Linking bosonic and fermionic descriptionsSuperconducting transition

Confinement transitions out of the Z2 FL* :

§ A Higgs transition induced by condensation of results in confinement

as B carries Z2 gauge charge

§ The pairing of f spinons induces a pairing of the c fermions, resulting in a superconductor

§ Non-trivial transformation of B under translation results in spatial modulation of the superconducting order parameter (Pair-Density Wave or FFLO state)

§ Topological properties of Z2 gauge theory:

§ 4 kinds of topologically distinct excitations: e, m, and 1 (topologically trivial)

§ Fusion rules:

§ In the context of Z2 spin liquids:

§ Symmetry fractionalization:

§ Symmetries act projectively on individual anyons

§ For each symmetry (space-group, time-reversal) combination equivalent to identity, we can associate a Z2 quantum number for each anyon

§ From the Z2 quantum numbers of the bosonic spinons and visons, we can determine the symmetry fractionalization quantum numbers of the fermionic spinons for a fully gapped Z2 spin liquid

§ Equivalence of bosonic and fermionic Z2 spin liquids on the rectangular lattice:

§ Can write down a Hamiltonian consistent with the projective symmetry realization for the fermionic spinons Pi-flux gapped spin liquid

Phys. Rev. B 94, 024502 (2016), Phys. Rev. B 94, 205117 (2016)

Superconductivity from confinement transition of FL* metals with Z2 topological order

Z2 spin liquids

Shubhayu Chatterjee1, Yang Qi2, Subir Sachdev1,2 and Julia Steinberg1

1Harvard University, USA and 2Perimeter Institute, Canada

§ Quantum disordered ground states of certain Mott insulators

§ No broken symmetry, topological degeneracy of ground states

§ Schwinger boson or Abrikosov fermion mean-field theory:

§

§

§ Excitations:

§ 1. Fractionalized spin-half spinons 2. Visons or vortices of the Z2 gauge field

§ Spinons and visons are mutual semions

§ J1-J2-J3 Heisenberg model on the square lattice:

§ Quantum fluctuations can drive a continuous phase transition from a spiral incommensurate antiferromagnet at (Q,0) or (0,Q) to a Z2 spin liquid with Isingnematic order broken C4 symmetry

Read and Sachdev, PRL 66, 1773 (1991)

Wen, PRB 44, 2662 (1991)

Kitaev, Annals of Physics 321 (2006)

Essin and Hermele, PRB 87, 104406 (2013)

Lu, Cho and Vishwanath, arXiv:1403.0575

Doped Mott insulator Holon-spinon pairing FL*

Punk, Allais and Sachdev, PNAS 112, 9552 (2015)

Uniform SC (2)

Incommensurate PDW (4)

Commensurate PDW (3)

0 2 4 6 8 10 120

2

4

6

8

10

Ty

T x+y

Hall effect measurements in YBCO

Badoux, Proust, Taillefer et al., Nature 531, 210 (2016)

14

0 0.1 0.2 0.3p

0

0.5

1

1.5

n H=

V /

e R

H

p

1 + p

SDW CDW FL

p*

a

b

Evidence for FL* metal with Fermi surface of size p ?!

Additional sectors of Z2-FL*

4

theory, the Schwinger boson itself becomes a bosonic, S = 1/2 spinon excitation which we identify

as belonging to the e sector. The vison, carrying Z2 magnetic flux, is spinless, and we label

this as belonging to the m sector. A fusion of the bosonic spinon and a vison then leads to a

fermionic spinon25, which belongs to the ✏ sector. We summarize these, and other, characteristics

of insulating Z2 spin liquids in Table I.

1 e m ✏ 1c ec mc ✏c

S 0 1/2 0 1/2 1/2 0 1/2 0

Statistics boson boson boson fermion fermion fermion fermion boson

Mutual semions � m, ✏, mc, ✏c e, ✏, ec, ✏c e, m, ec, mc � m, ✏, mc, ✏c e, ✏, ec, ✏c e, m, ec, mc

Q 0 0 0 0 1 1 1 1

Field operator � b � f c � � B

TABLE I. Table of characteristics of sectors of the spectrum of the Z2-FL* state. The first 4 columns are the

familiar sectors of an insulating spin liquid. The value of S indicates integer or half-integer representations

of the SU(2) spin-rotation symmetry. The “mutual semion” row lists the particles which have mutual

seminionic statistics with the particle labelling the column. The electromagnetic charge is Q. The last

4 columns represent Q = 1 sectors present in Z2-FL*, and these are obtained by adding an electron-like

quasiparticle, 1c, to the first four sectors. The bottom row denotes the fields operators used in the present

paper to annihilate/create particles in the sectors.

For a metallic Z2-FL* state, it is convenient to augment the insulating classification by counting

the charge, Q, of fermionic electron-like quasiparticles: we simply add a spectator electron, c, to

each insulator sector, and label the resulting states as 1c, ec, mc, and ✏c, as shown in Table I. It is

a dynamical question of whether the c particle will actually form a bound state with the e, m, or

✏ particle, and this needs to be addressed specifically for each Hamiltonian of interest.

