Emergent Coherence From Field-Induced Instabilities of a - PowerPoint PPT Presentation

Emergent Coherence From Field-Induced Instabilities of a Fractionalized Quantum Spin Liquid Mukul S. Laad (1), S. Acharya (2), A. Taraphder (2) (1) IMSc, Chennai, India, (2) IIT Kharagpur, India February 13, 2015

Outline Motivation. Kitaev Honeycomb Model: Fractionalization and Topological Order Our Model: Simplest Perturbed Kitaev Model Emergent Phenomena from a Perturbed Fractionalized Spin Liquid Applications Discussion and Open Issues.

Motivation Quantum Spin Liquids (QSL): Non-magnetic ground states of quantum spin models which do not spontaneously break any symmetries of the Hamiltonian. Elusive because (even quantum) spins generically “like to order”. Exception(s) (Any- S ) Heisenberg model on a Kagome lattice. Long-standing open problem. Quantum version (Herbertsmithite) shows finite- T signatures of a critical QSL (Helton et al, Mendels et al,...) nature of ground state (complex VBS, Z 2 QSL, U (1)-RVB) unsettled and controversial.

Susceptibility data

Motivation .... Increasing number of real Mott insulating TMOs with geometrically frustrated lattices (triangle, kagome) or with frustration induced by orbital degrees of freedom (Iridates). (Pseudo)spin frustration consequence of directionality of orbital hoppings in Mott-insulating TMO. Kugel-Khomskii spin-orbital Hamiltonian is frustrated in the orbital sector. However, care needed since crystal-field, spin-orbit, extended Heisenberg couplings can generically play spoilsport. But may it still be possible to consider these as perturbations over the idealized frustrated model???

Raman Shift data

� � � S a i S a H = J a j + J S i . S j + ( ... ) a < i , j > a < i , j > Artificially Engineered Kitaev Models with “Simple” Perturbations, e.g, Zeeman field! Explicit proposal of specially engineered Josephson Junction arrays (F. Nori’s group) For J = 0 rigorous topological order (TO). Mukul S. Laad (1), S. Acharya (2), A. Taraphder (2) ((1) IMSc, Chennai, India, (2) IIT Kharagpur, India) Emergent Coherence From Field-Induced Instabilities of a Fractionalized Quantum Spin Liquid February 13, 2015 7 / 35

Josephson Junction Array (Phys. Rev. B 81, 014505 (2010)). Φ 4 (a) V 4 4 4 4 4 z y x C m 1 1 1 1 E J C J E J C J M Φ 2 L Φ 3 C g V 2 V 3 V 1 Φ 1 C 2 2 3 3 2 2 3 3 (b) σ z σ y 3 2 σ x p 4 σ x 1 σ y σ z 5 6

Formalism � � � S a i S a H = J a j + J S i . S j + ( ... ) a < i , j > a < i , j > Deform honeycomb into brickwall lattice with“white” and“black” sites. Open BC: Consider JW transformation which threads the entire lattice by simple 1D path.

Formalism.. σ + i ′ j ] c † ij = 2[Π j ′ < j , i σ i ′ j ′ z ][Π i ′ < i σ z ij σ z ij = (2 c † ij c ij − 1) Majoranas: A w = ( c − c † ) w / i , B w = ( c + c † ) w and A b = ( c + c † ) b , B b = ( c − c † ) b / i , followed by the introduction of fermions c = ( A w + iA b ) / 2 , c † = ( A w − iA b ) / 2. H K = − i � � � 4[ J x A w A b − J y A b A w − J z α bw A b A w ] x − bonds y − bonds z − bonds Where, α bw = iB b B w defined on each Z bond. With [ α bw , H K ]= ± 1

Formalism.... 2 ( A w + iA b ), c † = 1 Applying the transformations c = 1 2 ( A w − iA b ), we get H K 1 = 1 � ( c † i + c i )( c † � ( c † i + c i )( c † 4[ J x i + e x + c i + e x ) + J y i + e y − c i + e y )] i i � α i (2 c † H K 2 = J z i c i − 1) i Local order parameters! i ( c † − c ) w σ z 2 b ( c † − c ) w Consider σ y 1 w σ z 2 b σ x 3 w = 1 2 b σ z 1 w σ z = i ( c † + c ) 1 w ( c † + c ) 3 w = iB 1 w B 3 w and σ x 5 w σ y 6 b σ z 4 b = iB 4 b B 6 b I h = σ y 3 w σ y 1 w σ z 2 b σ x 4 b σ z 5 w σ x 6 b = α 34 α 16 ; [ I h . H K ] = 0.

