Competing orders and field induction of d+id′ state

Physica C 388–389 (2003) 25–28 www.elsevier.com/locate/physc

Competing orders and field induction of d+id0 state A.V. Balatsky *, J.X. Zhu Theory Division, T-11 MS B 262, Los Alamos National Laboratory, Los Alamos, NM 87545, USA

Abstract The role of the magnetic field on the d-wave density wave as a model of pseudogap state of cuprates and on the dwave superconducting state will be addresses. We argue that in d-wave density state magnetic field can produce secondary gap components. This distortion by magnetic field offers a possibility to distinguish between different scenarios of pseudogap in normal state of high-Tc materials. Similarly we argue that magnetic field can distort the p-wave state and produce secondary component of the gap in p-wave superconductor. Ó 2003 Published by Elsevier Science B.V. Keywords: d+id0 superconductor; d-density wave; Pseudogap; Competing order; Field-induced component

1. Introduction We will argue that the ground state of a many body system can be distorted and a secondary component of the order parameter can be generated upon applying magnetic field. This distortion, if it produces a new nontrivial component of the order parameter, leads to the symmetry lowering of the state. We start with the general symmetry arguments on why magnetic field can lower the symmetry of the state. Consider a general many body ground state jW0 i. Now consider this state in the external magnetic field Bkz, where we take field to be along z-axis. Here we will focus on the orbital effect of magnetic field. The relevant interaction term in the Hamiltonian of the system is b zB Hint ¼ M ð1Þ b z ¼ glB b where orbital magnetic moment operator M Lz with g being the gyromagnetic ratio and lB is the Bohr magneton. The first-order perturbation theory gives for the correction to the ground state jW1 i ¼ Hint jW0 i ð2Þ There are three possibilities for jW1 i: (i) jW1 i is zero. In this case, there is no linear effect of the field on the

ground state. Symmetry of the state is not lowered. This is the case, as can be verified directly, when we apply magnetic field to s-wave superconductor. The angular momentum operator applied to ground state yields zero; (ii) jW1 i is collinear with jW0 i. The effect of the field to this order is only to change the amplitude of the state, e.g. the magnitude of the order parameter. Again symmetry is not lowered in this case; (iii) case when hW1 jW0 i ¼ 0. In this case, the newly acquired amplitude jW1 i is of different symmetry, the ground state is distorted by magnetic field jW0 i ! jW0 i þ jW1 i and symmetry of the ground state is lowered in the magnetic field. It is not surprising that magnetic field can produce a new component of the order parameter. For example d-wave state is time reversal invariant. In the magnetic field, since time reversal is explicitly broken, the state becomes a dx2 y 2 þ idxy with finite angular momentum. Below we will focus on the only nontrivial case (iii). We will specifically consider the case of d-wave density wave as a model for a pseudogap state of cuprates. We also mention the case of unconventional p-wave superconductors. 2. Pseudogap state and magnetic field induction of secondary component

*

Corresponding author. Tel.: +1-505-665-0077; fax: +1-505665-4063. E-mail address: [email protected] (A.V. Balatsky).

The nature of the competing orders in high-Tc phase diagram and the pseudogap phase of underdoped

0921-4534/03/$ - see front matter Ó 2003 Published by Elsevier Science B.V. doi:10.1016/S0921-4534(02)02606-0

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cuprates is an important issue that is one of the most strongly debated questions. Some models attribute PG to superconducting phase fluctuations above Tc [1–3]; or to a competing non-superconducting order parameter [4,5]. We propose here to use magnetic field as a test of the symmetry of pseudogap state. If indeed the normal state of cuprates is a d-wave density wave (DDW), as proposed in [5], it will have a nontrivial response to magnetic field. We argue that, in addition to the dominant dx2 y 2 ðdÞ component of the DDW order parameter, a subdominant dxy ðd0 Þ component can be generated by the magnetic field. The fully gapped particle spectrum and the magnetically active collective mode of the condensate are the experimentally relevant consequences of d þ id0 density wave state. Detailed discussion for DDW case is also given in [6]. Similar phenomenon of field induction of secondary component for d-wave superconductor was also considered, e.g. in [7]. The physical origin of this instability is the bulk orbital magnetic moment hMz i in the d þ id0 state. When an external magnetic field H is applied perpendicular to the plane of the two-dimensional (2D) system under consideration (namely, Hk^z), the resulting coupling of the magnetic induction B with the orbital magnetic moment, hMz iB, lowers the system free energy. For the DDW state, as opposed to superconducting case, there is no screening effect on the magnetic field, the magnetic induction B is homogeneous throughout the system and is close to the external magnetic field H . In the absence of the magnetic field, the pure d-density wave state can be regarded as the equal admixture of the orbital angular moment Lz ¼ 2 pairs W0 ðHÞ ¼ iW0 cosð2HÞ ¼

