Akira FURUSAKI (RIKEN) - Unconventional orders in frustrated ferromagnetic spin chains

Akira starts by thanking his collaborators for this work:
Toshiya Hikihara (Hokkaido Univ.)
Tsutomu Momoi (RIKEN)
Masahiro Sato (RIKEN)
Shunsuke Furukawa (Toronto)

And the work presented is published in these references:
PRB 78, 144404 (2008)
PRB 79, 060406 (2009)
PRB 81, 094430 (2010)

Introduction

We have the very general problem of the search for new states of matter in quantum spin systems. Geometric frustration suppresses conventional magnetic order, leaving two possibilities:
--No order at all (spin liquids, spinons, etc...) This was the topic of the previous seminar.
--Exotic unconventional orders (spin nematic, etc...) which will be the subject of this talk.

As a theorist, the strategy is to taken a simple minimal model, (in this case, essentially 1D), and solve it without any uncontrolled approximations.

Plan for talk:
--introduction to J1-J2 spin chain in magnetic field
--vector chiral order
--nematic and multipolar orders
--XXZ anisotropy (as an alternative to magnetic field)

Introduction to J1-J2 spin chain in magnetic field

H = J1 \sum_i S_i S_{i+1} + J2 \sum_i S_i S_{i+2} + h \sum_i S_i

This Hamiltonian can be viewed in two different ways: firstly as a spin chain with nearest neighbor (J1) and next-nearest-neighbor (J2) couplings, and a magnetic field h. Alternatively, one can consider the even sites on one leg of a 2-leg zig-zag ladder, and the odd sites on the other leg.

If J2>0 (anti-ferromagnetic AFM), then we get frustration, whether or not J1 is >0 (ferromagnetic - FM) or <0 (AFM). In this talk we will mainly consider J1<0 (FM), although later on we will also consider the other case. The J1-J2 model is a theorists simplification of reality, but there seems to be a pretty good experimental realization: a spin-1/2 edge-sharing Cu-oxide chain - where the nn is FM, but nnn is AFM due to presence of oxygens As specific examples of this, LiCuVO4, LiCu2O2, etc... different materials will have different ratios of J1/J2.  

More specific example: LiCuV04 J1=-1.6 meV, J2=5.6 meV
The phase diagram in mag field seems to show many different ordered phases (see Schrettle et al, 2007) This is one of the systems we will have in mind, and we will come back to it later.

J1-J2 model -- lets first discuss the classical limit, taking s as a classical variable.

In this case, we find that when -4 < J1/J2 < 4 (dominant n.n. AFM) gives the Neel state. Now, lets apply the magnetic field, still within the classical limit: in the helical state, we find the magnetic field leads to canting of spins forming an `umbrella' structure, before they eventually enter the FM phase at some critical h. This helical `umbrella' phase has a U(1) symmetry of overall rotation around the applied field, as well as a Z_2 (vector) chirality degree of freedom, related to the vector chiral operator (S_l x S_{l+1})^z. This chiral vector is a measure of which way around the spiral you are going. In 1D, the U(1) symmetry cannot be broken giving no helical long range order, however the Z_2 symmetry can be broken, giving long range order of vector chirality. Note it is very important to have the magnetic field breaking the SU(2) symmetry down to U(1) [see Kolezhuk & Vekua (2005)]. However, this breaking of SU(2) down to U(1) could also be done via easy-plane anisotropy (XXZ) [Nersesyan, Gogolin and Essler (1998)]. Question (Piers): Why can't we have this order without the magnetic field? Answer: One could get power law correlations functions, but not long range order. However, magnetic field may be infinitesimal. We will come back to this near the end. Lets look more closely at spin-nematic order:

Vector chirality: (S_l x S_{l+1})^z is antisymmetric p-type nematic [Chandra and Coleman (1991)]

But it is also possible to have symmetric products: i.e. n-type nematic order, which is in fact quadrupolar order with an order parameter Ql^-- = s_l^- s_l+1^-. In the classical picture, one sees that making a rotation of pi on the spins doesn't change Q - it is a director. It can also be related to bound states of magnons.

We can also look at operators for bounds states of higher numbers of magnons, e.g. 3-magnon, etc... These are multipolar orders. Back to the J1-J2 model, and we see the phase diagram in magnetic field (h>0): it is a very rich phase diagram, including a number of these multipolar orders such as anti-ferro-triatic, antiferro-nematic, nematic (IC) and vector chiral phases. We will now discuss aspects of this in more detail.

