Jordan-Wigner fermionization for spin-1/2 systems in two dimensions: A brief review

19 Pages·0.24 MB·English

by  Oleg Derzhko

Jordan–Wigner fermionization
1
for spin– systems in two dimensions:
2
A brief review
Oleg Derzhko
Institutefor Condensed Matter Physics
1 Svientsitskii Street, L’viv–11, 79011, Ukraine
and
1
0 Chair of Theoretical Physics, Ivan FrankoNational Universityof L’viv
0
12 Drahomanov Street, L’viv–5, 79005, Ukraine
2
(February1, 2008)
n
a WereviewthepapersontheJordan–Wignertransformationintwodimensionstocommenton
J
apossibility ofexaminingthestatisticalmechanicspropertiesoftwo–dimensional spin–1 models.
2 2
1 We discuss in some detail the two–dimensional spin–1 isotropic XY and Heisenberg models.
2

1
v PACS numbers: 75.10.–b
8
8
Keywords: 2D XY model, 2D Heisenberg model, Jordan–Wigner fermionization
1
1
0
Postal address:
1
0 Dr. Oleg Derzhko
/
t Institute for Condensed Matter Physics
a
m 1 Svientsitskii Street, L’viv–11, 79011,Ukraine
- tel/fax: [phone]
d
n email: [email protected]
o
c
:
v
i
X I. INTRODUCTORY REMARKS
r
a
A mapping of the spin–1 operators onto Fermi operators by means of the Jordan–Wigner transformation was
2
used by Lieb, Schultz and Mattis1 to introduce the exactly solvable one–dimensional spin–1 XY model. Later
2
the famous Onsager’s solution for the two–dimensional Ising model was reproduced by using the Jordan–Wigner
transformationforthetransfermatrixofthatmodel2. TheJordan–Wignertransformationistheessentialingredient
of the studies of the statistical mechanics properties of quantum spin chains3. Much effort has been devoted to
generalize the fermionization procedure for two4–14 and three15,16 dimensions. In the present paper we review the
Jordan–Wignertransformationintwodimensionsaswellassomeexistingapplicationsofthismappingforthespin
system theory.
We shall consider a spin model consisting of N =N N (N , N ) spins 1 on a square lattice of the
x y x →∞ y →∞ 2
size L L (L , L ) governedby the Heisenberg Hamiltonian
x y x y
→∞ →∞
H = Js s + hsz
i· j i
Xhi,ji Xi
∞ ∞ ∞ ∞
= J(s s +s s )+ hsz . (1)
i,j · i+1,j i,j · i,j+1 i,j
Xi=0Xj=0 Xi=0Xj=0
Here i,j denotesalldifferentnearestneighbouringsitesatthesquarelattice,J istheexchangeinteractionbetween
h i
the neighbouring sites, h is the external field. The isotropic Heisenberg interaction consists of the isotropic XY
part and the Ising part
s s = sxsx+sysy +szsz
i· j (cid:16) i j i j(cid:17) i j
1 1 1
= s+s−+s−s+ + s+s− s+s− , (2)
2(cid:16) i j i j (cid:17) (cid:18) i i − 2(cid:19)(cid:18) j j − 2(cid:19)
wherewehaveintroducedthespinraisingandloweringoperatorss± =sx isy andsz =s+s− 1. Itisimportant
± −2
to note that the operators s+, s− obey the Fermi commutation rules at the same site
s−,s+ =1, s+,s+ = s−,s− =0 (3)
i i i i i i
(cid:8) (cid:9) (cid:8) (cid:9) (cid:8) (cid:9)
and the Bose commutation rules at different sites i=j
6
s−,s+ = s+,s+ = s−,s− =0. (4)
i j i j i j
h i h i h i
The aim of the Jordan–Wigner trick is to transform the spin variables into pure fermion variables.
We start the paper by giving a short reminder of the Jordan–Wigner transformation in one dimension (Section
II).ThenwediscusstheextensionsfortwodimensionssuggestedbyM.Azzouz8(SectionIII),Y.R.Wang5(Section
IV), and E. Fradkin4 (Section V). The mean–field–like treatment of the Jordan–Wigner fermions for the isotropic
XY model is discussedin detail inSection VI. The considerationof a modelwith Ising term infermionic language
is given separately in Section VII. Finally, we summarize some of the results obtained using this approximate
approach and comment on a comparison with the results derived using other methods (Section VIII).
