Topological order in quantum matter

HARVARD Topological order in quantum matter Indian Institute of Science Education and Research, Pune Subir Sachdev November 13, 2017 Talk online: sachdev.physics.harvard.edu

1. Classical XY model in 2 and 3 dimensions 2. Topological order in the classical XY model in 3 dimensions 3. Topological order in the quantum XY model in 2+1 dimensions 4. Topological order in the Hubbard model

Z XY = Y i Z 2 0 d i 2 exp ( H/T ) H = J X hiji cos( i j ) Describes non-zero T phase transitions of superfluids, magnets with `easy-plane spins,..

In spatial dimension d = 3, in the low T phase, the symmetry i! i + c is spontaneously broken. There is (o -diagonal) long-range order (LRO) characterized by ( i e i i ) lim r i r j!1 i j = 0 2 6=0. We break the symmetry by choosing an overall phase so that h ii = 0 6= 0 h ii = 0 6= 0 h ii =0 LRO SRO Tc T

In spatial dimension d = 2, the symmetry i! i + c is preserved at all non-zero T. There is no LRO, and h ii = 0 for all T>0. Nevertheless, there is a phase transition at T = T KT, where the nature of the correlations changes lim r i r j!1 i 8 < j : r i r j, for T<T KT, (QLRO) exp( r i r j / ), for T>T KT,(SRO)

Kosterlitz-Thouless theory in d=2 QLRO Topological order SRO Vortices suppressed 8 < : TKT Vortices proliferate T The low T phase also has topological order associated with the suppression of vortices.

KT critical theory (Nobel Prize, 2016) Kosterlitz-Thouless theory in d=2 QLRO Topological order SRO Vortices suppressed 8 < : TKT Vortices proliferate T The low T phase also has topological order associated with the suppression of vortices.

Can we modify the XY model Hamiltonian to obtain a phase with topological order in d=3? h ii = 0 6= 0 h ii =0 SRO LRO J Jc

h ii =0 SRO Topological order K h ii =0 SRO h ii = 0 6= 0 LRO No topological order J Jc

h ii =0 SRO Topological order K Only even (4, 8...) vortices proliferate h ii =0 SRO h ii = 0 6= 0 LRO No topological order All (2, 4...) vortices proliferate J Jc

ez XY = Y i Z 2 0 d i 2 exp eh/t eh = J X hiji cos( i j )+... Add terms which suppress single but not double vortices..

X ez XY = { ij }=±1 eh = J X hiji Y i Z 2 0 d i 2 exp eh/t ij cos [( i j )/2] K X Y ij (ij)2 K

X ez XY = { ij }=±1 eh = J X hiji Y i Z 2 0 d i 2 exp eh/t ij cos [( i j )/2] K X Y ij (ij)2 With K = 0, we can explicitly sum over ij, and the theory reduces to the ordinary XY model.

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sha1_base64="f8adzy6vdaeuqbbjstqxd4xrkk0=">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</latexit> X ez XY = { ij }=±1 eh = J X hiji Y i Z 2 0 d i 2 exp eh/t ij cos [( i j )/2] K X Y ij (ij)2 The theory has a Z 2 gauge invariance: we can change i! i + (1 i ) ij! i ij j, with i = ±1, and the energy remains unchanged. The XY order parameter i = e i i is gauge invariant, as are all physical observables. So this is an XY model with a modified Hamiltonian, and no additional degrees of freedom have been introduced.

X ez XY = { ij }=±1 eh = J X hiji Y i Z 2 0 d i 2 exp eh/t ij cos [( i j )/2] K X Y ij (ij)2 A single (odd) 2 vortex in i has Q (ij)2 cos [( i j )/2] < 0. So for J>0, such a vortex will prefer Q (ij)2 ij = 1, i.e. a2 vortex has Z 2 flux = 1 in its core. So a large K>0 will suppress (odd) 2 vortices. There is no analogous suppression of (even) 4 vortices.

