Ground state of a spin-1/2 Heisenberg-Ising two-leg ladder with XYZ intra-rung coupling
The quantum spin-1/2 two-leg ladder with an anisotropic XYZ Heisenberg intra-rung interaction and Ising inter-rung interactions is treated by means of a rigorous approach based on the unitary transformation. The particular case of the considered model with X-X intra-rung interaction resembles a quan...
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irk-123456789-1210672017-06-14T03:05:15Z Ground state of a spin-1/2 Heisenberg-Ising two-leg ladder with XYZ intra-rung coupling Verkholyak, T. Strečka, J. The quantum spin-1/2 two-leg ladder with an anisotropic XYZ Heisenberg intra-rung interaction and Ising inter-rung interactions is treated by means of a rigorous approach based on the unitary transformation. The particular case of the considered model with X-X intra-rung interaction resembles a quantum compass ladder with additional frustrating diagonal Ising interactions. Using an appropriately chosen unitary transformation, the model under investigation may be reduced to a transverse Ising chain with composite spins, and one may subsequently find the ground state quite rigorously. We obtain a ground-state phase diagram and analyze the interplay of the competition between several factors: the XYZ anisotropy in the Heisenberg intra-rung coupling, the Ising interaction along the legs, and the frustrating diagonal Ising interaction. The investigated model shows extraordinarily diverse ground-state phase diagrams including several unusual quantum ordered phases, two different disordered quantum paramagnetic phases, as well as discontinuous or continuous quantum phase transitions between those phases. Квантову спiн-1/2 двоногу драбинку з анiзотропною XY Z взаємодiєю Гайзенберґа вздовж щаблiв i Iзинґовою взаємодiєю мiж спiнами на сусiднiх щаблях розглянуто в межах строгого пiдходу, який ґрунтується на унiтарному перетвореннi. Частковий випадок моделi з X–X взаємодiєю вздовж щаблiв вiдображає модель квантового компасу на драбинцi з додатковими дiагональними Iзинґовими взаємодiями. Використовючи унiтарне перетворення, модель можна звести до поперечного ланцюжка Iзинґа з композитними спiнами, i як наслiдок основний стан можна знайти строго. Ми отримуємо фазову дiаграму основного стану i аналiзуємо взаємовплив конкуренцiї кiлькох факторiв: XY Z анiзотропiї взаємодiї Гайзенберґа, Iзинґової взаємодiї вздовж ланцюжкiв та фрустрованої дiагональної Iзинґової взаємодiї. Розглянута модель демонструє складнi фазовi дiаграми основного стану, включаючи декiлька нетипових квантових впорядкованих станiв, два рiзних невпорядкованих квантових станiв, а також квантовi фазовi переходи першого i другого порядку мiж цими фазами. 2013 Article Ground state of a spin-1/2 Heisenberg-Ising two-leg ladder with XYZ intra-rung coupling / T. Verkholyak, , J. Strečka // Condensed Matter Physics. — 2013. — Т. 16, № 1. — С. 13601:1–8. — Бібліогр.: 11 назв. — англ. 1607-324X PACS: 05.30.Rf; 75.10.Jm DOI:10.5488/CMP.16.13601 arXiv:1303.5537 http://dspace.nbuv.gov.ua/handle/123456789/121067 en Condensed Matter Physics Інститут фізики конденсованих систем НАН України |
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The quantum spin-1/2 two-leg ladder with an anisotropic XYZ Heisenberg intra-rung interaction and Ising inter-rung interactions is treated by means of a rigorous approach based on the unitary transformation. The particular case of the considered model with X-X intra-rung interaction resembles a quantum compass ladder with additional frustrating diagonal Ising interactions. Using an appropriately chosen unitary transformation, the model under investigation may be reduced to a transverse Ising chain with composite spins, and one may subsequently find the ground state quite rigorously. We obtain a ground-state phase diagram and analyze the interplay of the competition between several factors: the XYZ anisotropy in the Heisenberg intra-rung coupling, the Ising interaction along the legs, and the frustrating diagonal Ising interaction. The investigated model shows extraordinarily diverse ground-state phase diagrams including several unusual quantum ordered phases, two different disordered quantum paramagnetic phases, as well as discontinuous or continuous quantum phase transitions between those phases. |
