Magnetic field induced phase transitions in spin ladders with ferromagnetic legs

0.21 MB·English

by  T. Vekua

Magnetic field induced phase transitions in spin ladders with ferromagnetic legs
T. Vekua,1 G.I. Japaridze,2 and H.-J. Mikeska1
1Institut fu¨r Theoretische Physik, Universit¨at Hannover, 30167 Hannover, Germany
2Andronikashvili Institute of Physics, Tamarashvili 6, 0177, Tbilisi, Georgia
We study the phase diagram of spin ladders with ferromagnetic legs under the influence of a
symmetry breaking magnetic field in the weak coupling effective field theory by bosonization. For
4
antiferromagnetic interleg coupling we identify two phase transitions introduced by the external
0
magnetic field. In order to establish theuniversality of the phases we supplement the bosonization
0
approach by results from a strong coupling (rung) expansion and from spin wave analysis.
2

n PACSnumbers: 75.10.JmQuantizedspinmodels
a
J

6 I. INTRODUCTION erties of the two-leg ladder systems is known from the
1 investigation by Schulz17 for the case of the correspond-
ing spin S = 1 Heisenberg chain model: In the case of
Recently there has been considerable interest in the
]
the gapless XY1 phase an external magnetic field leads
l studyofmagneticfield-inducedeffectsinlow-dimensional
e to the appearance of a magnetization in z direction for
- quantum spin systems, in particular devoted to the
r critical properties of spin S = 1/2 isotropic antiferro- arbitrarysmallmagneticfields. Inthecaseofthegapped
st magnetic two-leg ladders in an external magnetic field. rung-singlet phase the magnetization appears only at a
. finite critical value of the magnetic field which is equal
t In parallel, there was remarkable progress in recent
a to the spin gap8,17. This behavior is generic for the spin
m years in the fabrication of such ladder compounds1.
gappedU(1)symmetricsystemsinamagneticfield: The
Since antiferromagnetic two-leg ladder systems with
- magnetic field leaves the in-plane rotational invariance
d S = 1/2 have a gap in the spin excitation spectrum,
unchanged21 and the transition belongs to the univer-
n they reveal an extremely rich behavior, dominated by
salityclassofthecommensurate-incommensurate(C-IC)
o quantum effects, in the presence of a magnetic field.
c These quantum phase transitions were intensively in- transitions22,23.
[ On the other side, for the U(1) symmetric phase such
vestigated both theoretically, using different analytical
1 and numerical techniques8,9,10,11,12,13,14,15,16, and also astheXY1phase,theeffectofauniformtransversefield
v experimentally2,3,4,5,6,7. is highly nontrivial. In the case of classical anisotropic
1 spin chains this effect has been studied more than two
Laddermodelswithferromagneticlegshavebeenmuch
0 decades ago24. However, in the case of the antiferro-
less studied, although they exhibit many interesting
3 magnetic XXZ quantum chain this problem is still the
1 aspects17,18,19. It is true that up to now no materials subject of intensive recent studies25,26,27,28.
0 are available which realize these models. However, from
In this paper, we study the effect of a uniform trans-
4 thetheoreticalpointofviewthesesystemsareextremely
verse magnetic field on the ground state phase diagram
0 interesting, since they open up a new large class of sys-
/ of a two-leg ladder with anisotropic, ferromagnetically
at temsforthestudyofcomplicatedquantumbehavior,un- interactinglegscoupledbyantiferromagneticinterlegex-
expected in more conventional spin systems. The vari-
m change.
ety of possibilities is seen already from Fig. 1: here the
The outline of the paper is as follows: In section II
- groundstate phase diagram19 of a two-legferromagnetic
d we review the model and its bosonized version in the
ladder is presented in the variables intraleg exchange
n continuum limit. We discuss the phases and phase tran-
o anisotropy (∆) and (isotropic) interleg coupling (J⊥). sitions emerging from the XY1 and rung-singlet phases
c The ground state phase diagram contains, besides the
inatransversemagneticfieldinsectionsIII (intheweak
: fullygappedrung-singletandHaldanephases(commonly
v coupling approach) and IV (in the limit of strong rung
i known from the case of antiferromagnetic ladders17,20),
X exchange). We shortly summarize our results in section
the spin-liquid phase with easy-plane anisotropy(XY1),
V. In appendix A we present the spin-wave approach to
r the ferromagnetic and the stripe-ferromagnetic phases
a locatetheferromagnetictransitionline startingfromthe
which are realized only in the case of ferromagnetic legs
saturated phase.
