Excitation Spectrum Gap and Spin-Wave Stiffness of XXZ Heisenberg
  Chains: Global Renormalization-Group Calculation
Excitation Spectrum Gap and Spin-Wave Velocity of XXZ Heisenberg Chains:
Global Renormalization-Group Calculation
Ozan S. Sarıyer1, A. Nihat Berker2−4, and Michael Hinczewski4
1Department of Physics, Istanbul Technical University, Maslak 34469, Istanbul, Turkey,
2College of Sciences and Arts, Koç University, Sarıyer 34450, Istanbul, Turkey,
3Department of Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, U.S.A., and
4Feza Gürsey Research Institute, TÜBİTAK - Bosphorus University, Çengelköy 34684, Istanbul, Turkey
The anisotropic XXZ spin- 1
Heisenberg chain is studied using renormalization-group theory. The
specific heats and nearest-neighbor spin-spin correlations are calculated thoughout the entire tem-
perature and anisotropy ranges in both ferromagnetic and antiferromagnetic regions, obtaining a
global description and quantitative results. We obtain, for all anisotropies, the antiferromagnetic
spin-liquid spin-wave velocity and the Isinglike ferromagnetic excitation spectrum gap, exhibiting
the spin-wave to spinon crossover. A number of characteristics of purely quantum nature are found:
The in-plane interaction sxi s
j + s
induces an antiferromagnetic correlation in the out-of-plane szi
component, at higher temperatures in the antiferromagnetic XXZ chain, dominantly at low temper-
atures in the ferromagnetic XXZ chain, and, in-between, at all temperatures in the XY chain. We
find that the converse effect also occurs in the antiferromagnetic XXZ chain: an antiferromagnetic
szi s
j interaction induces a correlation in the s
component. As another purely quantum effect,
(i) in the antiferromagnet, the value of the specific heat peak is insensitive to anisotropy and the
temperature of the specific heat peak decreases from the isotropic (Heisenberg) with introduction
of either type (Ising or XY) anisotropy; (ii) in complete contrast, in the ferromagnet, the value and
temperature of the specific heat peak increase with either type of anisotropy.
PACS numbers: 67.40.Db, 75.10.Pq, 64.60.Cn, 05.10.Cc
I. INTRODUCTION
The quantum Heisenberg chain, including the possi-
bility of spin-space anisotropy, is the simplest nontrivial
quantum spin system and has thus been widely studied
since the very beginning of the spin concept in quan-
tum mechanics [1, 2, 3]. Interest in this model continued
[4, 5, 6, 7, 8, 9, 10] and redoubled with the exposition
of its richly varied low-temperature behavior [11, 12, 13]
and of its relevance to high-temperature superconductiv-
ity [14, 15, 16, 17, 18]. It has become clear that antifer-
romagnetism and superconductivity are firmly related to
each other, adjoining and overlapping each other.
A large variety of theoretical tools have been employed
in the study of the various isotropic and anisotropic
regimes of the quantum Heisenberg chain, including
finite-systems extrapolation [6, 19], linked-cluster [7] and
dimer-cluster [20] expansions, quantum decimation [21],
decoupled Green’s functions [22], quantum transfer ma-
trix [23, 24], high-temperature series expansion [25], and
numerical evaluation of multiple integrals [26]. The XXZ
Heisenberg chain retains high current interest as a theo-
retical model [27, 28] with direct experimental relevance
[29].
In the present paper, a position-space renormalization-
group method introduced by Suzuki and Takano [30, 31]
for d = 2 dimensions and already applied to a number
of d > 1 systems [30, 31, 32, 33, 34, 35, 36] is used
to compute the spin-spin correlations and the specific
heat of the d = 1 anisotropic quantum XXZ Heisenberg
model, easily resulting in a global description and de-
tailed quantitative information for the entire temperature
and anisotropy ranges including the ferromagnetic and
antiferromagnetic, the spin-liquid and Isinglike regions.
We obtain, for all anisotropies, the antiferromagnetic
spin-liquid spin-wave velocity and the Isinglike ferromag-
netic excitation spectrum gap, exhibiting the spin-wave
to spinon crossover. A number of other characteristics of
purely quantum nature are found: The in-plane interac-
tion sxi s
j induces an antiferromagnetic correlation
in the out-of-plane szi component, at higher temperatures
in the antiferromagnetic XXZ chain, dominantly at low
temperatures in the ferromagnetic XXZ chain, and, in-
between, at all temperatures in the XY chain. We find
that the converse effect also occurs in the antiferromag-
netic XXZ chain: an antiferromagnetic szi s
j interaction
induces a correlation in the s
i component. As another
purely quantum effect, (i) in the antiferromagnet, the
value of the specific heat peak is insensitive to anisotropy
and the temperature of the specific heat peak decreases
from the isotropic (Heisenberg) with introduction of ei-
ther type (Ising or XY) anisotropy; (ii) in complete con-
trast, in the ferromagnet, the value of the specific heat
peak is strongly dependent on anisotropy and the tem-
perature of the specific heat peak increases with either
type of anisotropy. This purely quantum effect is a pre-
cursor to different phase transition temperatures in three
dimensions [32, 36, 37, 38]. Our calculational method
is relatively simple, readily yields global results, and is
overall quantitatively successful.