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Verkholyak, T. Strečka, J. |
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Verkholyak, T. Strečka, J. Ground state of a spin-1/2 Heisenberg-Ising two-leg ladder with XYZ intra-rung coupling Condensed Matter Physics |
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Verkholyak, T. Strečka, J. |
| author_sort |
Verkholyak, T. |
| title |
Ground state of a spin-1/2 Heisenberg-Ising two-leg ladder with XYZ intra-rung coupling |
| title_short |
Ground state of a spin-1/2 Heisenberg-Ising two-leg ladder with XYZ intra-rung coupling |
| title_full |
Ground state of a spin-1/2 Heisenberg-Ising two-leg ladder with XYZ intra-rung coupling |
| title_fullStr |
Ground state of a spin-1/2 Heisenberg-Ising two-leg ladder with XYZ intra-rung coupling |
| title_full_unstemmed |
Ground state of a spin-1/2 Heisenberg-Ising two-leg ladder with XYZ intra-rung coupling |
| title_sort |
ground state of a spin-1/2 heisenberg-ising two-leg ladder with xyz intra-rung coupling |
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Інститут фізики конденсованих систем НАН України |
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2013 |
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http://dspace.nbuv.gov.ua/handle/123456789/121067 |
| citation_txt |
Ground state of a spin-1/2 Heisenberg-Ising two-leg ladder with XYZ intra-rung coupling / T. Verkholyak, , J. Strečka // Condensed Matter Physics. — 2013. — Т. 16, № 1. — С. 13601:1–8. — Бібліогр.: 11 назв. — англ. |
| series |
Condensed Matter Physics |
| work_keys_str_mv |
AT verkholyakt groundstateofaspin12heisenbergisingtwolegladderwithxyzintrarungcoupling AT streckaj groundstateofaspin12heisenbergisingtwolegladderwithxyzintrarungcoupling |
| first_indexed |
2025-07-08T19:08:13Z |
| last_indexed |
2025-07-08T19:08:13Z |
| _version_ |
1837106931859193856 |
| fulltext |
Condensed Matter Physics, 2013, Vol. 16, No 1, 13601: 1–8
DOI: 10.5488/CMP.16.13601
http://www.icmp.lviv.ua/journal
Ground state of a spin-1/2 Heisenberg-Ising two-leg
ladder with X Y Z intra-rung coupling
T. Verkholyak1, J. Strečka2
1 Institute for Condensed Matter Physics of the National Academy of Sciences of Ukraine,
1 Svientsitskii St., 79011 Lviv, Ukraine
2 Department of Theoretical Physics and Astrophysics, Faculty of Science, P. J. Šafárik University,
Park Angelinum 9, 040 01 Košice, Slovak Republic
Received July 3, 2012
The quantum spin-1/2 two-leg ladder with an anisotropic X Y Z Heisenberg intra-rung interaction and Ising
inter-rung interactions is treated by means of a rigorous approach based on the unitary transformation. The
particular case of the considered model with X–X intra-rung interaction resembles a quantum compass ladder
with additional frustrating diagonal Ising interactions. Using an appropriately chosen unitary transformation,
the model under investigation may be reduced to a transverse Ising chain with composite spins, and one may
subsequently find the ground state quite rigorously. We obtain a ground-state phase diagram and analyze the
interplay of the competition between several factors: the X Y Z anisotropy in the Heisenberg intra-rung cou-
pling, the Ising interaction along the legs, and the frustrating diagonal Ising interaction. The investigated model
shows extraordinarily diverse ground-state phase diagrams including several unusual quantum ordered phases,
two different disordered quantumparamagnetic phases, as well as discontinuous or continuous quantumphase
transitions between those phases.