(0 ∆ 1).
≤ ≤
In this paper we study the effect of an external mag-
netic field on the phase diagram of this system. In par- II. MODEL
ticular, we focus our attention on the study of new field
induced effects in the case of the easy-plane XY1 phase
Here we present a brief introduction to the model and
andinthe rung-singletphase inthe vicinity ofthe single
in particular to its bosonized version in the continuum
chain ferromagnetic instability point ∆=1.
limit. The Hamiltonian we consider is given by:
Theeffectoftheuniformmagneticfield,appliedparal-
lel to the anisotropy (z) axis, on the ground state prop- H =H(1)+H(2)+H , (1)
leg leg ⊥
2
∆
Now we introduce the symmetric and antisymmetric
"Stripe"
FERROMAGNET FM combinationsofthe bosonic fields φ± = 1/2(φ1±φ2),
1 RUNG oθ±bta=in fin1/a2lly(θa1s±effθ2e)ctaivnedbaofstoenrirceHscaamlinilgtpotnhieasne fields we
p
Spin−Liquid XY1 SINGLETS = ++ −+ ±, (8)
H H H H
where
Anisotropic Haldane RUNG J + = u+[(∂xθ+)2+(∂xφ+)2]
SINGLETS H 2
Phase
+ M cos 8πK φ (x), (9)
+ + +
u
FIG. 1: Schematic picture of the ground state phase dia- − = −[(∂xθp−)2+(∂xφ−(x))2]
gram of thetwo-leg ladder in thevariables intraleg exchange H 2
anisotropy (∆) and isotropic interleg coupling (J⊥). The 2π
limiting case of isotropic ferromagnetic legs corresponds to + cos θ (x), (10)
⊥ −
∆=1. J sK−
π π
± = hcos θ (x)cos θ (x) (11)
− +
where the Hamiltonian for leg α is H − 2K− 2K+
r r
N Here
H(α) = J Sx Sx +Sy Sy
leg − α,j α,j+1 α,j α,j+1 J 2K 1
Xj=1(cid:16) K± K 1 ⊥ − , (12)
N ≃ ∓ 2πJ sin(π/2K)
(cid:18) (cid:19)
+ ∆Sz Sz hext Sx , (2)
α,j α,j+1 − α,j J , h hext and u are the velocities of the
(cid:17) Xj=1 Jsy⊥mm∼etr⊥ic and∼antisymmetri±c modes.
and the interleg coupling is given by In deriving (8), several terms which are strongly ir-
N relevant in our case of a ladder with ferromagnetic legs
H =J S~ S~ . (3) andappliedtransversemagneticfield, wereomitted. For
⊥ ⊥ j,1 j,2
details of the full Hamiltonian we refer the reader to19.
j=1
X We note that at hext = 0 the effective theory
Here Sx,y,z are spin S =1/2 operatorson the j-th rung,
α,j of the original ladder model is given by two decou-
and the index α = 1,2 denotes the ladder legs. The
pled quantum sine-Gordon models which describe, re-
intraleg coupling constant is ferromagnetic, J > 0, and
spectively, the symmetric and antisymmetric degrees
thereforethelimiting caseofisotropic ferromagneticlegs
of freedom. The infrared properties of the antisym-
corresponds to ∆ = 1. We will restrict ourselves to the
metric field are governed by the strongly relevant op-
case 0 ∆ 1.