http://arxiv.org/abs/0704.1064v2
II. THE ANISOTROPIC QUANTUM
HEISENBERG MODEL AND THE
RENORMALIZATION-GROUP METHOD
A. The Anisotropic Quantum Heisenberg Model
The spin- 1
anisotropic Heisenberg model (XXZ model)
is defined by the dimensionless Hamiltonian
−βH =
sxi s
j + s
+ Jzs
, (1)
where β = 1/kBT and 〈ij〉 denotes summation over
nearest-neighbor pairs of sites. Here the sui are the quan-
tum mechanical Pauli spin operators at site i. The ad-
ditive constant G is generated by the renormalization-
group transformation and is used in the calculation of
thermodynamic functions. The anisotropy coefficient is
R = Jz/Jxy. The model reduces to the isotropic Heisen-
berg model (XXX model) for |R| = 1, to the XY model
for R = 0, and to the Ising model for |R| → ∞.
B. Renormalization-Group Recursion Relations
The Hamiltonian in Eq.(1) can be rewritten as
− βH =
{−βH(i, i+ 1)} . (2)
where βH(i, i+ 1) is a Hamiltonian involving sites i and
i + 1 only. The renormalization-group procedure, which
eliminates half of the degrees of freedom and keeps the
partition function unchanged, is done approximately [30,
Trodde
−βH =Trodde
{−βH(i,i+1)} (3)
=Trodde
{−βH(i−1,i)−βH(i,i+1)}
{−βH(i−1,i)−βH(i,i+1)}
′H′(i−1,i+1) ≃ e
i {−β′H′(i−1,i+1)} = e−β
′H′ .
Here and throughout this paper, the primes are used for
the renormalized system. Thus, at each successive length
scale, we ignore the non-commutativity of the operators
beyond three consecutive sites, in the two steps indicated
by ≃ in the above equation. Since the approximations
are applied in opposite directions, one can expect some
mutual compensation. Earlier studies [30, 31, 33, 34,
35] have been successful in obtaining finite-temperature
behavior on a variety of quantum systems.
The transformation above is summarized by
p s ms Two-site basis eigenstates
+ 1 1 |φ1〉 = | ↑↑〉
0 |φ2〉 =
{| ↑↓〉+ | ↓↑〉}
− 0 0 |φ4〉 =
{| ↑↓〉 − | ↓↑〉}
TABLE I: The two-site basis eigenstates that appear in
Eq.(8). These are the well-known singlet and triplet states.
The state |φ3〉 is obtained by spin reversal from |φ1〉, with the
same eigenvalue.
′H′(i,k) = Trj e
{−βH(i,j)−βH(j,k)}, (4)
where i, j, k are three successive sites. The operator
−β′H′(i, k) acts on two-site states, while the operator
−βH(i, j)−βH(j, k) acts on three-site states, so that we
can rewrite Eq.(4) in the matrix form,
〈uivk|e−β
′H′(i,k)|ūiv̄k〉 =
〈uiwj vk|e−βH(i,j)−βH(j,k)|ūiwj v̄k〉 , (5)
where state variables u, v, w, ū, v̄ can take spin-up or spin-
down values at each site. The unrenormalized 8× 8 ma-
trix on the right-hand side is contracted into the renor-
malized 4× 4 matrix on the left-hand side of Eq.(5). We
use two-site basis states vectors {|φp〉} and three-site ba-
sis states vectors {|ψq〉} to diagonalize the matrices in
Eq.(5). The states {|φp〉}, given in Table I, are eigen-
states of parity, total spin magnitude, and total spin z-
component. These {|φp〉} diagonalize the renormalized
matrix, with eigenvalues
J ′z +G
′, Λ2 = +
J ′xy −
J ′z +G
Λ4 = −
J ′zxy −
J ′z +G
′. (6)
The states {|ψq〉}, given in Table II, are eigenstates of
parity and total spin z-component. The {|ψp〉} diagonal-
ize the unrenormalized matrix, with eigenvalues
Jz + 2G, λ4 = 2G, (7)
λ2 = −
8J2xy + J
+ 2G,
λ3 = −
8J2xy + J
+ 2G.