iW0 ½expð2iHÞ þ expð2iHÞ: 2

ð3Þ

Here we have made an approximation to the order parameter W0 ðkÞ / hcykþQ;r ck;r i / W0 ðcos kx a  cos ky aÞ by confining the wave vector k near the Fermi surface and introduced H as the 2D azimuthal angle of the Fermi momentum, where ck;r annihilates an electron of spin r at k, W0 is the magnitude of the pure d-wave component. In the presence of an external magnetic field, the Lz ¼ 2 orbital wave functions becomes unequal and the coefficients for them are shifted linearly with the magnetic field H iW0 ½ð1 þ gBÞ expð2iHÞ þ ð1  gBÞ expð2iHÞ 2 ¼ i½W0 ðHÞ þ iBW1 ðHÞ; ð4Þ

W0 ðHÞ !

where B ¼ H and W1  g sinð2HÞ. Notice that the pure d-density wave order parameter is imaginary while the field generated d0 -wave component is real, the relative phase between the two components is still p=2 in the equilibrium. Microscopically, by focusing on the effect

of magnetic field on the order parameter, the system Hamiltonian can be written as X X H¼ nk cyk;r ck;r þ ½Wk cyk;r ckþQ;r þ h:c: k;r

 iglB B

X

k;r

cyk;r ½sin

ka  oka z ck;r :

ð5Þ

k;r

Here nk ¼ 2½cos kx a þ cos ky a  l with l the chemical potential is the single particle energy measured relative to the Fermi energy. The DDW order parameter is given by Wk ¼ iW0 u0 ðkÞ þ W1 u1 ðkÞ, where u0 ðkÞ ¼ cos kx a  cos ky a and u1 ðkÞ ¼ sin kx a sin ky a. The amplitude of the d- and d0 -wave components W0;1 are determined selfconsistently W0 ¼

iV0 X y hc ck;r iu0 ðkÞ 2N k kþQ;r

ð6Þ

and W1 ¼ 

2V1 X y hc ck;r iu1 ðkÞ; N k kþQ;r

ð7Þ

where V0;1 are, respectively, the d- and d0 -channel interaction, N is the number of 2D lattice sites. d-wave component is imaginary due to the equivalence of Q ¼ ðp; pÞ and Q, enforced by the underlying band structure. The notation ½sin ka  oka z represents sin ðkx aÞ oky a  sinðky aÞokx a . We define k ¼ 2½cos kx a þ cos ky a so that nk ¼ k  l. For Q ¼ ðp; pÞ, we have following symmetry properties: kþQ ¼ k , u0 ðk þ QÞ ¼ u0 ðkÞ, and u1 ðk þ QÞ ¼ u1 ðkÞ. In view of the fact that the DDW state breaks the translational symmetry with latpffiffiffi tice constant but conserves that by 2a along the diagonals of the square lattice, it is convenient to halve the Brillouin zone, by introducing two kinds of electron operators ck;r and ckþQ;r . The pairing of the particles and holes must cause correlations in their relative motions. According to the structure of the Hamiltonian and the self-consistency conditions for the DDW order parameter, we can introduce the following GreenÕs functions to describe the correlation: G11 ðk; k0 ; sÞ ¼ hTs ½ck;r ðsÞcyk0 ;r ð0Þi;

ð8Þ

G12 ðk; k0 ; sÞ ¼ hTs ½ckþQ;r ðsÞcyk0 ;r ð0Þi;

ð9Þ

G21 ðk; k0 ; sÞ ¼ hTs ½ck;r ðsÞcyk0 þQ;r ð0Þi;