≤ ≤ erator ⊥cos 2π/K−θ− with the scaling dimension
We use the following bosonizationexpressionsfor spin (2K )−J1 1/2. Therefore, the antisymmetric sector
−
operators,adaptedtothecaseofferromagneticexchange ≤ p
is gapped at arbitrary J = 0. Fluctuations of the field
(for details see19): ⊥ 6
θ (x) are completely suppressed and the field θ gets
− −
c π ordered with expectation values
Sx :cos θ :
j,α ≃ √2π K α
r K π/2 at >0
+ ( 1)j ib :sin√4πKφαsin π θα :, (4) hθ−i=(cid:26)0p − atJJ⊥⊥ <0 . (13)
− √2π K
r
Theinfraredpropertiesofthe symmetricfieldaregov-
c π
Sjy,α ≃ √2π :sinrKθα : eshrnoewdnbiny19th,ethmeasrygminmaletorpicermaotodreMre+mcaoinss g8aπpKles+sφf+or. feArs-
p
ib π romagnetic interleg exchange and arbitrary 0 ∆ 1,
( 1)j :sin√4πKφ cos θ :, (5) ≤ ≤
− − √2π α K α while in the case of antiferromagnetic interleg exchange
r
it is gapless in a finite regime in ∆,J parameter space
⊥
K
Sjz,α = ra π ∂xφα + Hgievreenγa1papnrodxiγm2aatreelypboysitγiv1eJ⊥co/nJst≤ant∆s (≤mor1e−rigγo2Jro⊥u/sJly.
( 1)j :sin√4πKφ (x): . (6) smoothfunctionsoftheanisotropy)oftheorderofunity.
α
− π Following Schulz17 who has discussed a similar phase in
Here, φ(x) and θ(x) are dual bosonic fields, ∂ φ = u∂ θ the context of the spin S = 1 chain we denoted this
t x
and K is the Luttinger liquid parameter phase as spin liquid XY1 phase for the spin ladder. At
π J⊥ > 0, outside of this regime, the symmetric mode is
K = . (7) also gapped and the field φ gets ordered and pinned in
2arccos∆ +
3
one ofits possible minima. We denoted the fully gapped (cid:0)(cid:1) h
(cid:0)(cid:1)
phase, realized in the case of antiferromagnetic interleg (cid:0)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1) (cid:0)(cid:1)(cid:0)(cid:1)π(cid:0)(cid:1) (cid:0)(cid:1)(cid:0)(cid:1)2(cid:0)(cid:1)π (cid:0)(cid:1)(cid:0)(cid:1)3(cid:0)(cid:1)π (cid:0)(cid:1)(cid:0)(cid:1)4(cid:0)(cid:1)π
exchange as rung-singlet phase. In this phase spins on h(cid:0)(cid:0)(cid:1)(cid:1)c(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)
(cid:0)(cid:1) (cid:0)(cid:1)(cid:0)(cid:1)
the same rung tend to form a singlet and the ground (cid:0)(cid:1) (cid:0)(cid:1)(cid:0)(cid:1)
(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1) (cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1) (cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1) (cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1) (cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)
statecorrespondstothe statewithasingletpaironeach (cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1) (cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1) (cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1) (cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1) (cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)
(cid:0)(cid:1) (cid:0)(cid:1)(cid:0)(cid:1) (cid:0)(cid:1)(cid:0)(cid:1) (cid:0)(cid:1)(cid:0)(cid:1) (cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1) (cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1) (cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1) (cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1) (cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)
rung. The ground state phase diagram of the system at (cid:0)(cid:1) (cid:0)(cid:1)(cid:0)(cid:1) (cid:0)(cid:1)(cid:0)(cid:1) (cid:0)(cid:1)(cid:0)(cid:1) (cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1) (cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1) (cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1) (cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1) (cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)
hexBte=low0 wisegsivtuendyscthheemgarotiucnadllystianteFpigh.a1s.ediagramofthe (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1) (cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1) (cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1) (cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1) (cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)
tpwreos-elengcelaodfdaermwaigtnheatincisfioetlrdo.pic ferromagnetic legs in the (cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1) (cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:0)(cid:1)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:0)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1) (cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1) (cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:1)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1) (cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:1)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)
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III. THE SPIN LADDER IN THE PRESENCE
OF AN IN-PLANE MAGNETIC FIELD
FIG.2: Evolutionofthesetofminimaoftheantisymmetric
field with increase of the applied transverse magnetic field.
In this Section we consider the effect of a transverse
magnetic field on the ground state phase diagram of the
spin-liquid XY1 phase. Since the magnetic field breaks At h = 0 only the antisymmetric field is gapped and
the in-plane rotational symmetry of the XY1 phase, it its set of availablevacua is givenby Θ =π/2,(mod π).
