spin liquids in insulators and metalsqpt.physics.harvard.edu/talks/samarkand19.pdfsamarkand, may 14,...

Spin liquids in

insulators and metalsICTP Summer School

Advances in Condensed Matter Physics: New trends and Materials in Quantum Technologies

Samarkand, May 14, 2019

Subir Sachdev

HARVARD

Talk online: sachdev.physics.harvard.edu

Band topology: free fermions in a bulk band have protected states on the edge (topological insulators, Majorana chain etc.)

Emergent gauge fields: “anyons” in the bulk, and ground state degeneracy dependent upon topology of space. Protected edge states may or may not exist

Combination of band topology and emergent gauge fields lead to exotic new possibilities (non-Abelian bulk anyons)

Topological quantum matter

Band topology: free fermions in a bulk band have protected states on the edge (topological insulators, Majorana chain etc.)

Emergent gauge fields: “anyons” in the bulk, and ground state degeneracy dependent upon topology of space. Protected edge states may or may not exist

Combination of band topology and emergent gauge fields leads to exotic new possibilities (non-Abelian bulk anyons)


1. Resonating valence bonds The Z2 spin liquid

2. SU(2) gauge theory of fluctuating antiferromagnetism on the triangular lattice The Z2 spin liquid

3. Electron-doped cuprates Higgs phase with topological order: Fermi surface reconstruction without translational symmetry breaking

Mott insulator: Triangular lattice antiferromagnet

H = J�

�ij⇥

⌅Si · ⌅Sj

Nearest-neighbor model has non-collinear Neel order

Spin liquid obtained in a generalized spin model with S=1/2 per unit cell

=

P. Fazekas and P. W. Anderson, Philos. Mag. 30, 23 (1974).


P. Fazekas and P. W. Anderson, Philos. Mag. 30, 23 (1974).

Spin liquid obtained in a generalized spin model with S=1/2 per unit cell

=


ssc

non-collinear Neel state


Z2 spin liquidwith neutral S = 1/2 spinonsand vison excitations

N. Read and S. Sachdev, Phys. Rev. Lett. 66, 1773 (1991) X.-G. Wen, Phys. Rev. B 44, 2664 (1991)

Excitations of the Z2 Spin liquid

=Spinon: S=1/2, charge 0 e (boson) or ✏ (fermion) particle

<latexit sha1_base64="78wssQVo5TUghvuaDRteBH/VidU=">AAACFXicdZDNSgMxFIUz/lv/qi7dBFtBQYZ02lrdiW5cKlgV2lIy6Z02NJMMSUYoxZdw46u4caGIW8Gdb2OmVlDRuzp8597k3hMmghtLyLs3MTk1PTM7N59bWFxaXsmvrl0YlWoGdaaE0lchNSC4hLrlVsBVooHGoYDLsH+c+ZfXoA1X8twOEmjFtCt5xBm1DrXzu0Uo4u1QGSV3sNK42ITEcKGkoxHomGc8odpyJqCdLxC/SirVoIaJT4LqXoVkorxfPiC45JNRFdC4Ttv5t2ZHsTQGaZmgxjRKJLGt4fi5m1wzNZBQ1qddaDgpaQymNRxddYO3HOngyC0VKWnxiH6fGNLYmEEcus6Y2p757WXwL6+R2mi/NeQySS1I9vlRlApsFc4iwh2ugVkxcIIyzd2umPWopsy6IHMuhK9L8f/iIvBLZT84CwqHR+M45tAG2kTbqIRq6BCdoFNURwzdonv0iJ68O+/Be/ZePlsnvPHMOvpR3usH4+Wddw==</latexit>

=

-1

Excitations of the Z2 Spin liquidA vison

m (boson) particle<latexit sha1_base64="np8d3jUlWyXIGBYwnWiTpcQvpZQ=">AAAB/XicdVDLTgIxFO34RHzhY+emEUxwM+kMILgjunGJiTwSIKRTCjR02knbMUFC/BU3LjTGrf/hzr+xPEzU6ElucnLOve29J4g40wahD2dpeWV1bT2xkdzc2t7ZTe3t17SMFaFVIrlUjQBrypmgVcMMp41IURwGnNaD4eXUr99SpZkUN2YU0XaI+4L1GMHGSp3UYSbMwGwgtRSnMMLKMMJpJ5VGbgHlC34RIhf5hbM8mpJcKXeOoOeiGdJggUon9d7qShKHVBjCsdZND0WmPV48N0m2Yk0jTIa4T5uWChxS3R7Ptp/AE6t0YU8qW8LAmfp9YoxDrUdhYDtDbAb6tzcV//KasemV2mMmothQQeYf9WIOjYTTKGCXKUoMH1mCiWJ2V0gGWGFibGBJG8LXpfB/UvNdL+f61366fLGIIwGOwDHIAg8UQRlcgQqoAgLuwAN4As/OvfPovDiv89YlZzFzAH7AefsEL7aUbw==</latexit>

