a Xi :cond-ma /0207180 1 6 Jul 2002
The In o ma ion Geome y
o he One-Dimensional
Po s Model
B.P. Dolan
Depa men o Ma hema ical Physics
Na ional Uni e si y o I eland
Maynoo h, I eland
D.A. Johns on
Dep . o Ma hema ics
He io -Wa Uni e si y
Ricca on
Edinbu gh, EH14 4AS, Sco land
and
R. Kenna
School o Ma hema ical and In o ma ion Sciences
Co en y Uni e si y
Co en y, CV1 5FB, England
Janua y 20, 2004
Abs ac
In a ious s a is ical-mechanical models he in oduc ion o a me ic on o he
space o pa ame e s (e.g. he empe a u e a iable, β, and he ex e nal ield a iable,
h, in he case o spin models) gi es an al e na i e pe spec i e on he phase s uc u e.
Fo he one-dimensional Ising model he scala cu a u e, R, o his me ic can
be calcula ed explici ly in he he modynamic limi and is ound o be R= 1 +
cosh(h)/qsinh2(h) + exp(−4β). This is posi i e de ini e and, o physical ields and
empe a u es, di e ges only a he ze o- empe a u e, ze o- ield “c i ical poin ” o
he model. In his no e we calcula e R o he one-dimensional q-s a e Po s model
inding an exp ession o he o m R=A(q, β, h) + B(q, β, h)/pη(q, β, h), whe e
η(q, β, h) is he Po s analogue o sinh2(h) + exp(−4β). This is no longe posi i e
de ini e, bu once again i di e ges only a he c i ical poin in he space o eal
pa ame e s. We ema k, howe e , ha a nai e analy ic con inua ion o complex
ield e eals a u he di e gence in he Ising and Po s cu a u es a he Lee-Yang
edge.
1 In oduc ion
The no ion o a dis ance be ween configu a ions in s a is ical-mechanical and la ice-field
models has been discussed by se e al au ho s [1, 2, 3, 4, 5, 6], and he geome y ha
his endows on he mani old, M, o he pa ame e s cha ac e ising he models explo ed.
This dis ance is measu ed using he Fishe -Rao me ic [7], which is calcula ed om he
Fishe in o ma ion ma ix o he sys em o in e es . When we conside a spin model in
field, Mis a wo-dimensional mani old pa ame ised by (θ1, θ2) = (β, h). In his case, he
componen s o he Fishe -Rao me ic ake he pa icula ly simple o m
Gij =−∂i∂j , (1)
whe e is he educed ee ene gy pe si e and ∂i=∂/∂θi.
I has been sugges ed ha in such a geome iza ion o s a is ical mechanics he scala
cu a u e, R, o Mplays a cen al ole [1, 2, 3]. Fo a spin model in field, wi h he me ic
gi en in equ.(1), Rmay be calcula ed succinc ly as
R=1
2G2
∂2
β ∂β∂h ∂2
h
∂3
β ∂2
β∂h ∂β∂2
h
∂2
β∂h ∂β∂2
h ∂3
h 
,(2)
whe e G= de (Gij). Since he only scale p esen nea c i icali y o a model displaying a
highe -o de ansi ion is he co ela ion leng h, ξ, i has been hypo hesised on dimensional
g ounds ha R ∼ ξd, whe e dis he dimensionali y o he sys em [1, 2, 3]. I we assume
ha hype scaling holds, νd = 2 −α, his leads o
R ∼ |ξ|(2−α)/ν.(3)
To es he beha iou o R, one equi es models which a e sol able in field in o de o
ob ain analy ic exp essions, and hese a e a he hin on he g ound. Indeed, Rhas been
calcula ed o he mean-field and Be he-la ice Ising models [4] and he abo e scaling
beha iou e ified. I has also been calcula ed o he one-dimensional Ising model [2]
whe e i akes he ema kably simple o m,
RIsing = 1 + cosh h
qsinh2h+e−4β
.(4)
In his case Ris posi i e defini e and di e ges only a he ze o- empe a u e, ze o-field
“c i ical poin ” o he model. The co ela ion leng h is gi en by
ξ−1=−ln( anh(β)),(5)
so ha ξ∼exp(2β) nea c i icali y, and (3) holds he e wi h α= 1, ν = 1 as expec ed1. A
second no ewo hy ea u e o he Be he-la ice and 1D Ising models conce ns he me ic
associa ed wi h (1). This me ic is diagonalized using he co esponding eno maliza ion
g oup in a ian .