−
is clear that it will eliminate the gaplessXY1 phase and At h = 0 the symmetric field gets pinned and the set
6
thatthe systemwillacquireagapintheexcitationspec- of possible vacua of the symmetric field does not change
trumforarbitrarysmallhext. Bothfields(symmetricand withfield. Ontheotherhand,atarbitrary0<h<4 ,
⊥
J
antisymmetric)willbepinnedintheirrespectiveminima. eachminimumintheantisymmetricsectorsplitsintotwo
In this case the very first direct approach to study the degenerate minima (see Fig. 2). At h h , ϑ π/2
c 0
→ →
ground state phase diagram of the model is to use the the split minima join each other and form a new set of
quasiclassical Ginzburg-Landau type analysis. This ap- possible vacua for the antisymmetric field which is fixed
proach does not cover the limiting case ∆ = 1 where for arbitrary h > h . At the critical point the effective
c
bosonization does not apply. The ∆ = 1 case will be potential transforms into the (Θ )4 potential, which is
−
discussed in section IV. common in describing the Ising universality class. Since
the location of the minima in the symmetric sector does
notchange,weconcludethatthetransitioninvolvesonly
A. Quasiclassical Ginzburg-Landau analysis the antisymmetric sector.
An analogous quasiclassical analysis was carried out
In the presence of an in-plane magnetic field the effec- by Fabrizio et al.29 in connection with the ionic Hub-
tive Hamiltonian (8) reduces to two free Gaussian fields bard model. There two transitions were identified for
coupled by the following effective potential finite values of alternating on-site energies. We note as
an important difference that assuming a vanishing order
eff(Θ−,Θ+)= ⊥cos2Θ− hcosΘ−cosΘ+, (14) parameter along the direction of an infinitesimally small
U J −
appliedmagnetic field wouldbe self-contradictoryin our
where Θ± = π/2K±θ±. To find the vacuum expecta- case,becauseherenoappropriatemarginaloperatorsare
tionvaluesofthepinnedfieldwesearchfortheminimaof present.
p
the effective potential eff with respect to Θ− and Θ+. Note that in the case of ferromagnetic interleg ex-
U
The straightforward analysis gives the following sets of change(J <0),thesetofminimaoftheeffectivepoten-
⊥
vacua: tialforarbitraryhext =0isgivenbytheset(15). There-
6
fore, no transitions with increasing field are expected in
h>h =4
• c J⊥ this case.
I. Θ = π, Θ =π, (mod 2π)
+ −
II. Θ = 0, Θ =0, (mod 2π) (15) B. Phase diagram
+ −
h<h =4
c ⊥ The quasi-classicalanalysisperformed aboveallowsto
• J
sketch qualitatively the ground state phase diagram of
I. Θ = π Θ =π ϑ , (mod 2π) the model under consideration.
+ − 0
±
As soon as we switch on an infinitesimally small mag-
II. Θ = 0, Θ = ϑ , (mod 2π) (16)
+ − 0
± netic field in X direction, breaking the in-plane rotation
where symmetry, the XY1 phase is expelled, the system be-
comesgappedandacquiresafinite orderparameter. Us-
ϑ =arccos(h/4 ). ing the bosonized expressions for the smooth parts of
0 ⊥
J
4
Y C. Double sine-Gordon analysis
0 < h < hc
(a)
h According to the quasiclassical analysis the magnetic
fieldinducedtransitionintheXY1phasefromthepartly
polarized stripe-ferromagnetic phase into the ferromag-
X Z netic phase takes place along the critical line h = 4
c ⊥
J
andbelongstotheIsinguniversalityclass. Inthissubsec-
tion we investigate the stability against quantum fluctu-
ationsofthe results obtainedsofar. For this purpose we
Y
h > hc decoupletheinitialtheoryofinteractingquantummulti-
(b)
frequency sine-Gordon fields (8) to get
h
= ++ −, (20)
X Z HMF H H
u
+ = +[(∂ θ )2+(∂ φ )2]
x + x +
H 2
h cosβ θ (x), (21)
FIG.3: Crossover from stripe-ferromagnetic (a)toferromag- + + +
−
u
netic (b) states. − = −[(∂ θ )2+(∂ φ (x))2]
x − x −
H 2
h cosβ θ (x)+ cos2β θ (x), (22)
− − − ⊥ − −
the spinoperatorsEqs.(4)-(5),expressedintermsofthe − J
symmetric and antisymmetric fields and the vacuum ex- where
pectationvaluesofthecorrespondingfields(15)-(16)one
easilyobtainsthatat0<h<h thespinladderacquires β± = π/2K± (23)
c
uniform magnetization in the direction of applied and p
•
field M
x
h =h cos π/2K θ (24)
± + ∓
h i
M = Sx (x) = Sx (x)
x j,1 j,2 Thusthesymmetricsectoprisdescribedbytheordinary
≃ (cid:10)hcosΘ+i(cid:11)hcos(cid:10)Θ−i=(cid:11)h/4J⊥ (17) sine-Gordon theory with β+2 ≤ π/2 for 0 ≤ ∆ ≤ 1, and
therefore with a strongly relevant massive term. On the
opposite magnetization of legs in the in-plane di-
• other hand, in the antisymmetric sector we arrive at the
rection perpendicular to the field:
double frequency sine-Gordonmodel with β2 π/2 and
− ≤
M = Sy (x) = Sy (x) therefore strongly relevant basic and double-field opera-
y j,1 − j,2 tors.
