Table Of ContentMay 1996
The Supersymmetric t-J Model with a Boundary
Fabian H.L. Eßler∗
7
9
9
1
n Department of Physics, Theoretical Physics
a
J Oxford University
6
2
1 Keble Road, Oxford OX1 3NP, Great Britain
2
v
0
8
1
5 ABSTRACT
0
6
An open supersymmetric t-J chain with boundary fields is studied by means of the
9
/ Bethe Ansatz. Ground state properties for the case of an almost half-filled band
t
a and a bulk magnetic field are determined. Boundary susceptibilities are calculated
m
as functions of the boundary fields. The effects of the boundary on excitations
-
d are investigated by constructing the exact boundary S-matrix. From the analytic
n structure of the boundary S-matrices one deduces that boundary bound states are
o
formed for sufficiently strong boundary fields.
c
:
v
i
X
r PACS numbers: 71.27.+a 75.10.Jm
a
∗e-mail: [email protected]
I. INTRODUCTION
Recently there has been renewed interest in one-dimensional impurity problems and the
related problem of one-dimensional Luttinger liquids with boundaries [1,2,3,4,5,6,7,8]. The
main focus of these investigations has been the effects of Kondo-like impurities and effects due
to potential scattering in Luttinger liquids. These impurity problems are closely related to
open1-dsystems with boundary fields. Some ofthese systems are integrable andcanbe solved
exactlybyBetheAnsatz[9,10,11,12,13,14]. Inparticular,in[1]ananisotropicHeisenbergchain
with open boundary conditions was studied. It is the purpose of the present work extend the
investigation of [1] to the case of the t-J model, which is a Luttinger liquid with both spin
and charge degrees of freedom.
In [15] a trigonometric generalization of the supersymmetric t-J model with open bound-
aries was constructed by means of the Quantum-Inverse Scattering Method (see e.g. [25]).
This generalized the previous work by F¨orster and Karowski on the quantum-group invariant
case [16]. Here we study this model at the rationalpoint, for which it reduces to the supersym-
metric t-J model with open boundaries and boundary fields. The reason for this restriction
is that the trigonometric model in general leads to a non-hermitian bulk hamiltonian. The
only exception is the hyperbolic regime, but there the spin excitations are gapped and thus
irrelevant for the low-energy physics of the model. The hamiltonian we consider in the grand
canonical ensemble is given by
L−1
H = c† c +c† c
−P j,σ j+1,σ j+1,σ j,σP
j=1 σ
X X
L−1 n n L−1
+2 S~ S~ j j+1 + n +n HSz µNˆ +H , (1)
j j+1 j j+1 αβ
· − 4 − −
j=1 j=1
X X
where projects out double occupancies, S~ are spin operators at site j, n = c† c +c† c ,
P j j j,↑ j,↑ j,↓ j,↓
and the four possible choices of boundary hamiltonians H compatible with integrability and
αβ
conservation of total spin in z-direction and particle number are given by
nh
H = h′n +h′ n , H = h (Sz 1)+h′ n
aa 1 1 L L ab 1 1 − 2 L L
nh nh nh
H = h′n +h (Sz L) , H = h (Sz 1)+h (Sz L) . (2)
ba 1 1 L L − 2 bb 1 1 − 2 L L − 2
Here nh = 1 n n is the number operator for holes (unoccupied sites) at site j. To
j − j,↑ − j,↓
simplify the computations we constrain ourselves to the regions h (0,2), h′ (0,1). It is
∈ ∈
straightforwardtoextend theanalysis belowtootherrangesofthefields. Forlaterconvenience
we define the quantities
2 2 for aa, ba 2 2 for aa, ab
S = − h′1 , S = − h′L . (3)
1 1 2 for ab, bb L 1 2 for ba, bb
− h1 − hL
In what follows we always assume that S and S are noninteger numbers. We note that
1 L
for zero boundary fields (1) exhibits a global sl(1 2) symmetry [16]. In the present work we
|
perform a detailed study of the boundary effects in the model defined by (1), paying particular
1
attention to the influence of the boundary fields. After some technical preliminaries we turn to
an analysis of the ground state properties. We find that the zero-temperature susceptibilities
exhibit some interesting singularities, which we argue to be related to the formation of bound
states near the boundaries. We then study the interaction of elementary excitations with
the boundaries by computing the exact boundary S-matrices. We find that boundary bound
states can be formed for sufficiently strong boundary fields. We concentrate on the case of
bandfillingsclosetoone(correspondingtotheparticularlyinteresting caseofthelightlydoped
Mott-Hubbard insulator) for which it is possible to obtain explicit analytical results. However
it is straightforward to extend the present analysis to arbitrary band-fillings by solving the
integral equations (14) numerically and then numerically integrating (13).
