5.5 Supersolid in the one-dimensional model – DMRG
5.5.1 The case of S=1
As the spin-1 anisotropic Heisenberg model has been studied earlier, it is useful to perform our variational calculation for theS = 1case and compare our results to the numerical quantum Monte Carlo findings of Ref. [Sengupta 2007a] The variational phase diagrams, forJ = 0 andJ = 0.2Jz are shown in Fig.5.10(a) and (b), respec-tively. In the Ising limit, we find two uniform phases, denoted by 00and 11, where
(a) (b)
zz J/h!zz
!
J
"/ z
"/!Jz J
/h!
1 3/4
1/2 1/4
0 3/2
1
1/2
0
1
1/2
0 3/2
1 3/4
1/2 1/4
0
× N/2
WN
WN
2 2
3
10
11
00 SF 10
SF
SS
11 11
11
00
Figure 5.10: The phase diagram of the anisotropic S=1 model in the (a) Ising–limit for a bipartite lattice with coordination number ζ and (b) for the square lattice (ζ = 4) whenJ = 0.2Jz, obtained from the variational calculation.
the numbers 0 and 1 correspond to the expectation value of the Sz. In addition,
there are two translational symmetry breaking phases: 1¯1with zero magnetization and 10 which is the 1/2 magnetization plateau. The saturation field is given by hsat = Λ +ζJz+ζJ, and from the stability analysis of the gapped phases, 1¯1, 00, and 10, we obtain the following phase boundaries
h2 = (ζJz−Λ−ζJ)(ζJz−Λ +ζJ), (5.32)
h2 = Λ(Λ−2ζJ), (5.33)
(h−Λ)(ζJz+ Λ−h)(h−ζJz+ Λ) = 2ζ2J2Λ, (5.34) respectively.
The XXZ–like physics can be identified for the transition between the phases 11,10, and 00, where the supersolid is fragile. The region between the10 and 1¯1 is of different nature, and we expect the supersolid phase to be robust in this part of the phase diagram. This is exactly the region where the supersolid phase was found in Ref. [Sengupta 2007a] . The nature of the phase transitions, is also in qualitative agreement with the numerical results; we recover the first order transition between the upper boundary of the10 and the superfluid phases.
Electromagnons and Ba 2 CoGe 2 O 7
They say a little knowledge is a dangerous thing, but it is not one half so bad as a lot of ignorance.
– Terry Pratchett, Equal Rites As we learned in the introductory chapter 2, spin systems of S > 1/2 sup-port tensor interactions and may exhibit unconventional, often non-magnetic or-ders, such as multipolar or nematic order. When one is interested in the dynam-ical properties of such systems, the conventional quasi particle approaches, such as the Holstein-Primakoff representation, fail and one needs to introduce general-ized bosonic operators related not only to the spin but also to higher order op-erators [Onufrieva 1985]. The idea of the extended spin wave approach is not entirely new; it has been introduced to study magnetic systems with single-ion anisotropy and/or higher order exchange terms [Onufrieva 1985,Papanicolaou 1984, Papanicolaou 1988,Shiina 2003], as well as for spin systems with orbital degeneracy in terms of flavor waves [Joshi 1999].
In this chapter we investigate the momentum and magnetic field dependent spin excitations in the multiferroic compound Ba2CoGe2O7 based on the generalized spin wave method, introduced in chapter 3. We will show that this approach is sufficient to describe the higher excitations observed in the far infrared absorption spectra, but would be beyond the reach of the conventional spin wave theory. A word will be added on the spin-induced polarization at zero and finite temperatures, with the aim to reproduce the experimentally observed peculiar behavior of the induced polarization in magnetic field in Ref. [Murakawa 2010] (see Fig.1.10in the introductory chapter 1).
