arXiv:cond-mat/0004269v1 [cond-mat.supr-con] 17 Apr 2000
ETL-xx-xxx
ETL Preprint 00-xxx
March 2000
The choice of the symmetry group for the cuprates
Sher Alam1, M. O. Rahman2, M. Ando2, S. B. Mohamed1 and
T. Yanagisawa1
1
2
Physical Science Division, ETL, Tsukuba, Ibaraki 305, Japan
GUAS & Photon Factory, KEK, Tsukuba, Ibaraki 305, Japan
Abstract
Following our recent conjecture to model the phenomenona of antiferromagnetism and superconductivity by quantum symmetry groups, we discuss in
the present note the choice of the classical symmetry group underlying the
quantum group. Keeping in mind the degrees of freedom arising from spin,
charge, and lattice we choose the classical group as SO(7). This choice is
also motivated to accomodate the several competing phases which are or may
be present in these and related materials, such as stripe phase [mesoscopically ordered phase], Luttinger liquids, nearly antiferromagnetic Fermi liquids,
charge-ordered Fermi liquids, glassy phase, stringy phase and perhaps more.
The existence and the behavior of pseudo-gap and lattice distortion are also
an important consideration. We have lumped the charge, spin and latticedistortion ordering and other orderings into the psuedogap.
1
In a previous work one of us [1] have advanced the conjecture that one should attempt
to model the phenomena of antiferromagnetism and superconductivity by using quantum
symmetry group. Following this conjecture to model the phenomenona of antiferromagnetism
and superconductivity by quantum symmetry groups, three toy models were proposed [2],
namely, one based on SOq (3) the other two constructed with the SOq (4) and SOq (5) quantum
groups. Possible motivations and rationale for these choices are were outlined. In [3] a model
to describe quantum liquids in transition from 1d to 2d dimensional crossover using quantum
groups was outlined.
In this short note we focus on the group choice since as mentioned before [1] that we
feel that the quantum groups arising from the classical orthogonal groups, i.e. SO(N) are a
good and worthwhile starting point, since they naturally incorporate the symmetry group of
the insulating antiferromagnetic state and are naturally rich enough to accomodate quantum
liquid behaviour.
The main purpose of this note is make a definite choice for the classical group underlying the quantum group. To this end we choose SO(7) for the purposes of this work. The
choice of SO(8) is also tempting from the point of view of its octonians. We note that we may
generally state that there are four major options confronting us towards the construction of
a model of the cuprates, viz:
• The choice of the underlying Hamiltonian
• Symmetry Group
• Nature of symmetry:- for example classical or quantum
• Broken and unbroken symmeries.
We choose a Hubbard-BCS Hamiltonian, and SOq (7) symmetry group. We take superconducting state to be dominated by a d-wave symmetry from experimental and theoretical
2
considerations. Contributions from a weak s wave component can be readily accomodated
and limits may be placed on it from experiment. Current experiments indicate that the
superconducting state is predominately d wave.
There are several reasons for choosing the classical group SO(7) which underlies directly the quantum group SOq (7):• As is well-known and already mentioned SO(3) [2] is a symmetry group for antiferromagnet insulator at the level of effective Hamiltonian.
• On the other hand effective Hamiltonian of a superconductor may be described by a
U(1) nonlinear sigma model [XY model], for example it was indicated in Ref. [4] that
the metal insulator transition may be described by the XY model.
• It was further pointed out in Ref. [5] that superconducting transition on the underdoped
side of oxides be described by a renormalized classical model. In addition it is worth
noting that SO(3) spin rotation and U(1) phase/charge rotation are symmetries of the
microscopic t-J model.
• At the level of effective Hamiltonian one needs at least a SO(3) to be representative of
charge ordering and local lattice distortions. This issue is a central one as recognized
in [6] for example and not dealt with is charge inhomogeneity and phase separation.
• In theories based on magnetic interactions [7] for modelling of HTSC, it has been
assumed that the CuO2 planes in HTSC materials are microscopically homogeneous.
