Close-packed dimers on nonorientable surfaces
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1 4 February 2002 Pysics Letters A wwwelseviercom/locate/pla Close-packed dimers on nonorientable surfaces Wentao T Lu FYWu Department of Pysics orteastern University Boston MA USA Received 19 September 2001; received in revised form 19 December 2001; accepted 19 December 2001 Communicated by CR Doering Abstract Te problem of enumerating dimers on an M net embedded on nonorientable surfaces is considered We solve bot te Möbius strip and Klein bottle problems for all M and wit te aid of imaginary dimer weigts Te use of imaginary weigts simplifies te analysis and as a result we obtain new compact solutions in te form of double products Te compact expressions also permit us to establis a general reciprocity teorem 2002 Elsevier Science BV All rigts reserved Keywords: Close-packed dimers; onorientable surfaces; Reciprocity teorem 1 Introduction A seminal development in modern lattice statistics is te solution of enumerating close-packed dimers or perfect matcings on a finite M simple-quartic net obtained by Kasteleyn 1 and by Temperley and Fiser 23 more tan 40 years ago In teir solutions te simple-quartic net is assumed to possess free or periodic boundary conditions 1 In view of te connection wit te conformal field teory 4 were te boundary conditions play a crucial role tere as been considerable renewed interest to consider lattice models on nonorientable surfaces 5 9 For close-packed dimers te present autors 6 ave obtained te generating function for te Möbius strip and te Klein bottle for even M and Independently Tesler 10 as solved te problem of perfect matcings on Möbius strips and deduced solutions in terms of a q-analogue of te Fibonacci numbers for all {M } It turns out tat te explicit expression of te generating function depends crucially on weter M and being even or odd and te analysis differs considerably wen eiter M or is odd In tis Letter we consider te general {M } problems for bot te Möbius strip and te Klein bottle by introducing imaginary dimer weigts Te use of imaginary weigts simplifies te analysis and as a result we obtain compact expressions of te solutions witout te recourse of Fibonacci numbers Te compact expressions of te solutions also permit us to establis a reciprocity teorem on te enumeration of dimers * Corresponding autor address: wtlu@neuedu WT Lu /02/$ see front matter 2002 Elsevier Science BV All rigts reserved PII: S
2 236 WT Lu FY Wu / Pysics Letters A Summary of results For te convenience of references we first summarize our main results Details of derivation will be presented in subsequent sections Consider an M simple-quartic net consisting M sites arranged in an array of M rows and columns Te net forms a Möbius strip if tere is a twisted boundary condition in te orizontal direction as sown in Fig 1 and a Klein bottle if in addition to te twisted boundary condition tere is also a periodic boundary condition in te vertical direction Let te dimer weigts be z and z v respectively in te orizontal and vertical directions We are interested in te close-form evaluation of te dimer generating function Z M z z v = z n zn v v 1 were n n v are respectively te number of orizontal and vertical dimers and te summation is taken over all close-packed dimer coverings of te net Our resultsare asfollows: For bot M and even we ave 6 M/2 ZM