Phases of spin chains with uniform Dzyaloshinskii-Moriya interactions

Transcription

1 Phases of spin chains with uniform Dzyaloshinskii-Moriya interactions Oleg Starykh, University of Utah Wen Jin (Univ of Waterloo, Canada) Yang-Hao Chan (IAMS, Taiwan) Hong-Chen Jiang (SLAC, SIMES) RIKEN condensed matter seminar, Nov.21, 2017

2 Outline Spin 1/2 chains with uniform DM interaction Electron spin resonance Phase diagram of the quantum model with orthogonal magnetic field Weakly coupled chains Conclusions

3 Dzyaloshinskii-Moriya (DM) interaction D M D ij S i S j Reduces symmetry to U(1) [rotations about D axis] Easy plane anisotropy (perp. to D) Promotes magnetic order Generically stabilizes incommensurate non-collinear (spiral) states

4 Cs2CuCl4: uniform Dzyaloshinskii-Moriya interaction ~D ij ~S i ~S j Focus on the in-chain DM: for a given chain (y,z) vector D is constant D y,z =( 1) y D c ĉ +( 1) z D a â

5 Spin chain material K2CuSO4Br2. Spin chain along a axis. Intra-chain exchange J=20.5K, Inter-chain exchange J =0.03K. Smirnov et al., Phys. Rev. B, 92,134417,

6 ESR - electron spin resonance Simple (in principle) and sensitive probe of magnetic anisotropies (and, also, q=0 probe: S = Σr Sr) I(h,w)= w 4L dte iwt h[s + (t),s ]i For SU(2) invariant chain in paramagnetic phase I(H,ω) ~ δ(ω-h) m(h) [Kohn s Th] Oshikawa, Affleck PRB 2002 eeman H along z, microwave radiation h polarized perpendicular to it. H=0

7 ESR data (Povarov and Smirnov, Kapitza Institute) Shift for H b Shift and splitting for H a, c H b H a Resonance line is significantly modified with lowering the temperature; modification is strongly anisotropic with respect to field. The lowest temperature T=1.3 К is still twice higher than ordering T N

8 Cs2CuCl4 ESR data: T=1.3 K Povarov et al, 2011 H along b-axis gb = 2.08 gap-like behavior for ν > 17 GHz 2p hn = p (g b µ B H) 2 + D 2 loss of intensity for ν < 17 GHz H along a-axis: splitting of the ESR line ga = 2.20

9 Continuum in magnetic field transverse structure factor Dender et al, PRL 1997 Oshikawa, Affleck, PRB (2002)

10 Explanation I: H along DM axis H = Â JS x,y,z S x+1,y,z D y,z S x,y,z S x+1,y,z gµ B H S x,y,z x,y,z chain uniform DM along the chain magnetic field! q Unitary rotation about z-axis S + (x)! S + (x)e i(d/j)x removes DM term from the Hamiltonian (to D 2 accuracy) boosts momentum to D/(J a0) = 0! q = D/(Ja 0 ) ) 2p hn R/L = gµ B H ± pd/2,s z (x)! S z (x) dotted lines: D=0 picture Oshikawa, Affleck 2002 rotated basis: q=0 original basis: q=d/j Chiral probe: ESR probes right- and leftmoving modes (spinons) independently

11 Spectrum shift due to the uniform DM Gangadharaiah, Sun, Starykh, PRB (2008) Spin transformation excludes DM interaction, but results in the spectrum shift by momentum D/J. This leads to two ESR peaks. Karimi, Affleck, PRB 2011

12 Explanation II: arbitrary orientation H = Field theory in terms of spin currents M 2pv 3 [( ~M R ) 2 +(~M L ) 2 ] vd J [Md R ML] d gµ B H[MR z + Mz L ] unperturbed chain uniform DM along the chain magnetic field BL Uniform DM produces internal momentum-dependent magnetic field along d-axis H BR -D +D Total field acting on right/left movers gµ B ~H ± hv~d/j Hence ESR signals at 2p hn R/L = gµ B ~H ± hv~d/j Polarization: for H=0 maximal absorption when microwave field hmw is perpendicular to the internal (DM) one. Hence hmw b is most effective. Povarov et al, PRL 2011 Gangadharaiah, Sun, OS, PRB 2008

