Perturbing the U(1) Dirac Spin Liquid State in Spin-1/2 kagome
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1 Perturbing the U(1) Dirac Spin Liquid State in Spin-1/2 kagome Raman scattering, magnetic field, and hole doping Wing-Ho Ko MIT January 25, 21
2 Acknowledgments Acknowledgments Xiao-Gang Wen Patrick Lee Tai-Kai Ng Ying Ran Zheng-Xin Liu Wing-Ho Ko (MIT) Perturbing the U(1) DSL State January 25, 21 2 / 34
3 Outline Outline 1 The Spin-1/2 Kagome Lattice 2 Derivation and Properties of the U(1) DSL State 3 Raman Scattering 4 External Magnetic Field 5 Hole Doping Reference: Ran, Ko, Lee, & Wen, PRL 12, 4725 (29) Ko, Lee, & Wen, PRB 79, (29) Ko, Liu, Ng, & Lee, PRB 81, (21) Wing-Ho Ko (MIT) Perturbing the U(1) DSL State January 25, 21 3 / 34
4 Outline The Spin-1/2 Kagome Lattice 1 The Spin-1/2 Kagome Lattice 2 Derivation and Properties of the U(1) DSL State 3 Raman Scattering 4 External Magnetic Field 5 Hole Doping Wing-Ho Ko (MIT) Perturbing the U(1) DSL State January 25, 21 4 / 34
5 The Spin-1/2 Kagome Lattice The nearest-neighbor antiferromagnetic Heisenberg model is highly frustrated on the kagome lattice. kagome lattice = 2D lattice of corner-sharing triangles. Simple Néel order does not work well on the kagome lattice. Classical ground states: i S i =. # classical ground states e N. [Chalker et al., PRL 68, 855 (1992)]. Wing-Ho Ko (MIT) Perturbing the U(1) DSL State January 25, 21 5 / 34
6 The Spin-1/2 Kagome Lattice The nearest-neighbor antiferromagnetic Heisenberg model is highly frustrated on the kagome lattice. kagome lattice = 2D lattice of corner-sharing triangles. Simple Néel order does not work well on the kagome lattice.? Classical ground states: i S i =. # classical ground states e N. [Chalker et al., PRL 68, 855 (1992)]. Wing-Ho Ko (MIT) Perturbing the U(1) DSL State January 25, 21 5 / 34
7 The Spin-1/2 Kagome Lattice In the quantum case, singlet formation is possible and may be favored. 1D chain: E Néel = S 2 J E singlet = 1 S(S + 1)J 2 per bond per bond Higher dimensions = singlet less favorable. kagome is highly frustrated = rare opportunity of realizing a singlet ground state. vs. Wing-Ho Ko (MIT) Perturbing the U(1) DSL State January 25, 21 6 / 34
8 The Spin-1/2 Kagome Lattice In the quantum case, singlet formation is possible and may be favored. 1D chain: E Néel = S 2 J E singlet = 1 S(S + 1)J 2 per bond per bond Higher dimensions = singlet less favorable. kagome is highly frustrated = rare opportunity of realizing a singlet ground state. vs. Wing-Ho Ko (MIT) Perturbing the U(1) DSL State January 25, 21 6 / 34
9 The Spin-1/2 Kagome Lattice In the quantum case, singlet formation is possible and may be favored. 1D chain: E Néel = S 2 J E singlet = 1 S(S + 1)J 2 per bond per bond Higher dimensions = singlet less favorable. kagome is highly frustrated = rare opportunity of realizing a singlet ground state. vs. Wing-Ho Ko (MIT) Perturbing the U(1) DSL State January 25, 21 6 / 34
10 The Spin-1/2 Kagome Lattice Two major classes of singlet states: valence bond solids (VBS) and spin liquid (SL). In a VBS, certain singlet bonds are preferred, which results in a symmetry-broken state. In a SL, different bond configurations superpose, which results in a state that breaks no lattice symmetry. A VBS state generally has a spin gap, while a SL state can be gapped or gapless. vs. = 1 2 ( ) Wing-Ho Ko (MIT) Perturbing the U(1) DSL State January 25, 21 7 / 34
11 The Spin-1/2 Kagome Lattice For S =1/2 kagome, the leading proposals are the 36-site VBS and the U(1) Dirac spin liquid. The 36-site VBS pattern is found in series expansion [Singh and Huse, PRB 76, 1847 (27)] and entanglement renormalization [Evenbly and Vidal, arxiv: ]. From VMC, the U(1) Dirac spin liquid (DSL) state has the lowest energy among various SL states, and is stable against small VBS perturbations [e.g., Ran et al., PRL 98, (27)]. Exact diagonalization: initially found small ( J/2) spin gap; now leaning towards a gapless proposal [Waldtmann et al., EPJB 2, 51 (1998); arxiv: ]. vs. Wing-Ho Ko (MIT) Perturbing the U(1) DSL State January 25, 21 8 / 34
12 The Spin-1/2 Kagome Lattice Experimental realization of S = 1/2 kagome: Herbertsmithite ZnCu 3 (OH) 6 Cl 2. Herbertsmithite: layered structure with Cu forming an AF kagome lattice. Caveats: Zn impurities and Dzyaloshinskii Moriya interactions. Experimental Results [e.g., Helton et al., PRL 98, 1724 (27); Bert et al., JP:CS 145, 124 (29)]: Neutron scattering: no magnetic order down to 1.8 K. µsr: no spin freezing down to 5 µk. Heat capacity: vanishes as a power law as T. Spin susceptibility: diverges as T. NMR shift: power law as T. Cu O H Cl Zn Wing-Ho Ko (MIT) Perturbing the U(1) DSL State January 25, 21 9 / 34
13 The Spin-1/2 Kagome Lattice Research motivation: Deriving further experimental consequences of the U(1) DSL state. All experiments point to a state without magnetic order. But more data is needed to tell if it is a VBS state or a SL state, and which VBS/SL state it is. Without concrete theory, the experimental data are hard to interpret. Without concrete theory, unbiased theoretical calculations are difficult. The U(1) Dirac spin-liquid state is a theoretically interesting exotic state of matter. Thus, our approach: Assume the DSL state and consider further experimental consequences. Wing-Ho Ko (MIT) Perturbing the U(1) DSL State January 25, 21 1 / 34