Mob z z v = /2 M/2 ZM Kln z z v = /2 4z 2 4n 1π sin2 + 4z 2 mπ v 2 cos2 M + 1 4z 2 sin2 4n 1π 2 + 4z 2 v 2m 1π sin2 M were te superscripts refer to te type of te nonorientable surface under consideration For M even and odd we ave M/2 ZM Mob z z v = Re 1 i 2i 1 M/2+m+1 4n 1π mπ z sin + 2z v cos 2 M + 1 M/2 ZM Kln z z v = Re 1 i 2i 1 M/2+m+1 4n 1π 2m 1π z sin + 2z v sin 2 M and for M odd and even we ave M z /2 z v = z Z Mob Z Kln M z z v = z /2 M+1/2 n=1 m=1 /2 M+1/2 /2 n=1 m=1 4z 2 sin2 4n 1π 2 For M and bot odd te generating function is zero + 4zv 2 mπ cos2 M + 1 4z 2 4n 1π sin2 + 4z 2 2m 1π v sin2 2 M Te Möbius strip It is well-known tat tere is a one-one correspondence between dimer coverings and terms in a Pfaffian defined by te dimer weigts However since terms in te Pfaffian generally do not possess te same sign te evaluation of te Pfaffian does not necessarily produce te desired dimer generating function Te crux of matter is to attac signs or more generally factors to te dimer weigts so tat all terms in te Pfaffian ave te same sign and te task is reduced to tat of evaluating a Pfaffian
3 WT Lu FY Wu / Pysics Letters A Fig 1 Edge orientations of Möbius strips wit twisted boundary conditions in te orizontal direction A B C D E denote repeated sites a M = 4 = 5 b M = 5 = 4 Tese tasks were first acieved by Kasteleyn 1 for te simple-quartic dimer lattice wit free and periodic boundary conditions wo sowed ow to attac signs to dimer weigts and ow to evaluate te Pfaffian Soon after te publication of 1 Wu 11 pointed out tat te structure of te Pfaffian and ence its evaluation can be simplified if a factor i is associated to dimer weigts in one spatial direction Indeed te Wu prescription requires only uniformly directed lattice edges wit te association of a factor i to dimer weigts in te direction in wic te number of lattice sites is odd If te number of lattice sites is even in bot directions ten te factor i can be associated to dimers in eiter direction For {M }={even odd} for example one replaces z by iz To see tat te Wu procedure is correct one considers a standard dimer covering C 0 in wic te lattice is covered only by parallel dimers wit real weigts Ten te two terms in te Pfaffian corresponding to C 0 and any oter dimer covering C 1 ave te same sign since te superposition polygon produced by C 0 and C 1 contains an even number of arrows pointing in one direction as well as a factor i 4n+2 = 1 were n is a nonnegative integer amely te superposition polygon is clockwise-odd 11 Tis use of imaginary dimer weigts is te starting point of our analysis 31 M = even Möbius strip For M even and = even or odd we write for definiteness M = 2M = werem are positive integers We orient lattice edges as sown in Fig 1a For te time being consider more generally tat te orizontal dimers connecting te first and t columns ave weigts z and te associated generating function Z Mob M z z v ; z = 2M m=0 z m T m were T m T m z z v is a multinomial in z and z v wit strictly positive coefficients Te desired result is ten obtained by setting z = z ote tat T 0 is precisely te generating function wit free boundary conditions Attac a factor i to all orizontal dimer weigts z Tis leads us to consider te antisymmetric matrix Az = iz F F T I2M + z v I F 2M F T 2M + z K + K T J2M were I is te identity matrix F T is te transpose of F andf 2M K and J 2M are matrices of te order given by te subscripts: F = K = J 2M =