Key words: quantum spin ladder, exact results, ground state, quantum phase transitions
PACS: 05.30.Rf, 75.10.Jm
1. Introduction
Quantum spin ladderswith frustrated interactions are intensively studied during the last few decades,
since they exhibit a rather complex ground-state behaviour to be reflected in extraordinarily rich low-
temperature thermodynamics as well [1, 2]. Quite recently, a number of exact solutions have been ob-
tained for several particular examples of quantum spin- 1
2
two-leg ladders [3–7]. The railroad ladder con-
sidered by Lai and Montrunich [6] has a quite specific configuration of inter-spin interactions, namely,
the staggering of X –X and Y –Y couplings along the legs is supplemented by the uniform Z –Z coupling
present along the rungs. An exact solution of this specific quantum spin ladder has been found by adopt-
ing themethod originally developed by Kitaev [8], which proved a striking spin-liquid ground state in this
quantum spin ladder. On the other hand, the railroad ladder with the uniform Z –Z interaction along the
legs and the uniform X –X interaction along the rungs has been rigorously solved by Brzezicki and Oleś
[3–5]. To a certain extent, this exactly solved quantum spin ladder can be regarded as an one-dimensional
analogue of the quantum compass model on a square lattice, which describes the orbital ordering in
transition-metal compounds [9].
In this work, we will examine a more general model of the quantum spin- 1
2
two-leg ladder, which
includes the fully anisotropic X Y Z -Heisenberg coupling between spins from the same rung and two dif-
ferent Ising (Z –Z ) interactions between spins from neighbouring rungs considered along the legs and
across the diagonals, respectively. Nevertheless, it should be mentioned that the investigated quantum
spin ladder extends our previous exact calculations for the spin- 1
2
Heisenberg-Ising ladder with the X X Z
intra-rung coupling [7], whereas the quantum compass ladder investigated in detail by Brzezicki and Oleś
© T. Verkholyak, J. Strečka, 2013 13601-1
http://dx.doi.org/10.5488/CMP.16.13601
http://www.icmp.lviv.ua/journal
T. Verkholyak, J. Strečka
[3–5] also represents a very special limiting case of the investigated model system. The main goal of the
present paper is to examine the simultaneous effect of two different kinds of frustration: the geomet-
ric frustration caused by the antiferromagnetic interaction between spins from an elementary triangle
plaquette and the competition between X –X and Y –Y intra-rung interactions with both Z –Z inter-rung
interactions.
The outline of the paper is as follows. In section 2, we define the model and show how to get the
ground state by a rigorous calculation based on the appropriate unitary transformation. The ground-
state phase diagram of the spin- 1
2
X Z -Ising and X Y -Ising ladders is explored in section 3. Finally, some
conclusions are drawn in section 4.
2. Model and solution
Consider the quantum spin- 1
2
Heisenberg-Ising ladder with an anisotropic X Y Z intra-rung coupling
and two different Ising-type couplings, which involve Z –Z spin-spin interactions along the legs and
across the diagonals of a two-leg ladder (see figure 1):
H =
N
∑
i=1
[(
J x
1 sx
1,i sx
2,i + J
y
1
s
y
1,i
s
y
2,i
+ J z
1 sz
1,i sz
2,i
)
+ J2
(
sz
1,i sz
1,i+1
+ sz
2,i sz
2,i+1
)
+ J3
(
sz
1,i sz
2,i+1
+ sz
2,i sz
1,i+1
)]
. (2.1)
Here, sα
j ,i
denote three spatial components α= x, y, z of the spin- 1
2
operator, the former subscript j = 1,2
determines the number of a leg and the latter subscript enumerates the lattice position in a particular leg.
Apparently, the interaction terms J x
1
, J
y
1
, J z
1
account for the quite anisotropic X Y Z -Heisenberg coupling
between two spins belonging to the same rung, while the interaction terms J2 and J3 take into consider-
ation the Ising-type interactions between the nearest-neighbor spins along the legs and across the diago-
nals of the two-leg ladder. It should be pointed out that the z-component of the total spin Sz
i
= sz
1,i
+ sz
2,i
Figure 1. (Color online) A schematic representation of the considered Heisenberg-Ising two-leg ladder.
The intra-rung X Y Z -Heisenberg coupling is represented by thick lines, while Z–Z Ising interactions
along the legs and across the diagonals are displayed by thin solid and broken lines, respectively.