14 The Spin-1/2 Kagome Lattice Research motivation: Deriving further experimental consequences of the U(1) DSL state. All experiments point to a state without magnetic order. But more data is needed to tell if it is a VBS state or a SL state, and which VBS/SL state it is. Without concrete theory, the experimental data are hard to interpret. Without concrete theory, unbiased theoretical calculations are difficult. The U(1) Dirac spin-liquid state is a theoretically interesting exotic state of matter. Thus, our approach: Assume the DSL state and consider further experimental consequences. Wing-Ho Ko (MIT) Perturbing the U(1) DSL State January 25, 21 1 / 34
15 Outline Derivation and Properties of the U(1) DSL State 1 The Spin-1/2 Kagome Lattice 2 Derivation and Properties of the U(1) DSL State 3 Raman Scattering 4 External Magnetic Field 5 Hole Doping Wing-Ho Ko (MIT) Perturbing the U(1) DSL State January 25, / 34
16 Derivation and Properties of the U(1) DSL State Deriving the U(1) DSL state: Slave boson formulation Start with Heisenberg (or more generally t J) model: H tj = ij J ( S i S j 1 4 n ) ) in j t (c iσ c jσ + h.c. ; σ c i c i 1 Apply the slave boson decomposition [Lee et al., RMP 78, 17 (26)]: S i = 1 2 α,β f iα τ α,β f iβ ; c iσ = f iσ h i ; f i f i + f i f i + h i h i = 1 Decouple four-operator terms by a Hubbard Stratonovich transformation, with the following ansatz: χ ij σ f iσ f jσ = χe iα ij ; ij f i f j f i f j = This results in a mean-field Hamiltonian... Wing-Ho Ko (MIT) Perturbing the U(1) DSL State January 25, / 34
17 Derivation and Properties of the U(1) DSL State Deriving the U(1) DSL state: Slave boson formulation Start with Heisenberg (or more generally t J) model: H tj = ij J ( S i S j 1 4 n ) ) in j t (c iσ c jσ + h.c. ; σ c i c i 1 Apply the slave boson decomposition [Lee et al., RMP 78, 17 (26)]: S i = 1 2 α,β f iα τ α,β f iβ ; c iσ = f iσ h i ; f i f i + f i f i + h i h i = 1 Decouple four-operator terms by a Hubbard Stratonovich transformation, with the following ansatz: χ ij σ f iσ f jσ = χe iα ij ; ij f i f j f i f j = This results in a mean-field Hamiltonian... Wing-Ho Ko (MIT) Perturbing the U(1) DSL State January 25, / 34
18 Derivation and Properties of the U(1) DSL State Deriving the U(1) DSL state: Slave boson formulation Start with Heisenberg (or more generally t J) model: H tj = ij J ( S i S j 1 4 n ) ) in j t (c iσ c jσ + h.c. ; σ c i c i 1 Apply the slave boson decomposition [Lee et al., RMP 78, 17 (26)]: S i = 1 2 α,β f iα τ α,β f iβ ; c iσ = f iσ h i ; f i f i + f i f i + h i h i = 1 spinon holon Decouple four-operator terms by a Hubbard Stratonovich transformation, with the following ansatz: χ ij σ f iσ f jσ = χe iα ij ; ij f i f j f i f j = This results in a mean-field Hamiltonian... Wing-Ho Ko (MIT) Perturbing the U(1) DSL State January 25, / 34
19 Derivation and Properties of the U(1) DSL State Deriving the U(1) DSL state: Slave boson formulation Start with Heisenberg (or more generally t J) model: H tj = ij J ( S i S j 1 4 n ) ) in j t (c iσ c jσ + h.c. ; σ c i c i 1 Apply the slave boson decomposition [Lee et al., RMP 78, 17 (26)]: S i = 1 2 α,β f iα τ α,β f iβ ; c iσ = f iσ h i ; f i f i + f i f i + h i h i = 1 Decouple four-operator terms by a Hubbard Stratonovich transformation, with the following ansatz: χ ij σ f iσ f jσ = χe iα ij ; ij f i f j f i f j = This results in a mean-field Hamiltonian... spinon holon Constraint enforced by Lagrange multiplier α i Wing-Ho Ko (MIT) Perturbing the U(1) DSL State January 25, / 34
20 Derivation and Properties of the U(1) DSL State Deriving the U(1) DSL state: Slave boson formulation Start with Heisenberg (or more generally t J) model: H tj = ij J ( S i S j 1 4 n ) ) in j t (c iσ c jσ + h.c. ; σ c i c i 1 Apply the slave boson decomposition [Lee et al., RMP 78, 17 (26)]: S i = 1 2 α,β f iα τ α,β f iβ ; c iσ = f iσ h i ; f i f i + f i f i + h i h i = 1 Decouple four-operator terms by a Hubbard Stratonovich transformation, with the following ansatz: χ ij σ f iσ f jσ = χe iα ij ; ij f i f j f i f j = This results in a mean-field Hamiltonian... spinon holon Constraint enforced by Lagrange multiplier α i Wing-Ho Ko (MIT) Perturbing the U(1) DSL State January 25, / 34
21 Derivation and Properties of the U(1) DSL State Deriving the U(1) DSL state: Slave boson formulation Start with Heisenberg (or more generally t J) model: H tj = ij J ( S i S j 1 4 n ) ) in j t (c iσ c jσ + h.c. ; σ c i c i 1 Apply the slave boson decomposition [Lee et al., RMP 78, 17 (26)]: S i = 1 2 α,β f iα τ α,β f iβ ; c iσ = f iσ h i ; f i f i + f i f i + h i h i = 1 Decouple four-operator terms by a Hubbard Stratonovich transformation, with the following ansatz: χ ij σ f iσ f jσ = χe iα ij ; ij f i f j f i f j = This results in a mean-field Hamiltonian... spinon holon Constraint enforced by Lagrange multiplier α i Wing-Ho Ko (MIT) Perturbing the U(1) DSL State January 25, / 34
