V.C The Niemeijer van Leeuwen Cumulant Approximation

Transcription

1 V.C The Nemejer van Leeuwen Cumulant Approxmaton Unfortunately, the decmaton procedure cannot be performed exactly n hgher dmensons. For example, the square lattce can be dvded nto two sublattces. For an RG wth b =, we can start by decmatng the spns on one sublattce. The nteractons between the four spns surroundng each decmated spn are obtaned by generalzng eq.(v.3). If ntally h = g =, we obtan, ζ 3, ζ Ks(ξ +ξ +ξ +ξ R(ζ, ζ 4) = e 3 4 ) = cosh [K(ζ + ζ + ζ3 + ζ4)]. (V.8) s=± Clearly the four spns appear symmetrcally n the above expresson, and hence are subject to the same two body nteracton. Ths mples that new nteractons along the dagonals of the renormalzed lattce are also generated, and the nearest neghbor form of the orgnal Hamltonan s not preserved. There s also a four pont nteracton, and R = exp [g + K (ζζ + ζζ 3 + ζ3ζ 4 + ζ4ζ + ζζ 3 + ζζ 4) + K ζ 3ζ 4 ζ ζ 4]. (V.9) The number (and range) of new nteractons ncreases wth each RG step, and some truncatng approxmaton s necessary. Two such schemes are descrbed n the followng sectons. One of the earlest approaches was developed by Nemejer and van Leeuwen (NvL) for treatng the Isng model on a trangular lattce, subject to the usual nearest neghbor Hamltonan λh = K j ζ ζ j. The orgnal lattce stes are grouped nto cells of three spns (e.g. n alternatng up pontng trangles). Labellng the three spns n cell as {ζ, ζ, ζ3 }, we can use a majorty rule to defne the renormalzed cell spn as [ ] = sgn ζ + ζ + ζ3. (V.3) ζ (There s no ambguty n the rule for any odd number of stes, and the renormalzed spn s two valued.) The renormalzed nteractons correspondng to the above map are obtaned from the constraned sum βh [ξ ] βh[ξ e = e ]. (V.3) {ξ ξ } 8

2 To truncate the number of nteractons n the renormalzed Hamltonan, NvL ntroduced a perturbatve scheme by settng λh = λh + U. The unperturbed Hamltonan λh ζ ζ 3 ζ = K ζ + ζ + ζ 3, (V.3) nvolves only ntra cell nteractons. Snce the cells are decoupled, ths part of the Hamltonan can be treated exactly. The remanng nter cell nteractons are treated as a perturbaton ( U = K ζ () ζ () + ζ () ζ (3) β β. (V.33) <,β> The sum s over all neghborng cells, each connected by two bonds. (The actual spns nvolved depend on the relatve orentatons of the cells.) Eq.(V.3) s now evaluated perturbatvely as e = e [ βh [ξ ] βh [ξ {ξ ξ } ] U + U The renormalzed Hamltonan s gven by the cumulant seres ]. (V.34) ( λh ] = ln Z [ζ [ζ ] + U + O(U 3 U ), (V.35) U where refers to the expectaton values wth respect to λh, wth the restrcton of fxed [ζ ], and Z s the correspondng partton functon. To proceed, we construct a table of all possble confguratons of spns wthn a cell, ther renormalzed value, and contrbuton to the cell energy: ζ ζ ζ ζ 3 exp [ λh ] e 3K e K e K e K + e K e 3K + e K + e K 83

3 The restrcted partton functon s the product of contrbutons from the ndependent cells, Z [ζ ] = {ξ ξ } e K(ξ ξ +ξ ξ3 +ξ3 ξ ( ) = e 3K + 3e [ ] 3 ζ + ζ β ζ = K ζ ζ j β, (V.37) K ) N/3. (V.36) It s ndependent of [ζ ], thus contrbutng an addtve constant to the Hamltonan. The frst cumulant of the nteracton s U = K <,β> ζ β <,β> where we have taken advantage of the equvalence of the three spns n each cell. Usng the table, we can evaluate the restrcted average of ste spns as K +e 3K e K + e for ζ = + K e 3K + 3e K e 3K + e ζ = e3k + e K e K e 3K + 3e K ζ. (V.38) e 3K for ζ + 3e = K Substtutng n eq.(v.37) leads to ( N e 3K + e K ) [ζ ] = ln e 3K + 3e + K ζ K ζ β + O(U ). (V.39) 3 e 3K + 3e λh K At ths order, the renormalzed Hamltonan nvolves only nearest neghbor nteractons, wth the recurson relaton. Eq.(V.4) has the followng fxed ponts: β ( e 3K + e K ) K = K. (V.4) e 3K + 3e K (a) The hgh temperature snk at K =. If K, K K(/4) = K/ < K,.e. ths fxed pont s stable, and has zero correlaton length. (b) The low temperature snk at K =. If K, then K K > K,.e. unlke the one dmensonal case, ths fxed pont s also stable wth zero correlaton length. (c) Snce both of the above fxed ponts are unstable, there must be at least one stable fxed pont at fnte K = K = K. From eq.(v.4), the fxed pont poston satsfes K e 3K + e 4K 4K = e 3K + 3e K, = e + = e + 3. (V.4) 84