Taking the rational limit of the Bethe Ansatz equations derived in [15] we obtain
Nh+N↓ Nh
η (λ )(e (λ ))2L = e (λ λ )e (λ +λ ) e (λ λ(1))e (λ +λ(1))
αβ k 1 k 2 k − j 2 k j −1 k − l −1 k l
j6=k l=1
Y Y
Nh+N↓
(1) (1) (1)
1 = ζ (λ ) e (λ λ )e (λ +λ ) , (4)
αβ l 1 l − j 1 l j
j=1
Y
x+in
where e (x) = 2 and α,β = a,b. The boundary terms are given by
n x−in
2
η (λ) = 1 , η (λ) = e (λ) , η (λ) = e (λ) , η (λ) = η (λ)η (λ)
aa ab − −S1 ba − −SL bb ab ba
ζ (λ) = 1 , ζ (λ) = e (λ) , ζ (λ) = e (λ) , ζ (λ) = ζ (λ)ζ (λ) . (5)
bb ab − −SL ba − −S1 aa ab ba
The restrictions imposed on h and h′ are chosen such that in all these expressions the label x
on e (λ) is positive with range (0, ). The energy of a state corresponding to a solution of
x
∞
(4) is (up to an overall constant, which we drop)
Nh+N↓ 1 H H
E = E +H(N +N )+(µ )N (µ+ )L , (6)
ij − 1 +λ2 ↓ h − 2 h − 2
j=1 4 j
X
where E = h′ + h′ , E = h1+hL and so on. The reference state used to derive (4) is the
aa 1 L bb 2
one with up-spin electrons at each site of the lattice. This leads to the constraint in (4) that
the number N of down-spins must be smaller than or equal to the number of up-spins N .
↓ ↑
Solutions of (4) violating this constraint can lead to vanishing wave-functions and must be
ignored. Eigenstates of (1) with N > N must be constructed by switching the reference state
↓ ↑
to the state with down-spin electrons at all sites. This leads to the same Bethe equations (4)
with N N and different values for the quantities S
↓ ↑ j
↔
2 2 for aa, ba 2 2 for aa, ab
S = − h′1 , S = − h′L . (7)
1 1+ 2 for ab, bb L 1+ 2 for ba, bb
h1 hL
Below we will mainly deal with situations for which N N . However, when considering
↓ ↑
≤
excitations over the antiferromagnetic ground states we will also consider the case N N ,
↓ ↑
≥
for which we will make use of the procedure outlined above.