22 Derivation and Properties of the U(1) DSL State Deriving the U(1) DSL state: Emergent gauge field H MF = f iσ iσ (iαi µ F )f iσ 3χJ 8 ij,σ (eiα ij f iσ f jσ +h.c.) + i h i (iαi µ B )h i tχ ij (eiα ij h i h j + h.c.) The α field is an emergent gauge field, corresponding to gauge symmetry f e iθ f, h e iθ h. At the lattice level α is a compact gauge field (i.e., monopoles are allowed). But with Dirac fermions, the system can be in a deconfined phase (i.e., monopoles can be neglected) [Hermele et al., PRB 7, (24)]. Wing-Ho Ko (MIT) Perturbing the U(1) DSL State January 25, / 34
23 Derivation and Properties of the U(1) DSL State Deriving the U(1) DSL state: Band Structure H MF = f iσ iσ (iαi µ F )f iσ 3χJ 8 ij,σ (eiα ij f iσ f jσ +h.c.) + i h i (iαi µ B )h i tχ ij (eiα ij h i h j + h.c.) Neglecting fluctuation of α, spinons and holons are decoupled. Mean-field ansatz for SL state can be specified by pattern of α flux. U(1) Dirac spin liquid state: flux per and flux per. flux = unit cell doubled in band structures. Wing-Ho Ko (MIT) Perturbing the U(1) DSL State January 25, / 34
24 Derivation and Properties of the U(1) DSL State Deriving the U(1) DSL state: Band Structure H MF = f iσ iσ (iαi µ F )f iσ 3χJ 8 ij,σ (eiα ij f iσ f jσ +h.c.) + i h i (iαi µ B )h i tχ ij (eiα ij h i h j + h.c.) Neglecting fluctuation of α, spinons and holons are decoupled. Mean-field ansatz for SL state can be specified by pattern of α flux. U(1) Dirac spin liquid state: flux per and flux per. flux = unit cell doubled in band structures. Enlarged Cell; Original Cell Wing-Ho Ko (MIT) Perturbing the U(1) DSL State January 25, / 34
25 Derivation and Properties of the U(1) DSL State Deriving the U(1) DSL state: Band Structure H MF = f iσ iσ (iαi µ F )f iσ 3χJ 8 ij,σ (eiα ij f iσ f jσ +h.c.) + i h i (iαi µ B )h i tχ ij (eiα ij h i h j + h.c.) Neglecting fluctuation of α, spinons and holons are decoupled. Mean-field ansatz for SL state can be specified by pattern of α flux. U(1) Dirac spin liquid state: flux per and flux per. flux = unit cell doubled in band structures. t eff = +t ; t eff = t Wing-Ho Ko (MIT) Perturbing the U(1) DSL State January 25, / 34
26 Derivation and Properties of the U(1) DSL State Properties of the U(1) DSL state: Band structure Real space k-space t eff = +t ; t eff = t Wing-Ho Ko (MIT) Perturbing the U(1) DSL State January 25, / 34
27 Derivation and Properties of the U(1) DSL State Properties of the U(1) DSL state: Band structure Real space k-space t eff = +t ; t eff = t x2 2/ 3 / 3 / 3 2/ 3 ky Wing-Ho Ko (MIT) Perturbing the U(1) DSL State January 25, / 34
28 Derivation and Properties of the U(1) DSL State Properties of the U(1) DSL state: Band structure Real space k-space x2 / 2 / 2 kx t eff = +t ; t eff = t x2 2/ 3 / 3 / 3 2/ 3 ky Wing-Ho Ko (MIT) Perturbing the U(1) DSL State January 25, / 34
29 Derivation and Properties of the U(1) DSL State Properties of the U(1) DSL state: Band structure Real space k-space x2 / 2 / 2 kx / 2 / 2 kx t eff = +t ; t eff = t x2 2/ 3 / 3 / 3 2/ 3 ky Wing-Ho Ko (MIT) Perturbing the U(1) DSL State January 25, / 34
30 Derivation and Properties of the U(1) DSL State Properties of the U(1) DSL state: Band structure Real space k-space x2 / 2 / 2 kx / 2 / 2 kx t eff = +t ; t eff = t x2 2/ 3 / 3 / 3 2/ 3 ky Wing-Ho Ko (MIT) Perturbing the U(1) DSL State January 25, / 34
31 Derivation and Properties of the U(1) DSL State Properties of the U(1) DSL state: Band structure Real space k-space x2 / 2 / 2 kx / 2 / 2 kx t eff = +t ; t eff = t x2 2/ 3 / 3 / 3 2/ 3 ky Wing-Ho Ko (MIT) Perturbing the U(1) DSL State January 25, / 34
32 Derivation and Properties of the U(1) DSL State Properties of the U(1) DSL state: Thermodynamics and correlation At low energy, the U(1) DSL state is described by QED 3. i.e., gauge field coupled to Dirac fermions in (2+1)-D. Thermodynamics of the U(1) DSL state is dominated by the spinon Fermi surface. Zero-field spin susceptibility: χ(t ) T. Heat capacity: C V (T ) T 2. U(1) DSL state is quantum critical many correlations decay algebraically [Hermele et al., PRB 77, (28)]. Emergent SU(4) symmetry among Dirac nodes = different correlations can have the same scaling dimension. Wing-Ho Ko (MIT) Perturbing the U(1) DSL State January 25, / 34
33 Outline Raman Scattering 1 The Spin-1/2 Kagome Lattice 2 Derivation and Properties of the U(1) DSL State 3 Raman Scattering 4 External Magnetic Field 5 Hole Doping Wing-Ho Ko (MIT) Perturbing the U(1) DSL State January 25, / 34
34 Raman Scattering Raman scattering in Mott-Hubbard system: the Shastry Shraiman formulation Raman scattering = inelastic scattering of photon. Good for studying excitations of the system. Probe only excitations with q. We are concerned with a half-filled Hubbard system, in the regime where ω i ω f U and ω i U. Both initial and final states are spin states. = T-matrix can be written in terms of spin operators. Intermediate states are dominated by the sector where i n i n i = 1. The T-matrix can be organized as an expansion in t/(u ω i ) [Shastry & Shraiman, IJMPB 5, 365 (1991)]. T (2) t2 U ω i S i S j +... Wing-Ho Ko (MIT) Perturbing the U(1) DSL State January 25, / 34