ð10Þ

G22 ðk; k0 ; sÞ ¼ hTs ½ckþQ;r ðsÞcyk0 þQ;r ð0Þi;

ð11Þ

where the factor Ts is a s-ordering operator as usual, ck;r ðsÞ ¼ eH s ck;r eHs is the operator in the Heisenberg representation. Given the Hamiltonian Eq. (5), by solving the equation of motion in the approximation up to the first-order in the orbital-magnetic field coupling,

A.V. Balatsky, J.X. Zhu / Physica C 388–389 (2003) 25–28

we obtain the Fourier transform: G21 ðk; k0 ; ixn Þ ¼ G021 ðk; k0 ; ixn Þ þ dG21 ðk; k0 ; ixn Þ, where G021 ðk; k0 ; ixn Þ ¼

 ðWk þ WkþQ Þdkk0 Dðk; ixn Þ

ð12Þ

and dG21 ðk; k0 ; ixn Þ ¼

iglB Bðixn  nkþQ Þ Dðk; ixn Þ  ½sinka  oka G021 ðk; k0 ; ixn Þ;

ð13Þ

where Dðk; ixn Þ ¼ ðixn  Ek;1 Þðixn  Ek;2 Þ with Ek;1ð2Þ ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2  k þ jWk þ WkþQ j2  l. We take the ansatz that V0 is bigger than V1 [8] such that in the absence of the magnetic field, the d-wave ordering is pure and no secondary phase transition for the appearance of the d0 ordering occurs. Therefore, the DDW gap appearing in the G0 is, Wk ¼ iW0 u0 ðkÞ. As a result, we find 4V1 X W1 ¼  Re½dG21 ðk; k; s ¼ 0Þu2 ðkÞ N k2rbz ¼ gBW0 ;

ð14Þ

where g¼ 

16glB V1 kB T X X k u21 ðkÞ N D2 ðk; ixn Þ k2rbz xn

16glB N ð0ÞV1 ; EF

ð15Þ

where N ð0Þ is the density of states at the Fermi energy EF . By taking the Fermi wavelength of a few lattice ) and Nð0ÞjV1 j  0:3, it is estimated constant a ( 4 A 2 jW1 =W0 j  10 at B ¼ 10 T, which makes the amplitude of the induced component jW1 j to be on the order of a few Kelvin. Eq. (14) suggests that the phenomenological Ginzburg–Landau (GL) free energy functional must contain the linear coupling between the original d-density wave order parameter and the field-induced d0 -density wave order parameter, fint ¼ i g2 ðiW0 ÞW1 B þ c:c: Consequently, we can write down the system GL functional of the form  Z a0 b F ¼ d2 r ðT  Tc0 ÞjW0 ðrÞj2 þ 0 jW0 ðrÞj4 2 4 K0 K1 2 þ jrðiW0 ðrÞÞj þ jrW1 ðrÞj2 2 2 a1 2 þ jW1 ðrÞj þ fint ðrÞ ; ð16Þ 2 where the first two terms describe the instability of the pure d-density wave state, with Tc0 being the transition temperature in the absence of magnetic field. The last two terms represent the energy shift of the d-wave state as a result of the field-induced d0 -wave order parameter, where a1 is positive. Notice that, unlike the supercon-

27

ducting order parameter, the gradient operator on the DDW order parameter is not shifted by the vector potential because the DDW pairs do not carry charge. It follows from Eq. (16) that, as far as the d0 -wave component is concerned, the coupling to the magnetic field term [i.e., fint ] is linear, while the stiffness term [i.e., the second last term in Eq. (16)] is quadratic. Therefore, at least at the weak field so that W1 is small, the linear term is dominant. Therefore, the system gains energy by having a nonzero equilibrium value of W1 . By treating W1 and W1 as independent variables, the GL functional F is minimized by enforcing dF=dW1 ¼ dF=dW1 ¼ 0, which leads to W1 ¼

