The effects of magnetic field on the d-density wave order in the cuprates

5 Pages·0.18 MB·English

by  Hoang K. Nguyen

The effects of magnetic field on the d-density wave order in the cuprates
Hoang K. Nguyen and Sudip Chakravarty∗
Department of Physics and Astronomy, University of California Los Angeles, Los Angeles, California 90095
(Dated: February 6, 2008)
2 Weconsidertheeffectsofaperpendicularmagneticfieldonthed-densitywaveorderandconclude
0 that if the pseudogap phase in the cuprates is due to this order, then it is highly insensitive to the
0 magnetic field in the underdoped regime, while its sensitivity increases as the gap vanishes in the
2 overdoped regime. This appears to be consistent with the available experiments and can be tested
furtherinneutronscatteringexperiments. WealsoinvestigatethenatureofthedeHaas-vanAlphen
n
effect in theordered state and discuss thepossibility of observing it.
a
J
PACSnumbers:
0
1
Recentlyitwasargued1,2 thattheobservedpseudogap form Landau levels,5 while those of a DDW state do.6
]
n in the cuprate superconductors is due to the develop- This implies that, in principle, it is possible to observe
o ment of a second type of order,the d-density wave order deHaas-vanAlphen(dHvA)oscillationsinaDDWstate,
c (DDW), which is a particle-hole condensate with an in- but not in a d-wave superconductor. The reason is that
-
ternal“angularmomentum”2. Thenotionofthisbroken quasiparticles in a superconductor do not couple simply
r
p symmetry has allowed us to understand an array of ex- to the vector potential A corresponding to the magnetic
u periments, although a priori there is no sense in which fieldB=∇ A,buttothesupercurrent(∇ϕ 2eA/~c),
s × −
such a state is close to, or adiabatically continuable to, whereϕisthephaseofthesuperconductingorderparam-
.
at a superconductor, or a Fermi liquid, which are the two eter, e is the electronic charge, and c is the velocity of
m most prominent states of matter. light. We calculate dHvA oscillations of the magnetiza-
- From considerations of the simplest Hartree-Fock tion in the DDW state and find that although ∆B/B
d Hamiltonian that captures the broken symmetry of the could be of reasonable magnitude, the amplitude of the
n DDW and the electromagnetic gauge invariance, we magnetization oscillations is weak, generically of order
o
c study the dependence of the DDW order parameter as 0.01µB. Thus, observation of these oscillations may be
[ a function ofthe applied magnetic field perpendicular to difficult.
CuO-planes. Wefindthattheeffectofthemagneticfield Consider the mean field Hamiltonian of a d-density
2 on the pseudogap is rather weak, if the DDW order is wave subjected to a magnetic field applied perpendic-
v
well-developed. For example, to destroy a DDW gap of ular to the plane. With the choice of the Landau gauge,
9
magnitude of order 20 meV would require fields of order the Peierls substituted Hamiltonian is written as:
3
0 1000T,orlarger. Moreover,thisgapishighlyinsensitive
1 to fields of order 25 T. This is, of course, a characteris- H = − µ c+m,ncmn−t c+m,ncm+1,n+h.c
0 tic of the underdoped regime. In the overdoped regime, Xm,n Xm,n(cid:0) (cid:1)
2 where the DDW gap drops rapidly to zero, the depen- t e2πiαmc+ c +h. c.
0 dence on the magnetic field could be substantial. − m,n m,n+1
/ Xm,n(cid:0) (cid:1)
t We suggest that the recent neutron scattering exper-
a ( 1)m+n∆ ic+ c +h.c
m iments, which report a tantalizing evidence of DDW − − − m,n m+1,n
order3 in the form of a resolution-limited elastic Bragg Xm,n (cid:0) (cid:1)
-
d peak at the in-plane wave vector (π/a,π/a), where a + ( 1)m+n∆ ie2πiαmc+ c +h. c.