35 Raman Scattering Spin-chirality terms in the Shastry Shraiman formulation Because of holon-doublon symmetry, there is no t 3 order contribution For the square and triangular lattice, there is no t 4 order contribution to spin-chirality because of a non-trivial cancellation between 3-site and 4-site pathways. But such cancellation is absent in the kagome lattice. Wing-Ho Ko (MIT) Perturbing the U(1) DSL State January 25, / 34
36 Raman Scattering Spin-chirality terms in the Shastry Shraiman formulation Because of holon-doublon symmetry, there is no t 3 order contribution ,2 1 2 i For the square and triangular lattice, there is no t 4 order contribution to spin-chirality because of a non-trivial cancellation between 3-site and 4-site pathways. But such cancellation is absent in the kagome lattice. Wing-Ho Ko (MIT) Perturbing the U(1) DSL State January 25, / 34
37 Raman Scattering Spin-chirality terms in the Shastry Shraiman formulation Because of holon-doublon symmetry, there is no t 3 order contribution. 2,3 ii i For the square and triangular lattice, there is no t 4 order contribution to spin-chirality because of a non-trivial cancellation between 3-site and 4-site pathways. 3 But such cancellation is absent in the kagome lattice. Wing-Ho Ko (MIT) Perturbing the U(1) DSL State January 25, / 34
38 Raman Scattering Spin-chirality terms in the Shastry Shraiman formulation Because of holon-doublon symmetry, there is no t 3 order contribution. iii 2 ii i For the square and triangular lattice, there is no t 4 order contribution to spin-chirality because of a non-trivial cancellation between 3-site and 4-site pathways. 3 But such cancellation is absent in the kagome lattice. Wing-Ho Ko (MIT) Perturbing the U(1) DSL State January 25, / 34
39 Raman Scattering Spin-chirality terms in the Shastry Shraiman formulation Because of holon-doublon symmetry, there is no t 3 order contribution. iii ii 1,2 i i For the square and triangular lattice, there is no t 4 order contribution to spin-chirality because of a non-trivial cancellation between 3-site and 4-site pathways. 3 But such cancellation is absent in the kagome lattice. Wing-Ho Ko (MIT) Perturbing the U(1) DSL State January 25, / 34
40 Raman Scattering Spin-chirality terms in the Shastry Shraiman formulation Because of holon-doublon symmetry, there is no t 3 order contribution. iii ii ii 1,2 3 i i For the square and triangular lattice, there is no t 4 order contribution to spin-chirality because of a non-trivial cancellation between 3-site and 4-site pathways. But such cancellation is absent in the kagome lattice. Wing-Ho Ko (MIT) Perturbing the U(1) DSL State January 25, / 34
41 Raman Scattering Spin-chirality terms in the Shastry Shraiman formulation Because of holon-doublon symmetry, there is no t 3 order contribution. iii ii 2 3 i i For the square and triangular lattice, there is no t 4 order contribution to spin-chirality because of a non-trivial cancellation between 3-site and 4-site pathways. iii 1 ii But such cancellation is absent in the kagome lattice. Wing-Ho Ko (MIT) Perturbing the U(1) DSL State January 25, / 34
42 Raman Scattering Spin-chirality terms in the Shastry Shraiman formulation Because of holon-doublon symmetry, there is no t 3 order contribution. iii ii iii ii + = i i For the square and triangular lattice, there is no t 4 order contribution to spin-chirality because of a non-trivial cancellation between 3-site and 4-site pathways. But such cancellation is absent in the kagome lattice. Wing-Ho Ko (MIT) Perturbing the U(1) DSL State January 25, / 34
43 Raman Scattering Spin-chirality terms in the Shastry Shraiman formulation Because of holon-doublon symmetry, there is no t 3 order contribution. iii ii iii ii + = i i For the square and triangular lattice, there is no t 4 order contribution to spin-chirality because of a non-trivial cancellation between 3-site and 4-site pathways. iii iii iv ii + ii iv + ii iii iv + ii iv iii i i = But such cancellation is absent in the kagome lattice. i i Wing-Ho Ko (MIT) Perturbing the U(1) DSL State January 25, / 34
44 Raman Scattering Spin-chirality terms in the Shastry Shraiman formulation Because of holon-doublon symmetry, there is no t 3 order contribution. iii ii iii ii + = i i For the square and triangular lattice, there is no t 4 order contribution to spin-chirality because of a non-trivial cancellation between 3-site and 4-site pathways. + + iv ii + iii i t4 S (U ω i ) 3 i S j S k +... But such cancellation is absent in the kagome lattice. ii iv iii i Wing-Ho Ko (MIT) Perturbing the U(1) DSL State January 25, / 34
45 Raman Scattering Raman T-matrix for the kagome geometry For kagome geometry, the Raman T-matrix decomposes into 3 irreps: T = T (A 1g )( xx+ȳy)+t (E g (1) )( xx ȳy)+t (E g (2) )( xy+ȳx)+t (A 2g )( xy ȳx) To lowest order in inelastic terms, T (E g (1) ( ) = 4t2 1 ω i U 4 ij,/ + ij,\ 2 ) ij, Si S j T (E g (2) ) = 4t2 3 ( ω i U 4 ij,/ ) ij,\ Si S j ( T (A 1g ) = t4 (ω i U) 2 ij + ) 3 ij Si S j ( T (A 2g ) = 2 3it 4 (ω i U) R ) k ( i j = S i S j S k, etc.) Wing-Ho Ko (MIT) Perturbing the U(1) DSL State January 25, 21 2 / 34