However, a number of experimental techniques have recently observed that the CuO2
are rather inhomogeneous, providing evidence for phase separation into a two component system. i.e. carrier-rich and carrier-poor regions [8]. In particular, extended x-ray
absorption fine structure [EXAFS] demonstrated that these domains forms stripes of
3
undistorted and distorted local structures alternating with mesoscopic length scale
comparable with coherence length in HTSC. The neutron pair distribution function
of Egami et al. [9] also provides structural evidence for two component charge carriers. Other techniques also seem to point that below a certain temperature T∗
∗
the
CuO2 planes may have ordered stripes of carrier-rich and carrier-poor domains [9]. The
emergence of experimental evidence for inhomogeneous structure has led to renewal
of interest, in theories of HTSC which are based on alternative mechanism, such as
phonon scattering, the lattice effect on high Tc superconductivity [10–12,9]. Polarized
EXAFS study of optimally doped YBa2 Cu3 Oy shows in-plane lattice anomaly [8] below
′
a characteristic temperature T∗ † which lies above Tc , and close to the characteristic
temperature of spin gap opening T∗ . It is an interesting question if the in-plane lattice
anomaly is related to the charge stripe or spin-phonon interaction. We note that it
has been attempted in [13,14] to relate the spin gap observed in various experiments
such as NMR, neutron scattering and transport properties to the short-range ordering
of spin singlets.
Thus from the above we see that there are several phases, namely antiferromagnetism,
superconductivity, charge ordering,.. etc. If we assign SO(3) to antiferromagnetism, U(1) to
superconductivity and SO(2) [or U(1)] to the pseudogap we are naturally led to SO(7). For
one of the simplest group to embed SO(3)×SO(2)×U(1) or alternatively SO(3)×U(1)×U(1)
is SO(7). We note that this only one way of spontaneously breaking SO(7) there are several
∗ The
following can be taken as a definition of T∗ : T∗ is an onset temperature of pseudogap opening
in spin or charge excitation spectra.
† T∗′
′
′
may be defined as follows: T∗ is an onset temperature of local phonon anomalies and T∗ <
T ∗.
4
~ = (φ1 , , , , , φ7)) on which SO(7)
others. The components of the seven-dimensional vector (Φ
acts can be assigned as follows: φ1 = ∆s +∆†s , and φ2 = i(∆s −∆†s ) for superconductivity, φ3 ,
φ4 , and φ5 for antiferromagnetism and φ6 = ∆p + ∆†p , and φ7 = i(∆p − ∆†p ) to represent the
pseudogap. We have lumped the phenomena of charge ordering, spin ordering and lattice
distortions into the psuedogap since we want to see if the ordering in these systems imply
′
the existence of the psuedogap and temperature T∗ . There are 21 symmetry generators
for SO(7). By breaking it down as above we have accounted for 5 generators, thus we are
left with 16. These 16 generators can be interpreted on the basis of experiments on High
Tc materials relating to spin [neutron scattering], charge ordering, and lattice-distortion
[polarized EXAFS].
We now turn to classical SO(7) group. The generators of SO(7) can be written as‡
[Lij , Lmp ] = iδim Ljp − iδip Ljm + iδjp Lim − iδjm Lip
(1)
We note that it is usual to denote the Lie algebra of SO(N) by so(N) which is the Lie algebra
of all N × N real antisymmetric matrices. It is useful to introduce N × N real antisymmetric
matrices Lpm defined by
(Lpm )jk = δpj δmk − δpk δmj ,
p, m, j, k = 1, 2, ....N,
(2)
It immediately follows from Eq. 2 that for p 6= m Lpm has zero elements everywhere except for
an entry +1 in (p, m) position and −1 in (m, p) position, Lpp = 0 and that Lpm = −Lmp [i.e.
antisymmetric]. Due to the antisymmetry one can immediately see that the the generators
L have which have 7 × 7 = 49 elements are reduced to (49 − 7)/2 = 21 elements. For SO(N)
‡ This
relation is not particular to SO(7) as is true for all SO(N). We are making this a point so
that readers not familar with group theory may not wrongly assume to the contrary.
5
the number of symmetry generators are N(N − 1)/2 due to antisymmetry. The familiar
rotation group in real three dimensions has 3 symmetry generators, this the reason why we
need three Euler angles in 3 dimensions to parametrize rotations.
It can be shown from Eq. 2 that the generators satisfy the following commutation
relation
[Lij , Lmp ] = iδim Ljp − iδip Ljm + iδjp Lim − iδjm Lip
(3)
which is nothing but Eq. 1 for the case N = 7.
As it is well-known any SO(N) represent orthogonal rotation in N-dimensions. Thus
they leave the norm of any N-dimensional vector invariant. If we choose an order parameter
~ = (φ1 , ..., φ7 ) we know that quantity φ2 + ... + φ2 is invariant
Φ with seven components viz Φ
7
1
under SO(7), we may normalize it and set it to unity. From standard group theory we know
that or using the definition of Lij we can immediately write down its commutation with any
~ namely φi
component of Φ,
[Lpm , φi ] = iδpi φm − iδmi φp
(4)
~
which is simply a statement of how rotation generator acts on the Φ.