II. THE JORDAN–WIGNER TRANSFORMATION IN ONE DIMENSION
With the help of the Jordan–Wignertransformationwe introduce instead of the operatorss+, s− satisfying (3),
(4) the operators c+, c satisfying the Fermi commutation rules (both at the same and different sites) in terms of
which the Hamiltonian of the isotropic XY chain is a bilinear form and the Ising interaction yields the products
of four Fermi operators. Explicitly the Jordan–Wigner transformationreads
c+ =eiαns+, c =e−iαns−, s+ =e−iαnc+, s− =eiαnc ,
n n n n n n n n
n−1
α =π n , n =c+c . (5)
n j j j j
Xj=0
Thesignsintheexponentsandtheorderofthemultipliers(firstlinein(5))arenotimportant. Togettheoperator
c (c+) we must multiply s− (s+) by the exponent containing a sum of n at all previous sites 0 j n 1 as
n n n n j ≤ ≤ −
can be seen from (5) and is shown symbolically in Fig. 1.
0 1 2 3 n-1 n n+1
FIG.1. Towards theJordan–Wigner transformation in one dimension.
To demonstrate how the transformation (5) works we need the following equations for Fermi operators
c+j eiπc+jcj =c+j , eiπc+jcjc+j =−c+j , cjeiπc+jcj =−cj, eiπc+jcjcj =cj, (6)
which can be easily obtained since
eiπc+jcj =1+iπc+j cj + 21!(iπ)2c+j cj +...=1−c+j cj +eiπc+j cj =1−2c+j cj.
From (6) one finds that
eiπc+ncncl =cleiπc+ncn, eiπc+ncnc+l =c+l eiπc+ncn, (7)
if n=l, but
6
eiπc+l clcl =−cleiπc+l cl, eiπc+l clc+l =−c+l eiπc+l cl. (8)
Besides,
e2iπc+l cl =1. (9)
Let us show that the commutation rules for the spin raising and lowering operators s+, s− (5) are given by (3),
(4) if c+, c are Fermi operators. At the same site we find
s−s+ =c eiαne−iαnc+ =c c+, s+s− =c+c , s+s+ =c+c+ =0, s−s− =c c =0, (10)
n n n n n n n n n n n n n n n n n n
and as a result Eq. (3) becomes evident. At different sites n and n+m (without any loss of generality m>0) we
have
s−ns+n+m =cneiπ nl=+nm−1c+l clc+n+m =cnc+n+meiπ nl=+nm−1c+l cl,
P P
s+n+ms−n =c+n+meiπPnl=+nm−1c+l clcn =−c+n+mcneiπPnl=+nm−1c+l cl, (11)
etc. which immediately yields Eq. (4).
Let us write down the transformed Hamiltonian. Besides s+s− =c+c (10) we have
j j j j
s+j s−j+1 =c+j eiπc+jcjcj+1 =c+j cj+1, s−j s+j+1 =cjeiπc+jcjc+j+1 =−cjc+j+1, (12)
and as a result the one–dimensional spin–1 Heisenberg Hamiltonian (1), (2) becomes
2
1 1 1 1
H = J c+c c c+ +J c+c c+ c + h c+c . (13)
(cid:18)2 j j+1− j j+1 (cid:18) j j − 2(cid:19)(cid:18) j+1 j+1− 2(cid:19)(cid:19) (cid:18) j j − 2(cid:19)
Xj (cid:0) (cid:1) Xj
It is clear now to what extent the Jordan–Wigner transformation simplifies further statistical mechanics cal-
culations. A nontrivial in spin language isotropic XY chain transforms into tight–binding spinless fermions and
further rigorous treatment becomes possible. For the anisotropic XY chain the operators c+c+ , c c enter
j j+1 j j+1
Eq. (13) besides, and therefore the Bogolyubovtransformationis required in addition. The Ising term leads to an
interaction between the Jordan–Wigner fermions, however, the low–energy properties may be analysed using the
bosonizationtechniques3. While interesting only in the low–energyphysics ofs> 1 spin chains one may represent
2
the spin–s operators as a sum of 2s spin–1 operators and then, making use of the Jordan–Wigner representation
2
for the latter operators, proceed in fermionic language.
