Nematicity As a Route to a Magnetic-Field–Induced Spin Density Wave Order: Application to the High-Temperature Cuprates

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Nematicity as a route to a magnetic-field–induced spin density wave order: Application to the high-temperature cuprates

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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.

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 rotational 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

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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. 1. x y SQ τ1 TQ τ1 The collective mode dispersions are shown by different lines in the boxes in fig. 1. The gapped magnon denoted by the thick line in the nematic state, as shown in fig. 1(a), Fig. 2: Diagram to compute the coupling constant γxy in eq. (5). can be detected by neutron scattering, and its intensity should be anisotropic in momentum at a given frequency. The anisotropic intensity of magnetic excitations was to fermions as reported in YBCO, and interpreted as evidence of the x † y † nematic state [4,10]. While the magnon can be detected H = g1 S c ck+Q↓ + g2 d(k)T c ck+Q↓ +h.c. Q k↑ Q k↑ by the neutron scattering, the tSF mode does not couple k k to the magnetic field directly. Therefore, the tSF mode † x y = ψk(g1SQτ1 + g2d(k)TQτ1)ψk, (7) shown in the dotted line in fig. 1(a) cannot be detected by k the neutron scattering technique. However, in the presence of a field, the two modes couple as we have shown above. where g1 and g2 are the interaction strengths in the AF and tSF channels, respectively. We have chosen x-and The magnon gap is pushed down, while the tSF gap is y-component of the AF and tSF fluctuations assuming pushed up, and the coupling also generates a finite but that there is a small anisotropy such as Dzyaloshinskii- small intensity of the tSF mode shown by the dashed Moriya interaction aligning the staggered moment in the line in fig. 1(b), so the two modes should be in principle (x, y)-plane. detectable by neutron scattering in a field. A further magnetic field decreases the magnon gap as shown in Then γxy in the nematic state and in the presence of a magnetic field is obtained by computing the Feymann fig. 1(c) leading a field-induced AF order, while the tSF diagram shown in fig. 2: gap becomes larger. d(k) γxy = g1g2 [nF (˜k + B − µk) Incommensurate SDW and tSF orders. – ˜k + B k Recently it was reported that the intensity of inelastic −nF (−˜k − B − µk)] , (8) neutron scattering peaks at the incommensurate wave vectors, Q =(π ± δ, π) in YBCO increases when the which is proportional to N0B. For example, for t =1, magnetic field is applied [11]. Here we show that the field t = −0.3, and µ = −0.867, γxy =2.36N0Bg1g2. induced SDW can be also applied to incommensurate Field-induced antiferromagnetism, collective orders for both collinear and spiral cases. We will discuss modes, and neutron scattering. – The free energy how to distinguish spiral and collinear orders at the end deep in the nematic state (static N0), expressed in terms of the section. of the AF and tSF fluctuations, is given by Let us first consider a collinear spin density wave order: F 1 N 2 −N 2 = (1 + 0)(∂xSQ) +(1 0)(∂ySQ) x 1 † † 2ms S Q Q Q = ck− Q ↑ck+ ↓ + ck+ Q ↑ck− ↓ +h.c. , (12) 2 2 2 2 2 1 2 2 k + (1 + N0)(∂xTQ) +(1−N0)(∂yTQ) 2mt 1 2 1 2 x + ∆sSQ + ∆tTQ − γSQ · TQ, (9) where we choose the spin density wave as S (r) ∝ cos 2 2 (Q · r). Taking into account broken symmetries as before, where ∆s and ∆t represent the gap of AF and tSF, the following incommensurate tSF couples to ISDW respectively, when a magnetic field is absent. linearly under the magnetic field: The above free energy becomes unstable to the formation of antiferromagnetism when the following condition y 1 ˜ † † is satisfied: TQ = d(k, Q) c Q ck+ Q ↓ − c Q ck− Q ↓ +h.c. , k− 2 ↑ 2 k+ 2 ↑ 2 |γ| = ∆s∆t. (10) 2 k (13) Strictly speaking, the condensed mode is not a pure AF, Qy Qx where d˜(k, Q) = 2(sin kx sin − sin ky sin ). We used a but a mixture of AF and tSF. However, if ∆t ∆s,the 2 2 ∝ − symmetric form of the order parameter for convenience. mode is dominated by AF. Assuming that ∆s (x xc), ˜ the doping dependence of the critical field for the onset of Note that the form factor of d(k, Q) has no longer the field-induced AF state is given by a d-waveness when Q deviates from (π,π) due to the constraint of the current conservation at each site [12]. (x − xc)∆t Note that due to the incommensubility δ, a deviation Bc ∝ , (11) N0 from π, there is a finite coupling between ISDW and ItSF