46 Raman Scattering Raman signals: spinon-antispinon pairs and gauge mode S i S j f f f f and S i (S j S k ) f f f f f f = contributions from spinon-antispinon pairs. = continuum of signal I α ( ω) = f Oα i 2 DOS( ω). At low energy, one-pair states dominates. For Dirac node, DOS 1pair E, and matrix element is suppressed in E g and A 1g, but not in A 2g. = Spinon-antispinon pairs contribute I A2g ( ω) E and I Eg /A 1g ( ω) E 3 at low energy. However, an additional collective excitation is available in A 2g : S i S j S k iχ 3 exp(i α dl) + h.c. χ3 b d 2 x = I A2g Φ b Φ b +... q 2 αα +... (Recall that f i f j χ exp(iα ij ) ) In our case (QED 3 with Dirac fermions), turns out that αα 1/ω when q [Ioffe & Larkin, PRB 39, 8988 (1989)]. Analogy: plasmon mode vs. particle-hole continuum in normal metal. Wing-Ho Ko (MIT) Perturbing the U(1) DSL State January 25, / 34
47 Raman Scattering Raman signals: spinon-antispinon pairs and gauge mode S i S j f f f f and S i (S j S k ) f f f f f f = contributions from spinon-antispinon pairs. = continuum of signal I α ( ω) = f Oα i 2 DOS( ω). At low energy, one-pair states dominates. For Dirac node, DOS 1pair E, and matrix element is suppressed in E g and A 1g, but not in A 2g. = Spinon-antispinon pairs contribute I A2g ( ω) E and I Eg /A 1g ( ω) E 3 at low energy. However, an additional collective excitation is available in A 2g : S i S j S k iχ 3 exp(i α dl) + h.c. χ3 b d 2 x = I A2g Φ b Φ b +... q 2 αα +... (Recall that f i f j χ exp(iα ij ) ) In our case (QED 3 with Dirac fermions), turns out that αα 1/ω when q [Ioffe & Larkin, PRB 39, 8988 (1989)]. Analogy: plasmon mode vs. particle-hole continuum in normal metal. Wing-Ho Ko (MIT) Perturbing the U(1) DSL State January 25, / 34
48 Raman Scattering Raman signals: spinon-antispinon pairs and gauge mode S i S j f f f f and S i (S j S k ) f f f f f f = contributions from spinon-antispinon pairs. = continuum of signal I α ( ω) = f Oα i 2 DOS( ω). At low energy, one-pair states dominates. For Dirac node, DOS 1pair E, and matrix element is suppressed in E g and A 1g, but not in A 2g. = Spinon-antispinon pairs contribute I A2g ( ω) E and I Eg /A 1g ( ω) E 3 at low energy. However, an additional collective excitation is available in A 2g : S i S j S k iχ 3 exp(i α dl) + h.c. χ3 b d 2 x = I A2g Φ b Φ b +... q 2 αα +... (Recall that f i f j χ exp(iα ij ) ) In our case (QED 3 with Dirac fermions), turns out that αα 1/ω when q [Ioffe & Larkin, PRB 39, 8988 (1989)]. Analogy: plasmon mode vs. particle-hole continuum in normal metal. Wing-Ho Ko (MIT) Perturbing the U(1) DSL State January 25, / 34
49 Raman Scattering Raman signals: spinon-antispinon pairs and gauge mode A 1g, E g A 2g Intensity Intensity ω 3 ω Energy shift??? Energy shift 1/ω Wing-Ho Ko (MIT) Perturbing the U(1) DSL State January 25, / 34
50 Raman Scattering Experimental Comparisons: Some qualitative agreements Wulferding and Lemmens [unpublished] have obtained Raman signal on herbertsmithite. At low T, data shows a broad background with a near-linear piece at low-energy. Roughly agree with the theoretical picture presented previously. Wing-Ho Ko (MIT) Perturbing the U(1) DSL State January 25, / 34
51 Outline External Magnetic Field 1 The Spin-1/2 Kagome Lattice 2 Derivation and Properties of the U(1) DSL State 3 Raman Scattering 4 External Magnetic Field 5 Hole Doping Wing-Ho Ko (MIT) Perturbing the U(1) DSL State January 25, / 34
52 External Magnetic Field External magnetic field and the formation of Landau levels In Mott systems, B-field causes only Zeeman splitting. This induces spinon and antispinon pockets near the Dirac node. However, with the emergent gauge field α, Landau levels can form spontaneously. From VMC calculations, e Prj FP.33(2)B3/2 +.(4)B 2 e Prj LL.223(6)B3/2 +.3(1)B 2 B b emf LL < efp MF to leading order in n = Landau level state is favored. Wing-Ho Ko (MIT) Perturbing the U(1) DSL State January 25, / 34
53 External Magnetic Field S z density fluctuation as gapless mode b is an emergent gauge field = its strength can fluctuate in space. The fluctuation of b is tied to the fluctuation of S z density. In long-wavelength limit, energy cost of b fluctuation = The system has a gapless mode! Derivative expansion = linear dispersion. Other density fluctuations and quasiparticle excitations are gapped = gapless mode is unique. Mathematical description given by Chern Simons theory. b Wing-Ho Ko (MIT) Perturbing the U(1) DSL State January 25, / 34
54 External Magnetic Field S z density fluctuation as gapless mode b is an emergent gauge field = its strength can fluctuate in space. The fluctuation of b is tied to the fluctuation of S z density. In long-wavelength limit, energy cost of b fluctuation = The system has a gapless mode! Derivative expansion = linear dispersion. Other density fluctuations and quasiparticle excitations are gapped = gapless mode is unique. Mathematical description given by Chern Simons theory. b Wing-Ho Ko (MIT) Perturbing the U(1) DSL State January 25, / 34