4 238 WT Lu FY Wu / Pysics Letters A ow te Pfaffian of Az gives te correct generating function T 0 in te case of z = 0 11 For general z we ave te following result: Teorem Te dimer generating function for te simple-quartic net wit a twisted boundary condition in te orizontal direction is Z Mob M z z v ; z = ipf Aiz ipf A iz = Re 1 ipf Aiz 11 Proof It is clear tat te term in 8 corresponding to te configuration C 0 m = 0 as te correct sign For any oter dimer configuration C 1 te superposition of C 0 and C 1 forms superposition polygons containing z edges We ave te following facts wic can be readily verified: i Te sign of a superposition polygon remains uncanged under deformation of its border wic leaves n z te number of z edges it contains invariant ii Deformations of te border of a superposition polygon can cange n z only by multiples of 2 and te sign of te superposition polygon reverses wenever n z canges by 2 iii Superposition polygons aving 0 or 1 z edges ave te sign + As a result we obtain Az Pf Az = X0 + zx 1 z 2 X 2 z 3 X 3 were denotes te determinant of and X α = T α + z 4 T α+4 + z 8 T α+8 + α= Te teorem is now a consequence of te fact ZM Mob z z v ; z = X 0 + z X 1 + z 2X 2 + z 3X Remarks Te teorem olds also for te Klein bottle wic in addition to a twisted boundary condition in te orizontal direction as a periodic boundary condition in te vertical direction see below For te Möbius strip we ave X 1 = X 3 = 0wen = even It now remains to evaluate Pf A±iz ToevaluatePfA±iz = A±iz we make use of te fact tat since F 2M F2M T commutes wit J 2Mte2M 2M matrix Az can be diagonalized in te 2M-subspace 6 Introducing te 2M 2M matrix U wose elements are U mm = i m 2 2M + 1 sin mm π 2M + 1 mm π 2M + 1 U 1 mm = i m 2 2M + 1 sin mm = 1 22M we find U F2M F2M T U 1 mm = 2i cos φ m δ mm U 1 J 2M U 15 mm = i 1 M+m δ mm mm = 1 22M were φ m = mπ/2m + 1 Tus we can replace te 2M 2M matrices in 9 by teir respective eigenvalues and express Az as a product of te replaced determinants namely Az 2M = i 2M A m z m=
5 WT Lu FY Wu / Pysics Letters A were we ave taken out a common factor i from eac element of te matrix A m z = 2z v cos φ m I + z F F T + 1 M+m z K + K T 17 Te matrix A m z can be evaluated for general z in terms of a q-analogue of Fibonacci numbers but for our purposes wen z =±iz te matrix can be diagonalized directly Define te matrix T = F + i 1 M+m K = i 1 M+m we can rewrite 17 wen z = iz as A m iz = 2z v cos φ m I + z T T 19 ow T and T commute so tey can be diagonalized simultaneously wit respective eigenvalues eiθ n and e iθ n were 1 θ n = 1 M+m+1 4n 1π/2 20 Tus we obtain A m iz = and as a result Aiz M = mπ 2z v cos 2M i 1M+m+1 z sin n=1 mπ 2z v cos 2M i 1M+m+1 z sin 4n 1π 2 4n 1π 2 2 were we ave made use of te fact tat cos φ 2M+1 m = cosφ m 1 2M+1 m = 1 m andi 2M = 1 M We tus obtain after taking te square root of 22: M Pf Aiz = 2z v cos 23 mπ 2M i 1M+m+1 z sin 4n 1π 2 Te substitution of 23 into 11 now yields 4 For = even te Pfaffian 23 is real and 4 reduces to 2 Tere is no suc simplification for = odd 32 M = odd Möbius strip For M = odd and = even or odd we write for definiteness M = 2M + 1 = Since te number of rows M is odd we now attac a factor i to dimers in te vertical direction Again we assign weigts z to orizontal dimers connecting te first and t columns and consider te generating function 2M+1 ZM Mob z z v ; z = z m T m m= A similar expression of θ n given in 17 of Ref 6 contains a typo were M + m + 1 in te exponent sould read M + m Tis does not alter te results of Ref 6
