Spin gap in chains with hidden symmetries

Spin gap in chains with hidden symmetries

M. N. Kiselev,¹D. N. Aristov,^2,*and K. Kikoin³

1Physics Department, Arnold Sommerfeld Center for Theoretical Physics,

and Center for NanoScience, Ludwig-Maximilians-Universität at München, 80333 München, Germany

2Max-Planck-Institut für Festkörperforschung, Heisenbergstraße 1, 70569 Stuttgart, Germany

3Ben-Gurion University of the Negev, Beer-Sheva 84105, Israel 共Received 23 December 2004; published 24 March 2005兲

We investigate the formation of a spin gap in one-dimensional models characterized by groups with hidden dynamical symmetries. A family of two-parametric models of isotropic and anisotropic spin-rotator chains 共SRC’s兲 characterized by SU共2兲⫻SU共2兲 and SO共2兲⫻SO共2兲⫻Z₂⫻Z₂symmetries is introduced to describe the transition from SU共2兲to SO共4兲 antiferromagnetic Heisenberg chains. The excitation spectrum is studied with the use of the Jordan-Wigner transformation generalized for o₄algebra and by means of the bosonization approach. Hidden discrete symmetries associated with invariance under various particle-hole transformations are discussed. We show that the spin gap in SRC Hamiltonians is characterized by the scaling dimension 2 / 3, in contrast to dimension 1 in the conventional Haldane problem.

DOI: 10.1103/PhysRevB.71.092404 PACS number共s兲: 75.10.Pq, 05.50.⫹q, 73.22.Gk More than 20 years ago Haldane¹made a conjecture that

the properties of spin-S Heisenberg antiferromagnetic 共AF兲 chains are different for integer and half-integer spins;

namely, the excitations in Heisenberg AF chains with half- integer spins are gapless, whereas for integer spins there is a gap in the spectrum共Haldane gap兲. While the first part of the Haldane conjecture has been proven a long time ago 共see Refs. 2 and 3兲, the second part, although confirmed by many numerical⁴and experimental⁵studies and tested by some ap- proximate analytical calculations,^6–10 remains a hypothesis.

The problem of SU共2兲 Heisenberg chains has been attacked by modern tools such as, e.g., bosonization^6–8共see also Ref.

11兲, various numerical methods,^4,12,13 and the recently pro- posed fermionization by means of the Jordan-Wigner trans- formation for higher spins.¹⁴ However, the main focus of interest has been put either on S = 1 chains characterized by SU共2兲 symmetry or on N-leg ladders described in terms of dynamic SO共N兲groups.¹⁶There have also been made several conjectures concerning spontaneous discrete symmetry breaking in S = 1 chain models associated with, e.g., exis- tence of hidden Z₂ and Z₂⫻Z₂ symmetries.^13,17 Neverthe- less, the general question about the nature of the spin gap is still open.

In this paper we propose yet another approach to the spin gap problem. It is based on investigation of a family of two- parametric Hamiltonians described by dynamical groups.¹⁸ This family includes the conventional two-leg ladder and several models intermediate between the ladder and the chain. Here we concentrate on the most instructive example of a “barbed-wire-like” chain with spins 1 / 2 in each site coupled by the ferromagnetic exchange J_⬜within a rung and the AF interaction J_储 along the leg 共Fig. 1兲. The model Hamiltonian is

H = J_储

兺

兺

This model is a natural extension of the S = 1 chain to a case where the states on a given rung form a triplet-singlet

pair. We call the chain shown in Fig. 1 the spin-rotator chain 共SRC兲 共in contrast to the spin-rotor model^10,19兲. Unlike ear- lier attempts to construct the representation of an S = 1 state out of s = 1 / 2 ingredients,^7,8we respect in this case the SO共4兲 symmetry of the spin manifold on each rung.²⁰ As a result, the singlet state cannot be projected out. Moreover, it plays an integral part in the formation of the spin gap. We show that the hidden Z₂symmetries in this model are an intrinsic property of the local SO共4兲group of the spin rotator on the rung, and the symmetry breaking due to nonlocal 共string兲 effects results in spin gap formation. These special symme- tries distinguish our model from 共N艌2兲-leg ladder models and SU共2兲chains. In particular we show also that the scaling dimension of a spin gap in a SRC differs from that in a two-leg ladder.

