Home Search Collections Journals About Contact us My IOPscience Nematicity as a route to a magnetic-field–induced spin density wave order: Application to the high-temperature cuprates This article has been downloaded from IOPscience. Please scroll down to see the full text article. 2009 EPL 86 57005 (http://iopscience.iop.org/0295-5075/86/5/57005) View the table of contents for this issue, or go to the journal homepage for more Download details: IP Address: 132.68.72.203 The article was downloaded on 03/10/2011 at 17:08 Please note that terms and conditions apply. June 2009 EPL, 86 (2009) 57005 www.epljournal.org doi: 10.1209/0295-5075/86/57005 Nematicity as a route to a magnetic-field–induced spin density wave order: Application to the high-temperature cuprates H.-Y. Kee1 and D. Podolsky1,2 1 Department of Physics, University of Toronto - Toronto, Ontario M5S 1A7 Canada 2 Physics Department, Technion - Haifa 32000, Israel received 11 March 2009; accepted in final form 21 May 2009 published online 22 June 2009 PACS 73.22.Gk – Broken symmetry phases Abstract – The electronic nematic order characterized by broken rotational symmetry has been suggested to play an important role in the phase diagram of the high-temperature cuprates. We study the interplay between the electronic nematic order and a spin density wave order in the presence of a magnetic field. We show that a cooperation of the nematicity and the magnetic field induces a finite coupling between the spin density wave and spin-triplet staggered flux orders. As a consequence of such a coupling, the magnon gap decreases as the magnetic field increases, and it eventually condenses beyond a critical magnetic field leading to a field-induced spin density wave order. Both commensurate and incommensurate orders are studied, and the experimental implications of our findings are discussed. Copyright c EPLA, 2009 Introduction. – Typically, conduction electrons in there is no strong influence between the two. We show that strongly correlated materials are either localized due to the situation can be dramatically different when the time interactions between them, or conduct with a uniform reversal symmetry is broken by an external perturbation and isotropic distribution. Can electrons in metals assem- such as a magnetic field. We find that there is a direct ble themselves and exhibit novel patterns seen in liquid coupling between the SDW and spin-triplet staggered flux crystals? If so, what is the mechanism behind such (tSF) orders inside the nematic phase when the magnetic a self-organizing pattern, and what are the effects of field is applied. The tSF phase, like the (spin-singlet) stag- such a metal on nearby phases? The electronic nematic gered flux (or sometimes called d-density wave) breaks the phase characterized by an anisotropic conduction has been translational and rotational symmetries due to circulating proposed as one such state. It was suggested that it plays currents with an alternating pattern. However, unlike the a relevant role in determining the superconducting tran- d-density wave state, it does not break the time reversal sition temperature in the cuprates [1], which motivated symmetry, because up- and down-spin circulations have several studies on the interplay between the nematic and opposite directions, which leads to the name of spin-triplet d-wave superconducting (dSC) states [2,3]. An evidence of staggered flux. A consequence of such a coupling is that its existence in the cuprates was found by neutron scat- the SDW order can be induced by the magnetic field via tering on YBa2Cu3O6.45 (YBCO) [4]. On the other hand, the nematicity. This result applies to both commensurate the interplay between the nematic and the antiferromag- and incommensurate orders —a finite coupling between net, another nearby phase of the cuprates, has not been incommensurate spin density wave (ISDW) and incom- addressed. mensurate triplet staggered flux (ItSF) orders occurs in In this paper, we study the relationship between the the presence of a magnetic field leading to ISDW order nematic and spin density wave (SDW) orders. It is triv- beyond a critical magnetic field. We will discuss the exper- ial to see that a direct coupling term between the two is imental implications of our result in the context of the not allowed in the free energy, since they break distinctly high-temperature cuprates. different symmetries —the nematic order breaks the rota- tional symmetry between x-andy-directions, while the Nematicity and commensurate SDW and tSF AF order breaks the translational, spin-rotational, and orders. – The nematic order, which breaks the 90 time reversal symmetries. Thus, a naive conclusion is that rotational symmetry between x-andy-directions, is 57005-p1 H.