III. THE JORDAN–WIGNER TRANSFORMATION IN TWO DIMENSIONS (M. AZZOUZ, 1993)
Consider the spin model (1) on a square lattice (Fig. 2).
i-1,j+1 i,j+1
i-1,j i,j
i-1,j-1 i,j-1
y
i-1,j-2 i,j-2
x
FIG.2. Towards the Jordan–Wigner transformation in two dimensions8.
Two coordinates i and j are taken at the x and y axes, respectively, to specify a given site.
M. Azzouz defined8 the extended Jordan–Wigner transformation as
s− =c eiαi,j =eiαi,jc , s+ =c+ e−iαi,j =e−iαi,jc+ ,
i,j i,j i,j i,j i,j i,j
i−1 ∞ j−1
αi,j =π nd,f + ni,f, nd,f =c+d,fcd,f (14)
Xd=0fX=0 fX=0
 
(compare with Eqs. (5)). The signs in the exponents in (14) (and the order of the multipliers in the first line in
(14)) are not important. Let us show that the introduced transformation (14) enables one to construct a fermion
representation for two–dimensional spin–1 models.
2
We start with the commutation rules. At the same site one has
s− s+ =c eiαq,pe−iαq,pc+ =c c+ , s+ s− =c+ c , s+ s+ =c+ c+ , s− s− =c c , (15)
q,p q,p q,p q,p q,p q,p q,p q,p q,p q,p q,p q,p q,p q,p q,p q,p q,p q,p
andthe Fermi type commutationrules remainunchanged. To illustrate how transformation(14)worksin the case
of different sites let us consider, for example, two sites q,p and q,p+m, m > 0. Then, similarly to Eq. (11) one
finds
s−q,ps+q,p+m =cq,peiπ fp+=mp−1nq,fc+q,p+m =cq,pc+q,p+meiπ pf+=mp−1nq,f,
P P
s+q,p+ms−q,p =c+q,p+meiπPfp+=mp−1nq,fcq,p =−c+q,p+mcq,peiπPpf+=mp−1nq,f (16)
(we haveusedananalogueof Eqs. (7), (8)intwo dimensions)andhence the operatorss− ands+ commute if
q,p q,p+m
theoperatorsc andc+ anticommute. Followingthesamereasoningonecanchecktherestofthecommutation
q,p q,p+m
rules.
Consider further the transformed spin Hamiltonian. Let us treat somewhat more general nearest neighbour
interactions than the one in Eq. (1). Namely, we assume different values of interaction in different directions on a
square lattice as shown in Fig. 3.
i,j+1
J
⊥
i-1,j J' i,j J i+1,j
J'
⊥
i,j-1
FIG.3. Nearest neighbour interactions on a square lattice.
(Evidently,wecanperformthefermionizationpresentedbelowforacompletely nonuniformmodelcharacterized
by a set of intersite interactions ...,J ,...;...,J ,... .) If J =J′ =J =J′ one faces the uniform
{ i,j;i+1,j i,j;i,j+1 } ⊥ ⊥
squarelattice. Toexaminetheeffectsofinterchaininteractionsinquasi–one–dimensionalsystemsonemayconsider
the case J , J′ J, J′ (with J not equal to J′ if the dimerised chain is considered). If J′ J,J′,J one has
⊥ ⊥ ≪ ⊥ ≪ ⊥
a model of interacting two–leg ladders. In the limiting case of J′ = 0 (the noninteracting two–leg ladders) the
⊥
model may be reduced to a one–dimensional system with interactions extending over the nearest sites.
We begin with the isotropic XY interaction (see Fig. 4).
i,j+1
i,j i+1,j i,j
FIG.4. Towards the fermionization of HXY.