55 External Magnetic Field S z density fluctuation as gapless mode b is an emergent gauge field = its strength can fluctuate in space. The fluctuation of b is tied to the fluctuation of S z density. In long-wavelength limit, energy cost of b fluctuation = The system has a gapless mode! Derivative expansion = linear dispersion. Other density fluctuations and quasiparticle excitations are gapped = gapless mode is unique. Mathematical description given by Chern Simons theory. vs. b b Wing-Ho Ko (MIT) Perturbing the U(1) DSL State January 25, / 34
56 External Magnetic Field S z density fluctuation as gapless mode b is an emergent gauge field = its strength can fluctuate in space. The fluctuation of b is tied to the fluctuation of S z density. In long-wavelength limit, energy cost of b fluctuation = The system has a gapless mode! Derivative expansion = linear dispersion. Other density fluctuations and quasiparticle excitations are gapped = gapless mode is unique. Mathematical description given by Chern Simons theory. vs. b Wing-Ho Ko (MIT) Perturbing the U(1) DSL State January 25, / 34
57 External Magnetic Field Gapless mode and XY-ordering Recap: we found a single linearly-dispersing gapless mode, which looks like... A Goldstone boson! And indeed it is. Corresponding to this Goldstone mode is a spontaneously broken XY order. Analogy: Superfluid: ˆψ = ˆρe i ˆθ, [ˆρ, ˆθ] = i, gapless ρ fluctuation = ordered (SF) phase; XY model: S ˆ + = e i ˆθ, [Ŝz, ˆθ] = i, gapless S z fluctuation = XY ordered phase. VMC found the q = order. S + in XY model V in QED 3 = in-plane magnetization M B γ. (V monopole operator, γ its scaling dimension).67(1).(1).67(1).65(1).1(1).65(1).66(1).65(1) Wing-Ho Ko (MIT) Perturbing the U(1) DSL State January 25, / 34
58 Outline Hole Doping 1 The Spin-1/2 Kagome Lattice 2 Derivation and Properties of the U(1) DSL State 3 Raman Scattering 4 External Magnetic Field 5 Hole Doping Wing-Ho Ko (MIT) Perturbing the U(1) DSL State January 25, / 34
59 Recap: Band structure Hole Doping Real space k-space x2 / 2 / 2 kx / 2 / 2 kx t eff = +t ; t eff = t x2 2/ 3 / 3 / 3 2/ 3 ky Wing-Ho Ko (MIT) Perturbing the U(1) DSL State January 25, / 34
60 Hole Doping Hole doping and formation of Landau levels Doping can in principle be achieved by substituting Cl with S. In slave-boson picture, hole doping introduces holons and antispinons. As before, an emergent b field can open up Landau levels in the spinon and holon bands. At mean-field, E spinon B 3/2 while E holon B 2 = LL state favored. At mean-field, b optimal when antispinons form ν = 1 LL state = holons form ν = 1/2 Laughlin state. dope b f, f h Wing-Ho Ko (MIT) Perturbing the U(1) DSL State January 25, 21 3 / 34
61 Hole Doping Charge fluctuation, Goldstone mode, and superconductivity b fluctuation holon density fluctuation charge density fluctuation. Long-wavelength b fluctuation cost E if real EM-field is turned off. = a single linearly-dispersing mode Goldstone boson. This time the Goldstone boson is eaten up by the EM-field to produce a superconductor. This superconducting state breaks T -invariance. Intuitively, four species of holon binded together = expects Φ EM = hc/4e for a minimal vortex. Confirmed by Chern Simons formulation. f 1,...,2 h 1,...,4 Wing-Ho Ko (MIT) Perturbing the U(1) DSL State January 25, / 34 b
62 Hole Doping Charge fluctuation, Goldstone mode, and superconductivity b fluctuation holon density fluctuation charge density fluctuation. Long-wavelength b fluctuation cost E if real EM-field is turned off. = a single linearly-dispersing mode Goldstone boson. This time the Goldstone boson is eaten up by the EM-field to produce a superconductor. This superconducting state breaks T -invariance. Intuitively, four species of holon binded together = expects Φ EM = hc/4e for a minimal vortex. Confirmed by Chern Simons formulation. f 1,...,2 h 1,...,4 Wing-Ho Ko (MIT) Perturbing the U(1) DSL State January 25, / 34 b
63 Hole Doping Quasiparticle Statistics Intuitions Quasiparticles are bound states of semions in holon sectors and/or fermions in spinon sector. For finite energy, bound states must be neutral w.r.t. the gapless mode. There are two types of elementary quasiparticles: 1 semion-antisemion pair in holon sector; 2 spinon-holon pair ( Bogoliubov q.p. in conventional SC). All other quasiparticles can be built from these elementary ones. Statistics can be derived by treating different species as uncorrelated. Semions from holon sector = existence of semionic (mutual) statistics. h 4 h 1 l 1 Wing-Ho Ko (MIT) Perturbing the U(1) DSL State January 25, / 34
64 Hole Doping Quasiparticle Statistics Intuitions Quasiparticles are bound states of semions in holon sectors and/or fermions in spinon sector. For finite energy, bound states must be neutral w.r.t. the gapless mode. There are two types of elementary quasiparticles: 1 semion-antisemion pair in holon sector; 2 spinon-holon pair ( Bogoliubov q.p. in conventional SC). All other quasiparticles can be built from these elementary ones. Statistics can be derived by treating different species as uncorrelated. Semions from holon sector = existence of semionic (mutual) statistics. h 4 h h 4 1 f l 1 l 2 Wing-Ho Ko (MIT) Perturbing the U(1) DSL State January 25, / 34
