Quantum phase transition in the one-dimensional compass model

Wojciech BrzezickiMarian Smoluchowski Institute of Physics, Jagellonian University, Reymonta 4, PL-30059 Kraków, Poland

Jacek DziarmagaMarian Smoluchowski Institute of Physics, Jagellonian University, Reymonta 4, PL-30059 Kraków, Poland

and Centre for Complex Systems, Jagellonian University, Reymonta 4, PL-30059 Kraków, Poland

Andrzej M. OleśMarian Smoluchowski Institute of Physics, Jagellonian University, Reymonta 4, PL-30059 Kraków, Poland

and Max-Planck-Institut für Festkörperforschung, Heisenbergstrasse 1, D-70569 Stuttgart, Germany�Received 29 December 2006; revised manuscript received 15 February 2007; published 16 April 2007�

We introduce a one-dimensional model which interpolates between the Ising model and the quantum com-pass model with frustrated pseudospin interactions �i

z�i+1z and �i

x�i+1x , alternating between even and/or odd

bonds, and present its exact solution by mapping to quantum Ising models. We show that the nearest-neighborpseudospin correlations change discontinuously and indicate divergent correlation length at the first-orderquantum phase transition. At this transition, one finds the disordered ground state of the compass model withhigh degeneracy 2�2N/2 in the limit of N→�.

DOI: 10.1103/PhysRevB.75.134415 PACS number�s�: 75.10.Jm, 03.67.Lx, 05.70.Fh, 73.43.Nq

I. INTRODUCTION

Recent interest in quantum models of magnetism with ex-otic interactions is motivated by rather complex superex-change models which arise in Mott insulators with orbitaldegrees of freedom.1 The degeneracy of 3d orbitals is onlypartly lifted in an octahedral environment in transition-metaloxides, and the remaining orbital degrees of freedom arefrequently described as T=1/2 pseudospins. They occur onequal footing with spins in spin-orbital superexchange mod-els and their dynamics may lead to enhanced quantum fluc-tuations near quantum phase transitions2 and to entangledspin-orbital ground states.3 The properties of such models arestill poorly understood, so it is helpful and of great interest toinvestigate first the consequences of frustrated interactions inthe orbital sector alone.

The orbital interactions are intrinsically frustrated—theyhave much lower symmetry than the usual SU�2� symmetryof spin interactions, and their form depends on the orienta-tion of the bond in real space,4 so they may lead to an orbitalliquid in three dimensions.5 Although such interactions are inreality more complicated,2–5 a generic and simple model ofthis type is the so-called compass model introduced longago,6 where the coupling along a given bond is of Ising type,but different spin components are active along different bonddirections, for instance, Jx�i

x� jx and Jz�i

z� jz along the a and b

axes in the two-dimensional �2D� compass model. The com-pass model is challenging already for the classicalinteractions.7 Although the compass model originates fromthe orbital superexchange, it is also dual to recently studiedmodels of p+ ip superconducting arrays.8 It was also pro-posed as a realistic model to generate protected cubits,9 so itcould play a role in the quantum information theory.

So far, the nature of the ground state and pseudospin cor-relations in the compass model was studied only by numeri-cal methods. It has been argued that the eigenstates of the 2Dcompass model are twofold degenerate.9 In contrast, a nu-

merical study of the 2D compass model suggests that theground state is highly degenerate and disordered.10 In fact,the numerical evidence suggests a first-order quantum phasetransition at Jx=Jz,

11 with diverging spin fluctuations and adiscontinuous change in the correlation functions.

The purpose of this paper is to show by an exact solutiona mechanism of a first-order quantum phase transition inquantum magnetism. Such a transition from quasiclassicalstates with short-range order to a disordered state occurs inthe one-dimensional �1D� XX-ZZ model, with antiferromag-netic interactions12 between �i

x and �iz pseudospin compo-

nents, alternating on even and/or odd bonds. The model issolved analytically by mapping its different subspaces to theexactly solvable quantum Ising model �QIM�,13 which playsa prominent role in understanding the paradigm of a quantumphase transition, including recent rigorous insights into thetransition through its quantum critical point at a finite rate.14