Inserting (14) into (1), (2) one finds
∞ ∞
1
H = J s− s+ +s+ s− +J s− s+ +s+ s−
XY 2 i,j;i+1,j i,j i+1,j i,j i+1,j i,j;i,j+1 i,j i,j+1 i,j i,j+1
Xi=0Xj=0(cid:0) (cid:0) (cid:1) (cid:0) (cid:1)(cid:1)
= 21 ∞ ∞ (cid:18)Ji,j;i+1,j(cid:18)ci,je−iπ(cid:0)P∞f=jni,f+Pfj−=10ni+1,f(cid:1)c+i+1,j +c+i,jeiπ(cid:0)P∞f=jni,f+Pfj−=10ni+1,f(cid:1)ci+1,j(cid:19)
Xi=0Xj=0
+J c e−iπni,jc+ +c+ eiπni,jc
i,j;i,j+1 i,j i,j+1 i,j i,j+1
(cid:16) (cid:17)(cid:17)
= 21 ∞ ∞ (cid:18)Ji,j;i+1,j(cid:18)−ci,je−iπ(cid:0)P∞f=j+1ni,f+Pfj−=10ni+1,f(cid:1)c+i+1,j +c+i,jeiπ(cid:0)P∞f=j+1ni,f+Pfj−=10ni+1,f(cid:1)ci+1,j(cid:19)
Xi=0Xj=0
+J c c+ +c+ c (17)
i,j;i,j+1 − i,j i,j+1 i,j i,j+1
(cid:0) (cid:1)(cid:1)
(to get the last equality we have used an analogue of Eq. (6) in two dimensions). After introducing the notations
∞ j−1
φi,i+1(j)=π ni,f + ni+1,f,
Xf=j fX=0
 
∞ j−1
φ˜i,i+1(j)=π ni,f + ni+1,f,
f=Xj+1 fX=0
 
ϕ (i)=πn (18)
j,j+1 i,j
the Hamiltonian (17) becomes as follows
∞ ∞
H = 1 J c e−iφi,i+1(j)c+ +c+ eiφi,i+1(j)c
XY 2 i,j;i+1,j i,j i+1,j i,j i+1,j
Xi=0Xj=0(cid:16) (cid:16) (cid:17)
+J c e−iϕj,j+1(i)c+ +c+ eiϕj,j+1(i)c
i,j;i,j+1 i,j i,j+1 i,j i,j+1
(cid:16) (cid:17)(cid:17)
∞ ∞
= 1 J c e−iφ˜i,i+1(j)c+ +c+ eiφ˜i,i+1(j)c
2Xi=0Xj=0(cid:16) i,j;i+1,j(cid:16)− i,j i+1,j i,j i+1,j(cid:17)
+J c c+ +c+ c . (19)
i,j;i,j+1 − i,j i,j+1 i,j i,j+1
(cid:0) (cid:1)(cid:1)
Eq. (19) can be viewed as the Hamiltonian of a two–dimensional tight–binding–like spinless fermions with the
hopping amplitudes
1J e∓iφ˜i,i+1(j) (20)
i,j;i+1,j
∓ 2
in the x direction and
1
J (21)
i,j;i,j+1
∓ 2
intheydirection. Thosehoppingsdependinacomplicatedwayonaconfigurationofthe‘intermediate’sites. Their
complexity explains how the isotropic XY model becomes difficult to examine in two dimensions in comparison
with an obvious analysis in one dimension.
There are no difficulties in rewriting the Ising interaction in fermionic language
∞ ∞
1 1 1 1
H = J s+ s− s+ s− +J s+ s− s+ s−
Z (cid:18) i,j;i+1,j(cid:18) i,j i,j − 2(cid:19)(cid:18) i+1,j i+1,j − 2(cid:19) i,j;i,j+1(cid:18) i,j i,j − 2(cid:19)(cid:18) i,j+1 i,j+1− 2(cid:19)(cid:19)
Xi=0Xj=0
∞ ∞
1 1 1 1
= J c+ c c+ c +J c+ c c+ c (22)
(cid:18) i,j;i+1,j(cid:18) i,j i,j − 2(cid:19)(cid:18) i+1,j i+1,j − 2(cid:19) i,j;i,j+1(cid:18) i,j i,j − 2(cid:19)(cid:18) i,j+1 i,j+1− 2(cid:19)(cid:19)
Xi=0Xj=0
and the interaction with an external field
∞ ∞ ∞ ∞
1 1
H = h s+ s− = h c+ c . (23)
f (cid:18) i,j i,j − 2(cid:19) (cid:18) i,j i,j − 2(cid:19)
Xi=0Xj=0 Xi=0Xj=0
Formulas (19), (22), (23) realize the fermionic representation of the spin–1 isotropic Heisenberg model on a
2
square lattice (1), (2).