2
In order to simplify (4) we make use of the ‘string-hypothesis’ †, which states that for
L all solutions are composed of real λ(1)’s whereas the λ’s are distributed in the complex
→ ∞ γ
plane according to the description
n+1
λn,j = λn +i j ,j = 1...n (8)
α α 2 −
(cid:18) (cid:19)
where α = 1...M labels different ‘strings’ of length n. This string hypothesis is naturally
n
identical to the one for the model with periodic boundary conditions. The imaginary parts of
the λ’scan now beeliminated from(4) via (8). Taking the logarithmof the resulting equations
(for M strings (8) of length n and N λ(1)’s (note that ∞ nM = N +N ) we arrive at
n h n=1 n ↓ h
2π 1 λn 1 P
In = (2+ )θ( α) θ (λn λm)+θ (λn +λm)
L α L n − L mn α − β mn α β
(Xmβ)
1 Nh λn λ(1) λn +λ(1) 1
+ θ( α − γ )+θ( α γ )+ κ(n)(λn) ,α = 1...M
L n n L ij α n
γ=1
X
2π 1 1
J = θ(λ(1) λn)+θ(λ(1) +λn)+ ω (λ(1)) ,γ = 1...M(1) (9)
L γ L γ − α γ α L ij γ
(Xnα)
where In and J are integer numbers, θ(x) = 2arctan(2x),
α γ
x x x x
θ (x) = (1 δ )θ( )+2 θ( )+...+2 θ( )+θ( ) ,
n,m m,n
− n m n m +2 n+m 2 n+m
| − | | − | −
(10)
and the boundary contributions are given by
n λ λ
(n)
κ (λ) = θ( ) , ω (λ) = θ( ) ,
ab n+1 2l S ab S
Xl=1 − − 1 − L
n λ λ
(n)
κ (λ) = θ( ) , ω (λ) = θ( ) ,
ba n+1 2l S ba S
Xl=1 − − L − 1
(n) (n) (n)
κ (λ) = κ (λ)+κ (λ), ω (λ) = 0 ,
bb ba ab bb
κ(n)(λ) = 0, ω (λ) = ω (λ)+ω (λ) . (11)
aa aa ba ab
The ranges of integers In and J are
α γ
∞
In = 1,2,...L+M 2 min m,n M +N , J = 1,2,...N +N 1 . (12)
α n − { } m h γ ↓ h −
m=1
X
There are two differences as compared to the case of periodic boundary conditions
[17,18,19,20,21,22,23,24]: first there are additional 1 terms (11), and secondly the integers In
L α
†As far as the present work is concerned we do not need to explicitly consider complex solutions of
the Bethe equations and all of our results are really independent of the string hypothesis.
3
andJ takedifferentvalues. TheallowedrangeoftheintegersIn andJ reflectsthefactthatall
γ α γ
(1)
solutions of (4) with one or more roots λ or λ having vanishing real parts must be excluded
j k
as they lead to vanishing wave-functions. This restriction leads to constraints on the allowed
values of the integers In and J : the In range from 1 to L+M 2 ∞ min m,n M +N ,
α γ α n− m=1 { } m h
the solution with In = 0 being excluded. Similarly J range from 1 to N +N 1 and 0 is
α γ P ↓ h −
again excluded.
Forzero boundaryfields (κ(n) = 0, ω = 0)we cancostruct a complete set of3L states from
ij ij
theBetheAnsatzstatesdefinedintheaboveway: themodel(1)withvanishingboundaryfields
is sl(1 2)-invariant and all Bethe states are highest weight states of sl(1 2) [16,15]. Additional
| |
linearly independent eigenstates of (1) can be constructed by acting with the sl(1 2) lowering
|
operatorsonthehighest-weight states. Thetotalnumber ofstatesobtainedinthisway is3L as
can be proved in the same way as for the periodic t-J chain [17] (the necessary combinatorics
are identical). Thus we obtain a complete set of eigenstates of (1).