Starting from the symmetry analysis in section 2.2.2 we consider the following Hamiltonian
H = JX
hi,ji
SixSjx+SiySjy
+JzX
hi,ji
SizSjz+ X
i
h
Λ (Siz)2+gzzhzSzi +gxx(hxSix+hySiy)i
, (6.1)
where hi, ji indicates nearest neighbor pairs, and the x, y, and z axes are parallel to the [110],[1¯10], and[001]crystallographic directions, respectively (See Fig. 1.7).
gxx = gyy and gzz are the principal values of the g tensor, and hα = µBBdc,α are the components of the magnetic field.
6.1 Zero field phase diagram
In this part we give a detailed study on the zero-field excitations of Ba2CoGe2O7, considering the effect of single-ion anisotropy, although, for simplicity we neglect the DM interaction which is in fact very small compared to the exchange interaction.
On the other hand, we introduce the exchange anisotropy which has a significant effect on the ground state, and subsequently on the excitations, as we have seen in the previous chapter.
We start our investigations with a short discussion of the variational phase dia-gram as the function of easy-plane and exchange anisotropies. Then, based on these variational findings, we introduce a suitable boson basis and perform the generalized spin wave approach to study the momentum dependent excitation spectrum. We examine the isotropic case(Λ = 0, Jz =J)separately and consider the limitΛ→ ∞ in detail. The latter case will be compared the effective spin-1/2 model, discussed in Ref. [Zheludev 2003]. Furthermore, we calculate the dynamical structure factor for the excitation modes so that we can compare our results qualitatively with that of neutron spectroscopy of Ref. [Zheludev 2003].
We search for the ground state in the site-factorized variational form introduced in5.3:
|Ψi=Y
i∈A
Y
j∈B
|ψAii|ψBij , (6.2)
where
|ψAi ∝ |3/2i+eiξ1u1|1/2i+eiξ2u2| −1/2i+eiξ2u3| −3/2i (6.3) and with a similar expression for |ψBi. The variational parameters, as usually, can be determined by minimizing the ground state energy E = hΨ|H|ΨihΨ|Ψi .
The phase diagram exhibits two, the completely and the partially aligned, axial antiferromagnetic states, A3andA1, already familiar from section5.1. In addition, a superfluid (U(1)) phase emerges between them; this is referred to as a planar state in Ref. [Sólyom 1984], for the spins are aligned in the lattice plane. However, one can call this phase superfluid for the spin-rotation symmetry breaking phases exhibit off-diagonal long-range order, or equivalently finite spin stiffness [Seabra 2011], that is the property of such phases. In the following we refer to this phase asSF0. Between the planar superfluid SF0 and the two axial antiferromagnetsA1 and A3 an other superfluid phase appears. The in-plane components of this conical antiferromagnet have the same properties as the planar superfluid but it exhibits finite staggered magnetization too, inheriting the property of the antiferromagnetic phases, A1 and A3, as well. Therefore we call this phase SFA.1 A schematic figure of the various phases is shown in the phase diagram in Fig. 6.1.
1As in a superfluid, or conical phase with finite staggered (antiferro) magnetization.
10
8
6
/JzJ 4
2
0
−2−10 −5 0 /J
!
5 10 15 20
0
F3
A1
A
A3 SF
SF
Figure 6.1: Variational phase diagram for h = 0 as the function of Λ/J and Jz/J. Solid lines stand for continuous (second order) phase boundaries, while the dashed lines denote the first order phase transition. The black dot represents the SU(2) symmetric isotropic Heisenberg model.