on i th rung commutes with the Hamiltonian, i.e. [(Sz
i
)2, H ] = 0, and this property allows us to present
the model in an integrable form. To obtain a ground state of the investigated quantum spin model quite
rigorously, it is advisable to closely follow the procedure worked out previously for the quantum spin- 1
2
Heisenberg-Ising ladder with X X Z intra-rung interaction. After the unitary transformation [7]
sx
1,i = s̃x
1,i , s
y
1,i
= 2s̃
y
1,i
s̃x
2,i , sz
1,i = 2s̃z
1,i s̃x
2,i ,
sx
2,i = 2s̃x
1,i s̃z
2,i , s
y
2,i
=−2s̃x
1,i s̃
y
2,i
, sz
2,i = s̃x
2,i , (2.2)
one may rewrite the Hamiltonian (2.1) into the following pseudospin representation:
H =
N
∑
i=1
{(
J x
1
2
− J
y
1
s̃z
1,i
)
s̃z
2,i +
J z
1
2
s̃z
1,i +
[
J2
(
1+4s̃z
1,i s̃z
1,i+1
)
+2J3
(
s̃z
1,i + s̃z
1,i+1
)]
s̃x
2,i s̃x
2,i+1
}
, (2.3)
which shows the symmetry of themodel in amore explicit way. It is quite obvious that only z-components
of spin operators from the first leg are present in the Hamiltonian (2.3), which means that s̃z
1,i
are good
quantum numbers. By contrast, different spatial components of spin operators from the second leg are
still involved in the Hamiltonian (2.3) and thus, they still represent quantum spins with regard to the
13601-2
X Y Z -Heisenberg-Ising two-leg ladder
presence of two non-commuting parts s̃x
2,i
and s̃z
2,i
of each spin operator. Altogether, the Hamiltonian
(2.3) can be identified as the Ising chain of composite spins in a transverse field, whereas the values of
the effective interaction and the effective transverse field locally depend on a particular choice of eigen-
values of the classical Ising spins s̃z
1,i
. Following [7], one may also establish the following correspondence
between new and initial states:
|↓̃1,i ↓̃2,i 〉 = |φi
0,0〉 =
1
p
2
(
| ↓1,i ↑2,i 〉− | ↑1,i ↓2,i 〉
)
, |↓̃1,i ↑̃2,i 〉 = |φi
1,0〉 =
1
p
2
(
| ↓1,i ↑2,i 〉+ | ↑1,i ↓2,i 〉
)
,
|↑̃1,i ↓̃2,i 〉 = |φi
1,−〉 =
1
p
2
(
| ↑1,i ↑2,i 〉− | ↓1,i ↓2,i 〉
)
, |↑̃1,i ↑̃2,i 〉 = |φi
1,+〉 =
1
p
2
(
| ↑1,i ↑2,i 〉+ | ↓1,i ↓2,i 〉
)
,
(2.4)
where |φi
0,0
〉 denotes the singlet state of the i th rung and the other three states |φi
1,±〉, |φ
i
1,0
〉 form the
triplet state of the i th rung. It should be stressed that the square of zth component of the total spin Sz
i
on the i th rung acquires two different values. It either equals (Sz
i
)2 = 0 for |φi
0,0
〉 and |φi
1,0
〉, or (Sz
i
)2 = 1
for |φi
1,±〉. To get the partition function one has to diagonalize the Hamiltonian (2.3) for all particular
configurations of s̃z
1,i
and sum up all contributions in the trace of statistical operator. However, it is quite
evident from the transformed Hamiltonian (2.3) that the chain decomposes into two independent parts
whenever two neighboring spins s̃z
1,i
and s̃z
1,i+1
have opposite orientation (i.e., take on different eigen-
values). In this respect, the composite chain is divided into a set of finite chains of different sizes for
any chosen configuration of s̃z
1,i
. Generally, this problem seems to represent a quite formidable task, but
the ground state of the investigated model can be found quite rigorously using the same arguments as
given in [7]. Since the ground-state energy of two finite but isolated spin- 1
2
Ising chains in a transverse
field is always higher than the ground-state energy of one unique spin- 1
2
Ising chain in a transverse field
obtained by joining both independent finite chains, the ground state of the model under investigation
should accordingly correspond only to the uniform configuration of all s̃z
1,i
. Therefore, one may single
out only two different uniform configurations with all s̃z
1,i
= 1
2
or all s̃z
1,i
= − 1
2
from which the ground
state of the Heisenber-Ising ladder can be derived. The effective Hamiltonian (2.3) for the two uniform
configurations acquires the following form:
H± =
N
∑
i=1
[
1
2
(
J x
1 ∓ J
y
1
)
s̃z
2,i ±
1
4
J z
1 +2(J2 ± J3) s̃x
2,i s̃x
2,i+1
]
. (2.5)
The ground state energies per site of both effective Hamiltonians can be exactly calculated using the
Jordan-Wigner fermionization [10, 11]:
e±0 ==−
(
J∓
1
+|J2 ± J3|
)
π
E
[
√
1−
(
γ±)2
]
±
J z
1
4
, (2.6)
where
γ± =
J∓
1
−|J2 ± J3|
J∓
1
+|J2 ± J3|
, J± =
(
J x
1
± J
y
1
)
2
and
E(κ) =
π
2
∫
0
dθ
√
1−κ2 sin2θ
is the complete elliptic integral of the second kind.