Note that the model introduced below is generic and by nomeans limited to the orbital physics. For instance, the QIM isrealized by the charge degrees of freedom in NaV2O5, whereit helped explain the temperature dependence of the opticalspectra.15

The paper is organized as follows: In Sec. II, we introducethe pseudospin �orbital� XX-ZZ model which realizes a con-tinuous interpolation between the classical Ising model andthe compass model in one dimension. This model is nextsolved exactly in its subspaces, which are separated fromeach other, by its mapping to quantum Ising models. Theproperties of the model and the mechanism of the quantumphase transition are elucidated in Sec. III. At the transitionpoint, we demonstrate the discontinuous change of correla-tion functions �Sec. III A� and the vanishing pseudospin gap�Sec. III B�. The paper is summarized in Sec. IV, where wealso argue that a similar transition to an orbital liquid stateoccurs in the 2D model.

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II. PSEUDOSPIN XX-ZZ MODEL

In order to understand the nature of a quantum groundstate with high degeneracy found in the 1D compass modelwith alternating superexchange interactions between �i

x and�i

z pseudospin components, it is important to investigate howthe pseudospin correlations develop when this state is ap-proached. Therefore, we start below from the classical Isingmodel with interacting �i

z pseudospins and modify graduallythe interactions on every second bond by replacing themwith the ones between �i

x. Although this does not correspondto any deformation of interacting orbitals in a crystal, in thisway we keep a constant value of the total pseudospin cou-pling constant J on each bond and demonstrate that the frus-tration of interactions gradually increases, and finally domi-nates the behavior of the compass model at the quantumcritical point.

A. Hamiltonian and its subspaces

We consider the pseudospin model with interactions de-pending on parameter �� �0,2�. When 0���1,

H��� � J�i=1

N�

��1 − ���2i−1z �2i

z + ��2i−1x �2i

x + �2iz �2i+1

z � ,

�1�

with N�=N /2, but when 1���2,

H��� � J�i=1

N�

��2i−1x �2i

x + �2 − ���2iz �2i+1

z + �� − 1��2ix �2i+1

x � .

�2�

Here ��ix ,�i

z� are Pauli matrices. We assume even number ofpseudospins N and periodic boundary conditions: �N+1=�1.The model reduces to the ZZ Ising model at �=0,

H�0� = J�i=1

N

�iz�i+1

z , �3�

and to the XX Ising model at �=2,

H�2� = J�i=1

N

�ix�i+1

x . �4�

It is frustrated between these two incompatible types of orderfor 0���2, and we focus on the consequences of increas-ing frustration. Right in the middle of the two classical cases�3� and �4�, i.e., for �=1, it becomes the proper quantumXX-ZZ model �called 1D compass model�,

H�1� = J�i=1

N�

��2i−1x �2i

x + �2iz �2i+1

z � , �5�

favoring antiferromagnetic order of x and z pseudospin com-ponents on odd and even bonds, respectively.

The Hamiltonians �1� and �2� are related by symmetry:Simultaneous exchange of �i

x↔�iz, 2i↔2i−1∀ i, and �1

−��↔ ��−1� maps the Hamiltonians into each other. Thanks

to this symmetry, we need to solve only the model �1� for�� �0,1�, with modulated interactions for odd pairs of pseu-dospins �2i−1,2i� �on odd bonds�. The Hamiltonian �1� canbe conveniently solved in the basis of eigenstates of the ZZIsing model at �=0. We start with one of the two degenerateground states,

↑↓↑↓↑↓↑↓↑↓↑↓ . . . , �6�

which is modified at finite ��0 by spin-flip operators: −�ix

��2i−1x �2i

x . Action of various �x’s on the state �6� generatesother eigenstates:

↓↑↑↓↑↓↑↓↑↓↑↓ . . . ,

↑↓↓↑↑↓↑↓↑↓↑↓ . . . , . . . ,

↓↑↓↑↑↓↑↓↑↓↑↓ . . . , . . . ,

↓↑↓↑↓↑↓↑↓↑↓↑ . . . , �7�

where the last state is the second degenerate ground state ofthe ZZ Ising model. This set of states �6� and �7� is a conve-nient basis in the subspace where all odd pairs of pseu-dospins are antiparallel, and gives a constant energy contri-bution of −�1−��J per bond. By construction, theHamiltonian �1� does not mix this subspace with the rest ofthe Hilbert space. Therefore, in this subspace the energy dueto the even �2i ,2i+1� bonds in Eq. �1� depends on the pseu-dospin orientation on the two neighboring odd bonds andmay be expressed by a product �i

z�i+1z of two spin operators,

with −�iz��2i−1

z �2iz . In this representation, it is clear that the

Hamiltonian �1� reduces in this subspace to the exactly solv-able QIM.13

In general, the Hilbert space can be divided into sub-spaces where different odd pairs of spins are either parallelor antiparallel. Each subspace can be labeled by a vector s�= �s1 , . . . ,sN��, with si=1 �si=0� when two pseudospins of theodd bond �2i−1,2i� are parallel �antiparallel�. In each sub-pace s�, the terms �1−�� for odd bonds in Eq. �1� give aconstant energy contribution

Cs��� � �1 − ��J�i=1

N�

�2i−1z �2i

z = − �1 − ��J�N� − 2s� , �8�

where s=�i=1N� si is the number of parallel odd pairs of spins.

With a convenient choice of spin operators for the antiparal-lel �2i−1,2i� pairs of pseudospins,

− �ix = �↑↓�↓↑ + ↓↑�↑↓� ,

− �iz = �− 1��j=1

i−1si �↑↓�↑↓ − ↓↑�↓↑� , �9�

and for the parallel pairs,

− �ix = �↑↑�↓↓ + ↓↓�↑↑� ,

− �iz = �− 1��j=1

i−1si �↑↑�↓↓ − ↓↓�↑↑� , �10�

the Hamiltonian �1� reduces to

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Hs���� = − J �i=1

N�−1

���ix + �i

z�i+1z � − J���N�

x + �− 1�s�N�z �1

z�

+ Cs��� . �11�

For even s, the Hamiltonian �11� is simply the ferromagneticQIM, but when s is odd, the interaction on the last �N� ,1�bond is antiferromagnetic.

B. Exact solution

Each effective model �11� can be solved using the Jordan-Wigner transformation for spin operators,

�ix = 1 − 2ci

†ci, �12�

�iz = − �ci + ci

†��j�i

�1 − 2cj†cj� . �13�

Here ci is a fermionic annihilation operator at site i. Afterthis transformation, the Hamiltonian �11� becomes

Hs���� = J �i=1

N�−1

��ci†ci − ci

†ci+1 − ci+1ci + H.c.� + J��cN�† cN�

− cN�† c̃1 − c̃1cN� + H.c.� + Cs��� , �14�

with

c̃1 = c1�− 1�1+s+�j=1N� cj

†cj , �15�

depending on the parity of the number of c quasiparticles.This parity is a good quantum number because c quasiparti-cles can only be created or annihilated in pairs, as can beseen in Eq. �14�.

In order to extend the sum in the first line of Eq. �14� toi=N� we split the Hamiltonian �14� as

Hs���� = P+Hs�+���P+ + P−Hs�

−���P−, �16�

where

P± =1

2 1 ± �

i=1

N�

�ix� =

1

2 1 ± �

i=1

N�

�1 − 2ci†ci�� �17�

are projectors on the subspaces with even �� and odd ���numbers of c fermions and

Hs�±��� = J�

i=1

N�

��ci†ci − ci

†ci+1 − ci+1ci + H.c.� + Cs���

�18�

are corresponding reduced quadratic Hamiltonians. Defini-tion of the Hamiltonians �18� is not complete without bound-ary conditions which depend both on the choice of subspace� and on the parity of s, see Eq. �15�. When both s and thenumber of c fermions have the same parity, then the bound-ary condition is antiperiodic, i.e., cN�+1�−c1, but when onthe contrary the two numbers have opposite parity—it is pe-riodic, i.e., cN�+1�c1. With these boundary conditions, theHamiltonians �14� and �18� are the same.