With these eigenstates, Eq.(5) is rewritten as
γp ≡ 〈φp|e−β
′H′(i,k)|φp〉 =
u,v,ū,
v̄,w,q
〈φp|uivk〉〈uiwjvk|ψq〉·
〈ψq|e−βH(i,j)−βH(j,k)|ψq〉〈ψq |ūiwj v̄k〉〈ūiv̄k|φp〉 . (8)
p ms Three-site basis eigenstates
+ 3/2 |ψ1〉 = | ↑↑↑〉
1/2 |ψ2〉 = µ{| ↑↑↓〉 + σ| ↑↓↑〉 + | ↓↑↑〉}
|ψ3〉 = ν{−| ↑↑↓〉 + τ | ↑↓↑〉 − | ↓↑↑〉}
− 1/2 |ψ4〉 =
{| ↑↑↓〉 − | ↓↑↑〉}
TABLE II: The three-site basis eigenstates that appear in
Eq.(8) with coefficients σ = (−Jz +
8J2xy + J
z )/2Jxy , τ =
(Jz +
8J2xy + J
z )/2Jxy and normalization factors µ, ν. The
states |ψ5−8〉 are obtained by spin reversal from |ψ1−4〉, with
the same respective eigenvalues.
Thus, there are three independent γp that determine the
renormalized Hamiltonian and, therefore, three renor-
malized interactions in the Hamiltonian closed under
renormalization-group transformation, Eq.(1). These γp
γ1 = e
J′z+G
= e2G−
Jz + cosh
8J2xy + J
Jz sinh
8J2xy + J
8J2xy + J
γ2 = e
J′xy−
J′z+G
=2e2G−
8J2xy + J
Jz sinh
8J2xy + J
8J2xy + J
γ4 = e
+G′ = 2e2G , (9)
which yield the recursion relations
J ′xy = ln
, J ′z = ln
, G′ =
γ21γ2γ4
As expected, J ′xy and J
z are independent of the additive
constant G and the derivative ∂GG
′ = bd = 2, where b =
2 is the rescaling factor and d = 1 is the dimensionality
of the lattice.
For Jxy = Jz, the recursion relations reduce to the spin-
isotropic Heisenberg (XXX) model recursion relations,
while for Jxy = 0 they reduce to spin-
Ising model re-
cursion relations. The Jz = 0 subspace (XY model) is
not (and need not be) closed under these recursion rela-
tions [30, 31]: The renormalization-group transformation
induces a positive Jz value, but the spin-space easy-plane
aspect is maintained.
In addition, there exists a mirror symmetry along
the Jz-axis, so that J
xy (−Jxy, Jz) = J ′xy (Jxy, Jz) and
J ′z (−Jxy, Jz) = J ′z (Jxy, Jz). The thermodynamics of
the system remains unchanged under flipping the in-
teractions of the x and y spin components, since the
renormalization-group trajectories do not change. In
fact, this is part of a more general symmetry of the XYZ
model, where flipping the signs of any two interactions
leaves the spectrum unchanged [8]. Therefore, with no
loss of generality, we take Jxy > 0. Independent of the
sign of Jxy, Jz > 0 gives the ferromagnetic model and
Jz < 0 gives the antiferromagnetic model.
C. Calculation of Densities and Response
Functions by the Recursion-Matrix Method
Just as the interaction constants of two consecutive
points along the renormalization-group trajectory are re-
lated by the recursion relations, the densities are con-
nected by a recursion matrix T̂ , which is composed of
derivatives of the recursion relations. For our Hamilto-
nian, the recursion matrix and density vector ~M are
∂J′xy
szi s
. (11)
These are densities Mα associated with each interaction
∂ lnZ
, (12)
where Nα is the number of α-type interactions and Z
is the partition function for the system, which can be
expressed both via the unrenormalized interaction con-
stants as Z( ~K) or via the renormalized interaction con-
stants as Z( ~K ′). By using these two equivalent forms,
one can formulate the density recursion relation [39]
Mα = b
M ′βTβα , Tβα ≡
∂K ′β
. (13)
Since the interaction constants, under renormalization-
group transformation, stay the same at fixed points such
as critical fixed points or sinks, the above Eq.(13) takes
the form of a solvable eigenvalue equation,
bd ~M∗ = ~M∗ · T̂ , (14)
at fixed points, where ~M = ~M ′ = ~M∗. The fixed point
densities are the components of the left eigenvector of the
recursion matrix with left eigenvalue bd [39]. At ordinary
points, Eq.(13) is iterated until a sink point is reached
under successive renormalization-group transformations.