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under the ﬁeld, even in the absence of the nematicity: 3 d˜(k, Q) γxy = g1g2 nF ˜k− Q + B − µk− Q 2 2(˜k− Q + B) 2 2 k 2 1 −nF −˜ Q − B − µ Q . (14) k− 2 k− 2 2 However, the coupling γxy is proportional to δ which 0 makes γxy to be 0.022B when δ =0.12π. On the other hand, when the nematicity is ﬁnite, γxy =2.4N0Bg1g2 Ϫ1 similar to the AF case.

Now let us study a non-collinear or spiral spin density Ϫ2 wave order. The spiral spin density wave can be written as Ϫ3 † † Ϫ3 Ϫ2 Ϫ10 1 2 3 SQ = c Q ck+ Q ↓ + c Q ck− Q ↑ , (15) k− 2 ↑ 2 k+ 2 ↓ 2 k Fig. 3: (Colour on-line) The topology of a Fermi surface in a where SQ(r) = cos Q · rxˆ +sinQ · ryˆ with spins lying in field-induced AF via the nematic order. Note the anisotropy the (x, y)-plane. A similar spiral incommensurate stag- between the electron pockets near (π, 0) and (0,π) due to the gered flux can be defined as nematic order. It is straightforward to generalize it for incommensurate SDW orders, where the electron pocket remains the ˜ † † 3 TQ = d(k, Q) c Q ck+ Q ↓ + c Q ck− Q ↑ , (16) same , while the hole pocket is sensitive to the incommensu- k− 2 ↑ 2 k+ 2 ↓ 2 k bility. Similar to the spiral spin density wave order, the spiral staggered flux can be viewed as a pattern of incommen- The full effect of the direct interaction between surate staggering current where a spin quantization axis magnetic field and AF order will be discussed later shifts from site to site determined by Q. Note that the in the context of high-temperature cuprates, as it coupling between the spiral spin density wave and the depends on circumstances. However, the effect of spiral staggered flux in the presence of a magnetic field the Zeeman coupling on the quasi-particles is rather is the same as that for the collinear case. straightforward to compute. When the staggered Our results in general support a magnetic-field–induced moment lying in the (x, y)-plane, and the magnetic spin density wave including both collinear and spiral field is alongz ˆ, it changes the electronic disper- orders. Which one of these can be finally stabilized in k+k+Q ± 1 − ± 2 2 the magnetic field requires a microscopic understanding of sion as Ek = 2 2 (k k+Q 2B) +4SQ, cuprates which is beyond the scope of our current study. where there is no linear B dependence. However, However, we offer an experimental way to distinguish the when the staggered moment and the magnetic field collinear and spiral orders. Note that the incommensurate are parallel, say along z ˆ, then the dispersion is collinear spin density wave couples to charge density wave k+k+Q±2B 1 2 2 Ek = 2 ± 2 (k − k+Q) +4SQ, with the order, which implies that there should be extra peaks at linear B dependence similar to the effect of the magnetic (±2δ, 0) or (0, ±2δ) as the spin density wave order sets in field on spin singlet condensate states such as a charge at (π ± δ, π)or(π,π ± δ) under the magnetic field. On the density wave. other hand, the spiral spin density wave does not induce a charge modulation. Therefore, a further neutron scattering Fermi surface and quantum oscillations. – The under a higher magnetic field would be a way to determine nematic phase in the absence of magnetic fields is metallic, the precise field-induced order. and thus the field-induced SDW phase is expected to 2 Other effects of the magnetic field. – The effects of be a metal . For example, as shown in fig. 3, in the the magnetic field on the order parameters deserves some field-induced AF coexisting with nematic order, there are discussion. In the weak-coupling approach, it was shown elongated pockets —one electron-like and one hole-like. that the magnetic field enhances the nematic order for It is straightfoward to generalize the Fermi surface for a one spin component, but weakens the other component, spiral order, where the electron pocket is qualitatively the and it shifts the phase boundary between the nematic same. and isotropic metal [13]. However, deep inside the nematic state, the field effect is negligible. The magnetic field does 2When the induced AF gap is large, it is possible to open a gap not directly couple to the tSF order, because the tSF does on a whole Fermi surface. However, such a large gap would require not break the time reversal symmetry. Thus the magnetic- an enormous magnetic field. 3For the collinear case, a higher order mixing of bands is ignored, field effect on the nematic and tSF orders does not affect assuming that magnetic breakdown occurs for such a small band our analysis. gap.

