Contents - Max-Planck-Institut für Physik komplexer Systeme

2.25 Magnon Pairing in a Quantum Spin Nematic
Introduction. Strong quantum fluctuations can destroy
conventional long-range magnetic order. A spin
system remains, then, in a disordered liquid-like state
down to zero temperature. Many theoretical studies in
the field of magnetism in the past two decades have
been devoted to the investigation of possible types of
quantum spin liquids. The enhanced fluctuations may
also stabilize magnetic analogues of liquid crystals—
states with partially broken rotational symmetry characterized
by the tensor order parameter:
Q αβ
ij
= 1
2 〈Sα i S β
j
MIKE ZHITOMIRSKY, HIROKAZU TSUNETSUGU
+ Sβ
i Sα j 〉 − 1
3 δαβ 〈Si · Sj〉 , (1)
where i,j belong to a nearest-neighbor bond. So far
such exotic spin-nematic states were considered either
phenomenologically [1] or by introducing an ad hoc biquadratic
exchange [2]. Identification of the relevant
microscopic mechanism for the spin-nematic order remains,
therefore, a challenging theoretical problem.
Recently, we have explored [3] a novel mechanism
for the spin-nematic ordering based on competition
between ferro- and antiferromagnetic interactions in
quantum magnets. The mechanism operates in strong
magnetic field and is based on the formation of bound
magnon pairs in the fully polarized state, see illustration
in Fig. 1. The high-field magnetization study of the
frustrated chain material LiCuVO4 [4] has indeed observed
a new phase in the predicted range of magnetic
fields. Note, that the phenomenon of magnon pair condensation,
discussed in more detail in the following,
has a close relationship to the old problem of particle
versus pair-superfluidity in an attractive Bose gas [5],
which again attracts significant interest in relation to
experiments in ultra-cold atomic gases.
Two-magnon bound states. We consider a quantum
Heisenberg antiferromagnet in an external magnetic
field:
ˆH = 1
2
i,r
J(r) Si · Si+r − H
i
S z i , (2)
In strong fields the Zeeman energy dominates over
the exchange interactions and stabilizes the fully polarized
state |0〉. This state is the vacuum for single
spin-flips or magnons with the excitation energy
εq = H +S(Jq −J0), where Jq is the Fourier transform
of J(r). In ordinary antiferromagnets spin-flips repel
each other. Then, once the band gap in εq vanishes at a
certain momentum Q, an antiferromagnet undergoes a
phase transition at the critical field Hs1 = S(J0 − JQ).
The conventional scenario for a high-field transition
may change if some of the exchange bonds are ferromagnetic.
In this case two spin-flips sitting on the same
bond with J(r) < 0 can lower the Ising part of their
interaction energy and may form a bound pair [6–8].
Note, that any gain in the potential energy competes
with a kinetic energy loss upon creation of the bound
state. Bound states are, therefore, typical for frustrated
models, where the single-magnon hopping is significantly
suppressed due to the competing exchange interactions.
Figure 1: Bound magnon pairs formed in the fully polarized state of
a quantum antiferromagnet in strong magnetic fields.
To treat exactly the problem of bound magnon pairs we
introduce a general two-magnon state
|2〉 = 1
fij S
2
−
i S− j |0〉 , (3)
i,j
with fij being the magnon pair wave-function, see also
Fig. 1. Separating the center of mass motion fij =
e ik(ri+rj)/2 fk(r) we obtain the Bethe-Salpeter equation
for bound magnon states:
EB − εk/2+q− εk/2−q fk(q) = (4)
= 1
2N
p
Jp+q+ Jp−q− J k/2+q− J k/2−q
fk(p),
where EB is the total energy of the pair. If bound
states are present (EB < 0), they start to condense at
Hs2 = Hs1 + 1
2 |EB| prior to the onset of the conventional
one-magnon condensation, see Fig. 2. Eq. (4)
provides a rigorous condition for the appearance of the
exotic spin-nematic state in the high-field region of a
quantum antiferromagnet.
Condensate of magnon pairs. Below Hs2 the bound
magnon pairs form a coherent condensate. Using the
Holstein-Primakoff bosons ai and a †
i instead of spin operators,
the many-body state with a macroscopic number
of the lowest-energy pairs below Hs2 can be expressed
as a coherent boson state of the pair creation
operator:
|∆〉 = e −N|∆|2
/2 1
exp
2 ∆ fija
i,j
†
ia†
j |0〉 . (5)
90 Selection of Research Results