The Hamiltonian �18� is simplified by a Fourier transfor-mation, cj �

1�N�

�kckeikj. Here k’s are quantized pseudomo-

menta. In the following, we assume for convenience that N�is even �i.e., N=4m , m integer�. For periodic boundary con-ditions, k’s take “integer” values �recall that N=2N��k=0, ± 2

N�, ±2 2

N�, . . . , , and in the antiperiodic case, they are

“half-integer,” i.e., k= ± 12

2

N�, ± 3

22

N�, . . . , ± 1

2 �N�−1� 2

N�. As a

result, the Hamiltonians �18� describe independent subspaceslabeled by k, with mixed k and −k quasiparticle states,

Hs�±��� = J�

k

�2�� − cos k�ck†ck + sin k�ck

†c−k† + H.c.�� + Cs��� .

�19�

Diagonalization of Hs�±��� is completed by a Bogoliubov

transformation,13 ck=uk�k+v−k* �−k

† , where the Bogoliubovmodes �uk ,vk� are eigenmodes of the Bogoliubov–de Gennesequations:

�uk = 2J�� − cos k�uk + 2J sin k vk, �20�

�vk = 2J sin k uk − 2J�� − cos k�vk. �21�

For each value of k, there are two eigenstates with eigenen-ergies �k and −�k. Positive eigenenergies

�k = 2J�1 + �2 − 2� cos k �22�

define quasiparticles �k for each k, which bring the Hamil-tonian �19� to the diagonal form

Hs�±��� = �

k

�k��k†�k −

1

2� + Cs��� , �23�

being a sum of fermionic quasiparticles. However, thanks tothe projection operators P± in Eq. �16� only states with evenor odd numbers of Bogoliubov quasiparticles belong to thephysical spectrum of Hs����.

In order to find if the number of Bogoliubov quasiparti-cles in a given subspace must be even or odd, we must findfirst the parity of the number of c quasiparticles in the Bo-goliubov vacuum in this subspace. This parity depends onthe boundary conditions. Indeed, when they are antiperiodic,then sin k�0 for any allowed k and the ground state of theHamiltonian �19� is a superposition over states with pairs ofquasiparticles ck

†c−k† . This Bogoliubov vacuum has even num-

bers of c quasiparticles. On the other hand, for periodicboundary conditions, we have two special cases with sin k=0 when k=0 or . When 0���1, the Bogoliubov vacuumcontains the quasiparticle c0 but not the quasiparticle c , andnumbers of c quasiparticles are odd. In short, �anti-�periodicboundary conditions imply �even� odd parity of the Bogoliu-bov vacuum.

Taking further into account that any �k† changes the parity

of the number of c quasiparticles, we soon arrive at thesimple conclusion that even �odd� s implies that only stateswith even �odd� numbers of Bogoliubov quasiparticles be-long to the physical spectrum of Hs����. Note that for odd s,the lowest energy state is not the Bogoliubov vacuum but astate with one Bogoliubov quasiparticle of minimal energy�k. Its pseudomomentum is k=0 in a subspace, but in a �

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subspace, there is a choice between two degenerate quasipar-ticle states with k= ± 1

22

N�.

III. QUANTUM PHASE TRANSITION

A. Correlation functions

For any �� �0,1�, the ground state is found in the s�=0�� subspace. Pseudospin correlators in the ground state canbe expressed by the spin correlators of the effective QIM Eq.�16�. For odd pairs of pseudospins,

��2i−1z �2i

z = − 1, �24�

��2i−1x �2i

x = − ��ix = − 1 +

2

N��

k

vk2. �25�

Surprisingly, in spite of decreasing interaction �1−�� inEq. �1�, the odd ��2i−1

z �2iz correlator �24� shows the same

perfect pseudospin order as that found at �=0, while the��2i−1

x �2ix one �Eq. �25��, gradually decreases from 0 at �

=0 to − 2 when �→1− �at N→��, see Fig. 1.

For different odd pseudospin pairs, one finds ���1�

��2i−mx �2j−n

x = 0, �26�

��2i−mz �2j−n

z = �− 1�m+n��iz� j

z , �27�

when i� j and for m ,n=0,1. As is well known in the QIM,the correlator on the right-hand side of Eq. �27� is theToeplitz determinant,16

��iz�i+r

z = � f1 f2 . . . fr

f0 f1 . . . fr−1

. . . . . . . . . . . .� , �28�

with constant diagonals fr���=�r,0−2ar���+2br���, givenby ar���= 1

N��kvk2 cos�kr� and br���= 1

N��kukvk

* sin�kr�.