Fornonvanishing boundaryfieldsthesituationismorecomplicatedasthesl(1 2)symmetry
|
is broken by the boundary conditions. Therefore we cannot use the symmetry generators
to construct additional states from the Bethe Ansatz states and are left with the a priori
incomplete set of eigenstates given by (9) and(12). Forthe present purposes this is inessential:
the ground state is always a Bethe Ansatz state, as are the states needed to extract the
boundary S-matrices. Ground state and excitations can be constructed from (9) in a standard
way (see e.g. [25]). The ground state is obtained by filling all allowed vacancies of integers I1
α
and J up to maximal values I and J , which corresponds to filling two Fermi seas of
γ max max
rapidities λ1 between 0 and A and λ(1) between 0 and B. The actual values of A, B (and thus
α γ
I and J ) depend on H and µ and are determined below. We are interested in the case
max max
of a small magnetic field H and a close to half-filled band (µ 2ln(2)), for which A 1 and
≈ ≫
B 1. As is shown in Appendix A the ground state energy per site (for the four possible
≪
sets of boundary fields) below half-filling is given by
E µN HSz 1 A 1 B
− e − = ε (0) 2µ+ dλ ε (λ)κ′ (λ)+ dλ ε (λ)ω′ (λ)
L c − 4πL Z−A s ij 4πL Z−B c ij
1 H
[ε (0)+µ 2E ]+o(L−1) , (13)
s ij
−2L − 2 −
where i,j = a,b and where the dressed energies ε (λ) and ε (λ) ‡ are given in terms of the
c s
coupled integral equations
A B
ε (λ) = 2πa (λ)+H dµ a (λ µ) ε (µ)+ dµ a (λ µ) ε (µ) ,
s 1 2 s 1 c
− −Z−A − Z−B −
H A
ε (λ) = µ + dµ a (λ µ) ε (µ) . (14)
c 1 s
− 2 Z−A −
Here a (λ) = 1 n . For later use we define
n 2πλ2+n2
4
‡These can be shown to be (minus) the energies of the order one contributions to the elementary
charge and spin excitations [22].
4
1 ∞ exp( x ω ) 1 1+x
G (λ) = dω exp( iωλ) − |2| = Re β( +iλ) , (15)
x 2π − 2cosh(ω) π 2
Z−∞ 2 (cid:18) (cid:19)
where x is real and where β(z) = 1 ψ(1+z) ψ(z) . Here ψ(z) is the digamma function. The
2 2 − 2
asymptotic behaviour of G (λ) for hlarge l 1 andiλ x is
x
≫ ≫
1 x
G (λ) + (λ−4) . (16)
x
∼ 4πλ2 O
Below we will also need the small-λ asymptotics of G (λ), which is given by
1
1 ∞
G (λ) = [2ln(2)+2 ( 1)n(1 2−2n)ζ(2n+1)λ2n] , λ < 1 . (17)
1
2π − − | |
n=1
X
II. WIENER-HOPF ANALYSIS FOR THE DRESSED ENERGIES
In this section we analyze the coupled integral equations (14) by means of Wiener-Hopf
techniques [30] (for detailed expositions see e.g. [31,29]). As (14) are similar to the analogous
equations for the densities in the periodic t-J chain the necessary steps are the same as in
[19]. However as we will need more explicit answers than are given in [19] for determining the
boundary contribution to the ground state energy we briefly summarize the most important
stepsbelow. AfterFourier-transforming,thefirstequationof(14)canbeturnedintoaWiener-
Hopf equation for y(λ) = ε (λ+A)
s
H ∞
y(λ) = 2πG (λ+A)+ + dν [G (λ ν)+G (λ+ν +2A)] y(ν)+CG (λ+A) ,
0 1 1 0
− 2 −
Z0
(18)
where C = B dν exp(πν) ε (ν). Here we have used the fact that A 1 and B 1 to
−B c ≫ ≪
approximate B dν G (λ ν + A) ε (ν) G (λ + A) B dν exp(πν) ε (ν). The quantity
R −B 0 − c ≈ 0 −B c
C is determined self-consistently below. Eqn (18) can now be solved by iteration y(λ) =
R R
y (λ)+y (λ)+..., where
1 2
H ∞
y (λ) = 2πG (λ+A)+ + dν G (λ ν) y (ν)+CG (λ+A) , (19)
1 0 1 1 0
− 2 −
Z0
∞ ∞
y (λ) = dν G (λ+ν +2A) y (ν)+ dν G (λ ν) y (ν) . (20)
2 1 1 1 2
−
Z0 Z0
These equations can be solved in a standard way through a Wiener-Hopf factorization. The
result for y is obtained in complete analogy with e.g. the Appendix of [28]
1
iHG−(0) 1 exp( πA)
y+(ω) = G+(ω) i(2π C)G−( iπ) − + (exp( 2πA)) . (21)
1 ( 2 ω +i0 − − − ω +iπ ) O −
e
Here the Fourier transform y+(ω) = ∞dλ y(λ)exp(iλω) is analytic in the upper half-plane
1 0
and G±(ω) are analytic functions in the upper/lower half-plane factorizing the kernel 1 +
R
exp( ω ) = G+(ω)G−(ω) e
−| |
5
√2π iω −2iπω iω
G+(ω) = G−( ω) = − exp( ) . (22)
− Γ(1 iω) 2π 2π
2 − 2π (cid:18) (cid:19)
Theequationfory (λ)ismoredifficulttosolve. Theykeyistousethefactthatλ+λ′+2A 1.