4 The fxed pont value K = 3 ln.3356, 4 (V.4) can be compared to the exactly known value of.747 for the trangular lattce.. Lnearzng the recurson relaton around the non-trval fxed pont yelds, ( δk e 4K + ) + 3K + ) e 4K (e4k δk =.64. (V.43) K e (e 4K + 3 4K + 3) 3 The fxed pont s ndeed unstable as requred by the contnuty of flows. Ths RG scheme removes /3 of the degrees of freedom, and corresponds to b = 3. The thermal egenvalue s thus obtaned as δk ln(.64) b y t =, = y t.883. (V.44) δk K ln( 3) Ths can be compared to the exactly known value of y t =, for the two dmensonal Isng model. It s certanly better than the mean-feld (Gaussan) estmate of y t =. From ths egenvalue we can estmate the exponents ξ = /y t.3 (), and = /y t =.6 (), where the exact values are gven n the brackets. 3. To complete the calculaton of exponents, we need the magnetc egenvalue y h, obtaned after addng a magnetc feld to the Hamltonan,.e. from λh = λh + U h Snce the fxed pont occurs for h =, the added term can also be treated perturbatvely, and to the lowest order λh = λh + U (ζ + ζ + ζ3 ), (V.46) h where the spns are grouped accordng to ther cells. Usng eq.(v.38), ζ. (V.45) e 3K + e K = ln Z ζ + K ζ β 3h e 3K + 3e K, (V.47) λh ζ <,β> 85

5 thus dentfyng the renormalzed magnetc feld as In the vcnty of the unstable fxed pont and e 3K + e K h = 3h. (V.48) e 3K + 3e K δh e 4K + 3 b y h = = 3 4K =, (V.49) δh K e + 3 ln 3/ y h = ln ( 3 ).37. (V.5) Ths s lower than the exact value of y h =.875. (The Gaussan value of y h = s closer to the correct result n ths case.) 4. NvL carred out the approach to the second order n U. At ths order two addtonal nteractons over further neghbor spns are generated. The recurson relatons n ths three parameter space have a non-trval fxed pont wth one unstable drecton. The resultng egenvalue of y t =.53, s tantalzngly close to the exact value of, but ths agreement s probably accdental. V.D The Mgdal Kadanoff Bond Movng Approxmaton Consder a b = RG for the Isng model on a square lattce, n whch every other spn along each lattce drecton s decmated. As noted earler, such decmaton generates new nteractons between the remanng spns. One way of overcomng ths dffculty s to smply remove the bonds not connected to the retaned spns. The renormalzed spns are then connected to ther nearest neghbors by two successve bonds. Clearly after decmaton, the renormalzed bond s gven by the recurson relaton n eq.(v.8), characterstc of a one dmensonal chan. The approxmaton of smply removng the unwanted bonds weakens the system to the extent that t behaves one dmensonally. Ths s remeded by usng the unwanted bonds to strengthen those that are left behnd. The spns that are retaned are now connected by a par of double bonds (of strength K), and the decmaton leads to K = ln cosh( K). (V.5). Fxed ponts of ths recurson relaton are located at (a) K = : For K, K ln( + 8K )/ 4K K,.e. ths fxed pont s stable. 86

6 (b) K : For K, K ln(e 4K /)/ K K, ndcatng that the low temperature snk s also stable. (c) The domans of attractons of the above snks are separated by a thrd fxed pont at 4K 4K + e e K = e, = K.35, (V.5) whch can be compared wth the exact value of K c.44.. Lnearzng eq.(v.5) near the fxed pont gves δk b y t = = tanh 4K.6786, = y t.747, (V.53) δk K compared to the exact value of y t =. The bond movng procedure can be extended to hgher dmensons. For a hypercubc lattce n d-dmensons, the bond movng step strengthens each bond by a factor of d. After decmaton, the recurson relaton s [ ] K = ln cosh d K. (V.54) The hgh and low temperature snks at K = and K, are stable, snce K, = K ln( + d K ) (d ) K K, (V.55) and e d K K, = ln d K K. (V.56) K (Note that the above result correctly dentfes the lower crtcal dmenson of the Isng model, n that the low temperature snk s stable only for d >.) The ntervenng fxed pont has an egenvalue δk y t = = d tanh d K. (V.57) δk K The resultng values of K.65 and y t.934 for d = 3, can be compared wth the known values of K c. and y t.59 on a cubc lattce. Clearly the approxmaton gets worse at hgher dmensons. (It fals to dentfy an upper crtcal dmenson, and as d, K ( d) and y t.) 87

7 The Mgdal Kadanoff scheme can also be appled to more general spn systems. For a one dmensonal model descrbed by the set of nteractons {K}, the transfer matrx method n eq.(v.7) gves the recurson relatons as T b ({K }) = T ({K}) b. For a d-dmensonal lattce, the bond movng step strengthens each bond by a factor of b d, and the generalzed Mgdal Kadanoff recurson relatons are T b ({K }) = T ({b d K}) b. (V.58) The above equatons can be used as a quck way of estmatng phase dagrams and exponents. The procedure s exact n d =, and does progressvely worse n hgher dmensons. It thus complments mean feld (saddle pont) approaches that are more relable n hgher dmensons. Unfortunately, t s not possble to develop a systematc scheme to mprove upon ts results. The RG procedure also allows evaluaton of free energes, heat capactes, and other thermodynamc functons. One possble worry s that the approxmatons used to construct RG schemes may result n unphyscal behavor, e.g. negatve values of response functons C and ν. In fact most of these recurson relatons (e.g. eq.(v.58)) are exact on herarchcal (Berker) lattces. The realzablty of such lattces ensures that there are no unphyscal consequences of the recurson relatons. 88