6 240 WT Lu FY Wu / Pysics Letters A defined similar to 8 It is readily verified tat wit lattice edge orientations sown in Fig 1b all terms in T 0 ave te same sign It follows tat we can use te teorem of te preceding subsection were Pf A is te Pfaffian of te antisymmetric matrix Az = z F F T I2M+1 iz v I F 2M+1 F T 2M+1 + zg H 2M+1 25 wit G = K K T and H 2M+1 = Apply to 25 te unitary transformation 14 wit 2M replaced by 2M + 1 Te transformation diagonalizes F 2M+1 F2M+1 T as in 15 and in addition produces 26 U 1 H 2M+1 U = 1 M J 2M+1 Tus we obtain Āz I U 1 AzI U= z F F T + 2zv cos φ m I I2M+1 zg J 2M were φ m = mπ/2m + 2 Writing B m z F F T + 2zv cos φ m I ten te matrix Āz assumes a form sown below in te case of 2M + 1 = 5: B 1 zg B 2 zg Āz = B zg zg B 4 zg B 5 Intercanging rows and columns tis matrix can be canged into a block-diagonal form aving te same determinant: B 1 zg zg B 5 zg zg B 4 B 2 B 3 + zg For general M we define te 2 2 matrix A m 2 z = B m 1 M+m+1 zg 1 M+m+1 zg B 2M+1 m and use te result B M+1 + zg 1 = z 2 z + z
7 WT Lu FY Wu / Pysics Letters A to obtain Az = Āz = z 2 z + z 2 M m=1 A m 2 z It terefore remains to evaluate A m 2 z Te matrix A m 2 z can be diagonalized for z =±iz To proceed it is convenient to multiply from te rigt by a2 2 matrix wose determinant is 1 to obtain Ā m iz A m iz I 0 1 = 2z v cos φ m I 2 + z F F T = 2z v cos φ m I 2 + z Q2 Q M+m+1 iz G were F Q 2 = i 1 M+m K 36 i 1 M+m K F ow Q 2 commutes wit Q 2 so tey can be diagonalized simultaneously wit te respective eigenvalues ei θ n and e i θ n were θ n = 2n 1/2 n = 1 22 Tus we obtain A m 2 iz = 1 Ā m 2 iz = 1 4z 2 sin 2 θ n + 4zv 2 cos2 φ m n=1 and taking te square root of 34 wit z replaced by iz Pf Aiz = /2 1 + iz /2 M n=1 m=1 4z 2 2n 1π sin zv 2 mπ cos2 2M + 2 Terefore te generating function vanises identically for = odd For = even we substitute 38 into 11 and make use of te fact tat te two sets sin 2 2n 1π/2 and sin 2 4n 1π/2 n = 1 2/2 are identical Tis leads to result Te Klein bottle A Klein bottle is constructed by inserting extra vertical edges wit weigt z v to te boundary of te Möbius strips of Figs 1a and b so tat tere is a periodic boundary condition in te vertical direction Te extra edges are oriented upward as te oter vertical edges Te consideration of te Klein bottle parallels to tat of te Möbius strip Again we need to consider te cases of even and odd M separately 41 M = even Klein bottle For a 2M Klein bottle wit orizontal edges connecting te first and t row aving weigts z weave as in 8 te generating function Z Kln M z z v ; z = 2M m=0 z m T m 39
8 242 WT Lu FY Wu / Pysics Letters A Te dimer weigts now generate te antisymmetric matrix A Kln z = Az z v I K 2M K T 2M were Az is given by 9 Following te same analysis as in te case of te Möbius strip since i iii still old we find te desired dimer generating function again given by teorem 11 or explicitly 40 Z Kln M z z v ; z = ipf A Kln iz ipf A Kln iz 41 To evaluate Pf A Kln z we note tat te 2M 2M matrices F 2M K 2M F2M T + KT 2M and J 2M commute and can be diagonalized simultaneously by te 2M 2M matrix V wose elements are V mm = 1 e im2m 1π/2M 2M V 1 mm = 1 2M e im 2m 1π/2M mm = 1 22M 42 We find V F2M K 2M F2M T + KT 2M V 1 mm = 2i sin α m δ mm U 1 J 2M U mm = i 1 M+m δ mm were α m = 2m 1π/2M Diagonalizing Az in te 2M-subspace we obtain 43 were A Kln z 2M = i 2M m=1 A m z A m z = 2z v sin α m I + z F F T + 1 M+m z K + K T 45 Tis expression is te same as 17 for te Möbius strip except wit te substitution of cos φ m by sin α m Tus following te same analysis we obtain Pf Aiz = M 2m 1π 2z v sin 2M + 2i 1 M+m+1 z sin 4n 1π 2 Te substitution of 46 into 41 now gives result 5 For = even Pfaffian 46 is real and 5 reduces to 3 42 M = odd Klein bottle For a 2M + 1 Klein bottle te inserted vertical edges ave dimer weigts iz v Te consideration ten parallels tat of te preceding sections Particularly te desired dimer generating function is also given by 41 but now wit A Kln z = Az + iz v I K 2M+1 K T 2M+1 47 To evaluate Pf A Kln z one again applies in te 2M + 1-subspace te unitary transformation 42 wit 2M replaced by 2M + 1 wic diagonalizes F 2M+1 K 2M+1 F2M+1 T + KT 2M+1 Using te result V 1 H 2M+1 V mm = e iᾱ m δ mm+1 m 48
9 WT Lu FY Wu / Pysics Letters A were ᾱ m = 2m 1π/2M + 1 m = 1 22M + 1 ten te matrix Āz = I U 1 AzI U assumes te form in te case of 2M + 1 = 5: B 1 ze iᾱ 1G B 2 ze iᾱ 2G Āz = B zg ze iᾱ 2G B 4 ze iᾱ 1G B 5 Here B m = z Q + 2z v sin ᾱ m Again intercanging rows and columns one canges Āz into te blockdiagonal form B 1 ze iᾱ 1G ze iᾱ 1G B 5 B 2 50 ze iᾱ 2G ze iᾱ 2G B 4 B 3 + zg Explicitly for general M temt block is A m 2 z = B m ze iᾱ m G ze iᾱ m G B 2M+1 m = z F F T e iᾱ m I2 + 2z v sin ᾱ m I zg e iᾱ m 0 We proceed as in 35 by multiplying a 2 2 matrix wose determinant is 1 from te rigt and obtain Ā m 2 iz A m iz I 0 1 = 2z v sin ᾱ m I 2 + z F F T e iᾱ m + iz 0 1 G e iᾱ m 0 were F Q 2 = ie iᾱ m K = 2z v sin ᾱ m I 2 + z Q2 Q 2 ie iᾱ m K F ow Q 2 commutes wit Q 2 and tey can be diagonalized simultaneously wit te respective eigenvalues ei β n and e i β n were β n = 2n 1/2 n = 1 22 Tus we obtain Ā m 2 iz = 4z 2 2n 1π sin2 2 n=1 Using 52 and 33 we get Pf Aiz = z 2 v /2 1 + iz /2 2m 1π sin2 2M + 1 M n=1 m=1 4z 2 2n 1π sin zv 2 2m 1π sin2 2M + 1 Tus te generating function 41 vanises identically for = odd For = even we replace as before sin 2 2n 1π/ by sin 2 4n 1π/ Te substitution of 55 into 41 now leads to result
10 244 WT Lu FY Wu / Pysics Letters A A reciprocity teorem Using te explicit expression of dimer enumerations on a simple-quartic lattice wit free boundaries Stanley 12 as sown tat te enumeration expression satisfies a certain reciprocity relation a relation rederived recently by Propp 13 from a combinatorial approac Here we sow tat our solutions of dimer enumerations lead to an extension of te reciprocity relation to enumerations on cylindrical toroidal and nonorientable surfaces We first consider solutions and 7 for = even Writing T Mob M = ZM Mob 1 1 T and using te identity 14 n 1 x 2 2x cos k=0 α + 2kπ n Kln M = ZM Kln = x 2n 2x n cosnα + 1 repeatedly we can rewrite our solutions for general z and z v in te form of were M z M /2 z v = z Z Mob M z M /2 z v = z Z Kln x m = G zv M+1/2 m=1 x m + x m 3 1 = M /2 M /2 y M+1 n zv 2 M+1/2 m=1 n=1 t m + t m + 1 M yn M 1 y n + yn = M /2 M /2 zv y M 2 n + y n M M mπ cos z M + 1 n=1 z y n = G sin z v 2n 1π zv t m = G sin 2 z and Gy = y + y Tus te following reciprocity relations are obtained by inspection: T Mob M = T Mob M = ɛ M T Mob M 2 2m 1π M T Kln M = T Kln M = ɛ M T Kln M 61 were { ɛ M = 1 M = 2 mod oterwise Tere are no reciprocity relations for = odd We