1The Be he la ice model also sa is ies he pos ula ed scaling, al hough he e a e some sub le ies
coming om he exponen αbeing ze o [4].
Gi en he sho age o explici ly calculable examples, any u he addi ions o he lis
would be e y wo hwhile in o de o see which ea u es in he models a e gene ic and
which a e pa icula o he models conce ned. In his pape we discuss ano he example,
he 1D q-s a e Po s model, whe e an exp ession o Rmay be ob ained in a e y simila
manne o he Ising model (which is he q= 2 case o he Po s model). We compa e
he p ope ies o he cu a u e, R, as well as he me ic in he Po s model and he Ising
model, highligh ing bo h hei s uc u al simila i ies and diffe ences in de ail.
2 The 1D Po s Model
The pa i ion unc ion o he 1D q-s a e Po s model is gi en by
ZN(y, z) = X
{σ}
exp 
β
N
X
j=1 δ(σj, σj+1)−1
q!+h
N
X
j=1 δ(σj,1) −1
q!
,(6)
whe e he spins, σj∈ {1, . . . , q}, a e defined on he si es, j∈ {1,...N}, o he la ice and
whe e we ha e defined y= exp(β) and z= exp(h) o la e calcula ional con enience.
The model may be sol ed by ans e ma ix me hods [8], jus as he 1D Ising model.
Fo gene al q he ull ans e ma ix T(y, z) may be w i en as q−2 diagonal elemen s,
(y−1)(yz)−1/q, and a 2 ×2 ac o (y, z):
(y, z) = 1
(yz)1/q yz z1/2√q−1
z1/2√q−1y+q−2.(7)
The pa i ion unc ion is ZN(y, z) = T T(y, z)Nand he eigen alues o T(y, z) a e
λ0, λ1,...,λq−1, whe e
λ0
λ1)=1
2y(1 + z) + q−2±q(y(1 −z) + q−2)2+ (q−1)4z(yz)−1
q(8)
and λ2,...λq−1= (y−1)(yz)−1/q. The educed ee ene gy pe si e in he he modynamic
limi , N→ ∞, is hus gi en by =−lnλ0.
I is s aigh o wa d o use his exp ession o he ee ene gy in equ.(2) o ob ain he
cu a u e, R. In he cu en no a ion we e-de i e he exp ession o he Ising model 2as
RIsing = 1 + y(1 + z)
√y2−2y2z+y2z2+ 4z.(9)
The exp ession o gene al qis simila in o m o his, and is
RPo s =A(q, y, z) + B(q, y, z)
qη(q, y, z),(10)
whe e he coefficien s may be u he b oken down as A(q, y, z) = α(q, y, z)/γ(q, y, z)2and
B(q, y, z) = β(q, y, z)/γ(q, y, z)2and a e smoo h unc ions o yand zand do no di e ge
o fini e (physical) empe a u e o field. Fu he mo e
η(q, y, z) = [y(1 −z) + q−2]2+ (q−1)4z. (11)
2The e is a ac o o wo di e ence in he de ini ions o β, h be ween he Ising and Po s no a ions
coming om he di e en spin de ini ions.