In algebraic form, this means
~M (0) = b−nd ~M (n)T̂ (n)T̂ (n−1) · · · T̂ (1) , (15)
where the upper indices indicate the number of iteration
(transformation), with ~M (n) ≃ ~M∗.
This method is applied on our model Hamiltonian.
The sink of the system is at infinite temperature J∗xy =
J∗z = 0 for all initial conditions (Jxy, Jz).
Response functions are calculated by differentiation
of densities. For example, the internal energy is U =
szi s
, employing T = 1/Jxy, and U =
szi s
, employing T = 1/|Jz|. The spe-
cific heat C = ∂TU follows from the chain rule,
C = J2xy
szi s
, for T = 1/Jxy,
C = J2z
szi s
∂|Jz|
, for T = 1/|Jz|.
III. CORRELATIONS SCANNED WITH
RESPECT TO ANISOTROPY
The ground-state and excitation properties of the XXZ
model offer a variety of behaviors [11, 12, 40, 41]: The
antiferromagnetic model with R < −1 is Isinglike and
the ground state has Néel long-range order along the z
spin component with a gap in the excitation spectrum.
For −1 ≤ R ≤ 1, the system is a ”spin liquid”, with a
gapless spectrum and power-law decay of correlations at
zero temperature. The ferromagnetic model with R > 1
is also Isinglike, the ground state is ferromagnetic along
the z spin component, with an excitation gap.
Our calculated
szi s
sxi s
nearest-neighbor spin-spin correlations for the
whole range of the anisotropy coefficient R are shown
in Fig.1, for various temperatures. The xy correlation is
always non-negative. Recall that we use Jxy > 0 with
no loss of generality. In the Isinglike antiferromagnetic
(R < −1) region, the z correlation is expectedly anti-
ferromagnetic. As the
szi s
correlation saturates for
large |R|, the transverse
correlation is some-
what depleted. In the Isinglike ferromagnetic (R > 1)
region, the
szi s
correlation is ferromagnetic, saturates
quickly as the
correlation quickly goes to zero.
In the spin-liquid (|R| < 1) region, the
szi s
correlation
monotonically passes through zero in the feromagnetic
side, while the
correlation is maximal. The re-
markable quantum behavior of
szi s
around R = 0 is
discussed in Sec.V below. It is seen in the figure that
these changeovers are increasingly sharp as temperature
is decreased and, at zero temperature, become discon-
tinuous at R = 1. As seen in Fig.1(b), at zero temper-
ature, our calculated
szi s
correlations
FIG. 1: (a) Calculated nearest-neighbor spin-spin correlations
szi s
(thick curves from lower left) and
(thin curves
from upper left) as a function of anisotropy coefficient R for
temperatures 1/Jxy = 0, 0.1, 0.2, 0.4, 0.8. (b) Calculated zero-
temperature nearest-neighbor spin-spin correlations (thin and
thick curves, as in the upper panel) compared with the exact
points of Ref.[4, 40, 42, 43, 44] shown with filled and open
symbols for
szi s
respectively. At R = 1, the
calculated
szi s
discontinuously goes from antiferromagnetic
to the exact result of 0.25 [40] of saturated ferromagnetism
and the calculated
discontinuously goes from ferro-
magnetic to the exact result of constant zero [40].
show very good agreement with the known exact points
[4, 42, 43, 44]. Also, our results for R > 1 fully overlap
the exact results of
szi s
= 0.25 and
= 0 [40].
We also note that zero-temperature is the limit in which
our approximation is at its worst.
FIG. 2: Calculated nearest-neighbor spin-spin corre-
lations
(upper panels) and
szi s
(lower
panels) for the antiferromagnetic XXZ chain, as a
function of temperature, for anisotropy coefficients
R = 0,−0.25,−0.50,−0.75,−1,−2,−4,−8,−∞ spanning the
spin-liquid (left panels) and Isinglike (right panels) regions.
Note that, in every one of the panels, the correlation curves
cross each other. This remarkable quantum phenomenon is
discussed in the text.
IV. ANTIFERROMAGNETIC XXZ CHAIN
For the antiferromagnetic XXZ chain, our calculated
szi s
nearest-neighbor spin-spin correla-
tions as a function of temperature are shown in Fig.2
for various anisotropy coefficients R. We find that when
Jxy is the dominant interaction (spin liquid), the corre-
lations are weakly dependent on anisotropy R. When Jz
is the dominant interaction (Isinglike), the correlations
are weakly dependent on anisotropy R only at the higher
temperatures. Our results are compared with multiple-
integral results [26] in Fig.3.