This correlator is positive for all r and finite when r→�,indicating long-range ferromagnetic order of �i

z moments.When r→�, it decays exponentially toward the finite long-range limit with a correlation length �, which diverges as ���1−��−1. As we have seen, in this limit the gap in thequasiparticle energy spectrum �22� tends to 0 and a quantumcritical point is approached in the universality class of theQIM.

For nearest-neighbor even pairs �when j= i+1, m=0, andn=1 in Eq. �27��, the ��2i

z �2i+1z pseudospin correlator is

negative, see Eq. �27�, as expected for an antiferromagnet.This correlator shows a complementary behavior to that ofthe ��2i−1

x �2ix correlator for odd bonds �25�—it interpolates

between −1 at �=0 and − 2 when �→1− �see Fig. 1�. In

contrast, the ��2i−1z �2i

z and ��2ix �2i+1

x correlators change in adiscontinuous way at the quantum critical point ��=1�,where the pseudospins become entirely disordered, and��2i−1

z �2iz = ��2i

x �2i+1x =0. Therefore, only ��2i−1

x �2ix

= ��2iz �2i+1

z =− 2 are finite and contribute to the ground-state

energy of the compass model.Using the symmetry of the XX-ZZ model �Eqs. �1� and

�2��, the ground-state correlators for �� �1,2� can be ob-tained by mapping the correlators for �� �0,1�. In this waywe obtain correlators that are well defined for any value of �except �=1, where most of the correlators are discontinuous�Fig. 2�. For example, when ��1, we have antiparallel oddpairs of pseudospins ���2i−1

z �2iz =−1�, but the same correlator

tends to 0 when �→1+ �Fig. 1�. This discontinuity of thecorrelators at the quantum phase transition is a manifestationof level crossing between ground states in different sub-spaces s� when �→1±.

B. Ground-state energy and excitations

The spectrum of the Hamiltonian �1� changes qualita-tively from a ladder spectrum with the width of 2JN at �=0 to a quasicontinuous spectrum with the width reduced to4 JN at the �=1 point �Fig. 3�. While the ground state has

FIG. 1. �Color online� Intersite pseudospin correlations on odd�2i−1,2i� and even �2i ,2i+1� bonds in the XX-ZZ model �Eqs. �1�and �2�� for increasing �. A crossover between two types of pseudo-order, with ��2i−1

z �2iz =−1 for ��1 and ��2i

x �2i+1x =−1 for ��1,

occurs at the quantum critical point �=1, where only ��2i−1x �2i

x = ��2i

z �2i+1z =− 2

are finite.

FIG. 2. Distance dependence of ��2iz �2�i+r�

z correlator betweeneven pseudospins belonging to different odd bonds. Two limits �→1± demonstrate the discontinuity at the quantum critical point��=1�. The correlators for �→1− approach the asymptotic valuefor large r in an algebraic way rather than in an exponential way,indicating divergent correlation length.

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degeneracy d=2 at �=0 �the and � lowest energy statesin the subspace with s=0� and d=1 for 0���1 �the ground state with s=0�, large degeneracy occurs at �=1.

The lowest energies in different subspaces s� are

Es�±��� = −

1

2�k

�k + Cs��� + O�1� = E0+��� + O�1� , �29�

where, for ��1,

E0+��� = −

1

2�1 − ��JN − JN

1

2 �

0

dk�1 + �2 − 2� cos k

�30�

is the energy of the ground state found in the s=0 �� sub-space in the limit of N→�. A similar formula to Eq. �30� iseasily obtained for ��1 by a substitution �→ �2−��. Forany s� the gap between the and � lowest energy states isO�1/N�, but the gaps 2�1−��s between s=0 and s�0 areO�1� and survive when N→�. However, when �→1−, thenall ground-state energies for /� and for any s� become de-generate, level crossing between all 2�2N/2 lowest energystates occurs when N→�. This ground-state degeneracy ismuch higher than the 2�2L found in the L�L 2D compassmodel,10 but the overall behavior is similar—it indicates thatfrustrated interactions, the discontinuity in the correlationfunctions, and the pseudospin liquid disordered ground stateare common features of the 1D and 2D compass models.