The exp essions o α(q, y, z), β(q, y, z) and γ(q, y, z) a e e y leng hy o gene al q(al-
hough s ill easily ob ained) so, o he sake o compac ness, we w i e down only hose
o q= 3, which al eady display he gene ic ea u es. We ha e
α(3, y, z) = −1
4h−6−12 z+ 116 z2+ 80 z3−16 z4
+y(−44 −68 z+ 400 z2+ 552 z3−192 z4)
+y2(−113 −64 z+ 138 z2+ 476 z3−1004 z4)
+y3(−178 + 30 z+ 384 z2−1884 z3−2896 z4+ 8 z5)
+y4(−188 + 458 z−222 z2−276 z3−3400 z4+ 64 z5)
+y5(−106 + 592 z−474 z2−68 z3+ 1232 z4+ 120 z5)
+y6(−17 + 294 z−368 z2+ 250 z3−127 z4−32 z5)
+y7(4 + 66 z−136 z2+ 60 z3+ 2 z5+ 4 z4)i,(12)
and
β(3, y, z) = −1
4h−30 + 12 z+ 468 z2+ 768 z3+ 240 z4
+y(−82 −82 z+ 664 z2+ 3604 z3+ 1744 z4−16 z5)
+y2(−171 + 12 z+ 1366 z2+ 2292 z3+ 1764 z4−160 z5)
+y3(−261 + 213 z+ 1710 z2+ 846 z3−7584 z4−756 z5)
+y4(−278 + 386 z+ 400 z2−3804 z3−8804 z4−2488 z5+ 8 z6)
+y5(−254 + 570 z−1094 z2−5654 z3−1584 z4−3712 z5+ 64 z6)
+y6(−165 + 504 z−1494 z2−1008 z3+ 891 z4+ 1152 z5+ 120 z6)
+y7(−51 + 263 z−866 z2+ 986 z3−203 z4−97 z5−32 z6)
+y8(−4 + 66 z−182 z2+ 188 z3−72 z4+ 2 z5+ 2 z6)i,(13)
o he nume a o s, and
γ(3, y, z) = −3−4z−2z2−4y−16 y z −16 y z2
−7y2+ 2 y2z−31 y2z2−4y3+ 4 y3z2,(14)
and
η(3, y, z) = 1 + 8 z+ 2 y−2y z +y2−2y2z+y2z2,(15)
o he e ms in he denomina o s.
In ze o-field (z= 1) he exp ession o Ris much mo e compac and is w i en o
gene al qas
R=(y+q−1)(4y2+ (q−2)y−(q−2)(q−1))
(q−1)(2y+q−2)2.(16)
We see ha as y anges om 1 o ∞,R anges om (4−q)/(q−1) o ∞. In pa icula , he
sign o he y= 1 (β= 0) limi o Rchanges a q= 4, al hough he gene al mo phology
o Ras a unc ion o yand z emains he same o all q > 2 as we see below.
The co ela ion leng h o he one-dimensional Po s model is defined in a simila
manne o ha o he Ising model,
ξ−1=−ln λ1
λ0!,(17)
so ξ∼y o z= 1, y → ∞. We hus e ie e he expec ed scaling o R o he one-
dimensional Po s model om equ.(3), namely R ∼ yas y→ ∞. The exponen s, as o
he Ising model, a e α= 1, ν = 1.
The gene al ea u es o Ra non-ze o empe a u e and field a e pe haps easies seen
in a con ou plo as a unc ion o yand z. In Fig.1 we show he Ising (q= 2) case which
has ce ain non-gene ic ea u es. The ±hsymme y o he Ising model is mani es as a
z→1/z symme y in he plo o R. In addi ion, one can see ha Ris posi i e o all y
and z. The maximum o R o a gi en y alue lies along he ze o field line a z= 1.
In Fig.2, Ris plo ed o he 3-s a e Po s model, using he exp essions gi en abo e
o A(3, y, z) and B(3, y, z). We see ha he e is no longe a z→1/z symme y and
ha Ris no posi i e defini e. This beha iou is ypical o he q > 3 case, as can also
be seen om he q= 10 esul s plo ed in Fig.3. I is also clea om Fig.3. ha he
z= 1, y = 1 limi is nega i e unlike he 3-s a e model, as indica ed by equ.(16). When
q > 2 he maximum o Rno longe lies on he ze o-field line hough i s locus is s ill simply
de e mined, as we discuss in he nex sec ion.