2
≫
Using the asymptotic behaviour (16) of G (λ) in the expression for the driving term D(λ) =
1
∞dλ′ G (λ+λ′ +2A)y (λ′) and then performing a Laplace transformation we obtain
0 1 1
R 1 ∞ x3
D(λ) dx exp( 2Ax)exp( λ x)y+(ix)[x+ +...] , (23)
∼ 4π − −| | 1 12
Z0
where the expansion in x corresponds to the asymptotic expansion of G (λ + λ′ + 2A). It
e 1
is clear that due to the strongly decaying factor exp( 2Ax) the leading contribution to the
−
integral comes from the small-x region. Inserting the expression (23) for the driving term into
(20) and then following through the same steps as in the analysis for y we arrive at
1
i ∞ x3 G+(ix)y+(ix)
y+(ω) G+(ω) dx exp( 2Ax)[x+ +...] 1
2 ∼ 4π − 12 ω +ix
Z0
iH√2 ∞ 1+ x ln(x)+...e
e G+(ω) dx exp( 2Ax) 2π . (24)
∼ 4π − ω +ix
Z0
By means of a similar analysis further corrections to y(λ) can be determined. As far as the
physical quantities determined below are concerned y , y etc give rise to contributions much
3 4
smaller than those due to y and y . We are now in a position to determine the limit of
1 2
integration A as a function of the magnetic field. By definition ε ( A) = 0 = y(0), which
s
±
leads to
ln(H) ln( 2π(2π C)) 1
A = + e − + +... . (25)
− π q π 4πln(H)
Using (21) and (24) we can now solve the integral equation (14) for ε (λ)
c
B 2acosh(πλ)
ε (λ) = ε (λ)+ dν[G (λ ν)+ exp(πν)]ε (ν) , (26)
c 0 1 c
Z−B − 2π
where a = π exp( 2πA), ε (λ) = µ 2πG (λ) 2acosh(πλ), and where we have neglected
e − 0 − 1 −
termsofordero(exp( 2πA)). HerethetermproportionaltoaoriginatesintheC-termin(21).
−
Equation (26) can now be solved by iteration as B 1 (corresponding to µ¯ = 2ln(2) µ
≪ | | | − | ≪
1) with the result
ε (λ) = µ (2π +g)G (λ) 2acosh(πλ)+O(µ¯a)+ (µ¯2)+ (a2) , (27)
c 1
− − O O
where
H2 1 8 1
3
a = (1 +...) , g = (µ¯+2a)2 . (28)
8π2 − 2ln(H) 3 6ζ(3)
q
The boundary of integration B defined via ε ( B) = 0 in this order is given by
c
±
2 8ln(2) 1
B2 = µ¯+2a+ (µ¯+2a)23 (29)
3ζ(3) 3π 6ζ(3)
q
C is determined self-consistently to be 2ζ(3)B3. The higher order (in B) contributions to ε
c
−
and B do not contribute to the singularities in the thermodynamic quantities and therefore
have been dropped.
6
III. GROUND STATE PROPERTIES
We are now in a position to determine bulk and boundary contributions to the energy (13).