(cid:10)cosΘ (cid:11) sinΘ(cid:10) = (cid:11)1 (h/4 )2. (18) In the double frequency sine-Gordon model the quan-
+ − ⊥
≃ h ih i − J tumphasetransitioninthegroundstatetakesplacewhen
q
This phase we denote as the ”stripe-ferromagnetic” the vacuum configurations of the two cosines compete
phase. with each other30, corresponding to the crossover from
Whenthemagneticfieldexceedsthe criticalvalueh> doublewelltosinglewellpotentialinquasiclassicalanal-
h , the new set of vacua (15) is reached and the system ysis. One easily verifies that this criterion can be ap-
c
passes into the ferromagnetic phase, where plied analogously to the case of antiferromagnetic rung
exchange (J > 0). The dimensional arguments based
⊥
on equating physical masses produced by the two cosine
M cosΘ =1and M sinΘ =0. (19)
x ≃h −i y ≃h −i termsseparatelyisusuallyusedtodefinethecriticalline:
Thus, the quasiclassical analysis of the ground state
m = (h)1/(2−dim[h])
phase diagram of the XY1 phase in the presence of a h (25)
transverse magnetic field shows a phase transition from (cid:26) mJ⊥ = J⊥1/(2−dim[J⊥])
the stripe-ferromagnetic to the ferromagnetic phase (we
Equating these two masses we obtain the following ex-
denoteby’ferromagneticphase’the phasewiththemag-
pression for the critical line external magnetic field vs.
netizationparalleltotheexternalfieldasonlynonvanish-
interchain coupling:
ing order parameter). This analysis also indicates, that
the transition belongs to the Ising universality class.
h =ηJµ(∆). (26)
Fromtheaboveitisalsoclearthatinthecaseofferro- cr ⊥
magnetic interleg exchange ( < 0) the ferromagnetic
J⊥ where
phase is realized for arbitrary h=0: This is so, since in
6
this case the vacua of two terms in (14) do not exclude 2 dim[h]
µ= − =(8K2 K)/2K (4K 1) (27)
each other. 2 dim[J ] − + −−
⊥
−
5
h /J (∆=1)
V(ϑ )
F e r r o m a g n e t i c
Luttinger
O r d e r Liquid

Rung Singlets
ϑ
J / J
0
FIG. 5: Phase diagram of a spin ladder with the SU(2) sym-
FIG.4: Appearanceofanadditionalminimum(dashedline)
metric ferromagnetic legs in a uniform magnetic field.
which could changethesecond orderphasetransition tofirst
order, depending on the sign of the higher order harmonic
generated.
laddercanbe mapped ontoa singleS =1/2chain3,10,33.
Forcompletenesswe brieflydiscussthe mapping here: A
given rung may be in the singlet or in the triplet state
and η is some numerical constant of the order of unity.
with energies given by
The self consistency of the mean field separation insures
that(27)followsfromequatingmassesproducedbymag-
J J 3J
neticfieldandinterchaincouplingsseparately,beforethe Et,± = ⊥ hext, Et,0 = ⊥, Es = ⊥.