The Kinetic energy of the system is also straightforward to write, since in analogy of
spinning top from elementary physics it has the form of angular-momentum squared divided
by the inertia J 2 /2I. Thus we can write
X
1
Lij Lij
i,j 2mij
X 1
=
(Lij )2
2m
ij
i,j
LK.E =
where mij represents the moment of inertia.
6
(5)
The potential energy term requires some discussion and there are several choice. We
first note that it is straightforward to see that one can interpret Φ as representing an order
parameter and write immediately the Landau-Ginzburg expression for free energy
F = m2 |Φ|2 + λ|Φ|4
(6)
near the mean field transition as is familair from ordinary field theory and statistical physics.
In the phenomenological picture of phase transition the generators of SO(7) rotate the order
parameter without changing its magnitude. Now symmetry will be spontaneously broken if
a fixed direction is chosen, this happens when for example the system settles in a particular
phase as is well-known. On the other hand we can explicitly break the SO(7) by introducing
terrms which don’t respect the SO(7) symmetry, for example if we write the following term
into the potential energy
L1P.E = −(a2 φ21 + b2 φ22 ).
(7)
This term clearly only includes unequal contributions from φ21 and φ22 and thus breaks SO(7)
explicitly. It even breaks SO(2) symmetry for a2 6= b2 . If we chose a2 = b2 this would respect
SO(2) symmetry whilst breaking the remainder of SO(7) symmetry. We note that it is useful
to know the subgroups for the purposes of symmetry breaking, hence we summarized some
information for the groups of interest for us namely SO(7) and SO(8) in the appendix.
Another type of term to first approximation that we must keep in the Lagarangian [see
Appendix] is the velocity-dependent terms
LvP.E ∼ φi φi ∂a φk ∂ a φk
a 2
∼ (vik
)
a
vik
= φi ∂a φk − φk ∂a φi .
Collecting all the terms we can write the total Lagrangian as
7
(8)
L=
X
i,j
1
ωij a 2
(Lij )2 +
(v ) + Lssb + Lesb
2mij
2 ij
(9)
where ssb is shorthand for spontaneous symmetry breaking , esb for explicit symmetry
breaking and ωij are some ‘velocity’ parameters .
Even if we start with classical groups we end up with quantum groups if we examine
their fixed points. This can be easily seen by examining the connection between Kac-Moody
algebra and Quantum groups which leads to the important relation A11 [see Appendix].
This supports the conjecture in [1,2] that we should start with SO(N)q since if we start with
their classical counterparts and examine their fixed points we arrive at particular k values [
which are related to q values]. For example for SO(5) one arrives at SO(5)k=1 as is obvious
from the discussion in the Appendix and as also noted in [16].
Another important point to note is that non-linear sigma models have connection with
strings. In turn non-linear sigma models leads us naturally to the notion of noncommutativity
via Kac-Moody algebra [quantum groups]. Yet another strong feature of quantum groups
is that they unify classical Lie algebras and topology. In general sense it is expected that
quantum groups will lead to a deeper understanding of the concept of symmetry in physics
in particular condensed matter physics.
In conclusion, we propose SO(7) as the classical group underlying the quantum symmetry. Moreover SO(7) is interesting in its own right for phenomenological studies of HTSC
materials. We have included the psuedogap into the SO(7) symmetry.
ACKNOWLEDGMENTS
The Sher Alam’s work is supported by the Japan Society for the Promotion of Science
[JST] via the STA fellowship.
8
APPENDIX:
As is well-known that orthogonal rotation in N-dimension is specified by the SO(N)
groups. By definition they leave the norm squared of N dimensional vector invariant. There
is a distinction between when N is odd and even, i.e. SO(2n) and SO(2n + 1), in fact
this is so in Cartan classification. We recall that in Cartan classification scheme groups
are classified into the categories: An , Bn , Cn , Dn , G2 , F4 , E6 , E7 , and E8 . For example
SU(n + 1) are in the category An , SO(2n) are in Dn , and SO(2n + 1) are in Cn .
The maximal subalgebra of classical simple Lie algebra of SO(7) reads:
SO(7) ⊃ SU(4),
SO(7) ⊃ SU(2) × SU(2) × SU(2),
SO(7) ⊃ Sp(4) × U(1),
SO(7) ⊃ G(2).
(A1)
We note that for Sp(4) the maximal subalgebra reads:
Sp(4) ⊃ SU(2) × SU(2),
Sp(4) ⊃ SU(2) × U(1),
Sp(4) ⊃ SU(2).