Comparing the ground state energies of A1 andA3 we find that the first order phase boundary between them isΛ = 2Jz. The ground state wave functions of sites A andB in the planar superfluid phaseSF0 can be expressed as
|ΨAi = e−iϕASˆAz|ΨSFi, (6.4)
|ΨBi = e−iϕBSˆBz|ΨSFi, (6.5) where in the absence of in-plane magnetic field the spins are antiparallel, that is ϕA=ϕ,ϕB =ϕ+π, and the energy depends only on the variational parameter η:
|ΨSFi= |32i −i√
3η|12i −√
3η| − 12i+i| −32i
p6η2+ 2 . (6.6)
The ground state energy as the function of parameterη reads E0SF0(η)
N = 3 4
η2+ 3
3η2+ 1Λ−18η2(η+ 1)2
(3η2+ 1)2 J . (6.7)
In the energy expression the Jz-term is missing, as this wave function has spin components only in the xy plane. The energy in this phase is minimal when the following equation is satisfied:
Λ
J = 3(η2−1)(3η+ 1)
3η2+ 1 . (6.8)
For small values of Λ the ground state energy can be approximated as:
E0SF0 =−9 2J+3
4Λ− Λ2
16J +O(Λ3) (6.9)
giving the phase boundary toward the antiferromagnetic phase A3 J =Jz−Λ
3 − Λ2
72Jz +O(Λ3) (6.10)
as plotted in Fig. 6.1. The parameter ϕcan take arbitrary value, carrying the U(1) symmetry breaking property of the ground state (we recall that the Hamiltonian commutes with the Sˆz operator), furthermore it determines the angle between the spins and the axis y:
hΨA|S|Ψˆ Ai = 3η(η+ 1)
3η2+ 1 (sinϕA,−cosϕA,0) (6.11) hΨB|S|Ψˆ Bi = 3η(η+ 1)
3η2+ 1 (−sinϕB,cosϕB,0) (6.12) WhenΛ →0, that is the single-ion anisotropy is absent, η →1 and the Hamil-tonian (6.1) has SU(2) symmetry and consequently the ground state corresponds to the spin-3/2 Néel-state. ForΛ>0theSz =±3/2 components in the wave function are suppressed. In the largeΛlimitη→ ∞, and the length of the spin is equal to 1.
Let us recall that starting from a more general wave function than (6.6), whenΛ>0 the Sz = ±3/2 components are suppressed and the wave function is composed of the Sz =±1/2 components. Then the tip of the spin can move within an ellipsoid and the length of the spin is maximal (equal to 1) when it lays in the xy-plane and minimal (equal to 1/2) along the z-axis. Therefore a finite antiferromagnetic exchange supports the planar ordering. When the exchange interaction becomes anisotropic, and the SˆizSˆjz term becomes strong, this energy can compensate the directional length dependence of the spin, and can choose a spin configuration with a finitez andxy component. This happens in the conical superfluid phase, denoted bySFAin Fig.6.1. The parameterηcan be expressed in the two limits as it follows:
η= (
1 +6JΛ +O
Λ2 J2
ΛJ
Λ
3J −13 +O ΛJ
ΛJ
(6.13) The phase boundary of the conical superfluid phase (SFA) towards the planar phase (SF0) and fully polarized AFM phase (A3) are beyond analytical reach, how-ever, the numerically obtained boundaries are shown in Fig. 6.1. Starting from phase A1 at a given Λ value, a second order phase transition takes place to the superfluid phase, SFA. When the exchange coupling J is large enough, in-plane spin components appear continuously as we reach SFA. The ground state of this conical superfluid phase can be expressed as it follows:
|ΨAi ∝ e−iϕSˆAz (|3/2i+u|1/2i+v| −1/2i+w| −3/2i) (6.14)
|ΨBi ∝ e−i(ϕ+π) ˆSBz (w|3/2i+v|1/2i+u| −1/2i+| −3/2i) (6.15)
where the variational parametersu,v and w are all real values.
The instability of the partially aligned AFM phase A1against canting gives the phase boundary between the phasesA1 and SFA
J = Jz(Jz−Λ)
Jz−4Λ . (6.16)
The same model for one dimension has been treated by mean field calculations in Ref. [Sólyom 1984] for quantum spins 1/2, 1 and 3/2. The phase diagram for the caseS= 3/2is similar to ours, however the conical superfluid phaseSFAis missing due to a less general variational wave function.