In every one of the panels of Fig.2, the correlation
curves cross each other, revealing a remarkable quantum
phenomenon. In a classical system, the correlation be-
tween a given spin component (e.g.,
) is expected
to decrease when the coupling of another spin component
(e.g., |Jz |) is increased. It is found from the antiferromag-
netic XXZ chain in Fig.2 that the opposite may occur
in a quantum system: In this figure, an increase in Jxy
causes an increase in |
szi s
| for 1/|Jz| > 0.9 and 0.4
in the spin-liquid and Isinglike regions respectively. Con-
versely, an increase in |Jz| causes an increase in
for 1/Jxy > 0.4 and 2.1 in the spin-liquid and Isinglike
regions respectively. This quantum effect can be called
cross-component spin correlation.
FIG. 3: Comparison of our results (thick lines) for the cor-
relation functions of the antiferromagnetic XXZ chain, with
the multiple-integral results of Ref.[26] (thin lines), for var-
ious anisotropy coefficients R spanning the spin-liquid and
Isinglike regions.
The antiferromagnetic specific heats calculated with
Eq.(16) are shown in Fig.4 for various anisotropy coeffi-
cients and compared, in Figs.5, 6, with finite-lattice ex-
pansion [6, 19], quantum decimation [21], transfer matrix
[24], high-temperature series expansion [25] results and,
for the R = 0 case, namely the XY model, with the exact
result [5] C = (1/4πT )
cosω/ cosh
dω. The
C(T ) peak temperature is highest for the isotropic case
(Heisenberg) and decreases with anisotropy increasing in
either direction (towards Ising or XY). The peak value
of C(T ) is only weakly dependent on anisotropy, espe-
cially for the Isinglike systems. A strong contrast to this
behavior will be seen, as another quantum mechanical
phenomenon, in the ferromagnetic XXZ chain.
The linearity, at low temperatures, of the spin liq-
uid (|R| ≤ 1) specific heat with respect to temperature
is expected on the basis of spin-wave calculations for
the antiferromagnetic XXZ model [45, 46]. This linear
form of C(T ) reflects the linear energy-momentum dis-
persion of the low-lying excitations, the magnons. The
low-temperature magnon dispersion relation is ~ω = ckn,
where c is the spin-wave velocity and n = 1 for the an-
tiferromagnetic XXZ model in d = 1 [40]. The internal
energy, given by U = (1/2π)
dk~ω(k)/(eβ~ω(k)−1), is
dominated by the magnons at low temperatures, yielding
U ∼ T 2 and C ∼ T for n = 1 in the dispersion relation.
From this relation, our calculated spin-wave velocity c as
a function of anisotropy R is given in Fig.7 and compares
well with the also shown exact result [47]. A simultaneous
fit to the dispersion relation exponent n, expected to be
1, yields 1.00± 0.02. However, for the Isinglike −R > 1,
the unexpected linearity instead of an exponential form
FIG. 4: Calculated specific heats C of the antiferromagnetic
XXZ chain, as a function of temperature for anisotropy coeffi-
cients R = 0,−0.25,−0.50,−0.75,−1,−2,−4,−8,−∞ span-
ning the spin-liquid (upper panel) and Isinglike (lower panel)
regions.
caused by a gap in the excitation spectrum, points to the
approximate nature of our renormalization-group calcu-
lation. The correct exponential form is obtained in the
large −R limit, where the renormalization-group calcu-
lation becomes exact.
Rojas et al. [25] have obtained the high-temperature
expansion of the free energy of the XXZ chain to order
β3, where β is the inverse temperature. The specific heat
from this expansion is
2 +R2
J2xy −
J3xy +
6− 8R2 −R4
J4xy. (17)
This high-temperature specific heat result is also com-
pared with our results, in Fig.6, and very good agreement
is seen. In fact, when in the high-temperature region of
0 < β < 0.1, we fit our numerical results for C(β) to the
fourth degree polynomial C = Σ4i=0Aiβ
i, and we do find
(1) the vanishing A0 < 10
−5 and A1 < 10
−7 for all R and
(2) the comparison in Fig.8 between our results for A2
and A3 and those of Eq.(17) from Ref.[25], thus obtaining
excellent agreement for all regions of the model.
V. FERROMAGNETIC XXZ CHAIN
For the ferromagnetic (i.e., R > 0) systems in Fig.1,
szi s
expectation value becomes rapidly negative
at lower temperatures for R < 1, even though for
R ≥ 0 all couplings in the Hamiltonian are ferromag-
FIG. 5: Comparison of our antiferromagnetic specific heat re-
sults (thick lines) with the results of Refs.[5] (open circles),
[6] (dotted), [19] (thin lines), [21] (dash-dotted), and [23, 24]
(dashed), for anisotropy coefficients R = 0,−0.5,−1,−2 span-
ning the spin-liquid and Isinglike regions.