New variables on a rung are introduced to keep track of S = 1 properties. We define Sជ_{i= s}ជ1,i+ sជ2,i, Rជ_{i= s}ជ1,i− sជ2,i, where Sជ_i stands for a triplet S = 1 ground state and singlet S = 0 excited state. The operator Rជ describes dynamical triplet-singlet mixing.^18,20Then

H =J_储

4

兺

兺

共2兲 where the set of operators Sជ_{i, R}ជ_ifully defines the o₄ algebra in accordance with the commutation relations

关S_i^␣,S^␤_j兴= i␦ij␧_␣␤␥S_i^␥, 关R_i^␣,R_j^␤兴= i␦ij␧_␣␤␥S_i^␥,

FIG. 1. Spin-rotator chain.

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关R_i^␣,S^␤_j兴= i␦ij␧_␣␤␥R_i^␥, 共3兲 where ⑀␣␤␥ is the totally antisymmetric Levi-Cività tensor and Casimir constraints on each sites are given by

共Sជ_i兲²+共Rជ_i兲²= 3, 共Sជ_i· Rជ_i兲= 0. 共4兲 In order to characterize low-lying excitations in the SRC we propose a fermionization procedure, which extends the Jordan-Wigner共JW兲transformation to the SO共4兲group, and a bosonization formalism based on this procedure. Our method incorporates the JW transformation for S = 1 pro- posed by Batista and Ortiz共BO兲in Ref. 14. The relationships between the SO共4兲JW representation and the BO represen- tation is discussed below in some detail.

We begin with a single-rung dimer problem. A two- component fermion共a^†b^†兲 basis representing Sជ operators is introduced as follows共S^±= S^x± iS^y兲:

S⁺= a^†+ e^i␲a†ab^†, S⁻= a + be^−i␲a†a, S^z= a^†a + b^†b − 1.

The complementary representation for Rជ generators is R⁺= a^†− e^i␲a†ab^†, R⁻= a − be^−i␲a†a, R^z= a^†a − b^†b.

This representation satisfies commutation relations 共3兲 for the SO共4兲group and preserves the Casimir operators共4兲. The advantage of the two-fermion formalism in comparison with two independent JW transformations for each s = 1 / 2 is that the latter requires an additional Majorana fermion to provide commutation of two spins on the same rung. Two-component spinless fermions may be combined into one spin fermion, which is most conveniently done by the definition

f_↑=共a − b兲/

冑

冑

K_j= exp

冉

^兺

n_␴k

冊

^兿

共1 − 2n_↑k兲共1 − 2n_↓k兲 共6兲 共n_␴= f_␴^†f_␴兲. As a result of the JW transformation the SO共4兲 generators acquire the following form:

S⁺_j=

冑

S_j⁻=共S_j⁺兲^†, S_j^z= n_↑j− n_↓j, 共7兲 R⁺_j=

冑

R⁻_j=共R⁺_j兲^†, R^z_j= f_↑^†_jf_↓^†_j+ f_↓jf_↑j. 共8兲 Part of the representation共7兲describing S = 1 coincides with the BO representation. Nevertheless, since Sជ² is no longer a conserved quantity, being defined by Sជ

j

2= 2关1 − n_↑jn_↓j兴, the projection of the SO共4兲group on the S = 1 representation of the SU共2兲group requires an additional Hubbard-like interac- tion responsible for the hidden constraint overlooked in the BO paper¹⁴ 共see also Ref. 15 where the unconstrained JW transformation is constructed for S = 3 / 2兲. When the S = 1 sector is fixed, three states共n_↑, n_↓兲, namely,共1,0兲,共0,0兲, and

共0,1兲, determine a threefold degenerate triplet state whereas the doubly occupied state共1,1兲stands for a singlet separated from the ground state by the gap⌬= J_⬜. The Hamiltonian共2兲 is fermionized by means of a purely one-dimensional 共1D兲 string operator K_j 关Eq. 共6兲兴 in contrast to the meandering strings proposed for the theory of two-leg ladders共see Ref.

21 and references therein兲.