-Y. Kee and D. Podolsky T where σ is the Pauli matrix, i = x, y, z,andQ =(π,π). It is straightforward to check that there is no direct coupling Ts To between the nematic and AF orders based on the distinctly different quantum numbers associated with their order Nematic ω(k) (a) parameters. SDW However, when we consider the following tSF order parameter, the situation changes: i † i T = i d(k)c σ ck+Qβ. (4) Q Q kα αβ k xc x ω (k) The tSF order is characterized by circulating currents (c) (b) with an alternating pattern like the d-density wave state, but up- and down-spins circulate in opposite directions. Thus, there is no net charge current, but a finite spin B fSDW current. It breaks the translational, x-y rotational, and Q Q spin-rotational symmetries, but preserves the time reversal symmetry. This state was identified as a component of the Fig. 1: (Colour on-line) Schematic phase diagram of SDW and 6-dimensional superspin which can be rotated under the nematic phases as a function of doping x, temperature, T and 15 generators forming the SO(6) group [7] which includes magnetic field B. The lines in the boxes are the dispersions as a subset the SO(5) group suggested for a unified theory of collective modes. (a) In the nematic state, both the SDW of the high-Tc cuprates [8]. It was also discussed as one of and tSF modes are gapped. (b) As the magnetic field increases, non-zero angular-momentum condensate states in ref. [9]. the magnon gap decreases, while the tSF gap gets pushed up. In a dSC state, both the tSF and AF modes are gapped Further magnetic field makes the magnon condense leading to a field-induced SDW (fSDW) in the yellow region. See the main while the AF gap, i.e. the magnon gap, is smaller than text for further discussions on the detection of the collective the tSF gap as shown in the box (a) in fig. 1, due to the modes for incommensurate and commensurate SDW orders via proximity of the AF phase in the phase diagram. k neutron scattering and on the anisotropic intensity between x Effects of a magnetic field in the nematic phase. and ky due to the nematicity. – Since both TQ and SQ carry charge 0, spin 1, and momentum Q, one may expect that there is a direct defined as [5,6]1 coupling between the two order parameters such as i j 1 † F = γij (N0, B)SQTQ, (5) N0 = d(k)ckσckσ, (1) 2 kσ where the coupling γij is a function of N0 and the where d(k) = cos kx − cos ky (setting the lattice constant magnetic field B to respect the discrete symmetries of a = 1). Inside the nematic state, marked by (a) in the the system. First, the tSF order breaks the x-y symme- phase diagram of fig. 1, the quasi-particle Green’s function try, while the AF does not, thus the coupling between is written as them vanishes, except inside the nematic phase. Therefore, γ(N0 =0, B) = 0. Second, the AF order breaks the time −1 G (k,iωn)=−iωn + k − µ, (2) reversal symmetry, while the tSF order does not, which implies γ(N0, B = 0) = 0. Therefore, the coupling between where the tSF and AF orders is finite only in the presence of both a magnetic field and the nematic order, γ(N0 =0 , B =0) = k = −2t(cos kx +cosky)+2td(k)N0 − 4t cos kx cos ky, 0. To linear order in N0 and B, the form of coupling is and µ is the chemical potential. t and t represent the found to be F∝N0B · (SQ × TQ), i.e. the index depen- k nearest-neighbor and second-nearest-neighbor hoppings, dence of γij is given by γij ∝ ijkB . It is straightforward respectively. to check that such a term is allowed in the free energy As shown in fig. 1, the commensurate antiferromagnetic based on the symmetry consideration discussed above. (AF) phase exists below a certain doping of xc, and its To compute γij in eq. (5), it is useful to introduce † † † order parameter with a moment alongx ˆ is given by ψk =(ck↑,ck+Q↓). In the basis of ψ, the Hamiltonian is i † i written as SQ = ckασαβck+Qβ, (3) 0 † k Hnem = ψk [(˜k + B)τ3 − µkI] ψk, (6) k 1While the electronic nematic order refers to a spontaneous broken symmetry, our current results can be also applied for a − k+k+Q k−k+Q − nematicity arising from an external perturbation such as orthorhom- where µk = µ 2 = µk+Q,˜k = 2 = ˜k+Q, bicity. and B = Bzˆ. Note that the AF and tSF fluctuations couple 57005-p2 Nematicity as a route to a magnetic-field–induced spin density wave order which determines the phase boundary between the pure nematic phase and the field-induced AF phase as shown in the dashed line in fig.