Both Hamiltonians H+ and H− imply a precise mapping correspondence between the spin- 1
2
Heisen-
berg-Ising two-leg ladder and the spin- 1
2
quantum Ising chain in a transverse field, which can be, how-
ever, characterized by different values of the effective interaction and transverse field. Bearing this
in mind, one should expect quantum phase transitions of two different types. The first kind of zero-
temperature phase transitions may correspond to a continuous (second-order) quantum phase transition
inherent to the transverse Ising chain, which arises for one particular ratio between the effective inter-
action and transverse field. Beside this, there may also occur discontinuous (first-order) quantum phase
13601-3
T. Verkholyak, J. Strečka
transitions whenever a crossing of the lowest-energy levels inherent to both effective Hamiltonians (2.5)
takes place. In the following two sections, we will illustrate all the aforementioned features of quantum
phase transitions on ground-state phase diagrams of two particular cases of the model under considera-
tion.
3. Ground state of X Z -Ising and X Y -Ising ladders
In this section, we will consider two particular cases of the investigated model (2.1) by switching off
either the y - or z-component of X Y Z -Heisenberg coupling (i.e., either J
y
1
= 0 or J z
1
= 0). It is notewor-
thy that the two aforementioned particular cases represent a direct extension of the quantum compass
ladder [3–5] to which the considered model reduces when neglecting the z-component of the Heisenberg
coupling (J z
1
= 0), one of the two transverse components of the Heisenberg coupling (i.e., either J x
1
= 0 or
J
y
1
= 0) and the frustrating Ising interaction across the diagonals (J3 = 0). Furthermore, the problem of
two-dimensional quantum compass model is quite complex and the exact solution for this model has not
been found yet.
Let us first consider all possible phases that may appear in the ground state of the model under in-
vestigation. Each uniform configuration of s̃z
1,i
corresponds to the transverse Ising chain, which has three
possible ground-state phases. The ground-state phases for all s̃z
1,i
=− 1
2
appear if e−
0
< e+
0
. It is worthwhile
to remark that the ground-state phases belonging to this effective model were thoroughly analyzed in our
preceding paper [7] and let us, therefore, give here just their definition for the sake of easy reference:
• Quantum paramagnetic (QPM1) state for 1
2
(J x
1
+ J
y
1
) > |J2 − J3|: the Heisenberg-Ising ladder is in
the gapped disordered state with no spontaneous magnetization. The rung singlet dimers dominate
on the Heisenberg bonds and in the very special case J2 = J3, the ground state factorizes to a set of
the completely non-correlated rung singlet dimers.
• Stripe Leg (SL) state for 1
2
(J x
1
+ J
y
1
) < J3 − J2: the Heisenberg-Ising ladder shows a ferromagnetic
order along the legs and antiferromagnetic order along the rungs, i.e., the magnetizations of two
chains are oriented opposite to each other (〈sz
1,i
〉 = 〈sz
1,i+1
〉 = −〈sz
2,i
〉 = −〈sz
2,i+1
〉, 0). The following
staggered magnetization can be defined as the relevant order parameter of this phase
mz
SL =
1
2N
N
∑
i=1
(
〈sz
1,i 〉−〈sz
2,i 〉
)
=
1
2
[
1−
(
J x
1
+ J
y
1
)2
4(J2 − J3)2
]
1
8
,
which undergoes the obvious quantum reduction of magnetization.