Now let us consider a confining phase transition in which the Z2 topological order is destroyed.

This can happen by the condensation of one of the non-trivial bosonic particles of the Z2-FL*

state. From Table I, we observe that there are 3 distinct possibilities:

1. Condensation of m: this was initially discussed in Refs. 2 and 4. For the case of insulating

antiferromagnets with an odd number of S = 1/2 spins per unit cell, the non-trivial space

group transformations of the m particle lead to bond density wave order in the confining

phase. The generalization to the metallic Z2-FL* state was presented recently in Ref. 26.

2. Condensation of e: now we are condensing a boson with S = 1/2, and this leads to long-range

antiferromagnetic order27–31.

4

theory, the Schwinger boson itself becomes a bosonic, S = 1/2 spinon excitation which we identify

as belonging to the e sector. The vison, carrying Z2 magnetic flux, is spinless, and we label

this as belonging to the m sector. A fusion of the bosonic spinon and a vison then leads to a

fermionic spinon25, which belongs to the ✏ sector. We summarize these, and other, characteristics

of insulating Z2 spin liquids in Table I.

1 e m ✏ 1c ec mc ✏c

S 0 1/2 0 1/2 1/2 0 1/2 0

Statistics boson boson boson fermion fermion fermion fermion boson

Mutual semions � m, ✏, mc, ✏c e, ✏, ec, ✏c e, m, ec, mc � m, ✏, mc, ✏c e, ✏, ec, ✏c e, m, ec, mc

Q 0 0 0 0 1 1 1 1

Field operator � b � f c � � B

TABLE I. Table of characteristics of sectors of the spectrum of the Z2-FL* state. The first 4 columns are the

familiar sectors of an insulating spin liquid. The value of S indicates integer or half-integer representations

of the SU(2) spin-rotation symmetry. The “mutual semion” row lists the particles which have mutual

seminionic statistics with the particle labelling the column. The electromagnetic charge is Q. The last

4 columns represent Q = 1 sectors present in Z2-FL*, and these are obtained by adding an electron-like

quasiparticle, 1c, to the first four sectors. The bottom row denotes the fields operators used in the present

paper to annihilate/create particles in the sectors.

For a metallic Z2-FL* state, it is convenient to augment the insulating classification by counting

the charge, Q, of fermionic electron-like quasiparticles: we simply add a spectator electron, c, to

each insulator sector, and label the resulting states as 1c, ec, mc, and ✏c, as shown in Table I. It is

a dynamical question of whether the c particle will actually form a bound state with the e, m, or

✏ particle, and this needs to be addressed specifically for each Hamiltonian of interest.

Now let us consider a confining phase transition in which the Z2 topological order is destroyed.

This can happen by the condensation of one of the non-trivial bosonic particles of the Z2-FL*

state. From Table I, we observe that there are 3 distinct possibilities:

1. Condensation of m: this was initially discussed in Refs. 2 and 4. For the case of insulating

antiferromagnets with an odd number of S = 1/2 spins per unit cell, the non-trivial space

group transformations of the m particle lead to bond density wave order in the confining

phase. The generalization to the metallic Z2-FL* state was presented recently in Ref. 26.

2. Condensation of e: now we are condensing a boson with S = 1/2, and this leads to long-range

antiferromagnetic order27–31.

bosonic chargon

Badoux, Proust, Taillefer et al, Nature 531, 9552 (2016)

Condense b SDW Condense v Bond-DW Condense B Superconductivity

Chubukov, Senthil and Sachdev, PRL 72, 2089 (1994)

Patel, Chowdhury, Allais and Sachdev, PRB 93, 165139 (2016)

§ Metallic state with charge-e spin-half c fermions in the background of a Z2 spin liquid

§ The size of the Fermi surface is determined by dopant density p

§ No low energy fractionalized excitations

§ The vortices of the internal Z2 gauge field (visons) survive in the FL* metal, hence its topological character

§ Violates Luttinger’s theorem due to presence of emergent gauge excitations

Fractionalized Fermi liquid (Z2 FL*)

§ Consider a ‘plain vanilla’ Z2 FL* with trivial PSG for fermionic spinons

§ D-wave spinon-pairing leads to uniform d-wave SC

§ Modified boson dispersions can lead to co-existing uniform SC with bond density waves and pair density waves at the same axial wave-vector Q, as observed in recent STM experiments.

§ Hall No. also shows a jump at the optimal doping critical point in the metallic state at T=0

§ Open questions:

§ How does the jump in the Hall No. smoothen at finite T?

§ Which experiments can distinguish the Z2 FL* from other candidate topological metals or field-induced magnetism?

SM

FL

Figure: K. Fujita and J. C. Seamus Davis

3 confinement transitions

from condensation of bosons in

Z2-FL*

Fujita et al, PNAS 111, 3026 (2014)Hamidian et al, Nature 532, 343 (2016)

D-wave SC, Density waves and Hall effect

Senthil, Voijta and Sachdev, PRL 90, 216403 (2003)

Paramekanti and Vishwanath, PRB 70, 245118 (2004)

0.06 0.11 0.16 0.21p

0.5

1.0

1.5

nH

1+ p

p