X ez XY = { ij }=±1 eh = J X hiji Y i Z 2 0 d i 2 exp eh/t ij cos [( i j )/2] K X Y ij (ij)2 SRO Topological order h ii =0 Only even (4, 8...) vortices proliferate K h ii = 0 6= 0 LRO h ii =0 SRO No topological order All (2, 4...) vortices proliferate J Jc

X ez XY = { ij }=±1 eh = J X hiji Y i Z 2 0 d i 2 exp eh/t ij cos [( i j )/2] K X Y ij (ij)2 Deconfined phase of Z2 gauge theory. Z2 flux is expelled Higgs phase of h ii =0 K Z2 gauge theory h ii =0 h ii = 0 6= 0 Confined phase of Z2 gauge theory. Z2 flux flucuates J Jc

A quantum Hamiltonian in 2+1 dimensions eh = J X hiji z ij cos [( i j )/2] K X Y (ij)2 z ij U X i @ 2 @ 2 i g X hiji x ij z z K z z

A quantum Hamiltonian in 2+1 dimensions eh = J X hiji z ij cos [( i j )/2] K X Y z ij (ij)2 U X i @ 2 @ 2 i g X hiji x ij In the topological phase, the suppressed Z 2 fluxes of -1 become well-defined gapped quasiparticle excitations ( visons ) above the ground state. -1

A quantum Hamiltonian in 2+1 dimensions eh = J X hiji z ij cos [( i j )/2] K X Y z ij (ij)2 U X i @ 2 @ 2 i g X hiji x ij In the topological phase, on a torus, inserting the Z 2 flux of -1 into one of the cycles of the torus leads to an orthogonal state whose energy cost vanishes exponentially in the size of the torus: there are 4 degenerate ground states on a large torus.

SM FL YBa 2 Cu 3 O 6+x Figure: K. Fujita and J. C. Seamus Davis

SM FL YBa 2 Cu 3 O 6+x d-wave superconductor Figure: K. Fujita and J. C. Seamus Davis

SM Insulating Antiferromagnet FL YBa 2 Cu 3 O 6+x Figure: K. Fujita and J. C. Seamus Davis

Undoped insulating antiferromagnet

Antiferromagnet with p mobile holes per square

Filled Band

Antiferromagnet with p mobile holes per square But relative to the band insulator, there are 1+ p holes per square

Antiferromagnet In a conventional metal (a Fermi liquid), with no broken symmetry, the area with p mobile holes per square enclosed by the Fermi surface must be 1+p But relative to the band insulator, there are 1+ p holes per square

M. Platé, J. D. F. Mottershead, I. S. Elfimov, D. C. Peets, Ruixing Liang, D. A. Bonn, W. N. Hardy, S. Chiuzbaian, M. Falub, M. Shi, L. Patthey, and A. Damascelli, Phys. Rev. Lett. 95, 077001 (2005) SM FL A conventional metal: the Fermi liquid with Fermi surface of size 1+p

S. Badoux, W. Tabis, F. Laliberté, G. Grissonnanche, B. Vignolle, D. Vignolles, J. Béard, D.A. Bonn, W.N. Hardy, R. Liang, N. Doiron-Leyraud, L. Taillefer, and C. Proust, Nature 531, 210 (2016). Pseudogap metal SM FL at low p Many indications that this metal behaves like a Fermi liquid, but with Fermi surface size p and not 1+p.

T. Senthil, M. Vojta and S. Sachdev, PRB 69, 035111 (2004) Pseudogap metal SM FL at low p Many indications that this metal behaves like a Fermi liquid, but with Fermi surface size p and not 1+p. If present at T=0, a metal with a size p Fermi surface (and translational symmetry preserved) must have topological order

The Hubbard Model H = X i<j t ij c i c j + U X i 1 1 n i" n i# 2 2 µ X i c i c i t ij! hopping. U! local repulsion, µ! chemical potential Spin index =", # n i = c i c i c i c j + c j c i = ij c i c j + c j c i =0 Will study on the square lattice

Electron spin D ~Si E = i ~ i where i = ±1 Fermi surface+antiferromagnetism p=0 h ~ i6=0 h ~ i =0 Insulator with AF order Metal with large Fermi surface of size 1 U/t

We can (exactly) transform the Hubbard model to the spin-fermion model: electrons c i on the square lattice with dispersion H c = X t c i, c i+v, + c i+v, c i, µ X c i, c i, + H int i, i are coupled to an antiferromagnetic order parameter ` = x, y, z X H int = `(i)c i i, ` c i, + V i `(i), where i = ±1 on the two sublattices. (For suitable V, integrating out the yields back the Hubbard model). When `(i) = (non-zero constant) independent of i, we have longrange AF order, which gaps out the fermions into an insulating state.