• Néel state for 1
2
(J x
1
+ J
y
1
) < J2 − J3: the nearest-neighbor spins both along the legs and rungs ex-
hibit predominantly antiferromagnetic ordering. The dependence of staggered magnetization as
the relevant order parameter is quite analogous to the previous case
mz
AF =
1
2N
N
∑
i=1
(−1)
i
(
〈sz
1,i 〉−〈sz
2,i 〉
)
=
1
2
[
1−
(
J x
1
+ J
y
1
)2
4(J2 − J3)2
]
1
8
.
The most fundamental difference between the ground states of the Heisenberg-Ising ladder with the
X X Z - and X Y Z -Heisenberg intra-rung interaction can be found in the phases arising from the uni-
form configuration with all s̃z
1,i
= 1
2
. While in the former model with the X X Z intra-rung interaction,
all ground-state phases are classical in their character [7], the emergent ground-state phases of the latter
model with the more anisotropic X Y Z intra-rung coupling display significant quantum features. One
may indeed identify the following three quantum ground states for a particular case e+
0
< e−
0
with all
s̃z
1,i
= 1
2
:
• Quantum paramagnetic (QPM2) state for 1
2
|J x
1
− J
y
1
| > |J2 + J3|: the equivalent transverse Ising
chain H+ (2.5) is in the gapped disordered state with no spontaneous magnetization 〈s̃x
i
〉 = 0 and
13601-4
X Y Z -Heisenberg-Ising two-leg ladder
non-zeromagnetization 〈s̃z
i
〉, 0 induced by the effective transverse field. For the initial Heisenberg-
Ising ladder, one consequently gets the ground state with the prevailing dimer state |φi
1,−〉 on the
rungs.
• Stripe Rung (SR) state for 1
2
|J x
1
−J
y
1
| < J2+J3: the equivalent transverse Ising chain exhibits a spon-
taneous antiferromagnetic orderingwith 〈s̃x
i
〉 = (−1)i mx , 0. Due to relationships (2.2), one obtains
for the Heisenberg-Ising ladder 〈sz
1,i
〉 = −〈sz
1,i+1
〉 = 〈sz
2,i
〉 = −〈sz
2,i+1
〉, 0. Thus, the Heisenberg-Ising
ladder shows an antiferromagnetic order along the legs and ferromagnetic order along the rungs.
The staggered magnetization as the relevant order parameter in this phase is non-zero and it ex-
hibits evident quantum reduction of the magnetization given by:
mz
SR =
1
2N
N
∑
i=1
(−1)
i
(
〈sz
1,i 〉+〈sz
2,i 〉
)
=
1
2
[
1−
(
J x
1
− J
y
1
)2
4(J2 + J3)2
]
1
8
.
• Ferromagnetic (FM) state for 1
2
|J x
1
− J
y
1
| < −(J2 + J3): the effective transverse Ising chain shows a
ferromagnetic ordering. Due to transformation relationships (2.2), all the spins of the ladder have
the same magnetization in the z-direction. The ground state corresponds to the ferromagnetic spin
state
mz
FM =
1
2N
N
∑
i=1
(
〈sz
1,i 〉+〈sz
2,i 〉
)
=
1
2
[
1−
(
J x
1
− J
y
1
)2
4(J2 + J3)2
]
1
8
.
Altogether, it could be concluded that the X Y anisotropy in the Heisenberg intra-rung coupling is re-
sponsible for quantum features of otherwise classical SR and FM states and, moreover, it may also lead
to the appearance of a new disordered phase QPM2. Two paramagnetic phases QPM1 and QPM2 have
quite similar features: they are both disordered states with the energy gap in their excitation spectrum
and, consequently, their pair spin-spin correlation functions decay exponentially. Both quantum param-
agnetic phases can be distinguished by the square of zth component of the total spin (Sz
i
)2 on i th rung,
which is equal to (Sz
i
)2 = 0(1) in QPM1 (QPM2).