n is the lattice spacing, should be carefully examined for Xm,n − (cid:0)− m,n m,n+1 (cid:1)
o its dependence on a perpendicular magnetic field. The
(1)
c
prediction is that the Bragg scattering intensity will be
:
v hardly affected, if the zero temperature DDW gap is
Here m and n are the site labels of a square lattice in
i substantial.4
X the x- and y-directions; α = φ/φ is the magnetic flux
0
Theeffectofmagneticfieldonasuperconductoriswell through each plaquette of the square lattice in terms of
r
a known. Since the Cooper pairs involve time reversed the flux quantum φ = hc/e, and c is the fermion
0 m,n
mates, the effect of any time reversalbreakingperturba- destruction operator at the site (m,n). Here on, the
tionisveryimportant. Incontrast,theparticle-holepairs hoppingmatrixelementtwillbesettounity. TheDDW
of the same spin orientation form the DDW condensate, gap satisfies the self-consistency condition:
and, as a consequence, such a condensate is inherently
less affected by a magnetic field. ∆= i( 1)m+nV c+ c h. c. , (2)
− − h m,n m+1,n− i
The contrasting response of a d-wave superconductor
(DSC)andaDDWtoappliedmagneticfieldisfurtheril- where the angular bracketsdenote the groundstateaver-
luminatedbyexaminingthenodalquasiparticlesofthese age. Due to the conservation of current, ensured by the
two systems. The nodal quasiparticles of a DSC do not gauge invariance of the Hamiltonian (1), the DDW gap
2
could as well be: One sees immediately that the states (k ,k ), (k +
x y x
2πα,k ), (k +4πα,k ), ..., (k +π,k +π), (k +π+
y x y x y x
∆=i(−1)m+nVhe2πiαmc+m,ncm,n+1−h. c.i. (3) 2πα,ky+π),(kx+π+4πα,ky+π),... arecoupled. For
rationalflux,α=p/q,wherepandq arerelativeprimes,
Since a uniform applied magnetic field cannot break
thesetofcoupledstatesisfinite, containing2q elements.
translational invariance, all physical quantities must be Atthis point,the connectiontothe Hofstadterproblem7
translationallyinvariant. HereV is anenergyparameter
becomes particularly obvious. The Hamiltonian can be
that controls the strength of DDW pairing.
written as the following quadratic form:
We rewrite the Hamiltonian in terms of the Fourier
transformedvariables,givenby(withlatticeconstantset
to unity)
H = Ψ†k(Hk µ)Ψk, (6)
c = 1 c ei(kxm+kyn), (4) Xk −
m,n √N kx,ky
kXx,ky
whereN isthetotalnumberofsitesonthelattice,toget where k [0,2πα) and k [0,π). The row vector ΨT
x y k
∈ ∈
is given by
H = µ c† c 2 cosk c† c
− kx,ky kx,ky − x kx,ky kx,ky
kXx,ky kXx,ky
− kXx,ky(cid:16)eikyc†kx+2πα,kyckx,ky +h. c.(cid:17) ΨTk =(cckkxx,+kπy,,kcyk+xπ+,2cπkαx,+kyπ,+c2kπxα+,4kπyα+,πky,c,k.x.+.,πc+k4xπ+α2,πkαy+(qπ−,1.).,k.y,,
+ ∆kXx,ky(cid:16)ieikxc†kx+π,ky+πckx,ky +h. c.(cid:17) ckx+π+2πα(q−1),ky+π),
∆ ieikyc† c +h. c. . (5)
− kXx,ky(cid:16) kx+π+2πα,ky+π kx,ky (cid:17)
and the matrix Hk is
2coskx e−iky 0 ... eiky 2i∆coskx i∆e−iky 0 ... i∆eiky
− − − −
(cid:12) eiky 2cos(kx+2πα) e−iky ... 0 i∆eiky 2i∆cos(kx+2πα) i∆e−iky 0 ... (cid:12)
(cid:12) − − − − (cid:12)
(cid:12) 0 eiky 2cos(kx+4πα) ... 0 i∆eiky ... ... ... (cid:12)
(cid:12) − − (cid:12)
(cid:12)(cid:12) ... ... ... ... (cid:12)(cid:12)
(cid:12) (cid:12)
(cid:12)(cid:12)2i∆coskx −i∆e−iky 0 ... −i∆eiky 2coskx e−iky 0 ... eiky (cid:12)(cid:12)
(cid:12) i∆eiky 2i∆cos(kx+2πα) i∆e−iky 0 ... eiky 2cos(kx+2πα) e−iky ... 0 (cid:12)
(cid:12) − − (cid:12)
(cid:12) 0 i∆eiky ... ... ... 0 eiky ... (cid:12)
(cid:12) − (cid:12)
(cid:12) .. .. (cid:12)
(cid:12) . . (cid:12)
(cid:12) (cid:12)
(cid:12) (7) (cid:12)
(cid:12) (cid:12)
The gap satisfies the equation obtained as
1
∆= ∆k, (9)
∆ = V ieikxc† c +h. c. 2qNxNy kX∈sBZ
N Xk hh− kx+π,ky+π kx,kyi i whereN =(qN )(2N ). Thewholeprocesswasrepeated
x y
V until ∆ converged.