The Hamiltonian of the anisotropic XXZ SRC model is H = H_储+兺iH_⬜,i, where

H_储= J^x_储 8

兺

i

关S_i⁺S_i+1⁻ + S_i⁺R_i+1⁻ +共S↔R兲+ H.c.兴+J_储^z 4

兺

i

关S_i^zS_i+1^z + S_i^zR_i+1^z +共S^z↔R^z兲兴, 共9兲

H_⬜,i= J_⬜^x

8 共R_i⁺R_i⁻+ R_i⁻R_i⁺兲+J_⬜^z

4 共R_i^z兲²−共Rជ_i↔Sជ_i兲. There exists a set of discrete transformations keeping the Hamiltonians共2兲 and共9兲intact and preserving the commu- tation relations 共3兲 and Casimir operators 共4兲. In general, these transformations are described by the matrix of finite rotations characterized by Euler angles␪^,␺^,␾^,␸for the case of the SU共2兲⫻SU共2兲 or SO共2兲⫻SO共2兲⫻Z₂⫻Z₂ groups.

An example of such a transformation is

S⁺→R⁺, S⁻→R⁻, S^z→S^z, R^z→R^z, 共10兲 which is a U共1兲⫻U共1兲 rotation in the “S^x-R^x” and “S^y-R^y” subspaces. This is in fact a particle-hole flavor transforma- tion f_↑→f_↓^†, f_↓→f_↑^†. On the other hand, it corresponds to the replacement b→−b, thus manifesting hidden Z₂ symmetry.

This means that an additional gauge factor exp共i␪兲 with

␪^{= ±}␲ appears in the fermion operator characterizing the

“free ends” of rungs in the SRC chain. Other examples are 共f_↑→f_↓兲 and 共f_↑→f_↑^†, f_↓→f_↓^†兲. The latter corresponds to a particle-hole transformation 共a→a^†, b→b^†兲 in the nonro- tated fermion basis.

After a JW transformation in the a-b basis 共5兲–共8兲 the Hamiltonian共9兲is written as follows:

H_储= J_储^x

兺

i

共a_i^†a_i+1+ a_i+1^† a_i兲cos共␲ⁿi b兲

+ J_储^z

兺

冉

冊冉

冊

and H_⬜=兺iH_⬜,iwith H_⬜,i= −J_⬜^x

2 共a_i^†b_i+ b_i^†a_i兲− J_⬜^z

冉

冊冉

冊

where the shorthand notations n^a= a^†a, n^b= b^†b, and cos共␲^nb兲= Re exp共±i␲^nb兲= 1 − 2n^b are used. Below we con- sider the domain J_⬜ⰆJ_储where the strongest deviations from the conventional Haldane gap regime^6,7 are anticipated. In the limit J_⬜= 0 our SRC model reduces to an s = 1 / 2 AF chain; the gauge factor cos共␲^nb兲= ± 1 is a fictitious random variable which can be eliminated by the transformation S^x→−S^xand S^y→−S^yon the corresponding site. This situa- tion is similar to the so-called Mattis disorder²² where ran-

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domness in an interaction is removed by proper redefinition of spin variables.

The kinematic factor⬃cos共␲ⁿi

b兲in H_储^x关Eq. 共11兲兴can be eliminated by a unitary transformation H˜ =U^†HU with U = exp共i␲兺l,j⬎ln_j^an_l^b兲. Then H_⬜^z and H_储^z remain unchanged and the J_⬜^x term acquires the string form

H˜

⬜,i

x = −1

2J_⬜^x

冋

冉

^兺

关a^†_ja_j+ b_j^†b_j兴

冊

册

共13兲 The s = 1 / 2 chain is represented in terms of a half-filled band of fermions. Since interactions共11兲and共12兲 do not change the occupation numbers for each color, we expect that the interacting case is also represented by two half-filled bands 共see below兲. We note that the Hamiltonian H_储 in Eq. 共9兲 possesses U共1兲⫻U共1兲 symmetry whereas only one local U共1兲associated with b fermions exists in Eq.共11兲due to the nonlocal character of the JW transformation.