FIG. 6: Comparison of our antiferromagnetic specific heat
results (thick lines) with the high-temperature J → 0 behav-
iors (thin lines) obtained from series expansion in Ref.[25],
for anisotropy coefficients R = 0,−0.50,−0.75,−1,−2,−∞
spanning the spin-liquid and Isinglike regions.
netic. This is actually a real physical effect, not a nu-
merical anomaly. In fact, we know the spin-spin corre-
lations for the ground state of the one-dimensional XY
model (the R = 0 case of our Hamiltonian), and we can
compare our low-temperature results with these exact
values. The ground-state properties of the spin- 1
model are studied by making a Jordan-Wigner trans-
FIG. 7: Our calculated antiferromagnetic spin-wave veloc-
ity c versus the anisotropy coefficient R. The dashed line,
2π sin(γ)/γ where γ ≡ cos−1(−R), is the exact result [47].
FIG. 8: Comparison of our results with the high-temperature
expansion of Ref. [25] for all regions: antiferromagnetic (outer
panels) and ferromagnetic (inner panels), spin-liquid (left
panels) and Isinglike (right panels). Triangles and circles de-
note our results, while solid and dashed lines denote the re-
sults of Ref.[25] for A2 and A3, respectively. The error bars,
due to the statistical fitting procedure of the coefficients A2
and A3, have half-heights of 1.7×10
−4 and 2.6×10−3 respec-
tively.
formation, yielding a theory of non-interacting spinless
fermions. Analysis of this theory yields the exact zero-
temperature nearest-neighbor spin-spin correlations [4]
shown in Table III. Our renormalization-group results
in the zero-temperature limit, also shown in this table,
compare quite well with the exact results, as with the
other exact points in Fig.1(b), although in the worst re-
gion for our approximation. Finally, by continuity, it is
reasonable that for a range of R positive but less than
one, the z component correlation function is as we find,
intriguingly but correctly negative at low temperatures.
Thus, the interaction sxi s
j (irrespective of its sign,
due to the symmetry mentioned at the end of Sec.IIB)
induces an antiferromagnetic correlation in the szi com-
ponent, competing with the szi s
j interaction when the
FIG. 9: Calculated nearest-neighbor spin-spin correlations
(upper panels) and
szi s
(lower panels) for the
ferromagnetic XXZ chain, as a function of temperature,
for anisotropy coefficients R = 0, 0.25, 0.50, 0.75, 1, 2, 4, 8,∞
spanning the spin-liquid (left panels) and Isinglike (right pan-
els) regions.
FIG. 10: Left panel: Calculated nearest-neighbor spin-
spin correlations
szi s
for the ferromagnetic XXZ chain,
as a function of temperature 1/Jxy in the spin liquid, for
anisotropy coefficients R = 0, 0.25, 0.50, 0.75, 1. Right panel:
The sign-reversal temperature T0 of the nearest-neighbor cor-
relation 〈szi s
j 〉: our results (full curve) and the analytical re-
sult from the quantum transfer matrix method (dashed) [23].
latter is ferromagnetic.
For finite temperatures, our calculated nearest-
neighbor spin-spin correlations are shown in Figs.9, 10,
for different values of R. These results are compared with
Green’s function calculations [22] in Fig.11. As expected
from the discussion at the beginning of this section, in
the spin-liquid region, the correlation 〈szi szj 〉 is negative
at low temperatures. Thus, a competition occurs in the
FIG. 11: Comparison of our ferromagnetic R = 1, 5
results
with Green’s function calculations [22] .
FIG. 12: Calculated specific heats C of the ferromagnetic
XXZ chain, as a function of temperature for anisotropy coef-
ficients R = 0, 0.25, 0.50, 0.75, 1, 2, 4, 8,∞ spanning the spin-
liquid (upper panel) and Isinglike (lower panel) regions.
correlation 〈szi szj 〉 between the XY-induced antiferromag-
netism and the ferromagnetism due to the direct cou-
pling between the sz spin components. In fact, the rein-
forcement of antiferromagnetic correlations of 〈szi szj 〉 by
increasing Jxy (and also its converse) was seen in the
antiferromagnetic XXZ chain discussed in the previous
section. Thus, we see that whereas this cross-component
effect is dominant at low temperatures in the ferromag-
netic XXZ chain, it is seen at higher temperatures in the
antiferromagnetic XXZ chain and, in-between, through-
out the temperature range in the XY chain.