= ∆k. (8)
N Typical magnetic fields in laboratories are of order
Xk
20T,equivalent to (1/1225)φ for the cuprates with lat-
0
tice spacing of 4 ˚A. In our computation, we were able
For each value of (k ,k ), the Hamiltonian was diag- to diagonalize matrices of size 4000 4000, or q = 2000
x y
×
onalized, giving rise to a new set of fermion opera- corresponding to the smallest field of 12.5T. Since the
tors di , i = 1,...,2q. Then, ∆ , ∆ , gapis soinsensitive to the applied field, goingto smaller
{ k} kx,ky kx+2πα,ky
..., ∆ , ∆ , ∆ , ..., fields appeared unnecessary.
kx+2πα(q−1),ky kx+π,ky+π kx+π+2πα,ky+π
∆ were computed and so was the en- In Fig. 1, we show the dependence of the DDW gap
kx+π+2πα(q−1),ky+π
ergy spectrum. Several values of k and k were chosen versus magnetic field at absolute zero for µ= 0.1t as a
x y
−
in the sub-Brillouin zone (sBZ) and finally the gap was function of V. We observe a striking insensitivity of the
3
0.20
0.8
0.15
0.6
0
0.10
t f /fc
D / 0.4
0.05
0.2
0.00
0.00 0.5 1.0 1.5 2.0
V/ t
0.0 0.05 0.1 0.15 0.2 FIG. 2: Thecritical field as a function of V for µ=−0.1t.
f/f
0
ing the system area and a cutoff energy. The above for-
FIG. 1: The DDW gap magnitude ∆ as a function of the mula is expected to hold only for small fields such that
applied flux φ/φ0 for various values of the coupling strength ~ωc = ~eB/m∗c << W. This appears to be a surpris-
V/t ((cid:13) : 0.5; 2 : 0.75; 3 : 1.0; △ : 1.25; ⊳ : 1.5; ▽ : 1.75; inglyrobustresult,asshowninFig.3,especiallybecause
⊲:2.0) for µ=−0.1t. we made no approximations involving nodal Fermions.
The fit at the smallest field gave a sensible W = 1.6t.
The quantity W was interpreted in Ref. 6 as the cutoff
gapwithrespecttothemagneticfield. Thegaponlycol-
energy below which the Dirac fermion description holds
lapses, through a first-order transition at a flux strength
for the DDW state.
φ /φ ,correspondingtoalargelaboratorymagneticfield
c 0
(φ 25,000T for cuprates). The gaps at zero field was
0
≈
calculatedseparately,and the results for finite field were
seen to converge to them as φ 0. It is important to
→
note that because the chemical potential is finite, the 10- 1
DDW transitiondoes nottake place atV =0+, but ata
finitethresholdvalueV . ThisisclearlyseeninFig.2. In
c
the samefigure,weseethatφ /φ dropsquite rapidlyto
c 0
zeroasV is approached. Thus,althoughφ corresponds
c c
titnoheqthugeiatpeunliasdresgrmedmoapllaegidnnerttehigceimfioeevl,edristdwocphoeuednldtrhebgeeiDmquDei.WteFosgmrapaalilsgwalaphregonef 0 E ( ) ] / t0 10- 2
oTr.der20 meV, the criticalmagnetic fieldis oforder1250 fE ( ) - 0
[ 10- 3
In order to test the correctness of our computations,
wealsocalculatedthegroundstateenergydensityathalf-
filling(µ=0). Weobserveanincreaseinthegroundstate
energy as the magnetic field is applied; this increase in-
dicates a diamagnetismin the DDW system. This result 10- 4
has, in fact, been analytically derived6 using the nodal 10 - 3 10 - 2 10 - 1
f /f
fermigas formalismof the DDW state for µ=0. In par- 0
ticular, it was shown that the DDW groundstate energy,
FIG. 3: The groundstate energy as a function of the applied
expressed in terms of lattice parameters, behaves as
magnetic field at µ=0.