For fluctuating antiferromagnetism, we transform to a rotating reference frame using the SU(2) rotation R i ci" i,+ = R i, c i# in terms of fermionic chargons s and a Higgs field H a (i) i, ` `(i) =R i a H a (i) R i The Higgs field is the AFM order in the rotating reference frame. Note that this representation is ambiguous up to a SU(2) gauge transformation, V i! V i i,+ i, i,+ R i! R i V i a H a (i)! V i b H b (i) V i. i,

Fluctuating antiferromagnetism The simplest e ective Hamiltonian for the fermionic chargons is the same as that for the electrons, with the AFM order replaced by the Higgs field. H = X i, t i,s i+v,s + i+v,s i,s H int = X i i H a (i) i,s µ X i a ss 0 i,s 0 + V H i,s i,s + H int IF we can transform to a rotating reference frame in which H a (i) = a constant independent of i and time, THEN the fermions in the presence of fluctuating AFM will inherit the insulating nature of the electrons in the presence of static AFM.

Fluctuating antiferromagnetism We cannot always find a single-valued SU(2) rotation R i to make the Higgs field H a (i) a constant! n-fold vortex in AFM order A.V. Chubukov, T. Senthil and S. Sachdev, PRL 72, 2089 (1994); S. Sachdev, E. Berg, S. Chatterjee, and Y. Schattner, PRB 94, 115147 (2016)

Fluctuating antiferromagnetism We cannot always find a single-valued SU(2) rotation R i to make the Higgs field H a (i) a constant! n-fold vortex in AFM order R ( 1) n R A.V. Chubukov, T. Senthil and S. Sachdev, PRL 72, 2089 (1994); S. Sachdev, E. Berg, S. Chatterjee, and Y. Schattner, PRB 94, 115147 (2016)

Topological order We cannot always find a single-valued SU(2) rotation R i to make the Higgs field H a (i) a constant! n-fold vortex in AFM order R ( 1) n R Vortices with n odd must be suppressed: such a metal with fluctuating antiferromagnetism has BULK Z 2 TOPOLOGICAL ORDER and fermions which inherit the insulating behavior of the antiferromagnetic metal.

Electron spin D ~Si E = i ~ i where i = ±1 Fermi surface+antiferromagnetism p=0 h ~ i6=0 h ~ i =0 Insulator with AF order Metal with large Fermi surface of size 1 U/t

Fermi surface+antiferromagnetism+topological order Insulator with Z2 topological order Higgs phase of a SU(2) gauge theory (SU(2) is broken down to Z2) h ~ i =0 p=0 h ~ i6=0 h ~ i =0 Insulator with AF order Metal with large Fermi surface of size 1 U/t

Topological order in the pseudogap metal Mathias S. Scheurer, 1 Shubhayu Chatterjee, 1 Wei Wu, 2, 3 Michel Ferrero, 2, 3 Antoine Georges, 2, 4, 3, 5 1, 6, 7 and Subir Sachdev We compute the electronic Green s function of the topologically ordered Higgs phase of a SU(2) gauge theory of fluctuating antiferromagnetism on the square lattice. The results are compared with cluster extensions of dynamical mean field theory, and quantum Monte Carlo calculations, on the pseudogap phase of the strongly interacting hole-doped Hubbard model. Good agreement is found in the momentum, frequency, hopping, and doping dependencies of the spectral function and electronic self-energy. We show that lines of (approximate) zeros of the zero-frequency electronic Green s function are signs of the underlying topological order of the gauge theory, and describe how these lines of zeros appear in our theory of the Hubbard model. We also derive a modified, non-perturbative version of the Luttinger theorem that holds in the Higgs phase. to appear..