In the ferromagnetic XXZ chain, as a consequence
of the competition mentioned above, a sign reversal in
〈szi szj 〉 occurs from negative to positive correlation, at
temperatures J−1xy = T0(R).[48] At this temperature, by
cancelation of the competing effects, the nearest-neighbor
FIG. 13: Comparison of our ferromagnetic specific heat re-
sults (thick lines) with the high-temperature J → 0 behaviors
(thin lines) obtained from series expansion [25], for anisotropy
coefficients R = 0.25, 0.50, 0.75, 1, 2,∞ spanning the spin-
liquid and Isinglike regions.
Zero-temperature
correlations of the
spin- 1
XY chain
Exact values
from Ref. [4]
results
〉 0.15915 0.17678
〈szi s
j 〉 −0.10132 −0.12500
TABLE III: Zero-temperature nearest-neighbor correlations
of the spin- 1
XY chain.
correlation 〈szi szj 〉 is zero. Our calculated T0(R) curve is
shown in Fig.10, and has very good agreement with the
exact result T0 = (
3 sin γ/4γ) tan[π(π − γ)/2γ] where
γ ≡ cos−1(−R) [23].
The calculated ferromagnetic specific heats are shown
in Fig.12 for various anisotropy coefficients and com-
pared, in Figs.13, 14, with finite-lattice expansion [6],
quantum decimation [21], decoupled Green’s functions
[22], transfer matrix [23, 24], high-temperature se-
ries expansion [25] results and, for the R = 0 case,
namely the XY model, with the exact result [5] C =
(1/4πT )
cosω/ cosh
dω. In sharp contrast
to the antiferromagnetic case in Sec.IV, the peak C(T )
temperature is highest for the most anisotropic cases
(XY or Ising) and decreases with anisotropy decreas-
ing from either direction (towards Heisenberg). In the
same contrast, the peak value of C(T ) is dependent
on anisotropy, decreasing, eventually to a flat curve, as
anisotropy is decreased. This contrast between the fer-
romagnetic and antiferromagnetic systems is a purely
quantum phenomenon. Specifically, the marked contrast
between the specific heats of the isotropic antiferromag-
netic and ferromagnetic systems, seen in the full curves
of Figs.4 and 12 respectively, translates into the different
critical temperatures of the respective three-dimensional
systems.[32, 36, 37, 38] Classical ferromagnetic and anti-
FIG. 14: Comparison of our ferromagnetic specific heat re-
sults (thick lines) with the results of Refs.[5] (dash-double-
dotted), [6] (dotted), [21] (dash-dotted), [22] (open cir-
cles), and [23, 24] (dashed), for anisotropy coefficients R =
0, 0.5, 1, 5
, 2, 5 spanning the spin-liquid and Isinglike re-
gions.
ferromagnetic systems are, on the other hand, identically
mapped onto each other.
The low-temperature specifics heats are discussed in
detail and compared to other results in Sec.VI.
VI. LOW-TEMPERATURE SPECIFIC HEATS
Properties of the low-temperature specific heat of
the ferromagnetic XXZ chain have been derived from
the thermodynamic Bethe-ansatz equations [40]. For
anisotropy coefficient |R| ≤ 1, the model is gapless
[11, 12] and, except at R = 1, the specific heat is lin-
ear in T = J−1xy in the zero-temperature limit, C/T =
2γ/(3 sinγ) where again γ ≡ cos−1(−R). Note that
this result contradicts the spin-wave theory prediction of
C ∼ T 1/2 for the ferromagnetic chain (n = 2 for the fer-
romagnetic magnon dispersion relation of the kind given
above in Sec.IV). The spin-wave result is valid only for
R = 1, the isotropic Heisenberg case. From the expres-
sion given above, we see that C/T diverges as R → 1−,
and at exactly R = 1 it has been shown that C ∼ T 1/2
[40].
In the Isinglike region R > 1, the system exhibits a
FIG. 15: The calculated excitation spectrum gap ∆ versus
anisotropy.
FIG. 16: Calculated specific heat coefficient C/T as a function
of anisotropy R, for T = 0.10, 0.05, 10−10 .
gap in its excitation spectrum and the specific heat be-
haves as C ∼ T−a exp(−∆/T ), with ∆ being the excita-
tion spectrum gap [11, 12, 40]. There exist two gaps for
the energy, called the spinon gap and the spin-wave gap,
given by ∆spinon =
1−R−2 and ∆spinwave = 1−R−1.
These are the minimal energies of elementary excita-
tions [10, 40]. A crossover between them occurs atR = 5
below this value, the spinon gap is lower, while above
this value the spin-wave gap is lower. We have double-
fitted our calculated specific heats with respect to the
gap ∆ and the leading exponent a, for the entire range
of anisotropy R between 0 < R−1 < 1 (Fig.15). Our
calculated gap ∆ behaves linearly in R−1 for R−1 close
to 1, and crosses over to 1/2 at R−1 = 0, as expected.