ζ(3/2) W φ 3/2
E (φ)=E (0)+2 At , (10)
GS GS √π r t (cid:18)φ (cid:19) As a consequence of the nonanalytic field dependence
0
of the energy at µ = 0, the diamagnetic susceptibility
where ζ(x) is Riemann’s zeta function, A and W be- diverges at zero field: χ B−1/2. Obviously, this di-
∝
4
vergence poses a serious question about the stability of
- 0.04
the DDW state at µ = 0. It is clear, however, that this
divergence will be cut off for a finite chemical potential,
µ = 0. Similarly, the divergence will also be removed
6
at finite temperatures even for µ = 0. Moreover, three
dimensionalcoupling betweenthe layerswillalsomodify )B
the resultfor the susceptibility.6 Itisimportanttostress me (- 0.06
sit
that the divergence in the susceptibility is intimately re- er
p
latedto the appearanceofzero-energymodes,whichis a n
o
direct consequence of the nodal points and the doubling ati
z
of the Dirac fermions. eti
n
It has recently been proposed in Ref. 8 that a finite g
a
M - 0.08
magnetic field may induce an additional d gap. Such
xy
a gap can efficiently remove the nodes and hence the di-
vergence in the diamagnetic susceptibility. However,the
small d gap will not be manifest at a finite temper-
xy
ature and the dominant cutoff will be provided by the
temperature or the chemical potential. Therefore our - 0.10
computation for the DDW gap retains its relevance to 1.0 1.2 1.4 1.6 1.8 2.0
experimental situations. 1/B (T- 1 )
AsaninterestingcontrasttoDSC,anessentialfeature
oftheDiracfermionpictureoftheDDWistheformation FIG.4: ThedeHaas-van Alphenoscillations of themagneti-
of Landaulevels in a magnetic field:6 E = n(~ω )∆ , zation in theDDW state.
n c 0
where ∆0 is the amplitude of the gap, and pωc is the cy-
clotron frequency, eB/m∗c, m∗ being the effective mass
hole-doped cuprates, with a finite chemical potential µ,
correspondingtothenodalregion. Asaresult,theDDW
theformula(10)forthegroundstateenergymustbeaug-
system away from half-filling is expected to exhibit de
mented by the following additional term:
Haas-van Alfven (dHvA) oscillations in the magnetiza-
tion. eB nµ
For a free electron system at chemical potential µ, ~ωc∆0 √n (13)
hc
wheretheenergyspacingis~ωc,theoscillationfrequency p nX=0
satisfies
where[n ]isthelargestintegersmallerthanµ2/(~ω ∆ ).
µ c 0
∆B 1 ~ωc In other words, nµ is the number of filled Landau lev-
= B∆ = . (11)
els. With the above choice of parameters, the magneti-
B − (cid:18)B(cid:19) µ
| | zation per unit cell, in units of Bohr magneton, µ , is
B
Taking B = 1T, µ 0.25eV, one would get ∆B showninFig.4. Inplottingthis figure,wehaveremoved
5G. For nodal fermi≈gas at chemical potential µ, th≈e the spurious δ-function singularities arising from taking
oscillation frequency is derivatives of the step functions. At finite temperature,
the sharpness of the oscillations will be smoothed out,
∆B 1 ~ω ∆ but the oscillation amplitude will still roughly be of or-
c 0
= B∆ =( )( ). (12)
B − (cid:18)B(cid:19) µ µ der 0.01µB, which is quite weak. As is evident from
| | | | Fig.≈4, as the magnetic field swept between 0.5 T and 1
To make an estimate, we can choose B =1T. It is more T, the system undergoes about 14 oscillations.