The ground-state energy E0+��� �30� increases with � for �

��0,1� �Fig. 3�, as pseudospin interactions are graduallymore frustrated when �→1.17 The ground state is separatedfrom the lowest energy pseudospin excitation by a pseu-dospin gap �=2�0=4J 1−�, which vanishes at �=1 �seeFig. 4�. Note that this excitation corresponds to reversing a�i

z pseudospin component for ��1 and a �ix pseudospin

component for ��1. We have verified that the linear de-crease of � when �→1 is well reproduced by the energyspectra obtained by exact diagonalization of finite systems

�as obtained, for instance, from the data of Fig. 3�.

IV. CONCLUSIONS

We have shown by an exact solution of the 1D XX-ZZmodel �Eqs. �1� and �2�� that pseudospin disordered states aretriggered already by an infinitesimal admixture of the inter-actions between other spin components than those used toconstruct classical Ising models at �=0 �Eq. �3�� or �=2�Eq. �4��. The properties of the XX-ZZ model are summa-rized in Fig. 4, with two different types of pseudospin corre-lations, dictated by the “dominating” interactions. Thesepseudospin correlations and finite excitation gap may be seenas a precursor of the antiferromagnetic order induced by thetype of interactions in the respective classical Ising model,with either �i

z or �ix pseudospins, at �=0 or �=2, respec-

tively. These opposite trends become frustrated in the 1Dcompass model, lying precisely at the quantum critical point.We anticipate that a similar quantum critical point deter-mines the properties of the orbital liquid state in a 2D com-pass model.10

In conclusion, the 1D XX-ZZ model provides a beautifulexample of a first-order quantum phase transition betweentwo different disordered phases, with hidden order of pairs ofpseudosins on every second bond. When the pseudospin in-teractions become balanced at the quantum critical point, thepseudospin �orbital liquid� disordered state takes over. It ischaracterized by �i� a high 2�2N/2 degeneracy of the groundstate and �ii� a gapless excitation spectrum.

ACKNOWLEDGMENTS

We thank Peter Horsch for insightful discussions. W.B.acknowledges the kind hospitality of Laboratoire de Phy-sique des Solides, Université Paris Sud, Orsay, where part ofthis work was done. J.D. was supported in part by the KBNGrant No. PBZ-MIN-008/P03/2003 and the Marie Curie ToKproject COCOS No. �MTKD-CT-2004-517186�. A.M.O. ac-knowledges the support by the Polish Ministry of Scienceand Education under Project No. N202 068 32.

FIG. 3. Eigenenergies En of the XX-ZZ model �Eqs. �1� and �2��as obtained for a chain of N=8 sites with periodic boundary condi-tion for increasing �. Level crossing at �=1 marks the quantumcritical point of the compass model �Eq. �5��.

FIG. 4. �Color online� Pseudospin excitation gap � in theXX-ZZ model �Eqs. �1� and �2�� for increasing � �solid lines�. Thegap collapses at the quantum critical point �QCP� �in the compassmodel obtained at �=1�, which separates the disordered phase withfinite pseudospin ��2i

z �2�i+r�z correlations �left� from the one with

finite ��2ix �2�i+r�

x correlations �right�.

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romagnetic �Ref. 6�, but the present analysis can also be gener-alized to the ferromagnetic case.

13 E. Lieb, T. Schultz, and D. Mattis, Ann. Phys. 16, 407 �1961�; S.Katsura, Phys. Rev. 127, 1508 �1962�.

14 J. Dziarmaga, Phys. Rev. Lett. 95, 245701 �2005�.15 M. V. Mostovoy, D. I. Khomskii, and J. Knoester, Phys. Rev. B

65, 064412 �2002�; M. Aichhorn, P. Horsch, W. von der Linden,and M. Cuoco, ibid. 65, 201101 �2002�.

16 E. Barouch and B. M. McCoy, Phys. Rev. A 3, 786 �1971�.17 Note that E0

+�1� defined by Eq. �30� is half the ground-state energyof the full 1D XY model, see Ref. 16.

BRZEZICKI, DZIARMAGA, AND OLEŚ PHYSICAL REVIEW B 75, 134415 �2007�

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