IV. THE JORDAN–WIGNER TRANSFORMATION IN TWO DIMENSIONS (Y. R. WANG, 1991)
Let us turn back to the Jordan–Wigner transformation in one dimension (5) using this case as a guideline and
define a particle–annihilation operator as
di =e−iαis−i , αi = Bijnj, (24)
jX(6=i)
where B is the c–number matrix element, n =d+d . A particle–creationoperator is given by
ij j j j
d+ =s+eiαi =eiαis+, (25)
i i i
whereas the inverse to Eqs. (24), (25) formulas read
s−i =eiαidi =dieiαi, (26)
s+ =e−iαid+ =d+e−iαi. (27)
i i i
We want the introduced operators d+, d to obey the Fermi type commutation rules. They are indeed the Fermi
operators at the same site due to Eq. (3). Consider further two different sites i = j. Assuming that d+, d are
6
Fermi operators one finds
hs+i ,s−j i=e−iαid+i eiαjdj−eiαjdje−iαid+i
=e−injBijd+i einiBjidj−einiBjidje−injBijd+i . (28)
Since
e−injBij =1+ e−iBij 1 nj, einiBji =1+ eiBji 1 ni
(cid:16) − (cid:17) (cid:16) − (cid:17)
one can easily proceed in calculation of the r.h.s. of Eq. (28) finally arriving at
s+i ,s−j = 1+e−iBijeiBji d+i dj. (29)
h i (cid:16) (cid:17)
The result equals to 0 (as it should for the commutator of the operators s+ and s− attached to different sites) if
eiBij = eiBji. (30)
−
Thus we assume that B ’s in (24), (25), (26), (27) satisfy relation (30) that yields the spin commutation relations
ij
for s+, s (3), (4) if d+, d are Fermi operators.
Y. R. Wang suggested5 the following choice for B . Consider two complex numbers
ij
τ =i +ii (31)
i x y
and
τ =j +ij , (32)
j x y
which correspond to the sites i = i n +i n and j = j n +j n , respectively. Here n and n are the unit
x x y y x x y y x y
vectors directed along x and y axes, respectively. Assume that
B =arg(τ τ ). (33)
ij j i
−
Evidently
eiBji =eiarg(τi−τj) =ei(arg(τj−τi)±π) = eiBij, (34)
−
that is the required condition (30). Since τi τj = τi τj eiarg(τi−τj) Eq. (33) can be rewritten in the form
− | − |
B =Imln(τ τ), (35)
ij j i
−
and hence the introduced transformations (24), (25), (26), (27) contains
α = Imln(τ τ)n . (36)
i j i j
−
jX(6=i)
It is worth noting that the transformation of M. Azzouz (14) can also be written as Eqs. (26), (27) with
B =π Θ(i j )(1 δ )+δ Θ(i j ) 1 δ (37)
ij x− x − ix,jx ix,jx y− y − iy,jy
(cid:0) (cid:0) (cid:1)(cid:1)
where Θ(x) is the step function, and a necessary condition for having spin to fermion mapping (30) fulfilled.
The advantage of the choice of Y. R. Wang (36) becomes clear when one tries to introduce an approximate
treatment of the transformed Hamiltonian. After inserting (26), (27) into (1), (2) one finds
H =Xhi,ji(cid:18)21Ji,j(cid:16)d+i ei(αj−αi)dj+diei(αi−αj)d+j (cid:17)
1 1 1
+J d+d d+d + h d+d . (38)
i,j(cid:18) i i− 2(cid:19)(cid:18) j j− 2(cid:19)(cid:19) (cid:18) i i− 2(cid:19)
Xi
Note that
j
α α = dr A(r) (39)
j i
− Z ·
i
where A(r)= α and hence
r r
∇
∂
A(r)= ∂r  nr′Imln rx′ −rx+i ry′ −ry nx
x r′X(6=r) (cid:0) (cid:0) (cid:1)(cid:1)
 
∂
+∂r  nr′Imln rx′ −rx+i ry′ −ry ny
y r′X(6=r) (cid:0) (cid:0) (cid:1)(cid:1)
 
r′ r n (r′ r )n
= nr′(cid:0) y − y(cid:1)(rx′ −r)2x− x y
r′X(6=r) −
n (r′ r)
=− nr′ z(r×′ r−)2 . (40)
r′X(6=r) −
The crucial approximation to proceed is to make the change in (40)
1 1
n n = sz + . (41)
r →h ri h ri 2 → 2
HeretheangularbracketsdenotethethermodynamicalcanonicalaveragewiththeHamiltonian(1). Thus,wehave
postulatedthe mean–fielddescriptionofthe phase factorsin(38). A similartreatmentwasadoptedinRefs. 17,18.