The bulk energy per site is found to be
ζ(3)
e = [µ+2ln(2)+2a+2ln(2) B3] , (30)
bulk
− π
from which we can determine the leading contributions to the zero-temperature magnetization
per site, magnetic susceptibility, density and compressibility close to half-filling in a weak
magnetic field
∂e H 1 ln(2) 8(µ¯+2a)
bulk
m = = (1 ) 1+ +... ,
bulk − ∂H 2π2 − 2ln(H) π v 3ζ(3)
u
u
1 1 2ln(2) H2 1t 1
χ = (1 )+ (1 ) +... ,
H,bulk
2π2 − 2ln(H) π 4π4 − ln(H) 6ζ(3)(µ¯+2a)
q
∂e 2ln(2) 2(µ¯+2a)
bulk
D = = 1 +... ,
bulk − ∂µ − π v 3ζ(3)
u
u
1 ∂D 2ln(2)t 1
bulk
χ = = +... , (31)
c,bulk Db2ulk ∂µ π 6ζ(3)[µ¯+2a]
q
in agreement with the expressions for periodic boundary conditions [19,20]. We note that both
the magnetic susceptibility and the compressibility diverge when we approach half-filling.
Contributions to the surface energy, i.e. all terms proportional to L−1 in (13), can be
divided into boundary-field dependent ones E(αβ) and contributions due to the “geometry”
i.e. openess of the chain E0 so that we can write for the four permitted sets of boundary
conditions
E = E0 +E(αβ) , α,β = a,b . (32)
boundary
The boundary field independent contributions are easily determined
1 H ln(ln(H)) 8
E0 = 1+ +µ π (µ¯+2a)23 +... . (33)
−2 (−2ln(H) 2ln(H) ! − −s27ζ(3) )
We note that for zero bulk magnetic field H = 0 and half-filling µ = 2ln2 we obtain E0 =
π ln2, which is the correct result for the surface energy of the open XXX Heisenberg chain
2 −
[13,35]. By differentiating the surface energy with respect to the thermodynamic parameters
H and µ we can evaluate the surface contributions to particle number and magnetization in
analogy with e.g. the treatment of the Kondo model [31] (see also [1]). It is reasonable to
assume that these contributions are concentrated in the boundary regions, e.g. we interpret
the surface contribution to the particle number to lead to a depletion/increase of electrons in
the “vicinity” of the boundaries.
The leading contributions to the boundary magnetization, particle number and suscepti-
bilities due to E0 are
7
1 ln(ln(H)) 1 ln(ln(H))
M0 = 1+ +... , χ0 = 1+ +... ,
−4ln(H) 2ln(H) ! H 4H(ln(H))2 2ln(H) !
1 2(µ¯+2a) 4ln2+π
N0 = 1+ , χ0 = . (34)
2 vu 3ζ(3) c −2π 6ζ(3)(µ¯+2a)
u
t q
We first note that boundary region exhibits a stronger magnetization as compared to the bulk,
i.e. M0 = π2 , which is much larger than one for the small magnetic fields considered
mbulk −2HlnH
here. The boundary magnetic susceptibility is seen to diverge for H 0. Following [1] we
→
interpret this as an indication for the presence of a magnetic bound state in the boundary
region. The magnetic behaviour is similar to the one for the XXZ spin chain with an open
boundary studied in [1].
The leading contribution to the boundary particle number is 1. Because of the constraint
2
of at most single occupancy at any site this increase in particle number (recall that we are very
close to half-filling) must be spread out over large regions neighbouring the boundaries. This
indicates that boundary effects spread deeply into the bulk. The boundary compressibility
for zero boundary fields is seen to be negative and to diverge as we approach half-filling. The
type of singularity is the same as for the bulk. Combining the results for magnetization and
particle number we see that there is a tendency for spin-up electrons to get pushed towards
the boundary.