4 ± 4 − 4
mean field separation. From (27) we see h J in
cr ⊥
≃
the limit K , i.e. for the single chain ferromagnetic At hext J , one component of the triplet becomes
instability po→in∞t. This will be seen to be consistent with closer to≤the⊥singlet ground state such that for a suffi-
thelargerungcouplingaswellaswithspinwaveanalysis ciently strong magnetic field we have a situation where
(see respectively eqs. (34) and (A3)). the singlet and the S = +1 component of the triplet
z
Finally we want to mention that the operator prod- form a new effective spin 1/2 system. One can easily
uct expansion of the two lowest frequencies of the sine- project the original ladder Hamiltonian (1) on the new
Gordon theory for the range of anisotropy parameter singlet-triplet subspace
0 ∆ 1 does not close, i.e. higher relevant harmonics
ar≤e gen≤erated in the RG procedure. In some situations t+ = (28)
|⇑i ≡ | i |↑↑i
it is knownthat higher harmonicscandestabilize second 1
order phase transitions and make it weakly first order31. s = [ ]
|⇓i ≡ | i √2 |↑↓i−|↓↑i
This happens when generated harmonics introduce new
minima (e.g. the dashed minimum in Fig. (4)) which This leads to the definition of the effective spin 1/2 op-
at the phase transition point coexists with the minima erators
before the transition.
1
For example in the case of only the third harmonic S+ = ( 1)α τ+ (29)
retained, the position of the new minimum depends on n,α=1,2 − √2 n
the sign of the generated harmonics. We have followed 1
Sz = [I+2τz] (30)
thesignofthegeneratedharmonicalongtheoneloopRG n,α=1,2 4 n
flows and checked that in our case there is no indication
When expressed in terms of the effective spin operators
ofanupsettingofthesecondorderphasetransition. The
(29)-(30), the original Hamiltonian (1) becomes
newminimagetpinnedinbetweentheminimaofthefirst
two harmonics; they thus may shift the critical line (26)
1
to larger values of the magnetic field but otherwise do Heff = −J [2τizτiz+1+τiyτiy+1+∆τixτix+1]
not introduce any qualitatively new effects. i
X
h τz +Constant, (31)
− eff i
i
X
IV. LARGE RUNG COUPLING RESULTS
where the effective magnetic field
Inthissectionweconsidertheeffectofanappliedmag- 1
netic fieldonthe groundstatephasediagramofthe two- heff =hext J⊥ J. (32)
− − 2
leg ladder system with ferromagnetic legs (1) in the lim-
iting case of strong rung coupling J J. In this limit Note that in writing (31) for convenience we have ex-
⊥
≫
it is convenient to discuss the model by representing the changed X and Z axis in effective spin space.
site-spinalgebraintermsofon-bond-spinoperators32. In The Hamiltonian (31) describes a spin 1/2 fully
particular, in the case of isotropic interleg exchange the anisotropic XYZ chain in an effective magnetic field.
6
There exists a special case, ∆ = 1, which allows for rig- h / J (0< ∆< 1)
orousanalysis. Inthis casethe effective problemreduces
to the theory of the XXZ chain with a fixed ferromag-
netic XY anisotropyof1/2inaneffectivemagneticfield
Ferro magnetic
h . The gapped phase at h < hc1 for the ladder
eff eff eff Order Rung
corresponds to the fully saturated magnetization phase

for the effective spin chain pointing in the direction op- Singlets
Stripe
posite to the applied field, whereas the massless phase
FM
for the ladder corresponds to the finite magnetization J / J
phase of the effective spin-1/2 chain3. The critical field 0
hc2 where the ladder is fully magnetized corresponds
eff
to the fully magnetized phase of the effective spin chain XY1 Spin−Liquid Phase
pointingalongthedirectionofappliedfield. Fromtheex-
act groundstate phase diagramof the anisotropicXXZ FIG.6: Phasediagramofaladderwithanisotropicferromag-
chain in a magnetic field34 using (32) we get that the netic legs in transverse magnetic field.
isotropic ferromagnetic ladder in a magnetic field shows
two second order phase transitions: at
we obtain the result that the phases appearing in large
hext =J J (33)
c1 ⊥− rung coupling and in weak rung coupling are identical.
On general grounds one expects an Ising transition to
a transition occurs from the rung dimer to a Luttinger
take place at critical strengths of the magnetic field also
liquid phase and at
for large couplings. Thus for ∆ = 1 the Luttinger liq-
6
hext =J (34) uid phase will be replaced by the striped ferromagnetic
c2 ⊥
phase (as in the weak coupling limit), and the transi-
tions become of Ising type due to the reduced symmetry
a transition from a Luttinger liquid phase into the fully
(Fig. 6).
polarized phase.