Apparently, assuming further in (41) that sz = 0 one should suppose that h = 0 (although in Refs. 10,11 the
h ri
uniformmagneticfieldwasincludedintothe Hamiltonian). InprincipleEq. (41)simplifiesthe problemdrastically
since one faces a tight–binding spinless fermions on a square lattice. However, in practice it is hard to proceed
because of nonuniformity of that model.
TheHamiltonian(38),(39),(40),(41)describesthe(charged)spinlessfermionsmovinginaplaneinanexternal
uniform(classical)magneticfield whichis perpendicular to the plane. Due to (41) nr′ canbe takenout fromthe
h i
summation and in the continuum limit the vector potential of the field A(r) can be written7 in the following form
n (r′ r)
A(r)=−hnr′i z(×r′ r−)2
r′X(6=r) −
=−hnr′iS10 Z−LL2xx drx′ Z−LL2yy dry′ nz(×r′(r′r−)2r)
2 2 −
π
=...= nr′ nz r, (42)
h iS ×
0
where S is the area of the elementary plaquette in the plane (see Fig. 5)
0
H
S
0
n i,j
z n A(r)
y 2
r
n A(r) 2
x r
1
1
FIG. 5. Fermions in the magnetic field, which appears within the mean–field treatment of the phase factors in the
Hamiltonian (38).
and L =L =L . The corresponding magnetic field H(r) immediately follows from Eq. (42)
x y
→∞
n n n
x y z
π (cid:12) (cid:12) 2π
H(r)=rotA(r)=hnr′iS (cid:12)(cid:12) ∂∂x ∂∂y ∂∂z (cid:12)(cid:12)=hnr′iS nz, (43)
0 (cid:12) (cid:12) 0
(cid:12) ry rx 0 (cid:12)
(cid:12)− (cid:12)
(cid:12) (cid:12)
so that A(r)= 1H(r) r. The flux per elementary p(cid:12)laquette equa(cid:12)ls to
2 ×
Φ0 =H(r) S0nz =2π nr′ =π. (44)
· h i
The vector potential A(r) (42) shown schematically in Fig. 5 is not convenient to be concerned with. Because
of the gauge invariance one may perform a gauge transformation introducing a new vector potential A˜(r) which
yields the same flux per elementary plaquette Φ =π. Namely, assume that A˜(r) is such that
0
i+1,j
α α = dr A˜(r)=π, (45)
i+1,j i,j
− Z ·
i,j
whereas
α α =α α =α α =0 (46)
i+1,j+1 i+1,j i,j+1 i+1,j+1 i,j i,j+1
− − −
(see Fig. 6). From Eqs. (45), (46) one finds that
dr A˜(r)=π, (47)
I ·
and on the other hand
dr A˜(r)= dS rotA˜(r)= dS H(r)=Φ , (48)
0
I · Z · Z ·
and hence the flux per elementary plaquette Φ remains equal to π.
0
Letus turnbackto the Hamiltonian(38). Only nowwe areinpositionto proceedwiththe statisticalmechanics
analysis. Withintheframesoftheintroducedmean–fieldtreatmentofthephasefactors(45),(46)theHamiltonian
(38) can be rewritten as
1
H = J d+d dd+ +H +H , (49)
Xhi,ji2 i,j(cid:16) i j− i j (cid:17) Z f
where even for initially uniform lattice J =J′ =J =J′ in the Hamiltonian H one has
⊥ ⊥ XY
J = J, J =J, J =J, J =J (50)
i,j;i+1,j i,j;i,j+1 i+1,j;i+2,j i+1,j;i+1,j+1
−
etc. (see Fig. 6).
i,j+1 i+1,j+1
i,j i+1,j
FIG.6. Towards Eqs. (45), (46) and Eqs. (49), (50).