Quantum Lifshitz point and a multipolar cascade for frustrated ferromagnets

Transcription

1 Quantum Lifshitz point and a multipolar cascade for frustrated ferromagnets Leon Balents, KITP, UCSB Progress and Applications of Modern Quantum Field Theory, Aspen, Feb. 015

2 Collaborators Oleg Starykh U. Utah Teddy Parker UCSB

3 What this talk is not (almost) nothing topological No gauge fields Nothing fractional No anyons. Not even fermions No CFT, no bootstrap

4 What this talk is not (almost) nothing topological No gauge fields Nothing fractional No anyons. Not even fermions No CFT, no bootstrap Not even a complete solution

5 What it is about I will discuss the simplest example of a frustrated ferromagnet, and argue that there is a simple QFT description of such systems, with surprisingly rich phenomenology It is clear that this description extends to higher dimensions and perhaps the phenomenology does as well

6 Outline Introduction and phenomena: a QCP, and multipolar phases QFT: what we need Lifshitz NLsM Limits and analysis

7 Frustrated ferromagnet 1d S=1/ chain J 1 <0 J >0 AF H = J 1 i S i S i+1 + J i S i S i+ h i S i z, Compound J 1, J Cu-O-Cu T N H s (K) (deg) (K) (T) Li ZrCuO 4 [1, 13] 151, Rb Cu Mo 3 O 1 [14, 15] 138, , < , PbCuSO 4 (OH) [16 18] 100, , LiCuSbO 4 [19] 75, , 95.0 < , 96.8 LiCu O [0 ] 69, 43 9., LiCuVO 4 [3 31] 19, NaCuMoO 4 (OH) 51, , LiCuVO 4 K. Nawa et al, arxiv:

8 Frustrated ferromagnet 1d S=1/ chain J 1 <0 J >0 AF H = J 1 i S i S i+1 + J i S i S i+ h i S i z, M PM gap 0 1/5 1 J /( J 1 +J )

9 Frustrated ferromagnet 1d S=1/ chain J 1 <0 J >0 AF H = J 1 i S i S i+1 + J i S i S i+ h i S i z, M PM gap 0 1/5 1 J /( J 1 +J )

10 Frustrated ferromagnet 1d S=1/ chain J 1 <0 J >0 AF H = J 1 i S i S i+1 + J i S i S i+ h i S i z, M PM gap 0 1/5 1 J /( J 1 +J ) very weakly dimerized

11 Multipolar phases H/( J 1 +J ) Hikihara et al, 008 Sudan et al, VC 0 1/5 1 J /( J 1 +J )

12 Multipolar phases H/( J 1 +J ) Hikihara et al, 008 Sudan et al, VC 0 1/5 1 J /( J 1 +J )

13 Magnon BEC 1-magnon E-E = ε 1 + h E h S z =-1 Magnon (quasi-)bec hs i i e iqx i

14 Magnon BEC 1-magnon E-E = ε 1 + h MCE C (T) φ = 1.55(5) φ = 0.15 T (K) T (K) B (T)- B c 0.05 'B c ' (T) T w (K) B (T) ð Þ T. Radu et al, 007

15 Magnon BEC 1-magnon E-E = ε 1 + h T. Giamarchi et al, 008

16 Magnon binding 1-magnon -magnon bound state E-E = ε 1 + h E-E = ε + h E S z =- S z =-1 ε < ε 1 : molecular bound state h Formation of molecular fluid For d>1 at T=0 this is a molecular BEC = true spin nematic

17 Hidden order No dipolar order hs z i i M =0 hs ± i i =0 + hsi S j i e i j / S z =1 gap Nematic order hs + i S+ i+a i6=0 nematic director Magnetic quadrupole moment Symmetry breaking U(1) Z can think of a fluctuating fan state

18 Multipolar phases H/( J 1 +J ) Hikihara et al, 008 Sudan et al, VC 0 1/5 1 spin nematic quadrupolar J /( J 1 +J ) A progression of higher and higher multipolar phases on approaching the QCP!

19 Multipolar phases H/( J 1 +J ) Hikihara et al, 008 Sudan et al, VC 0 1/5 1 J /( J 1 +J ) Is there a QFT that describes this region?

20 A QFT? H/( J 1 +J ) 4 3 VC 0 1/5 1 n h S J /( J 1 +J ) n i Is this behavior generic? Is the cascade infinite, or does it terminate? Can a single QFT describe an infinite number of order parameters? Is this specific to one dimension?

21 A QFT? H/( J 1 +J ) 4 3 VC 0 1/5 1 n h S J /( J 1 +J ) n i A strong constraint: Entire green area including the QCP itself has exact trivial ground state Not a CFT

22 Lifshitz Point Z S = Effective action - NLσM ˆm =1 dxd isa B x ˆm + x ˆm + x ˆm 4 Z h ˆm z WZW/Berry phase term A B = ˆm 1@ ˆm ˆm 1 1+ ˆm 3. tunes QCP two symmetry allowed interactions at O(q 4 ) A All properties near Lifshitz point obey one parameter universality dependent upon u/k ratio

23 Lifshitz Point S = Z dxd isa B x ˆm + x ˆm + x ˆm 4 h ˆm z S = Intuition: behavior near the Lifshitz point should be semi-classical, since close to state which is classical r K Z x! s K x! K dxd isa B [ˆm] + sgn( x ˆm x ˆm + x ˆm 4 h ˆm z Large parameter: saddle point! v = u K h = hk