3 Rema ks on Co-o dina es and Geodesics
We ha e concen a ed so a on he cu a u e R, which is o cou se a geome ic in a ian
and independen o he pa icula co-o dina e scheme we use in he calcula ion. The
p ope ies o he pa icula me ics and line elemen s used in he calcula ion a e also o
some in e es [5]. The me ic o he 1D Ising model calcula ed om equ.(1) gi es he
line elemen
ds2=e−4β
(sinh2h+ e−4β)3/2×

4e−4βcosh h+ 8 sinh2h(cosh h+qsinh2h+ e−4β)
(cosh h+qsinh2h+ e−4β)2dβ2
+ 4 sinh h dβdh + cosh h dh2i.(18)
Al hough his is clea ly non-diagonal, he choice o βand has co-o dina es has he
ad an age o gi ing he simplified exp ession, equ.(2), o Rsince hey appea as linea
mul iplie s in he Hamil onian.
The me ic can be diagonalized by inspec ion in his case, and his p e e ed choice
o co-o dina es u ns ou o be a he in e es ing om he physical poin o iew. I one
conside s a decima ion ype eno maliza ion ans o ma ion on a 1D Ising chain (o ing)
whe e e e y o he e ex is decima ed and he leng h scale sui ably adjus ed, hen one
can exac ly de e mine he ela ions be ween he eno malized pa ame e s β′, h′and he
o iginals β, h om
ZN/2(β′, h′) = ANZN(β, h) (19)
whe e ZN/2is he decima ed pa i ion unc ion and ZN(β, h) is he o iginal one. In
addi ion o he pa ame e ans o ma ion he e is also an unimpo an o e all scaling

ac o , AN. This ans o ma ion has an in a ian ρ= e2βsinh h, which is in effec encoding
he in a iance o he magne iza ion unde he ans o ma ion since
M=ρ
√1 + ρ2.(20)
I he co-o dina es a e ans o med om β, h o β, ρ hen, using
dρ = 2e2βsinh hdβ + e2βcosh hdh, (21)
one finds he diagonalized exp ession
ds2=1
q(1 + ρ2)(e4β+ρ2)

4 e4βdβ2
(√1 + ρ2+qe4β+ρ2)2+dρ2
(1 + ρ2)
.(22)
The use o he eno maliza ion g oup in a ian , o al e na i ely he magne iza ion,
also diagonalizes he me ic o he Ising model on a Be he la ice. I is he e o e na u al
o ask whe he his ea u e pe sis s in he Po s models conside ed he e. Al hough he
Po s me ic and line elemen s a e a he mo e complica ed han hose o he Ising model,
he same gene al s uc u e is once again appa en
ds2=(q−1)
η(q, y, z)(3/2) ×

C(q, y, z) + D(q, y, z)qη(q, y, z)
˜
λ(q, y, z)2dβ2+ 4[yz(z−1)]dβdh
+z[y(1 + z) + q−2]dh2i(23)
whe e η(q, y, z) is defined in equ.(11) and
˜
λ(q, y, z) = y(1 + z) + q−2 + q(y(1 −z) + q−2)2+ (q−1)4z(24)
is p opo ional o λ0, he dominan eigen alue. The unc ions C(q, y, z) and D(q, y, z) a e
bo h easily calculable, bu a he long, and a e no ep oduced he e.
I we ca y ou a decima ion on he 1D Po s model
ZN/2(y′, z′) = ANZN(y, z),(25)
hen he in a ian is gi en by
˜ρ=[y(1 −z) + q−2]
√z,(26)
which is again ela ed o he magne iza ion. Following he Ising p ocedu e, we can change
a iables β, h o β, ˜ρusing
d˜ρ=y(1 −z)
√zdβ −1
2
[y(1 + z) + q−2]
√zdh (27)
and find ha he me ic is, indeed, diagonalized.