In the limit J1/J2-> 0, we can bosonize by taking two AF chains + weak J1 coupling (this is thinking in the zig-zag ladder picture). The spin operators represented in terms of the bosonic fields as smooth+staggered parts in the usual way. Taking all of the relevant inter-chain coupling terms gives us a nasty looking sine-Gordon like model with two cosines- but this has already been discussed in the literature by Nersesyan, Gogolin and Essler (1998). Depending on which cosine is the most relevant (which in turn depends on the scaling dimensions, which in turn depends on some non-trivial combination of the microscopic parameters), can find either vector chiral order, or nematic order.

In the Vector Chiral phase, find the vector chiral order parameter has long range order. This corresponds to a phase with alternating orbital spin currents in the zig-zag picture, - but note however that there is no net spin current flow. On the other hand, the spin-spin correlation functions have an incommensurate power-law decay. This is a quantum counterpart of the classical helical state.

This field-theory (for J1/J2->0) can be supplemented with DMRG (which is essentially numerically exact) - calculation of correlation functions here agrees with the analytic results.


Now, let us look at energy epectrum of magnon excitations at saturation field (h=hc). Can compute 1 magnon, 2 magnon, etc... and find depending on ration J1/J2 that 2 magnon, 3 magnon, etc... bound states may have the lowest energy state. This will lead us to multipolar order.

How can we write an effective theory for multi-polar Tomonaga-Luttinger (TL) liquids? Treat the p-magnon bound-state as a hard-core boson (with some extra residual interaction), then use Haldane's hydrodynamic approach (bosonization of bosons, [Haldane 1980]). This allows easy calculation of certain correlation functions, e.g. power law decays of bound magnons, as well as of SzSz correlation functions. However, depending on the Luttinger Liquid (LL) parameter (which depends in complicated way on microscopic parameters as usual), can find which of these correlation functions is the dominant one, giving both SDW phase, or nematic phase. Note that these `phases', unlike the Z2 chiral order, correspond to no broken symmetries however, and simply label the dominant correlation function in quasi-LRO.

DMRG results also confirm existence of these phases (nematic, triatic, SDW2 and SDW3), and clarifies positions of phase boundaries, which are not always reliably obtained within the field theory.

How to detect nematic, triatic etc... order?
One way is to look at dynamical spin structure factors, or in the NMR relaxation rate.

We then see a load of plots of these quantities in different phases, which I'm not sure I can describe in words. Do we have any photos of this slide?


Back to LiCuVO4
We know the ratio J1/J2, so can identify the experimentally determined phases with those of the theoretical phase diagram.

Question (Andrey): How well are the phases really understood experimentally?
Answer: Not well, but there is some evidence for the spiral order (I missed exactly what this evidence is).

Question (blogger): What about inter-chain couplings in this material?
Answer: We didn't really study that, although they must of course be important to give real phase transitions. There is a recent paper (unfortunately I missed the authors) which address inter-chain couplings in nematic type phases.

AFM J1-J2 model

Now, lets look at a slight variation of the model, where all couplings are AFM (but still in a magnetic field). As references, see Okunishi & Tonegawa (2003); McCuolloch et al (2008); Okunishi (2008); Hikihara, Momoi, Furusaki and Kawamura (2010).

We now have a picture comparing the phase diagrams for J1<0>0 ... hopefully we can add this picture to the blog as it is rather too complicated to describe in real time...
We learn that this phase diagram was obtained mostly by numerics.

XXZ J1-J2 model

Another variation of the model: the case of no magnetic field, but breaking the SU(2) symmetry down to U(1) by XXZ easy-plane anisotropy.

Many previous studies of AF J1-J2 case with anistropy \Delta - showing regions of gapless chiral order. However, (more recent work), if J1 is FM, find much larger region (in phase space) of the gapless chiral order. For more details, see Furukawa, Sato and Onoda arXiv:1003.3940. We are running out of time during the talk however, and there is no time to discuss this more fully.






 

Summary:
--We study the spin-1/2 frustrated FM J1-J2 spin chain in a magnetic field.
--Many interesting `phases' seen, including conventional SDW forms, but also nematic and multipolar phases, as well as a vector chiral phase.

Question time!!

Question (Andrey): what happens in isotropic chain in zero field - there seems to be an accumulation point of many transitions?
Answer: this is an open question - some recent numerics seem to show a dimerized state, but it could be different to that seen in the anisotropic phase. This is such an unstable point, it's hard to say.

Comment (Shura): As long as SU(2) is unbroken, there is no room for phases with local currents (as the SU(2) cannot be spontaneously broken). So what happens? Seems we don't really know. This is either unfortunately or fortunately (depending on your point of view) an open question.

Question (Piers): Any hope for this physics to be seen in non-1D systems?
Answer: Maybe LiCuVO4 (note: this was also mentioned briefly within the main body of the talk, but the blogger missed it)
Comment (Chubukov): This is essentially 1D physics - any real material will be quasi-1D, and there will be a critical inter-chain coupling where this physics is killed.

Andrey Chubukov with Blogger, Sam Carr.