gB W0 : a1

ð17Þ

Upon substituting the above result into Eq. (16), we find the energy gained by the system with d0 R 2the2 induced 2 2 density wave component: dF ¼  d rg jW0 j B =2a1 . Therefore, the transition temperature which is now field dependent, and is renormalized by the magnetic field as Tc ðBÞ ¼ Tc0 þ dTc ðBÞ, where dTc ðBÞ ¼ g2 B2 =2a0 a1 . It then follows that coupling of the magnetic field with the orbital angular momentum shifts the instability of the ddensity wave ordering to the high temperature at higher fields. Here we note that, since the particle–hole pairing takes place with the equal spin, the coupling between the magnetic field and the electron spin (i.e., the spin Zeeman coupling) will not depress the induction of d0 component in the DDW metal. We do not address here the case of strong field. Up to now our analysis of the induction of the secondary d0 component has been focused on the equilibrium solution. If we assume that this secondary d0 order parameter has been created, we can write in general iW0 ¼ jW0 jei/0 and W1 ¼ jW1 jei/1 , and study the dynamics of the relative phase / ¼ /1  /0 , which is governed by [9] o2 / dF ; ¼ q1 ot2 d/

ð18Þ

where q1  N ð0Þ. With Eq. (16), we find o2 / ¼ q1 gBjW0 jjW1 j cos /  s2 r2 /  x20 /; ot2

ð19Þ

which leads to the clapping mode with dispersion x2 ðB; kÞ ¼ x20 ðBÞ þ s2 k 2 with x20 ðBÞ ¼ gB2 jW0 j2 =q þ x20 , where x20 is the zero field gap and s2 ¼ jW0 j2 ðK0 þ g2 B2 K1 Þ=4q. Alternative approach to the clapping mode in d+id0 state is presented in [10]. This mode represents the oscillation of the relative phase between the d and d0 components of the DDW order parameter, and is tunable by the magnetic field. We thus have argued that (a) the applied magnetic field can generate the dxy order parameter in the

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d-density wave metal, whose amplitude is linearly proportional to the field strength, (b) the transition into the d þ id0 -density wave state occurs at a higher transition temperature, and (c) there exists a new clapping mode corresponding to the oscillation of the relative phase between the two components.

Acknowledgements

3. p + ip0 superconductor

References

Here we consider the case of quasi-2-dimensional pwave superconductor in an external field. We identify here jW0 i / px and jW1 i / py . Assume that the zero field state has real order parameter that transforms as jW0 i / px . We argue then that external magnetic field will induce secondary component jW1 i ¼ ipy . The state jW0 i þ jW1 i will have a finite magnetic moment hMz i that can couple to magnetic field. The free energy term driving the secondary component is again given by Eq. (16) and we find jW1 i / BjW0 i in case of p-wave superconductor. Here we do not address the spin part of the order parameter that can also couple to magnetic field. The results for the clapping mode are similar to the case of d-wave superconductor. We note that the similar clapping mode for the p-wave state that violates time reversal in zero field, e.g. px þ ipy , was considered in [11].

We thank J.C. Davis, M.J. Graf, R.B. Laughlin, K.K. Loh, and D. Morr for stimulating discussions. This work was supported by the Department of Energy at Los Alamos National Laboratory.

[1] [2] [3] [4] [5] [6] [7] [8]

[9]

[10] [11]

V.J. Emery, S.A. Kivelson, Nature 374 (1995) 434. B. Janko et al., Phys. Rev. Lett. 82 (1999) 4304. I. Martin, A.V. Balatsky, Phys. Rev. B 62 (2000) R6124. I. Martin, G. Ortiz, A.V. Balatsky, A.R. Bishop, Int. J. Mod. Phys. B 14 (2000) 3567. S. Chakravarty et al., Phys. Rev. B 63 (2001) 094503. J.X. Zhu, A.V. Balatsky, Phys. Rev. B 65 (2002) 132502. A.V. Balatsky, Phys. Rev. B 61 (2000) 6940. Because the pre-existing of the dx2 y2 -wave order parameter, the transition temperature for the appearance of the dxy component can be suppressed to be negative for V1 < V0 . It is unnecessary for V1 to have a different sign than V0 . The detailed derivation for the clapping mode here is similar to that for a d-wave superconductor, see A.V. Balatsky, P. Kumar, J.R. Schrieffer, Phys. Rev. Lett. 84 (2000) 4445. S. Sachdev et al., cond-mat/0110329. L. Teword, Phys. Rev. Lett. 83 (1999) 1007; H.Y. Kee, Y.B. Kim, K. Maki, cond-mat/9911131.