In he Ising model i can be shown ha he line ρ= 0 (i.e. h= 0 o z= 1) is a
geodesic o he me ic (1). In Fig.1 his is he line unning along he idge o R. Fo
q6= 2 Po s models he local maximum in Rno longe lies a z= 1, as is clea om Fig.2
and Fig.3, howe e an analysis o he geodesic equa ions
dV β
ds + Γβ
ββVβVβ+ 2Γβ
βhVβVh+ Γβ
hhVhVh=λ(s)Vβ
dV h
ds + Γh
ββVβVβ+ 2Γh
βhVβVh+ Γh
hhVhVh=λ(s)Vh(28)
whe e spa ame e izes he flow lines, Vβ=dβ/ds,Vh=dh/ds, he Γ a e he a ious
Ch is offel symbols o he me ic o equ.(1), and λ(s) allows o non-affine pa ame e s,
shows ha he line z= 1 is s ill a geodesic. To p o e his conside a ec o field wi h a
flow line along z= 1 (h= 0). This flow line has Vh= 0 so equa ions (28) become
dV β
ds +Γβ
ββh=0VβVβ=λ(s)Vβ(29)
Γh
ββh=0VβVβ= 0.(30)
The fi s o hese equa ions is a second o de o dina y diffe en ial equa ion o β(s) which
always has a solu ion. The second equi es Γh
ββh=0 = 0 and i can easily be shown,
using he me ic (23), ha his is indeed he case. Hence he line z= 1 is a geodesic o
he me ic (23) o any alue o q.
A diffe en change o a iable in he Po s models b ings one back o some hing e y
simila o he Ising pic u e. I we ans o m z o
w=yz
y+q−2(31)
and lea e yun ouched, we find ha he local maximum in Rlies on he line w= 1.
We show Rplo ed agains win Fig.4 o he 3-s a e Po s model o compa ison wi h
Fig.2. The pic u e is simila o highe q. This ans o ma ion does no g ea ly simpli y
α(q, y, z), β(q, y, z) o γ(q, y, z) and we do no ep oduce he exp essions he e.
In summa y, al hough he choice o β, h as pa ame e s is na u al in any calcula ion
o R, he me ic is diagonalized o bo h he 1D Ising and Po s models when he eno -
maliza ion g oup in a ian , o equi alen ly he magne iza ion, is used as a co-o dina e
ins ead o h. The line z= 1 is a geodesic o he me ic (1) o any q.
4 A Lee-Yang Di e gence
Some yea s ago Lee and Yang [9] add essed he ques ion o how he singula i ies associ-
a ed wi h field-d i en phase ansi ions in Ising-like spin models on la ices a ose in he
he modynamic limi . This was la e ex ended by a ious au ho s o o he models and
o empe a u e-d i en ansi ions [10, 11]. Lee and Yang obse ed ha he ze oes o he
pa i ion unc ion o a spin model in a complex ex e nal field on a fini e la ice would gi e
ise o singula i ies in he ee ene gy. In he he modynamic limi hese complex ze oes
mo e in o pinch he eal axis, signalling he he onse o a physical phase ansi ion.
Typically, he loci o ze oes a e lines in he complex field o empe a u e plane and when
he endpoin s o such lines occu a non-physical (i.e. complex) ex e nal field alues hey
can be conside ed as o dina y c i ical poin s wi h an associa ed edge c i ical exponen ,
usually dubbed he Lee-Yang edge exponen [10].
The Lee-Yang ze oes o he one-dimensional Po s model on a pe iodic chain wi h N
si es a e gi en by he solu ions [8, 12] o
ZN= (λ1)N+ (λ0)N= 0 ⇔λ1= exp(inπ
N)λ0(32)
whe e λ0,1a e he eigen alues gi en in equ. (8) and nis odd. In he he modynamic limi
he locus o ze oes is de e mined by |λ0|=|λ1|o
η(q, y, z) = [y(1 −z) + q−2]2+ (q−1)4z= 0 (33)
which can be sa isfied o complex (in he q= 2 Ising case, pu ely imagina y) alues o h.
Unlike he Ising model case, he Lee-Yang ze oes o he 1D q6= 2 Po s models do no
lie on he uni ci cle in he complex zplane. Howe e , p ecisely he change o co-o dina es
we used in discussing he Po s me ics in he p e ious sec ion, w=yz/(y+q−2), places
hem on he uni ci cle in he complex wplane. I we associa e a “field” wi h w ia
w=exp(˜
h), hen pu ely imagina y alues o ˜
hgi e he ze oes, as in he Ising case3.