65 Hole Doping Quasiparticle Statistics Intuitions Quasiparticles are bound states of semions in holon sectors and/or fermions in spinon sector. For finite energy, bound states must be neutral w.r.t. the gapless mode. There are two types of elementary quasiparticles: 1 semion-antisemion pair in holon sector; 2 spinon-holon pair ( Bogoliubov q.p. in conventional SC). All other quasiparticles can be built from these elementary ones. Statistics can be derived by treating different species as uncorrelated. Semions from holon sector = existence of semionic (mutual) statistics. h 4 h h 4 1 f l 1 l 2 θ = 2 l 1 l 2 Wing-Ho Ko (MIT) Perturbing the U(1) DSL State January 25, / 34
66 Hole Doping Quasiparticle Statistics Intuitions Quasiparticles are bound states of semions in holon sectors and/or fermions in spinon sector. For finite energy, bound states must be neutral w.r.t. the gapless mode. There are two types of elementary quasiparticles: 1 semion-antisemion pair in holon sector; 2 spinon-holon pair ( Bogoliubov q.p. in conventional SC). All other quasiparticles can be built from these elementary ones. Statistics can be derived by treating different species as uncorrelated. Semions from holon sector = existence of semionic (mutual) statistics. h 4 h h 4 1 h 1 l 1 l 1 θ = + = 2 l 1 l 1 Wing-Ho Ko (MIT) Perturbing the U(1) DSL State January 25, / 34
67 Hole Doping Quasiparticle Statistics Intuitions Quasiparticles are bound states of semions in holon sectors and/or fermions in spinon sector. For finite energy, bound states must be neutral w.r.t. the gapless mode. There are two types of elementary quasiparticles: 1 semion-antisemion pair in holon sector; 2 spinon-holon pair ( Bogoliubov q.p. in conventional SC). All other quasiparticles can be built from these elementary ones. Statistics can be derived by treating different species as uncorrelated. Semions from holon sector = existence of semionic (mutual) statistics. h 4 h 1 h 1 l 1 l θ = l 1 l 1 h 2 1 Wing-Ho Ko (MIT) Perturbing the U(1) DSL State January 25, / 34
68 Hole Doping Crystal momenta of quasiparticle projective symmetry group study For spinon-holon pair, k is well-defined on the original Brillouin zone. These can be recovered using Projective symmetry group (PSG). In contrast, the semion is fractionalized from a holon. = semion-antisemion pair may not have well-defined k. = Wing-Ho Ko (MIT) Perturbing the U(1) DSL State January 25, / 34
69 Hole Doping Crystal momenta of quasiparticle projective symmetry group study For spinon-holon pair, k is well-defined on the original Brillouin zone. These can be recovered using Projective symmetry group (PSG). In contrast, the semion is fractionalized from a holon. = semion-antisemion pair may not have well-defined k. * * * * * * * = k of spinon-holon pairs * * Wing-Ho Ko (MIT) Perturbing the U(1) DSL State January 25, / 34
70 Conclusions Conclusions The U(1) Dirac spin-liquid state possess many unusual properties and may be experimentally realized in herbertsmithite. The Raman signal of the DSL state has a broad background (contributed by spinon-antispinon continuum) and a 1/ω singularity (contributed by collective [gauge] excitations). Under external magnetic field, the DSL state forms Landau levels, which corresponds to a XY symmetry broken state with Goldstone boson corresponding to S z density fluctuation. When the DSL state is doped, an analogous mechanism give rise to an Anderson Higgs scenario and hence superconductivity. But minimal vortices carry hc/4e flux and the system contains exotic quasiparticle having semionic mutual statistics. Reference: Ran, Ko, Lee, & Wen, PRL 12, 4725 (29) Ko, Lee, & Wen, PRB 79, (29) Ko, Liu, Ng, & Lee, PRB 81, (21) Wing-Ho Ko (MIT) Perturbing the U(1) DSL State January 25, / 34
71 Appendix Appendix (a.k.a. hip pocket slides) Wing-Ho Ko (MIT) Perturbing the U(1) DSL State January 25, / 34
72 Appendix Comparison of ground-state energy estimate Method Max. Size Energy State Ref. Exact Diag [1] DMRG (7) SL [2] VMC (2) U(1) Dirac SL [3] Series Expan..433(1) 36-site VBS [4] Entang. Renorm site VBS [5] [1] Waldtmann et al., EPJB 2, 51 (1998) [2] Jiang et al., PRL 11, (28) [3] Ran et al., PRL 98, (27) [4] Singh and Huse, PRB 76, 1847 (27) [5] Evenbly and Vidal, arxiv: Wing-Ho Ko (MIT) Perturbing the U(1) DSL State January 25, / 34
73 Appendix Magnified band structure of DSL state, with scales 2/ 3 / 3 Energy (χj) Energy (χj) 2 x / 3 / 2 / 2 k y / x2 k x k x = k y = Energy (χj) -2.8 / 2 / k x Brillouin zone Wing-Ho Ko (MIT) Perturbing the U(1) DSL State January 25, / 34
74 Appendix Raman signals contributed by spinon-antispinon: full scale Intensity (arb. units) Intensity (arb. units) Intensity (arb. units) Energy shift (χj ) Energy shift (χj ) Energy shift (χj ) E g A 1g A 2g (χj 56 cm 1 ) Wing-Ho Ko (MIT) Perturbing the U(1) DSL State January 25, / 34
75 Appendix Chern Simons description: B-field case For the B-field case, introduce two species of gauge fields to describe the current of up/down spins: J µ ± = 1 2 ɛµνλ ν a ±,λ The Lagrangian in terms of α and a ± : L = ± 1 4 ɛµνλ a ±,µ ν a ±,λ ɛµνλ α µ ν a ±,λ +... higher derivative terms (e.g., Maxwell term a a for a ± ) omitted L yields the correct equation of motion J µ ± = 1 2 ɛµνλ ν α λ Let c = (α, a +, a ) T, can rewrite L as L = 1 4 ɛµνλ c T µ K ν c λ K has one null vector c gapless mode argued previously. Dynamics of c is driven by Maxwell term = linearly dispersing. Wing-Ho Ko (MIT) Perturbing the U(1) DSL State January 25, / 34