The transition from the rung-singlet phase into the Inordertodeterminethenatureofthetransitionfrom
Luttingerliquidphaseinthecaseoftheisotropic antifer- rung singlets to the striped ferromagnetic state in the
romagnetic ladder was studied in detail in several recent weak coupling limit we can use the fact that in the rung
publications10,11,12,33. It was shown that in the case of singlet phase the operator
thegappedrung-singletphasethemagnetizationappears
onlyatafinitecriticalvalueofthemagneticfieldequalto J cos 8πK φ (x)
⊥ + +
the spin gap. Since this behavior is generic for isotropic
systems with spin gap21, and the gap in the ladder sys- is relevant and the grounpdstate consists of nonmagnetic
tem is governedby J⊥ we conclude that the rung-singlet singlets situated along the rungs of the ladder. On the
toLuttingerliquidphasetransitionlinesmoothlyreaches other hand, while the in-plane magnetic field couples to
zero at J⊥ 0 (see Fig.5). the dual fields (disorder operators) we expect an Ising
→
Notethatthislargerungcouplinganalysisrevealsthat phasetransitiontotakeplacewiththeappearanceofthe
the phase transitions in the antiferromagnetically cou- magnetization perpendicular to the applied field (stripe-
pled ladder with ferromagnetic legs in uniform magnetic FM) as dominant order parameter.
field are connected with those in a ladder with antiferro-
magnetic legs but in staggered magnetic field16. In both
casesthemagneticfieldtriestopromotetripletsonrungs,
V. CONCLUSIONS
while the antiferromagnetically coupled ladder supports
on-rung singlets.
Away from the isotropic point ∆ = 1 the effective WehaveinvestigatedthephasediagramoftheS =1/2
Hamiltonian (31) describes the fully anisotropic ferro- ladder with ferromagnetic legs under the influence of a
magnetic XYZ chainin a magnetic fieldthat is directed uniform magnetic field breaking the in-plane rotational
perpendicular to the easy axes. For the particular value symmetry. Incaseofantiferromagneticcouplingbetween
of magnetic field heff = 0, the effective XYZ chain is legs we identified two phase transitions in the plane of
long range orderedin Y direction,correspondingto the magnetic field vs interchaincoupling. We have extended
−
originalladdersystembeingorderedinthedirectionper- our analysis to the strong rung coupling limit and have
pendicular to the applied magnetic field with opposite identified a Luttinger liquid phase which replaces the
magnetization on the legs (stripe-ferromagnetic phase). stripe-ferromagneticphaseinthecaseofSU(2)symmet-
For larger values of the effective field it is clear that this ric legs. The phase transition line in the case of SU(2)
stripedferromagneticorderwillbereplacedeitherbythe ferromagnetic legs was confirmed also by the spin wave
rung singlet phase or the phase with only one order pa- calculation starting from the saturated limit.
rameter - magnetization along the applied field. Thus
7
VI. ACKNOWLEDGEMENTS without magnetic field. In the case of SU(2) symmetric
legs it is straightforwardto add the magnetic field term,
TVisgratefultoA.A.Nersesyanforinterestingdiscus- sinceitcouplestothe diagonaloperator. Inthiscasethe
sions. GIJ acknowledges support by the SCOPES grant two sets of spin wave excitation frequencies19 read:
N7GEPJ62379. This workwassupportedinpartbythe
J J
DFG-Graduiertenkolleg No. 282, ”Quantum Field The- ω−(q) = hext ⊥ Jcosq ⊥ (A1)
ory Methods in Particle Physics, Gravitational Physics, − 2 − − 2
J J
and Statistical Physics”. ω+(q) = hext ⊥ Jcosq+ ⊥ . (A2)
− 2 − 2
For J >0 we have
⊥
APPENDIX A: FERROMAGNETIC
INSTABILITY ω−(q)<ω+(q)
Inthis appendix we use the spin waveapproachto de- and from the instability condition ω−(q = 0) = 0 we
termine the critical line corresponding to the ferromag- obtain
neticinstabilityforthecaseofSU(2)symmetriclegs. We
refer the reader to the appendix of (19) where we have hext =J⊥ (A3)
considered in detail the analogous analysis of a ladder
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