(A2)
Sp(4) is isomorphic to SO(5), and SO(4) ∼ SU(2) × SU(2). SU(2) ⊃ U(1) and Sp(2),
SO(3), and SU(2) are all isomorphic.
The maximal subalgebra of classical simple Lie algebra of SO(8) is given by
SO(8) ⊃ SU(4) × U(1),
SO(8) ⊃ SU(2) × SU(2) × SU(2) × SU(2),
SO(8) ⊃ Sp(4) × SU(2),
9
SO(8) ⊃ SU(3),
SO(8) ⊃ SO(7).
(A3)
In a simple sense one may say that non-linear sigma model is like a Taylor expansion in
field theory. Let us explain what this means. We can write the action of string [1-dimensional]
propagating in a manifold with metric Gµν as [15]
L ∼ Gµν (X)∂a X µ ∂b X ν g ab + ....
(A4)
gab is the two-dimensional metric generated by the ‘motion’ of string in the background
manifold Gµν (X). A crucial observation is that X’s play a dual role of coordinate of the
string in the background space and scalar field in the 2-dimensional space specifield by the
metric gab [i.e. the space generated by the motion of the string]. Different choices for the
background metrics lead to different conformal field theories. Of interest to us is the choice
that the string is propagating on a manifold specified by a Lie Group[for e.g., SU(N), SO(N),
etc] in other words group manifold. We thus let g be an element of the Lie group. From
Eq. A4 we can guess that a string propagating on this group manifold has an action of form
L ∼ tr(∂a g −1 ∂ a g)
(A5)
where g is some function of the string field X. Simple differentiation gives
∂a g = ∂a Xµ f aµ
(A6)
for some function f , thus the metric G can be expressed in terms of f . The exact form of
action is
Z
1
tr(∂a g −1 ∂ a g) + kΓ(g)
4λ2 Z
1
d3 Xǫαβγ tr[(g −1 ∂ α g)(g −1∂β g)(g −1∂γ g)]
Γ(g) =
24π
S=
10
(A7)
where Γ(g) is the Wess-Zumino term which is integrated over 3-dimensional disk whose
boundary is two-dimensional space. For k = 0 it reduces to ordinary sigma-model, which
is not conformally invariant [it is asymptotically free and massive]. For special values of
k = 1, 2, 3, .. the theory becomes effectively massless and has an infrared-stable fixed point
at the values of parameters λ and k related via
λ2 = 4π/k
(A8)
Thus at these special values of k we have a conformally invariant σ model where the theory
is defined on the group manifold. This theory is called Wess-Zumino-Witten [WZW] model.
The symmetry generators J satisfy a special case of Kac-Moody algebra, viz
1
b
c
[Jna , Jm
] = f abc Jn+m
+ knδ ab δn+m,0
2
(A9)
In Eq. A9 we note the following the generators J carry two indices, namely a, b, c... which
are the Lie group indices and n, m.. which are arise in the decomposition of the generator
J = J(z) in terms of its moments, viz,
J(z) =
∞
X
Jn z n−1 .
(A10)
n=−∞
In some sense the Kac-Moody algebra smears the generators of ordinary Lie algebra around
a circle or string.
Finally we recall [15,1] that
q ↔ e2πi/(k+2)
(A11)
If we make the above correspondence it can be shown by examining various identities of
WZW model that the braiding properties of WZW model at level k are determined by the
representation theory of quantum groups. As a trivial check if one sets q = 1 in A11, which
11
is the limit in which quantum group reduces to the ordinary classical group, then the righthand side of A11 we must set k → ∞, which is precisely the limit in which Kac-Moody
algebra reduces to ordinary classical algebra. We recall that the symmetry generators of the
WZW model obey a special case of Kac-Moody algebra.
Non-linear sigma models have been extensively used in particle theory to describe
interactions phenomenologically between strongly interacting particle. For example the Lagarangian for non-linear sigma-model for the special case of SU(2) × SU(2) spontaneously
broken to SU(2)
L=−
1 ∂µ~π · ∂ µ~π
2 (1 + ~π 2 /F 2 )2
(A12)
where the factor 1/F acts as the coupling term that comes with the interaction of each
additional pion. Expanding the expression in Eq. A12 and keeping only the first two terms
we get
1
L = − ∂µ~π · ∂ µ~π + (1/F 2)~π 2 ∂µ~π∂ µ~π + ....
2
(A13)
The first term is the simple kinetic energy term, the second being the potential energy term
of the form πi π i ∂µ~πj ∂ µ π j [in words it is a velocity dependent potential term]. We must also
retain in general in our phenomenological modelling [as in this paper] of HTSC material keep
such terms.
12
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