ave carried out similar analyses for dimer enumerations on a simple-quartic net embedded on a cylinder and a torus using te solutions given in 115 and ave discovered universal rules of associating reciprocity relations to specific boundary conditions Generally tere are 3 different boundary conditions or matcings tat can be imposed between 2 opposite boundaries of an M net Te conditions can be twisted suc as tose sown in te orizontal direction in Fig 1 periodic suc as on a torus or free wic means free standing To establis te convention we sall refer to te boundary condition between te first and te t columns as te boundary condition in te direction and similar tat between te first and te Mt rows as in te M direction Ten our findings togeter wit tose of Ref 13 lead to te following teorem applicable to all cases:
11 WT Lu FY Wu / Pysics Letters A Reciprocity teorem Let TM be te number of close-packed dimer configurations perfect matcings on an M simple-quartic lattice wit free periodic or twisted boundary conditions in eiter direction Te case of twisted boundary conditions in bot directions is excluded If te twisted boundary condition if occurring is in te M direction we restrict to M = even Ten we ave 1 TM = ɛ M T 2 M if te boundary condition in te M direction is free 2 TM = ɛ M T M if te boundary condition in te M direction is periodic or twisted 6 Summary and discussion We ave evaluated te dimer generating function 1 for an M simple-quartic net embedded on a Möbius strip and a Klein bottle for all M Te results are given by 2 7 Our results can also be written in terms of te q-analogue of te Fibonacci numbers F n q defined by 1 1 qs s 2 = F n qs n n=0 Using te first line of 58 for example and te identity F n q + F n 2 q = x n + x n q x x 1 one can verify tat our results 2 and 6 for te Möbius strip are te same as tose given by Tesler 10 Details of te proof wic also lead to some new product identities involving te Fibonacci numbers will be given elsewere 16 We ave also deduced a reciprocity teorem for te enumeration TM of dimers on an M lattice under arbitrary including free periodic and twisted boundary conditions Finally we point out tat results 2 7 can be put in a compact expression valid for all cases as M /2 Z M z z v = z Re 1 i M+1/2 m=1 were x is te integral part of xand { zv z cos mπ M+1 for te Möbius strip X m = zv z sin 2m 1π M for te Klein bottle n=1 2i 1 M/2+m+1 4n 1π sin 2 + 2X m Acknowledgement Work as been supported in part by ational Science Foundation Grant DMR References 1 PW Kasteleyn Pysica HV Temperley ME Fiser Pilos Mag ME Fiser Pys Rev HWJ Blöte JL Cardy MP igtingale Pys Rev Lett R Srock Pys Lett A WT Lu FY Wu Pys Lett A
12 246 WT Lu FY Wu / Pysics Letters A W-J Tzeng FY Wu Appl Mat Lett WT Lu FY Wu Pys Rev E K Kaneda Y Okabe Pys Rev Lett G Tesler J Combin Teory B TT Wu J Mat Pys R Stanley Discrete Appl Mat J Propp matco/ IS Gradsteyn IM Ryzik Table of Integrals Series and Products Academic Press ew York or specifically and BM McCoy TT Wu Te Two-Dimensional Ising Model Harvard University Press Cambridge WT Lu FY Wu in preparation
arxiv:cond-mat/ v3 [cond-mat.stat-mech] 18 Feb 2002
Close-packed dimers on nonorientable surfaces arxiv:cond-mat/00035v3 cond-mat.stat-mec 8 Feb 2002 Wentao T. Lu and F. Y. Wu Department of Pysics orteastern University, Boston, Massacusetts 025 Abstract
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