24 S = r K Z Lifshitz point dxd isa B [ˆm] + sgn( x ˆm x ˆm + x ˆm 4 h ˆm z v derives from quantum fluctuations Need it be positive? ˆm ˆm x x ˆm = x ˆm x ˆm Theory is stable for v>-1 In fact, v<0 Semiclassical large s limit: v 3/s s=1/ exact -magnon calculation v = 5/8

25 Saddle point r K Z S = dxd isa B [ˆm] + sgn( x ˆm x ˆm + x ˆm 4 h ˆm z h K first order local instability of state IC cone ˆm = 0 cos(qx + ) ± sin(qx + ) p 1 1 A spiral 1 <v< 1 4 h c = 8K p v (1 p v ) > K

26 Multipolar phases H/( J 1 +J ) Sudan et al, VC 0 1 first order metamagnetism J /( J 1 +J )

27 Saddle point r K Z S = dxd isa B [ˆm] + sgn( x ˆm x ˆm + x ˆm 4 h ˆm z h K first order h K second order IC cone IC cone 0 spiral 0 spiral 1 <v< 1/4 v> 1/4 N.B.: at saddle point level there is no scale for δ

28 Beyond saddle point Issues: Fate of ordered saddle points? Endpoint of metamagnetic line? Multipolar orders?

29 Zero field Saddle point is a spiral phase ˆm(x) =ê 1 cos qx +ê sin qx (ê 1, ê, ê 3 =ê 1 ê ) form an SO(3) matrix Fluctuations are described by an SO(3) NLsM Z S e = 1 g d x Tr (@ µ O) + is topo

30 Zero field Z S e = 1 g d x Tr (@ µ O) + is topo NLsM is asymptotically free gap e 1/g 1 (SO(3)) = Z S topo Z vortex instanton carries phase factor (-1) x dimerization

31 Multipolar phases H/( J 1 +J ) 4 3 VC 0 dimerized 1 J /( J 1 +J )

32 Multipolar phases H/( J 1 +J ) dimerized 1 ˆm = VC?? cos(qx + ) ± sin(qx + ) p 1 1 A J /( J 1 +J ) S e = 1 g Z d x (@ µ ) c=1 broken TR symmetry ẑ hs i S i+1 i6=0

33 Metamagnetic endpoint? h K E = E cone 1 =0 0 1 <v< 1/4

34 Metamagnetic endpoint? h K E = E cone 1 =0 E E cone a Quantum corrections penalize E cone but not E 0

35 Metamagnetic endpoint? h K E = E cone 1 =0 E E cone a Quantum corrections penalize E cone but not E 0 E cone =+f(v) 5/

36 S = Z Metamagnetic endpoint? dxd isa B x ˆm + x ˆm + x ˆm 4 h ˆm z ˆm = q n 1 n (n 1ê 1 (x)+n ê (x)) + (1 n 1 n )ê 3 (x) ê 1 ê =ê 3 =ˆm sp (x) = n 1 + in = n 1 in Z S = S sp + dx d + H(, )} +O( 3 ) Bogoliubov transformation gives correction to GS energy

37 Metamagnetic endpoint? h K E = E cone 1 =0 E E cone a f(v) 5/ 0 Corrected first order curve bends slightly downward to intersect second order line

38 Metamagnetic endpoint? h K E = E cone 1 =0 E E cone a f(v) 5/ 0 c Control? v = 1/4 " E E cone 1 =0 "3 " 5/ c " 1

39 Multipolar phases H/( J 1 +J ) 4 3 VC 0 dimerized 1 J /( J 1 +J ) Still need to understand multipolar phases!

40 Instabilities h K Choose E =0 E cone =0 1 =0 0 What about multi-particle instabilities?

41 Low density limit ˆm x + i ˆm y = 1/ ˆm z =1 Low energy 1 e iqx + e iqx L a (@ + h K x) a + 1 [( 1 1 ) +( ) ] = 4K (1 + 4v) = (5 + 4v) K " < 0 +

42 Low density limit H = 4 i + 1 X i<j (x i x j ) 1 " < 0 attractive delta-function gas! n = b n(n 1) 6 b = collapse: bound states have size `n n 1 1 n" 1 8 = " 3 8

43 Instabilities h K Choose E =0 4 =0 E cone =0 3 =0 =0 1 =0 0 At low density level it appears higher bound state instabilities dominate

44 Instabilities h K Choose E =0 4 =0 E cone =0 3 =0 =0 1 =0 0 But the bound states cannot get arbitrarily deep - low density approximation is violated

45 A guess Scaling Matching? n " 3 n 3 F(n 1/, 1/ " ) n 1/ 1 F(X, Y ) 1/X f(y ) Suggests maximum bound state n max 1/ 1/" (at this scale, 3-body interactions enter)

46 Choose E =0 Instabilities h K E cone =0 0 Expect that n-boson bound states bend with increasing n to approach continuum line

47 Choose E =0 Instabilities h K E cone = Expect that n-boson bound states bend with increasing n to approach continuum line

48 Summary H S = Z 0 n max 1/ 4 c " 3 VC Lifshitz point is a parent of many phases dxd isa B x ˆm + x ˆm + x ˆm 4 h ˆm z

49 Other frustrated ferromagnets In 1+1d, we could figure out (nearly) everything by numerically exact methods (DMRG) But in d>1, we have fewer tools but plenty of experiments

50 Eg. a frustrated ferrimagnet Single crystals of volborthite crystal 1. natural leaf crystals, long time ago. low-quality polycrystalline samples by precipitation, high-quality polycrystalline samples by hydrothermal annealing, small single crystals, large arrowhead-shaped crystals, 013 volborthite Ishikawa s crystal F chains J AF 1 mm 0. mm Hiroi group S= Z dxdd 1 yd isab [m ] m + m + m + m 4 same saddle point analysis applies... hm z