Now, let us pay our attention to the ground-state phase diagram established for the particular case of
X Z -Ising ladder as depicted in figure 2 by considering J
y
1
= 0. Assuming the X Z intra-rung interaction,
one gets a striking competition between the X –X interaction along the rungs and the Z –Z interaction
along the legs, while the additional Z –Z interaction along rungs acts generally against the X –X inter-
action. It should be also mentioned that one may recover some known examples from the ground-state
phase diagram of the X Z -Ising ladder presented in figure 2. In fact, figure 2 (a) shows the particular lim-
iting case of a quantum compass ladder with an additional diagonal frustrating Ising interaction. Let us
follow the known results of a simple quantum compass ladder [3] to be obtained from our model by dis-
regarding the frustrating Ising interaction J3 = 0. Both Hamiltonians H+ and H− (2.5) become identical
under this special condition and, consequently, the ground state of the model is always two-fold degen-
erate due to the equality e−
0
= e+
0
. The investigated model is either in the SL or FM state for 1
2
J x
1
< −J2,
either in QPM1 or QPM2 state for 1
2
J x
1
> |J2|, either in SR or Néel state for 1
2
J x
1
< J2. Note that the quantum
phase transition from the disordered to the long-range ordered state takes place at |J2| = 1
2
J x
1
.
It is quite evident that the diagonal interaction J3 removes the two-fold degeneracy of the ground state
[see figure 2 (a)] by changing the effective spin interaction in the effective Hamiltonians H+ and H−. The
effect of the additional Z –Z intra-rung interaction is shown in figure 2 (b)–(d), where the results for a
different relative strength of J z
1
are presented. It could be understood from (2.5) that J z
1
lowers initially
the energy of the s̃z
1,i
= − 1
2
configuration by the amount 1
2
N J z
1
with respect to the energy of the s̃z
1,i
=
1
2
configuration. Therefore, the regions of the phases, which correspond to the s̃z
1,i
= 1
2
configuration,
become smaller with increasing J z
1
. For sufficiently strong J z
1
, QPM2 phase can completely disappear
from the ground-state phase diagram. Another distinctive feature is that J z
1
makes the singlet-dimer phase
favorable along the line J2 = J3 inside the region of QPM1 state.
The relevant ground-state behavior can be supported by the dependencies of respective order pa-
rameters as displayed in figure 3 (a). If there is no frustrating diagonal interaction, the model may stay
13601-5
T. Verkholyak, J. Strečka
Figure 2. (Color online) Ground-state phase diagram of the X Z -Ising ladder for J x
1
= 1, J
y
1
= 0 and four
different values of J z
1
: (a) J z
1
= 0; (b) J z
1
= 0.05; (c) J z
1
= 0.1; (d) J z
1
= 0.5. QPM1 is the paramagnetic phase
with the prevailing rung states |φi
0,0
〉. QPM2 is the paramagnetic phase with the prevailing rung states
|φi
1,−〉. Dotted lines indicate the rung singlet-dimer state.
Figure 3. (Color online) Order parameters (a) and nearest-neighbor correlation function (b) as a function
of intra-leg interaction J2 for J x
1
= 1, J z
1
= 0.1 and different J3 = 0,0.25,0.5,0.75. (a) The curves on the left
correspond to mSL, the curves on the right for J3 = 0.25,0.5,0.75 (J3 = 0) correspond to mSR (mAF).
13601-6
X Y Z -Heisenberg-Ising two-leg ladder
in the disordered QPM1 or the ordered SL and Néel phases. The corresponding nearest-neighbor corre-
lation function along the legs shows a continuous change with a weak singularity at the quantum critical
points indicated by filled circles in figure 3 (b). The curve for another particular case J3 = 0.25 looks sim-
ilar except that the diagonal interaction of this strength leads to a direct phase transition between two
disordered quantum paramagnetic states QPM1 and QPM2. This unusual transition can be recognized
from the relevant dependence of the nearest-neighbor correlation function,which sustains a jump at this
special critical point. It is interesting to note that the further increase of a frustrating Ising interaction
J3 demolishes both disordered phases QPM1 and QPM2. Thus, one may also detect the quantum phase
transition between two ordered SL and Néel phases, whereas the order parameters do not reach zero
continuously in this particular case.
Figure 4. (Color online) Ground-state phase diagram of the X Y -Ising model for J x
1
= 1, J z
1
= 0 and four
different values of J
y
1
: (a) J
y
1
= 0; (b) J
y
1
= 0.05; (c) J
y
1
= 0.1; (d) J
y
1
= 0.5. Dotted lines indicate the rung
singlet-dimer state.