We also obtain the exponent a = 1.99± 0.02 in the Ising
FIG. 17: Calculated specific heat coefficient C/T as a func-
tion of temperature for anisotropy coefficient R = −5 (thin
grey), R = −2 (thick grey), -1 (dotted), -0.5 (dash-dotted),
0.5 (dashed), and 2 (thin black).
limit R−1 ≤ 0.2 and a = 1.52 ± 0.10 in the Heisenberg
limit R−1 ≥ 0.9. These exponent values are respectively
expected to be 2 and 1.5 [9, 10].
We now turn to the discussion of our specific heat re-
sults for the entire ferromagnetic and antiferromagnetic
ranges. Our calculated C/T curves are plotted as a func-
tion of anisotropy and temperature in Figs.16 and 17
respectively. We discuss each region of the anisotropy R
separately:
(i) R > 1 : The specific heat coefficient C/T vanishes in
the T → 0 limit and has the expected exponential form as
discussed above in this section. The spin-wave to spinon
excitation gap crossover is obtained.
(ii) R ≈ 1 : The double-peak structure of C/T in Fig.16
is centered at R = 1. As temperature goes to zero, the
peaks narrow and diverge.
(iii) −1 ≤ R < 1 : The specific heat coefficient is
C/T = 2γ/(3 sinγ) in this region [11, 40], and our calcu-
lated specific heat is indeed linear at low temperatures.
The C/T curves for R = −1,−0.5, 0.5 in Fig.17 all ex-
trapolate to nonzero limits at T = 0. The spin-wave
dispersion relation exponent and velocity, for the antifer-
romagnetic system, is correctly obtained for the isotropic
case and for all anisotropies, as seen in Fig.7. Fig.18
directly compares C/T = 2γ/(3 sinγ) with our results:
The curves have the same basic form, gradually rising
from R = −1, with a sharp divergence as R nears 1.
At R = 1+, we expect C/T = 0. Our T = 10−10
curve diverges at R = 1 and indeed returns to zero at
R = 1.0000001.
(iv) R < −1 : We expect a vanishing C/T , which we do
find as seen in Fig.16 and in the insets of Fig.17. The
exponential behavior of the specific heat is clearly seen
in the Ising limit.
VII. CONCLUSION
A detailed global renormalization-group solution of
the XXZ Heisenberg chain, for all temperatures and
anisotropies, for both ferromagnetic and antiferromag-
netic couplings, has been obtained. In the spin-liquid
region, the linear low-temperature specific heat and, for
the antiferromagnetic chain, the spin-wave dispersion re-
lation exponent n and velocity c have been obtained. In
the Isinglike region, the spin-wave to spinon crossover of
the excitation spectrum gap of the ferromagnetic chain
has been obtained from the exponential specific heat,
as well as the correct leading algebraic behaviors in the
Heisenberg and Ising limits. Purely quantum mechani-
cal effects have been seen: We find that the xy corre-
lations and the antiferromagnetic z correlations mutu-
FIG. 18: Calculated specific heat coefficient C/T as a func-
tion of anisotropy coefficient R in the spin-liquid region,
−1 ≤ R ≤ 1, at constant temperature T = 10−10 . Our
renormalization-group result (grey curve) is compared to the
zero-temperature Bethe-Ansatz result (black curve). Inset:
our calculation (grey curve) at constant T = 10−2 is again
compared to the zero-temperature Bethe-Ansatz result (black
curve).
ally reinforce each other, for different ranges of temper-
atures and anisotropies, in ferromagnetic, antiferromag-
netic, and XY systems. The behaviors, with respect to
anisotropy, of the specific heat peak values and locations
are opposite in the ferromagnetic and antiferromagnetic
systems. The sharp contrast found in the specific heats
of the isotropic ferromagnetic and antiferromagnetic sys-
tems is a harbinger of the different critical temperatures
in the respective three-dimensional systems. When com-
pared with existing calculations in the various regions of
the global model, good quantitative agreement is seen.
Even at zero temperature, where our approximation is at
its worst, good quantitative agreement is seen with exact
data points for the correlation functions (Fig.1(b)), which
we extend to all values of the anisotropy. Finally, the rel-
ative ease with which the Suzuki-Takano decimation pro-
cedure is globally and quantitatively implemented should
be noted.
Acknowledgments
This research was supported by the Scientific and
Technological Research Council (TÜBİTAK) and by the
Academy of Sciences of Turkey. One of us (O.S.S.) grate-
fully acknowledges a scholarship from the Turkish Sci-
entific and Technological Research Council - Scientist
Training Group (TÜBİTAK-BAYG).
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