Let us consider the XY_储-XY_⬜model共J_储^z= J_⬜^z = 0兲. We split the first term in Eq.共11兲into the bare hopping and the kine- matic term⬃J_储^xn_i^b共a_i^†a_i+1+ H.c.兲playing the part of the effec- tive interaction H_int^XY. One gets after diagonalization of the hopping term

H₀=_p,

兺

␭=±

␧_␭共p兲c_␭^†_,pc_␭,p 共14兲

with c₊= u₊a + u₋b, c₋= u₊b − u₋a,

u_±²共p兲= ±␧±共p兲/关␧+共p兲−␧−共p兲兴, 共15兲

␧±共p兲= J_储^xcos p ±关共J_储^xcos p兲²+共J_⬜^x兲²兴^1/2. 共16兲 The chemical potential␩= 0 is pinned in the gap. Thus, the mixing term fixes the global phase difference for a-b fields.

The remaining symmetry is local Z₂⫻Z₂.

We represent H_int^XYin terms of new variables c_±by expand- ing the Hamiltonian共11兲in the vicinity of two Fermi points of the nonhybridized system:

H_int^XY=1 2_兵␮,

兺

␯,␣其=±1,q

g_␮␮^␯␯⬘_⬘共q兲␳␮␮⬘^,␣共q兲⌳_␯␯⬘,␣⬘共− q兲 共17兲 where the operator␳_␮␮⬘is given by

␳␮␮⬘^,␣共q兲=

兺

k

c_␣^†_,␮,k−q/2c_{␣,␮⬘,k+q/2}. 共18兲 Its diagonal part is the quasiparticle density. The operator

⌳_␯␯⬘is defined as

⌳_{␯␯⬘,␣}共q兲= −␣

兺

k

k c_␣^†_,␯,k−q/2c_{␣,␯⬘,k+q/2}, 共19兲 while its diagonal part is⌳_␯␯= div j_␯␯= −⳵t␳_␯␯^.

In expressions共18兲and共19兲the index␣= ± stands for the

“old” Fermi surface points k_F^±= ±␲^{/ 2} 共we take unit lattice spacing兲, and k is measured from k_F. The indices

␮^,␮

⬘

⬘

冑

g_␮␮

␯␯⬘⬘= J_储^x共␦␮␮⬘+␴_␮␮^x _⬘兲共␦_␯␯⬘⁻␴_␯␯^x_⬘兲. 共20兲 We analyze Eq. 共17兲in terms of the g-ology approach²³ classifying various terms in g_␮␮^␯␯⬘_⬘共q兲 as forward and back- ward scattering and umklapp processes. We see, first, that if 兩q兩Ⰶ␲^{/ 2 and g}⬃± J_储^x, both diagonal and off-diagonal matrix elements of ⌳_␯␯⬘ vanish in accordance with Adler’s principle.²⁴Thus, the forward scattering processes leading to small renormalization of the coupling ⬃共J_⬜^x兲²/ J_储^x are irrel- evant. The backward scattering processes 共±␲^{/ 2}→⫿␲^{/ 2}兲 result in a reduction J_储→J_储^1−␥J_⬜^␥ of the effective coupling 共0⬍␥⬍1 is a constant兲. To get this estimate we cut off the logarithmic corrections to the coupling constant at

⌬min⬃共J_⬜^x兲²/ J_储where⌬mindetermines the gap in the density of spin-fermion states␧±. However, there is yet another en- ergy scale ⌬⬃J_⬜^x associated with the gap in a two-point particle-hole correlator with zero total momentum of the pair.

This energy scale is attributed to the gap separating the S = 0 excited state on a rung from the triplet state. The cross- over between the two energy scales will be discussed else- where. The Hamiltonian 共17兲 allows also “interband” um- klapp processes determined by the off-diagonal elements of

␳␮␮⬘ and ⌳_␯␯⬘ and responsible for the periodicity Q = 2␲^. These processes, associated with the transfer of a pair of quasiparticles over the gap, do not change the leading term in Eq.共16兲.

The above arguments are complemented by bosonization calculations for the strongly asymmetric two-leg ladder with finite Fermi velocity u_bin the b subsystem, which may be set to zero at the end of the scaling procedure. The continuum representation for spin operators sជ1, sជ2 in Eq. 共1兲 reads^11,25 关i = a共1兲, b共2兲兴

s_i^±共x兲 ⬃e^±i␪i关cos共␲^x兲+ cos共2␾i兲兴,

s_i^z共x兲 ⬃␲⁻¹⳵x␾i+ cos共␲^{x + 2}␾i兲 共21兲 with canonically conjugated variables␾iand⌸i=⳵x␪i. Keep- ing only the most relevant terms in the rung interaction J_⬜^␣, we arrive at the conventional equations of Abelian bosoniza- tion for the spin Hamiltonian共2兲,

FIG. 2. Dispersion law for hybridized spin fermions c_±.