In figure 4, the ground-state phase diagram of the X Y -Ising ladder is depicted by considering another
particular casewith J z
1
= 0. The effect of the Y –Y intra-rung interaction has some similaritieswith the one
of Z –Z intra-rung interaction, although the origin is completely different. The ground-state energy of H−
is generally lowered with respect to that of H+, because J
y
1
> 0 increases the effective transverse field in
H− and decreases it in H+. Owing to this fact, the regions of SL, QPM1, and Néel states that correspond to
the uniform configuration with all s̃z
1,i
=− 1
2
get extended with an increase of J
y
1
. One may also generally
conclude that J
y
1
> 0 acts against the QPM2 phase, which gradually shrinks with increasing J
y
1
until it
completely disappears at J
y
1
= 1
2
J x
1
. Similarly to the case with the Z –Z intra-rung interaction, the Y –Y
intra-rung interaction also induces the presence of the rung singlet-dimer state along a special line J2 = J3.
13601-7
T. Verkholyak, J. Strečka
4. Conclusions
In the present paper, the effect of the most general X Y Z anisotropy in the intra-rung interaction
on the ground state of the spin- 1
2
Heisenberg-Ising two-leg ladder was investigated in detail. It has been
shown that the most general kind of anisotropy,which breaks the rotational symmetry of the Heisenberg
interaction, may lead to the appearance of new quantum phases in the ground-state phase diagram. We
have also considered the special case of quantum compass ladder with an additional frustrated diagonal
interaction and showed that the singlet-dimer phase cannot appear in this particular case. The order
parameters and the nearest-neighbor correlation function were calculated and analyzed in detail in the
ground state. It has been demonstrated that the relevant behavior of the correlation function can help us
to reveal the quantum phase transition between two different disordered quantum paramagnetic phases.
Acknowledgements
J.S. acknowledges the financial support provided by ERDF EU (European Union European regional
development fund) grant under the contract ITMS 26220120005 (activity 3.2).
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Основний стан спiн-1/2 двоногої драбинки
Гайзенберґа-Iзинґа з X Y Z взаємодiєю вздовж щаблiв
Т. Верхоляк1, Й. Стречка2
1 Iнститут фiзики конденсованих систем НАН України, вул. Свєнцiцького, 1, 79011 Львiв, Україна
2 Кафедра теоретичної фiзики i астрофiзики, Унiверситет П.Й. Шафарика, парк Ангелiнум, 9, 04001 Кошицi,
Словацька республiка
Квантову спiн-1/2 двоногу драбинку з анiзотропною X Y Z взаємодiєю Гайзенберґа вздовж щаблiв i Iзин-
ґовою взаємодiєю мiж спiнами на сусiднiх щаблях розглянуто в межах строгого пiдходу, який ґрунтується
на унiтарному перетвореннi. Частковий випадок моделi з X –X взаємодiєю вздовж щаблiв вiдображає
модель квантового компасу на драбинцi з додатковими дiагональними Iзинґовими взаємодiями. Вико-
ристовючи унiтарне перетворення, модель можна звести до поперечного ланцюжка Iзинґа з композит-
ними спiнами, i як наслiдок основний стан можна знайти строго. Ми отримуємо фазову дiаграму основ-
ного стану i аналiзуємо взаємовплив конкуренцiї кiлькох факторiв: X Y Z анiзотропiї взаємодiї Гайзенбер-
ґа, Iзинґової взаємодiї вздовж ланцюжкiв та фрустрованої дiагональної Iзинґової взаємодiї. Розглянута
модель демонструє складнi фазовi дiаграми основного стану, включаючи декiлька нетипових квантових
впорядкованих станiв, два рiзних невпорядкованих квантових станiв, а також квантовi фазовi переходи
першого i другого порядку мiж цими фазами.
Ключовi слова: квантова спiн-1/2 драбинка, точнi результати, основний стан, квантовий фазовий
перехiд
13601-8
http://dx.doi.org/10.1103/PhysRevB.80.014405
http://dx.doi.org/10.1088/1742-6596/200/1/012017
http://dx.doi.org/10.1063/1.3518903
http://dx.doi.org/10.1103/PhysRevB.84.235148
http://dx.doi.org/10.1016/j.aop.2005.10.005
http://dx.doi.org/10.1088/0305-4470/36/35/307
http://dx.doi.org/10.1016/0003-4916(61)90115-4
http://dx.doi.org/10.1016/0003-4916(70)90270-8
Introduction
Model and solution
Ground state of XZ-Ising and XY-Ising ladders
Conclusions
|