13 Explanation III: arbitrary orientation General orientation of H and D 4 sites/chains in unit cell a-b plane D a /(4 h)=8 GHz D c /(4 h)=11 GHz 0.3 Tesla 0.4 Tesla D ~ J/10 b-c plane for H along b-axis only: the gap is determined by the DM interaction strength D = p 2 q D 2 a + D 2 c! (2p h)13.6 GHz T-dependent line width S. C. Furuya Phys. Rev. B 95, (2017)

14 This explanation suggests: 1) ESR absorption in the absence of H 2) strong polarization dependence in zero field p D 2 a + D 2 c H The largest absorption occurs when microwave field h(t) is lined along crystal b-axis, h b [so that it is perpendicular to the D vector in a-c plane]

15 Extension to 2d spin liquids

16 Outline Spin 1/2 chains with uniform DM interaction Electron spin resonance Phase diagram of the quantum model with orthogonal magnetic field Weakly coupled chains Conclusions

17 Model Hamiltonian H = J X i S x i S x i+1 + S y i Sy i+1 + Sz i S z i+1 X Dẑ (S i S i+1 ) h X i S x i, i J > 0, the spins antiferromagnetically interact with nearest neighbor. The DM interaction is uniform along chain. The external magnetic field is perpendicular to the DM vector. Small anisotropy. 1 D -y x h 17

18 Classical version: chiral soliton lattice In classical version, the spin can be treated as vectors with fixed length. H = J X i S i = S(sin i cos i, sin i sin i, cos i ) S x i Si+1 x + S y i Sy i+1 + X Sz i Si+1 z Dẑ (S i S i+1 ) i h X Si x, i h=0 helical structure L(0) h<hc L(h) L(0) J D chiral soliton lattice h hc -x y z h h fully polarized 18 Togawa et al., Phys. Rev.Lett, 2012.

19 Quantum version In quantum mechanics, spins are operators satisfy commutation relation, [S,S ]=i S. Spin-1/2 operator can be mapped to fermionic one by Jordan-Wigner transformation: S + j = j exp(i X k<j k k), S j =exp( i X k<j k k) j, S z j = j j 1/2, Spin Hamiltonian transforms to interaction fermions: H = J X hi,ji apple 1 2 (S+ i S j + S i S+ j )+ Sz i S z j. H F = J X i apple 1 2 ( + i i+1 + i + i+1 )+ ( i i 1 2 )( i+1 i ). 19

20 Bosonization E (x) =R(x)e ip F x + L(x)e ip F x L R p p F p F q = X p p p+q (2p F = ). E(p + p F ) ' vp, E(p p F ) '= vp, p p F. a is lattice spacing with Here v is Fermi velocity. 20

21 Hamiltonian in terms of bosonization Low-energy theory H = H 0 + V + H bs, H = J X i i S x i Si+1 x + S y i Sy i+1 + X Sz i Si+1 z Dẑ (S i S i+1 ) h X i S x i, Isotropic Heisenberg chain H 0 = 2 v 3 dx(j R J R + J L J L ), V contains DM interaction and eeman term L p F E R p F p V = D R dx(j z R J z L ) h R dx(j x R + J x L ), Back-scattering interaction between right- and left- spin current. H bs = g bs R dx[j x R J x L + J y R J y L +(1+ )J z R J z L ], with coupling constant Exchange and DM-induced anisotropy: Garate and Affleck (2010) = c(1 + D2 2J 2 ). 21

22 Chiral rotation Perform a chiral rotation J R/L = R( R/L )M R/L, R( R/L )= Simplified Hamiltonian, 1 cos R/L 0 sin R/L A, sin R/L 0 cos R/L. H = H 0 + V + H bs,! H = H 0 + H bs, H bs = H A + H B + H C + H, H A = vy A dx(m z RM + L eit 'x H B = vy B H C = vy C H = 2 vy dx(m + R M L e i2t 'x +h.c.), dx(m + R M + L +h.c.), dxmrm z L. z R = 0 + /2, L = 0 + /2, D 0 arctan. h Oscillating factors M + R M z Le it 'x +h.c.), t ' p D2 + h 2 v 22