F om hese conside a ions, i is clea ha Rwill also di e ge as he locus o ze oes
is app oached o bo h Ising and Po s models i we allow ou sel es he libe y o an
analy ic con inua ion o he field o complex h alues once Rhas been calcula ed, since
R=A+B/√ηand A, B a e fini e as η→0. The p esence o he squa e oo means
ha he di e gence is cha ac e ised by an exponen σ=−1/2 which is he Lee-Yang edge
exponen o he one-dimensional Po s (and Ising) model [10].
The s a us o hese obse a ions a e a li le unclea o us, since he calcula ion o R
has assumed a eal me ic geome y h oughou and such an a bi a y con inua ion in he
final exp ession migh be a he dange ous. One is on sligh ly fi me g ound wi h he Ising
model, since in ha case he equi ed con inua ion is o pu ely imagina y fields which
co espond simply o a change in he signa u e o he me ic in β, ρ co-o dina es. The use
o he wco-o dina e in he Po s case does, howe e , sugges a simila in e p e a ion since
he e he ze oes occu o imagina y alues o ˜
h. Wi h hese ca ea s, i is none heless
in e es ing ha he Lee-Yang edge ansi ion is s ill isible as a di e gence in R.
5 Conclusions
We ha e seen ha he scala cu a u e, R, o he Fishe -Rao me ic may be ob ained o
he 1D q-s a e Po s model in a e y simila ashion o he 1D Ising model since he ee
ene gy can be calcula ed in field in bo h cases using ans e ma ix me hods.
Al hough R o he q-s a e Po s model canno be exp essed as succinc ly as o he
Ising model i s ill has he same gene al o m, R=A+B/√η, and di e ges only a
he ze o empe a u e c i ical poin o he Po s model o physical empe a u e and field
alues, i.e. y= 1 ...∞and z= 0 ...∞. We w o e down Rexplici ly he e o q= 3, bu
no ed i was a simple ma e o calcula e i o a bi a y q. We obse ed ha he e we e
some ea u es o he Ising model Rwhich did no pe sis o gene al q, as was clea om
3This is no , o cou se, he ield happea ing in he Hamil onian.
con ou plo s. In pa icula , Rwas no longe posi i e defini e and he z→1/z symme y
o he Ising model was no longe p esen .
The choice o co-o dina e scheme used in he calcula ion o Rwas also discussed. We
no ed ha al hough β, h we e a na u al choice om he poin o iew o simpli ying his
calcula ion, he use o he eno maliza ion g oup in a ian in place o hdiagonalized he
me ic o bo h he Ising and Po s models. A diffe en choice o coo dina es, in ol ing a
escaling o zwas also employed o iden i y he local maximum in R. Fo any alue o q
he line z= 1 (h= 0) is a geodesic o he Fishe -Rao me ic. Finally, we obse ed ha i
complex field alues a e pe mi ed he e is a di e gence in Ra he Lee-Yang edge wi h
an exponen σ=−1/2.
We ha e confined ou discussion he e o physical alues o q, i.e. 2,3,..., in he Po s
models. Since qappea s as a pa ame e in he ans e ma ix solu ion he e is in p inciple
no ba ie o ex ending i o gene al non-in ege qand also o q < 2. The q→1 limi o
he Po s model is ela ed o pe cola ion, o example, so he beha iou o Rmigh be
o in e es in his limi oo. I would also be in e es ing o calcula e R o models wi h
genuine ansi ions a fini e β, an exe cise ha has so a been confined o mean-field o
mean-field-like spin models. Possibili ies which sugges hemsel es in his con ex a e he
Ising model on plana andom g aphs [13] and he sphe ical model.
6 Acknowledgemen s
D.J. was pa ially suppo ed by EC IHP ne wo k “Disc e e Random Geome ies: F om
Solid S a e Physics o Quan um G a i y” HPRN-CT-1999-000161. D.J. and R.K. we e
also pa ially suppo ed by an En e p ise I eland/B i ish Council Resea ch Visi s Scheme
g an .
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