76 Appendix Chern Simons description: B-field case For the B-field case, introduce two species of gauge fields to describe the current of up/down spins: J µ ± = 1 2 ɛµνλ ν a ±,λ The Lagrangian in terms of α and a ± : L = ± 1 4 ɛµνλ a ±,µ ν a ±,λ ɛµνλ α µ ν a ±,λ +... higher derivative terms (e.g., Maxwell term a a for a ± ) omitted L yields the correct equation of motion J µ ± = 1 2 ɛµνλ ν α λ Let c = (α, a +, a ) T, can rewrite L as L = 1 4 ɛµνλ c T µ K ν c λ K has one null vector c gapless mode argued previously. Dynamics of c is driven by Maxwell term = linearly dispersing. Wing-Ho Ko (MIT) Perturbing the U(1) DSL State January 25, / 34
77 Appendix Chern Simons description: B-field case For the B-field case, introduce two species of gauge fields to describe the current of up/down spins: J µ ± = 1 2 ɛµνλ ν a ±,λ The Lagrangian in terms of α and a ± : L = ± 1 4 ɛµνλ a ±,µ ν a ±,λ ɛµνλ α µ ν a ±,λ +... higher derivative terms (e.g., Maxwell term a a for a ± ) omitted L yields the correct equation of motion J µ ± = 1 2 ɛµνλ ν α λ Let c = (α, a +, a ) T, can rewrite L as L = 1 4 ɛµνλ c T µ K ν c λ K has one null vector c gapless mode argued previously. Dynamics of c is driven by Maxwell term = linearly dispersing. Wing-Ho Ko (MIT) Perturbing the U(1) DSL State January 25, / 34
78 Appendix Chern Simons description: doped case In the doped case: L = 1 4 ɛµνλ c T µ K ν c λ + e 2 ɛµνλ (q c µ ) ν A λ + (l c µ )j µ V +... where c = [α; a 1,... a 4, a 5, a 6; b 1,... b 4 ] spinon spinon* holon K describes the self-dynamics of the system; has null vector p = [2; 2,..., 2, 2, 2; 1,..., 1] corresponding to gapless mode c. q = [;...,, ; 1,..., 1] is the charge vector. l is an integer vector with α-component and characterizes vortices. Varying L w.r.t. c gives B = 2 l p jv e q p = 2n 4e j V Wing-Ho Ko (MIT) Perturbing the U(1) DSL State January 25, 21 4 / 34
79 Appendix Chern Simons description: doped case In the doped case: L = 1 4 ɛµνλ c T µ K ν c λ + e 2 ɛµνλ (q c µ ) ν A λ + (l c µ )j µ V +... where c = [α; a 1,... a 4, a 5, a 6; b 1,... b 4 ] spinon spinon* holon K describes the self-dynamics of the system; has null vector p = [2; 2,..., 2, 2, 2; 1,..., 1] corresponding to gapless mode c. q = [;...,, ; 1,..., 1] is the charge vector. l is an integer vector with α-component and characterizes vortices. Varying L w.r.t. c gives B = 2 l p jv e q p = 2n 4e j V Wing-Ho Ko (MIT) Perturbing the U(1) DSL State January 25, 21 4 / 34
80 Appendix Chern Simons description: doped case In the doped case: L = 1 4 ɛµνλ c T µ K ν c λ + e 2 ɛµνλ (q c µ ) ν A λ + (l c µ )j µ V +... self dynamics EM coupling vortices where c = [α; a 1,... a 4, a 5, a 6; b 1,... b 4 ] spinon spinon* holon K describes the self-dynamics of the system; has null vector p = [2; 2,..., 2, 2, 2; 1,..., 1] corresponding to gapless mode c. q = [;...,, ; 1,..., 1] is the charge vector. l is an integer vector with α-component and characterizes vortices. Varying L w.r.t. c gives B = 2 l p jv e q p = 2n 4e j V Wing-Ho Ko (MIT) Perturbing the U(1) DSL State January 25, 21 4 / 34
81 Appendix Chern Simons description: doped case In the doped case: L = 1 4 ɛµνλ c T µ K ν c λ + e 2 ɛµνλ (q c µ ) ν A λ + (l c µ )j µ V +... self dynamics EM coupling vortices where c = [α; a 1,... a 4, a 5, a 6; b 1,... b 4 ] spinon spinon* holon K describes the self-dynamics of the system; has null vector p = [2; 2,..., 2, 2, 2; 1,..., 1] corresponding to gapless mode c. q = [;...,, ; 1,..., 1] is the charge vector. l is an integer vector with α-component and characterizes vortices. Varying L w.r.t. c gives B = 2 l p jv e q p = 2n 4e j V Wing-Ho Ko (MIT) Perturbing the U(1) DSL State January 25, 21 4 / 34
82 Appendix Quasiparticles and their statistics L = 1 4 ɛµνλ c T µ K ν c λ + e 2 ɛµνλ (q c µ ) ν A λ + (l c µ )j µ V +... Vortices with l p = does not couple to c = They can exist when B = and corresponds to quasiparticles. When particle l 1 winds around another particle l 2, the statistical phase θ = 2l T 1 K 1 l 2 K is part of K that s p. Derived by integrating out all gauge fields having non-zero Chern Simons term. { l1 = [;,...,,, ; 1,,, 1] Taking l 2 = [;,...,,, ; 1,, 1, ], found: θ 11 = θ 22 = 2, θ 12 = = Fermions with semionic statistics! Wing-Ho Ko (MIT) Perturbing the U(1) DSL State January 25, / 34
83 Appendix The full form of K-matrix for doped case K = (Recall c = [α; a 1,... a 4, a 5, a 6; b 1,... b 4 ]) spinon spinon* holon Wing-Ho Ko (MIT) Perturbing the U(1) DSL State January 25, / 34
84 Spinon and holon PSG Appendix ϕ 2 ϕ 3 = η 1 = η 3 ϕ 4 ϕ 1 = η 2 = η 4 T x [ϕ 2 1 ] = ϕ2 4 T x [ϕ 2 2 ] = ϕ2 3 T x [ϕ 2 3 ] = e i 3 ϕ 2 2 T x [ϕ 2 4 ] = e i 3 ϕ 2 1 T x [η 1 ] = e i 12 η 2 T x [η 2 ] = e 11i 12 η 1 T x [η 3 ] = e i 12 η 4 T x [η 4 ] = e 11i 12 η 3 Wing-Ho Ko (MIT) Perturbing the U(1) DSL State January 25, / 34
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