23 Renormalization group (RG) analysis RG equations are well-known Kosterlitz-Thouless equations, dy C d` = y Cy, H C = vy C H = 2 vy dy d` = y2 C. dx(m + R M + L +h.c.), dxmrm z L. z Ground state is determined by the initial values. y C (0) / [(1 + 2 ) cos 1+ 2 ], y (0) / [(1 + 2 ) cos C = y 2 (0) y 2 C(0). The initial values are determined by ratio D/h, and also influenced by anisotropy. cos =(h 2 D 2 )/(h 2 + D 2 ) = c(1 + D2 2J 2 ). 2 ], 23

24 h/d phase diagram There are three distinct phases. hs(x)i = M ˆx +( 1) x hn(x)i. N z : hn(x)i /( 1) k1+1 z. N y : hn(x)i /cos 0 ( 1) k 2 y, Phase boundaries. N z - N y : N z -LL: N z -LL: c1 = (D J )2 2 c (D h )2. c2 = (D J )2 2 c c3 = p (D/h) 2. Luttinger-liquid (LL): hn(x)i =0, hn a (x)n a (0)i Ca x. Compared to classical phase diagram, LL replaces soliton lattice. D/J=0.1 N y Soliton Lattice p 2 Quantum phase diagram 24 Classical phase diagram Garate and Affleck (2010)

25 h/d phase diagram Large DM interaction promotes the N z state. Phase boundary between LL and N y is independent of DM interaction. Phase diagrams are verified by numerical study (DMRG and itebd). D/J=0.1 D/J=0.01 N y N y p 2 D/J=0.05 D/J= Hong-Chen Jiang SLAC & Stanford University Yang-Hao Chan IAMS, Taiwan

26 Outline Spin 1/2 chains with uniform DM interaction Electron spin resonance Phase diagram of the quantum model with orthogonal magnetic field Weakly coupled chains Conclusions

27 h? D Model Hamiltonian H = X x,y [JS x,y S x+1,y + J 0 S x,y S x,y+1 ] y D + D X ( 1) y S x,y S x+1,y h X x,y x,y S x,y, y y - 1 J J D J > 0. Also J >0, the interchain exchange interaction, and J << J. x - 1 x x + 1 The DM interaction is uniform along chain, but antiparallel in neighboring chains. Real magnetic insulator K2CuSO4(Br/Cl)2. There is no exchange anisotropy, =1. Here we only consider magnetic field perpendicular to DM interaction. Intra-chain exchange J=20.5K, Inter-chain exchange J =0.03K, DM interaction D=0.28K. 27 Smirnov et al., Phys. Rev. B, 92,134417, 2015

28 Hamiltonian in terms of bosonization Low-energy theory S x,y! J yl (x)+j yr (x)+( 1) x/a N y (x), H = X y [H 0 + V + H bs + H inter ], H = X x,y [JS x,y S x+1,y + J 0 S x,y S x,y+1 ]+D X ( 1) y S x,y S x+1,y h X S x,y, x,y x,y H 0 = 2 v dx(j yr J yr + J yl J yl ), 3 V = h z dx(jyr z + JyL) z h x dx(jyr x + JyL) x +( 1) y D dx(jyr z JyL), z H bs = g bs dx[jyrj x yl x + J y yr J y yl +(1+ H inter = J 0 dxn y N y+1, S x,y S x,y+1! N y (x) N y+1 (x) )J z yrj z yl] Isotropic Heisenberg chain. Contains DM interaction and eeman term. Back-scattering interaction between rightand left- spin current. Most relevant part of interchain coupling. DM-induced anisotropy Garate and Affleck (2010) Ground state is determined by H inter, which is strongly affected by H bs. 28

29 h? D Hamiltonian in terms of bosonization Perform a chiral rotation J R/L = R( R/L )M R/L, R( R/L )= 1 cos R/L 0 sin R/L A, sin R/L 0 cos R/L. R = 2 + y 0, L = 2 y 0, y 0 ( 1)y tan 1 [ D h ]. Simplified Hamiltonian, H = X y h H0 + H bs + H inter i [H 0 + V + H bs + H inter ]! H = X y H bs = H A + H B + H C + H, H A = vy A dx(m z RM + L eit 'x H B = vy B H C = vy C H = 2 vy dx(m + R M L e i2t 'x +h.c.), dx(m + R M + L +h.c.), dxmrm z L. z t ' M + R M z Le it 'x +h.c.), p D2 + h 2 v H inter = H x + H y + H ' H x =2 vg x dxny x Ny+1, x H y =2 vg y dxny y N y y+1, H inter,' / g '1 dx cos[ p 2 (' y ' y+1 )]. 29

