the sngle partcle ground state wll be macroscopcally occuped. (a) Usng Tr N Â = ] π dθ e π Tr (N ˆN)θ Â, where ˆN = m b mb m and the trace on the rght-hand sde runs over the entre Hlbert space contanng any number of partcles (ths s the so-called Foc space), show that the dmensonless Landau functon L BEC N (q) s gven by π ] L BEC N (q) = N ln dθ N( q) eθ m ln( e (ɛm+θ) ). π Here, ɛ m E m /T. What s the physcal nterpretaton of L BEC N (q)? (b) Show that alternatvely, we can wrte Z N = d φ e NLSSB N (φ), where d φ = dreφ] dimφ] and the Landau functon for spontaneous symmetry breang s gven by N (φ) = φ N ln N π L SSB N (N φ ) n n= n! ] e NLBEC N (n/n). Hnt: Carry out the partal trace over m = by usng coherent states z = e zb+,.e., egenstates of b, b z =z z. Recall that Tr m= Â = d z π e z

30 5 Mean-Feld Theory and the Gaussan Approxmaton z  z (see e.g., Shanar 994, Chap. ). You wll also need the overlap n z = zn n! wth a state of fxed boson number. (c) Show the recurson relaton Z N = N N = Z N Z (), where Z () = m e ɛ m = e /τ ] D, and the dmensonless temperature s τ = T/( ω). Hnt: Show for the generatng functon Z(u) N Z N u N = m ( ue ɛ m ), by expressng t as a trace over Foc space. Tae dervatves on both sdes of ths equalty and express the rght-hand sde agan n terms of Z(u). (d) Show that Z N (n) = Z N n Z N n. Hnt: Derve a relaton between the generatng functons Z(u) and h(u) := N Z N (n)u N. (e) Plot L BEC N (q) and LSSB N (φ) forn = and D = 3 as a functon of q and φ, respectvely, evaluatng the expressons derved above numercally for dfferent temperatures τ. What s the temperature at whch the Landau functons start to develop nontrval mnma? More detals and a dscusson of N can be found n Snner et al. (6). References Amt, D. J. (974), The Gnzburg crteron-ratonalzed, J. Phys. C: Sold State Phys. 7, Gnzburg, V. L. (96), Some remars on second order phase transtons and mcroscopc theory of ferroelectrcs,fz.tverd.tela, 3. Gnzburg, V. L. and L. D. Landau (95), Concernng the theory of superconductvty, Zh. Esp. Teor. Fz., 64. Huang, K. (987), Statstcal Mechancs, Wley, New Yor, nd ed. Hubbard, J. (959), Calculaton of partton functons, Phys. Rev. Lett. 3, 77. Kopetz, P. (997), Bosonzaton of Interactng Fermons n Arbtrary Dmensons, Sprnger, Berln. Matts, D. C. (6), The Theory of Magnetsm Made Smple, World Scentfc, New Jersey. Onsager, L. (944), A Two-Dmensonal model wth an order-dsorder transton, Phys. Rev. 65, 7. Shanar, R. (994), Prncples of Quantum Mechancs, Plenum Press, New Yor, nd ed. Snner, A., F. Schütz, and P. Kopetz (6), Landau functons for nonnteractng bosons, Phys. Rev. A 74, 368. Stratonovch, R. L. (957), On a method of calculatng quantum dstrbuton functons,sov.phys. Dol., 46. Wanner, G. H. (966), Statstcal Physcs, Dover Publcatons, New Yor.