Formalism..... Vortex variables products of z consecutive Ising bond variables α r [ α i , c i ] = 0 = [ α i , c † i ], G.S:- All α i = 1( − 1). After Fourier transformation we get, q c q + i ∆ q � q c † [ ǫ q c † 2 ( c † H K = − q + h . c )] q ǫ q = 1 4[2 J z − 2 J x cosq x − 2 J y cosq y ] ∆ q = 2 J x sinq x + 2 J y sinq y wave function: | G > = Π k ( u k + v k c † k c † − k ) | 0 � )

Perturbations i S z Simplest perturbation: “External” Zeeman field, H z = − h z � i ; H = H K + H z Naive expectation: field induced magnetization, perhaps metamagnetic transition. In Kitaev case, however, S z = ib z c , [ b x j , H ] = 0 = [ b y i b x i b Y j , H ] ∀ ( ij ) � xx , yy = ⇒ emergent, local Z 2 symmetries. Topological order (TO) only partially lifted, as [ b z i b z j , H ] � = 0

Nature of the remnant TO! Focus on the XX-YY part. For a single chain, can solve exactly! For J x � = J y , ǫ ± � ( J 2 x + J 2 q = ± y + 2 J x J y cosq x ); lower band full, energy gap. For J x = J y ; gap closes continuously. Transition does not involve change of symmetry, but of TO. i , S y l = i τ y We can write, S x i = τ x i − 1 τ x i = Π 2 N l H K = � N 2 i + J y τ y i =1 ( J x τ x 2 i − 2 τ x 2 i ), 1D QIM! 2 i > ∼ [1 − ( J y J x ) 2 ] 1 / 4 For J x > J y :- Lim i →∞ < τ x o τ x = Lim i →∞ < Π 2 i +1 l =2 S Y � = 0 > l

Hence string orders both melt at QCP ( J x = J y ). Due to emergent d=1 GLS partial topological order survives. The QCP is easy to characterize in dual variables, where two spin j − S y i S y nematic ordered states < S x i S x j > = ± < n > , simultaneously vanish at J x = J y (“spin liquid”!) How does the field induced magnetization along ZZ-bonds“interplay” with remnant TO above? Consequences?

Our work starts here Clearer picture from JW fermion language! i ( c † � H z = 2 h z i α i + h . c ) i q c q + i ∆ q − q + h . c )] + J z � q c † � (2 n α, i − 1)(2 c † [ ǫ q c † 2 ( c † H K = i c i − 1) 4 q i “Hubbard like” model of JW fermions. p-wave BCS pairing. onsite “Hubbard” U = J z . local “spin-flip” or hybridization. = ⇒ orbital selectivity on a 2-D square lattice.

JW .. However, as [ n i α , H ] � = 0, local Z 2 gauge symmetry is lost. Gauge field becomes truly dynamical. Now, no exact solution. However can still be solved almost exactly as: xx,yy spin correlations exactly subsumed into bilinears of JW fermions. At h z = 0 spin correlations rogorously only 1 lattice spacing long. Problem is similar to Anderson OC ; however, singular behavior cut off by “Dirac” JW fermion spectrum, and by non-zero ( J z / 2) α fermion energy (Baskaran et al. 2007, Knolle et al. 2014). For h z � = 0, an approximation, however is expected to be adequate.

JW ... Impurity solver: Two-band IPT. Works quantitatively for related spinful Anderson Lattice model. Expect p-wave BCS+ field induced magnetizaton, perhaps metamagnetic quantum criticality. Surprises in Store!!

Susceptibility : anisotropic Kitaev limit Small h z : spin liquid remains stable (symmetry protected TO). J x = J y > J z , For J z < 0 . 25 J x m ( h z ) smooth function of h z .

Susceptibility : anisotropic Kitaev limit The nature is similar to conventional field-induced magnetization in a “free e − ” paramagnet. However, m ( h z ) ∝ h α , where 0.78 < α < 1.0. Using exact GFs of KM, can show that < S z i ; S z j > ∝ ( i − j ) − 4 (Feigelman et al.).

Plateaus and jumps: isotropic Kitaev limit However for h > h c z , we find a remarkable series of magnetization plateaus in m h z vs h z at m sat = 1 m 16 , 1 12 , 1 11 , 1 9 , 1 8 , 1 7 , 2 13 , 1 6 , 2 11 , 1 5 , 2 9 , 1 4 , 2 7 , 1 3 , 3 8 , 3 7 , 2 5 , 4 7 , 5 8 , 3 4 .

Both even and odd denominator plateaus. Odd denominator plateaus in σ xy well-known in FQHE, which is also the odd only other example of a real system showing rigorous TO. Also the possibility of even denominators is observed in Shastry-sutherland models (Mila’s talk): crystals of two triplon bound states. due to competition between frustration and field induced magnetization.

So no relation to FQHE but to “incompressible” solids of kink-dipole crystals (excitonic solids), sandwiched between BECs of kink-dipoles. Oscillations in χ h z as dHvA or SdH quantum oscillations of JW fermions in partially magnetized spin liquid phase. Hidden coherence in a spin liquid (Anderson, 1973). Here due to nodal Bogoliubov (p-wave) fermions in H K the TO phase of KM unstable to an intricate sequence of partial ordered “solids” co-existing with remnant of TO state (before reaching saturation).

Spectral functions : different plateaus Clear orbital selectivity: G αα ( ω ) = ⇒ Kondo Insulator. G cc ( ω ): “spin-metal” of “nodal” JW-Bogoliubov qps.

Large-scale spectral weight reshuffling across energy scales O(2 J x ) occurs. Thus the plateaus originate from between Mott localization ( J z ) and hybridization + hopping (h, J x ). Can be mssed by static HF. Alike FL* (c.f f electron QCP, Senthil et al.; PRL 2003) Topological FL ∗ state!

Dispersion and “Fermi surface” G − 1 cc ( k , ω ) = 0