30 h? D Backscattering RG equations, dy C dl = y C y, dy dl RG equations = y 2 C, Interchain interaction RG equations, H C = vy C H = 2 vy dx(m + R M + L +h.c.), dxmrm z L. z dg x = g x (1 + y C + 1 dl 2 y ), dg y = g y (1 y C + 1 dl 2 y ), dg '1 1 = g '1 (1 dl 2 y ).! g(l) / g(0)e l Ground state is determined by the fasted growing g, y B / y C y C (0) / [(1 + 2 ) cos 1+ 2 ], c < 0 2 y (0) / [(1 + ) cos 2 C = y 2 (0) yc(0). 2 2 ], 3 1 c > 0 y z / y σ 4 6 is function of h and D. 5 30

31 h? D (h/d) phase diagram There are three distinct phases. SDW(z): hs x,y i Mx +( 1) x+y sdw(z)z, SDW(y): hs x,y i Mx +( 1) x+y h p h2 + D 2 sdw(y)y. SDW(y) distorted-cone: hs x,y i Mx +( 1) x+y dist cone sin[ p 2 ˆ' + t ' x]x ( 1) y D p h2 + D cos[p 2 ˆ' + t ' x]y. 2 Distinguish strong coupling by the initial values of y C and y. 0.2 II III IV distorted-cone y c (0) V 0 y (0) C(0) h x /D h x /D 31

32 h? D D h phase diagram SDW(y) / (D/J) 2 SDW(y) real material Phase boundary between two SDWs is independent of D, when D/J is small. Phase boundary between SDW(y) and distorted-cone is linear with h/d~1.5. Distorted-cone is unlikely to realize in real materials with small D/J y c y g x g y -0.2 g 1 g 2 h x /D=0.1, = L Typical RG flows for SDW(z). 32

33 h D z DM-induced frustration Identical-between-chains DM interactions stabilize 2D helical structure. Staggered-between-chains DM interactions effectively cancel the transverse interchain coupling. Hälg et al., Phys. Rev. B, Intra-chain exchange J=20.5K, Inter-chain exchange J =0.03K, DM interaction D=0.28K. 33 Smirnov et al., Phys. Rev. B, 92,134417, 2015

34 Model: h D z Weakly coupled S = 1/2 AFM Heisenberg chains with a uniform DMI and magnetic field, : Intra-chain and inter-chain exchange, and : staggered between chains. y+1 y h Low-energy theory: y-1 Spin operator in spin current J R/L and staggered magnetization N, x -1 x x+1 DMI ( a is lattice spacing ) can be absorbed into H 0 by a shift of bosonic field, Cone: spiral in XY plane Rapidly oscillates when D is large. Collinear SDW

35 Strong DMI: quantum fluctuations generate interaction between next-nearest (NN) chains Effective interaction between NN chain, no frustration due to DMI, J 2 ~ - (J ) 2 /D with y+1 x -1 x x+1 : scaling dimension of, modulated by magnetic field y y-1 Induced state: Cone2N h Magnetic field enhances Cone2N. J 2 < 0

36 Strong DMI: competition between SDW and Cone2N RG flows: for D/J =10, Ordering temperatures: for D/J =10, Small field, h/d= Large field, h/d=5. T cone2n SDW - cone transition h c ~ J

37 Summary h D h Strong DMI promotes collinear SDW. Magnetic field enhances Cone2N, quantum fluctuation generate J 2 among NN chains. C-IC phase transition between SDW and Cone2N. Strong DMI K 2 CuSO 4 Br 2 1 st chain NN chain T cone2n Weak DMI K 2 CuSO 4 Cl 2 1 st chain

38 Conclusions Simple model but many interesting properties Interesting dependence on the angle between h and D Strong in-chain DM frustrates interchain exchange Experimental checks are available