57005-p4 Nematicity as a route to a magnetic-ﬁeld–induced spin density wave order

In a metal with a closed Fermi surface, magnetization which supports that the primary oscillation is from a M and conductivity oscillate in 1/B as pocket insensitive to the ortho-II potential. Within our picture, the electron pocket is not affected by the ortho-II A n( F + φ) potential. On the other hand, the hole pockets are modi- M ∝ An cos , (17) n B fied by the ortho-II potential as shown in ref. [26], and the topologies of the modified hole pocket are different for A where F is the area of a closed Fermi surface at the the AF and d-density wave orders. Thus, a detailed study chemical potential. Since there are two types of pocket, of the magnetic breakdown due to the ortho-II potential the primary periodicities (n = 1) are determined by the would be another way to differentiate a field-induced AF size of both the hole and electron pockets. Using the via the nematicity and the d-density wave order [27]. same parameter set of t =1, t = −0.3, µ = −0.867 for In summary, we show that nematicity is a route to x γ, and setting SQ =0.07 and N0 =0.05, we obtain achieve a field-induced spin density wave order in the the periodicity of 540T and 900T for the electron and presence of a magnetic field. An induced spin density hole-pocket, respectively, as a unit cell of square lattice wave is obtained via the coupling between the spin density ∼ 3.82 × 3.89A2. Due to the nematic order, the electron wave mode and the spin-triplet staggered flux mode pockets around (π,0) and (0,π) are elongated along which are both gapped in the nematic phase. Such a different directions, but their area is the same yielding a direct coupling between the two orders occurs only when single frequency of the quantum oscillations. As discussed a magnetic field is applied, and the coupling strength above, in a Fermi liquid, the Zeeman effect generates increases linearly to the magnetic field. A further increase two different phases φ↑ and φ↓ (up and down spins have of the magnetic field eventually condenses the magnon different phases), due to the linear B dependence in the leading to a field-induced spin density wave order, while electronic dispersion. However, in the field-induced AF the gap of the triplet staggered flux mode becomes larger. with a staggered moment lying in the (x, y)-plane, there A neutron scattering study under a high magnetic field is no linear B dependence, implying a single phase of will offer a precise order among possible commensurate oscillation. and incommensurate orders discussed above. Our study It is tempting to argue that our finding is relevant indicates that the nematicity plays an important role to the quantum oscillations observed in high-temperature in understanding the phenomena observed in the high- YBa2Cu3O6.5 and YBa2Cu4O8 [14–19]. Recent quantum temperature cuprates, in particular in the context of oscillations reported in YBCO around 10–12% doping anisotropic magnetic excitations and quantum oscillations were striking and contrast with the Fermi arc —a trun- observed in YBCO materials. cated Fermi surface— reported by the angle-resolved photoemission spectroscopy. [20,21] There have been a ∗∗∗ few theoretical proposals which attempt to explain the observed quantum oscillations, including the d-density This work is supported by NSERC of Canada, Canadian wave as a pseudogap state [22], the staggered flux in a Institute for Advanced Research, and Canada Research vortex liquid [23], stripe formation [24], and a field-induced Chair. antiferromagnet [25]. As shown in ref. [25], the competition between the dSC REFERENCES and AF orders also leads to a field-induced AF phase, where the coexistence of AF and dSC states between [1] Kivelson S. A., Fradkin E., Oganesyan V., Bindloss 0

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