The leading order boundary-field dependent contributions E(αβ) are expressed (see (13))
in terms of the quantities
A B
ǫ (S) = dν a (ν) ε (ν) , ǫ (S) = dν a (ν) ε (ν) . (35)
b S s a S c
Z−A Z−B
where according to (13) for the four possible sets of boundary conditions
1
E(αβ) = [ǫ ( S )+ǫ ( S )]+E , α,β = a,b . (36)
β 1 α L αβ
2 − −
The leading contribution to the quantity ǫ (S) can be easily determined for the case S
a
≫
B, in which we can expand a (ν) in a power series in ν and then perform the elementary
S
integrations using the expression (26) for ε (ν). For S B we instead expand ε (ν) in an
c c
≪
infinite power series (using that G (ν) is a smooth function around zero), then perform the
1
integrations, resum the result, and then retain only the leading terms. This results in
4ζ(3)B3 1 +... if S B
ǫ (S) = − π S ≫ (37)
a
εc(0)+ π2BS(µ¯+2a+...)+... if S ≪ B .
The analogous computations for ǫ (S) are more involved, so that we give a brief summary of
b
the necessary steps in Appendix B. We find the following result for the leading behaviour
S−1 H +... if S A
ǫ (S) = G (0)(2π +2ζ(3)B3)+ H + 2 ln(H) ≪ . (38)
b S
− 2 H 2 Hln(H)+... if S A
−2 − π2S ≫
8
A. Contributions due to small boundary fields of type a
The contribution to theboundary energy isgiven by 1ǫ (2 2) with 2 2 B = 2(µ¯+2a).
2 a h− h− ≫ 3ζ(3)
r
(This defines what we mean with “small” boundary field). Here h is a boundary chemical
potential. We define the quantity σ = h µ¯+2a, which in the present case is much smaller
1−h 6ζ(3)
than one. We obtain the following contribuqtions to boundary magnetization/particle number
and susceptibilities
H 1 2
Ma = (1 )σ , Na = σ ,
π3 − 2ln(H) −π
1 1 H2 1
χa = (1 )σ + (1 )σ(µ¯+2a)−1 ,
H π3 − 2ln(H) 4π5 − 2ln(H)
σ
χa = (µ¯+2a)−1 . (39)
c π
We see that a small boundary chemical potential leads essentially to the same type of diver-
gences as are present in the bulk. As expected electrons get pushed away from the boundary
althoughtheeffect is small. Bydifferentiation with respect totheboundary chemical potential
we can evaluate the average number of electrons at the boundary site
2ζ(3) B3
n = 1 , h 0 , (40)
e
h i − π (1 h)2 →
−
where B is given by (29). We see that the electron number is larger than the bulk value.
This is consistent with the above observation that an open boundary without field leads to an
increase in the elctron density in the boundary region.
B. Contributions due to large boundary fields of type a
Heretheboundarychemicalpotentialistakenlarge, bywhichwemeanthat0 < 2 2 B.
h− ≪
We again use the notation σ = h µ¯+2a, but now σ 1. We find
1−h 6ζ(3) ≫
q
H 1 1 1 2ln(2) 2(µ¯+2a) 1
Ma = (1 )(1 ) , Na = (1+ )+ ,
4π2 − 2ln(H) − πσ −2 π v 3ζ(3) 2πσ
u
u
1 1 1 1 H2 1 t
χa = (1 )(1 )+ (1 )(µ¯+2a)−1 +... ,
H 4π2 − 2ln(H) − πσ πσ(2π)4 − ln(H)
3ln(2) 1
χa = + (µ¯+2a)−1 . (41)
c π 6ζ(3)(µ¯+2a) 4πσ
q
The magnetization is againproportional toH and the magnetic susceptibility can only diverge
at half-filling. The large boundary field yields however a contribution of 1 to the boundary
−2
particle number, which indicates a strong depletion of electrons in the boundary region. This
is in accordance with the expectation based on a naive analysis of the hamiltonian (1) that
large boundary chemical potentials (with our choice of sign in (1)) should favour the presence
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