problematic to estimate µ in the underdoped regime in There are already some experiments that appear to
which the DDW state is likely to be found, and inwhich shed light on the effect of magnetic field on the pseu-
the typical non-interacting band picture may not hold. dogap. Nuclear magnetic resonance experiments in un-
There is some evidence, however, that µ tx2, where derdoped YBa Cu O clearly support the view that
2 4 8
≈ −
x is the doping.9 Choosing x 0.15, ∆ of the order the pseudogap is hardly affected by an applied per-
0
≈
of the pseudogap 20 meV, and t 0.25 eV, results pendicular magnetic field,10 and so does experimets in
∼ ≈
in ∆B 0.05 T, which is clearly within the detectable nearly optimally doped11 YBa Cu O . In contrast,
2 3 7−δ
≈
range of dHvA oscillation experiments. nuclear magnetic resonance experiments in overdoped
The aboveestimatefor∆B is notsufficientforthe de- TlSr CaCu O provides contrasting evidence that in
2 2 6.8
tectability of dHvA effect. We must also estimate the this regime the pseudogap is sensitive to the magnetic
magnitude of the magnetization oscillations. For this field.12 At first sight, the experiments in YBa Cu O
2 3 7−δ
purpose, it is convenient to consider energy spectrum of appear to be controversialbecause there arealso reports
the nodal fermions of the DDW state. The oscillation to the contrary,13,14 but as was pointed out by the au-
in energy, and thus of the magnetization, occurs when- thors of Ref. 12, the later reports by the authors of
ever a Landau level passes through the Fermi level. For Ref. 13,14 seem to be in agreementwith Ref. 11. There-
5
fore, our theoretical results would be consistent with ex- rents need not be in the perfect staggered pattern, but
periments, if the pseudogap is indeed caused by DDW. canbeflippedaslongasthecurrentconservationsatthe
We hope thatour workwillcatalysefurther experiments vertices are satisfied.15 We hope to report on this issue
on this important question. On the theoretical front, it in the future.
will be necessaryto go beyond the Hartree-Fockapprox- WethankA.V.Balatsky,PengchengDai,H.A.Mook,
imation, especially at finite temperatures. To go beyond and C. Nayak for interesting discussions. This work was
Hartree-Fockapproximations,wemustrecognizethefact supportedby a grantfrom the NationalScience Founda-
that the DDW gap, which characterizes the orbital cur- tion: NSF-DMR-9971138.
∗ First author: [email protected]; second author: 9 The choice of this chemical potential was discussed in S.
[email protected] Tewari, H. -Y. Kee, C. Nayak, and S. Chakravarty, Phys.
1 S. Chakravarty and H. Y. Kee, Phys. Rev. B 61, 14821 Rev.B 64, 224516 (2001).
(2000). 10 Guo-quing Zheng, W. G. Clark, Y. Kitaoka, K. Asayama,
2 S.Chakravarty,R.B.Laughlin,D.K.Morr,andC.Nayak, Y.Kodama, P. Kuhns,and W.G. Moulton, Phys.Rev.B
Phys. Rev.B 64, 094503 (2001). 60, R9947 (1999).
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012502 (2001), and H. A. Mook et al., in preparation. C.H.Pennington,P.C.Hammel,W.L.Hults,J.L.Smith,
For a theoretical analysis of these experiments, see S. P.L.Kuhns,A.P.Reyes,andW.G.Moulton,Phys.Rev.
Chakravarty,H.-Y.Kee,andC.Nayak,Int.J.Mod.Phys. Lett. 82, 177 (1999).
15, 2901 (2001)and cond-mat/0112109. 12 Guo-quing Zheng, H. Ozaki, W. G. Clark, Y. Kitaoka, P.
4 Preliminaryresultsindicatethatthisisindeedmaybethe Kuhns, A. P. Reyes, W. G. Moulton, T. Kondo, Y. Shi-
case (H.A. Mook, private communications). makawa, and Y.Kubo, Phys.Rev. Lett. 85, 405 (2000).
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Phys. 77, 293 (1989); see also, X. Yang and C. Nayak, R99682 (1996).
cond-mat/0108407. 15 The finite temperature behavior is that of the six-vertex
7 D. R.Hofstadter, Phys.Rev. B 14, 2239 (1976). model (S. Chakravarty,unpublished).
8 J-H. Zhu and A.V. Balatsky, cond-mat/0110214.