magnetic ratio shape measure along the c axis the line shape i viously measur 17 using O NMR measurements of an isotopenmr line shap Candidate forsingle an insulator 2Iztopo(OH) Clto We enriched crystal of with ZnCu3Z =6m +1 2. m nuclear spin an logicaldemonstrated order whichthat hasthecharge neutral, spin spin excitation spectrum tum number m exhibits a finite gap D = 0.03 J to 0.07 J bes = 1/2, spinon excitations. which are sep Kagome lattice antiferromagnet tween a S = 0 spin-liquid ground state and the S. Sachdev, Physical Review B 45, 12377 (1992) excited states, where J ~ 200 K represents the Cu-Cu superexchange interaction (5, 21). 295K (9T See Atalk tomorrow byusing H. Changlani major advantage of a single crystal for NMR is that we can achieve high resolution by applying an external magnetic field Bext along specific crystallographic directions. In Fig. 2A, we present the 17O (nuclear spin I = 5/2; gyromagnetic ratio gn/2p = 5.772 MHz/T) NMR line shape measured at 295 K in Bext = 9 T, applied along the c axis; the temperature dependence51.5of the line shape is presented in Fig. 2C. Unlike preand 17O viously measured powder-averaged 35Cl 2.0 NMR line shapes (22, 23), we can resolve the five Iz = m to m + 1 transitions (Iz, z component 1.5of the ZnCu (OH) nuclear angular momentum; magnetic quan3 spin 6 Cl 2 tum number m = 5/2, 3/2, 1/2, +1/2,1.0+3/2), Spin echo intensity (arb. units) ES E A RC H R E PO R TS Herbertsmithite:

Evidence for a gapped spin-liquid ground state in a kagome Heisenberg antiferromagnet Mingxuan Fu, 1 Takashi Imai, 1,2 * Tian-Heng Han, 3,4 Young S. Lee 5,6 The kagome Heisenberg antiferromagnet is a leading candidate in the search for a spin system with a quantum spin-liquid ground state. The nature of its ground state remains amatterofactivedebate.weconductedoxygen-17single-crystalnuclearmagnetic resonance (NMR) measurements of the spin-1/2 kagome lattice in herbertsmithite [ZnCu 3 (OH) 6 Cl 2 ], which is known to exhibit a spinon continuum in the spin excitation spectrum. We demonstrated that the intrinsic local spin susceptibility c kagome,deduced from the oxygen-17 NMR frequency shift, asymptotes to zero below temperatures of 0.03J, where J ~ 200 kelvin is the copper-copper superexchange interaction. Combined with the magnetic field dependence of c kagome that we observed at low temperatures, these results imply that the kagome Heisenberg antiferromagnet has a spin-liquid ground state with a finite gap. Gap to topological S =1/2 spinon excitations: J/20. Science 350, 655 (2015) Neutron scattering observations T Han, Young Lee, et al, Nature 492, 406 (2012) SCIENCE 6NOVEMBER2015 VOL 350 ISSUE 6261 655 Theory by Punk, Chowdhury, Sachdev Nature Physics, 2013

Gapped Spin-1/2 Spinon Excitations in a New Kagome Quantum Spin Liquid Compound Cu Zn(OH) FBr 3 6 Zili Feng ( ) 1, Zheng Li ( ) 1,2, Xin Meng ( ) 1, Wei Yi ( ) 1, Yuan Wei ( ) 1, Jun Zhang ( ) 3, Yan-Cheng Wang ( ) 1, Wei Jiang ( ) 4, Zheng Liu ( ) 5, Shiyan Li ( ) 3,6 2017 Chinese Physical Society and IOP Publishing Ltd Chinese Physics Letters, Volume 34, Number 7 * Gap to topological S =1/2 spinon excitations: J/20. (c) (K) 8 4 0 19 K-Kchem (%) 1E-2 1E-3 1E-4 S = 1 S = 1/2 0.0 0.2 0.4 0.6 7.864 T 5.026 T 3 T 0.914 T 0 4 8 12 B (T) 1/T (K -1 )

New classes of quantum states with topological order Can be understood as: (a) defect suppression in states with fluctuating order (b) Higgs phases of emergent gauge fields A metal with bulk topological order (i.e. long-range quantum entanglement) can explain existing experiments in cuprates, and agrees well with cluster- DMFT Direct evidence for topological order in kagome antiferromagnets