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H =

兺

i=a,b

冕

冉

冊

with K = 1 / 2 and J_⬜Ⰶu_bⰆu_a=␲^J储/ 2 for J_储= J_储^x= J_储^z.

To find the scaling dimension of the gap we start with the case J_⬜^x= 0, J_⬜^z = J_⬜⫽0. Using the scaling procedure 共x→⌳x , t→⌳t兲, one has J˜

⬜→J_⬜⌳^2−␤ where ␤/ 2 = K / 2 is the scaling dimension of cos 2␾i. The renormalization of the b component stops when the renormalized J_⬜becomes comparable with the lower scale of the energy u_b. The corresponding scale ⌳=␰b defines the first correlation length ␰b=共u_b/ J_⬜兲^{1/共2−␤兲} and the first energy gap

⌬b= u_b␰b

−1= u_b共J_⬜/ u_b兲^{1/共2−␤兲}. At the second stage of the renor- malization, with frozen具cos 2␾b典⬃␰b

−␤/2

, the factor cos 2␾a

undergoes further enhancement. The procedure halts when the renormalized amplitude J˜

⬜ is comparable with u_a at J_⬜⌳^2−␤/2␰b

−␤/2⬃u_a, which defines a second correlation length

␰a=共␰b␤/2

u_a/ J_⬜兲^{1/共2−␤/2兲} and a second gap ⌬a= u_a␰a

−1. In our particular case␤= 1 these formulas simplify as follows:

␰b=共u_b/J_⬜兲, ⌬b= J_⬜,

␰a=␰b共u_a/u_b兲^2/3, ⌬a= J_⬜共u_a/u_b兲^1/3. 共23兲 One may decrease u_b in the regime of frozen ␾b down to u_b⬃J_⬜. Then⌬a⬃J_储共J_⬜/ J_储兲^2/3. Further decrease of u_bdoes not change the exponent 2 / 3 of the spin gap fully determined by the scattering on the random potential cos 2␾a.²⁶The two-

stage renormalization procedure is essential for understand- ing the SRC model. In the limit u_a⬃u_b, Eq. 共23兲 leads to standard scaling of the spin gap⌬⬃J_⬜共see, e.g., Ref. 27兲.

In the case J_⬜^x ⫽= 0, J_⬜^z = 0 the scaling behavior of the spin gap⌬⬃J_储共J_⬜^x / J_储兲^2/3is determined by the backward scat- tering processes of the field a on the random potential asso- ciated with fluctuations of cos␪a.

The fully isotropic case, J_⬜^x= J_⬜^z = J_⬜, might be expected to yield the same estimate⌬⬃J_储^1/3共J_⬜兲^2/3. A refined analysis 共see, e.g., Ref. 28兲 including the less relevant terms in Eq.

共17兲 may correct the gap values, but does not change this estimate.

To summarize, we introduced a 1D model intermediate between the spin S = 1 chain and the two-leg ladder. Our SRC possesses special hidden Z₂symmetries connected with dis- crete transformations in a 6D space of the SO共4兲group char- acterizing the spin rotator. The SRC chain is mapped on the two-component unconstrained interacting fermions by means of an o₄JW transformation. The two fermion fields are char- acterized by sharply different dynamics demanding a two- stage renormalization procedure. One of the two fields is frozen at k→±␲/ 2 and the scaling dimension␤of the rung operator exchange J_⬜is␤= 1 / 2 instead of␤^{= 1,27,29as in the} conventional Haldane problem. As a result, the formation of massive excitations in the isotropic SRC model is character- ized by a “two-thirds” scaling law.

We are grateful to N. Andrei, A. Finkel’stein, and A. Ts- velik for useful discussions. This work is supported by SFB- 410, Grant No. BSF-1999354 from the U.S.-Israel Binational Science Foundation, the A. Einstein Minerva Center, and the Transnational Access Program No. RITA-CT-2003-506095 at Weizmann Institute of Sciences.

*On leave from Petersburg Nuclear Physics Institute, Gatchina 188300, Russia.

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