=

-1

-1

Excitations of the Z2 Spin liquidA vison

m (boson) particle<latexit sha1_base64="np8d3jUlWyXIGBYwnWiTpcQvpZQ=">AAAB/XicdVDLTgIxFO34RHzhY+emEUxwM+kMILgjunGJiTwSIKRTCjR02knbMUFC/BU3LjTGrf/hzr+xPEzU6ElucnLOve29J4g40wahD2dpeWV1bT2xkdzc2t7ZTe3t17SMFaFVIrlUjQBrypmgVcMMp41IURwGnNaD4eXUr99SpZkUN2YU0XaI+4L1GMHGSp3UYSbMwGwgtRSnMMLKMMJpJ5VGbgHlC34RIhf5hbM8mpJcKXeOoOeiGdJggUon9d7qShKHVBjCsdZND0WmPV48N0m2Yk0jTIa4T5uWChxS3R7Ptp/AE6t0YU8qW8LAmfp9YoxDrUdhYDtDbAb6tzcV//KasemV2mMmothQQeYf9WIOjYTTKGCXKUoMH1mCiWJ2V0gGWGFibGBJG8LXpfB/UvNdL+f61366fLGIIwGOwDHIAg8UQRlcgQqoAgLuwAN4As/OvfPovDiv89YlZzFzAH7AefsEL7aUbw==</latexit>

4-fold degeneracy on the torus

Topological order in the Z2 spin liquid ground state


vison


• 3 non-trivial particles: e (boson), ✏ (fermion), m (boson).

• e and m are mutual ‘semions’: the e particle acquires a phase(�1) upon encircling the m particle (and vice versa).

• ✏ and m are also mutual semions.

• The bound state e and m (if it exists) is an ✏. Fusion rule:e⇥m = ✏.

• The bound state of ✏ and m is an e. Fusion rule: ✏⇥m = e.

• The bound state of e and ✏ is a m. Fusion rule: e⇥ ✏ = m.

• There is a 4-fold degeneracy on the torus.

• Protected edge states do not exist in general, but could ap-pear in the presence of symmetries.

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Properties of the Z2 spin liquid

Properties of the Z2 spin liquid• 3 non-trivial particles: e (boson), ✏ (fermion), m (boson).

• e and m are mutual ‘semions’: the e particle acquires a phase(�1) upon encircling the m particle (and vice versa).

• ✏ and m are also mutual semions.

• The bound state e and m (if it exists) is an ✏. Fusion rule:e⇥m = ✏.

• The bound state of ✏ and m is an e. Fusion rule: ✏⇥m = e.

• The bound state of e and ✏ is a m. Fusion rule: e⇥ ✏ = m.

• There is a 4-fold degeneracy on the torus.

• Protected edge states do not exist in general, but could ap-pear in the presence of symmetries.

<latexit sha1_base64="vZDjxrtELp7+6vP67i4065JpSsM=">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</latexit>

ssc




N. Read and S. Sachdev, Phys. Rev. Lett. 66, 1773 (1991) X.-G. Wen, Phys. Rev. B 44, 2664 (1991)

H = �X

i<j

tijc†i↵cj↵ + U

X

i

✓ni" �

1

2

◆✓ni# �

1

2

◆� µ

X

i

c†i↵ci↵

tij ! “hopping”. U ! local repulsion, µ ! chemical potential

Spin index ↵ =", #

ni↵ = c†i↵ci↵

c†i↵cj� + cj�c

†i↵ = �ij�↵�

ci↵cj� + cj�ci↵ = 0

The Hubbard Model

First study on the triangular lattice

We use the operator equation (valid on each site i):

U

✓n" �

1

2

◆✓n# �

1

2

◆= �2U

3~S2 +

U

4

Then we decouple the interaction via

exp

2U

3

X

i

Zd⌧ ~S2

i

!=

ZD~�i(⌧) exp

�X

i

Zd⌧

3

8U~�2i � ~�i

~Si

�!

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In this manner, we obtain the “spin-fermion” model

Z =

ZDc↵D~� exp (�S)

S =

Zd⌧

X

k

c†k↵

✓@

@⌧� "k

◆ck↵

� �

Zd⌧

X

i

c†i↵~�i · ~�↵�ci�

+ V (~�)<latexit sha1_base64="oRQahBzkrD/WeCWl0gwtUhzqbjo=">AAAENHicdVNLb9QwEE6zPEp4tXDkMqKiD0FX2UdVekCqgAPcikofom63jjPJWnWcYDttV1H6E/g1iBv8DyRuiCs37jj76HZpGcnS6JvxfJ++sYNMcG18//uUW7t2/cbN6Vve7Tt3792fmX2wrdNcMdxiqUjVbkA1Ci5xy3AjcDdTSJNA4E5w9Kqq7xyj0jyV700vw/2ExpJHnFFjoc6su0BkymWI0nhvJZgu15BQKVE9gxOENDCUVzDC4aHOuFyOUCX25sICJGmIwiMBxlwW+FFSpWiv9ObngSTUdBkVxYcSXgDh0oyh1yWwTkGoyLq0nIDJMbKCbHS5TfE0AyIwMouwPG7atBXF465ZArCyZZ4EqICQCc7NinNAGgIxNAei86RTFCSI4KgsAdhBQUIax6jKcxjOFQ1YSaSoVZNRZTgV5XnWn1h6VtQxVZhpLqyL49kjeeyKwZOCzwZhBwm7rZBeVsw9O4WP7h8MJV+wqWOLLEzNENI8Tmg58tbuxdC+17ZrkF+S8PTsDGB7cTxxadziEZThhbV2Zub8eru50mqtgV/3W63VRtsmjcZqe60NjbrfjzlnGBudmT8kTFme2JfFBNV6r+FnZr+obGQCS4/kGjPKjmiMRf8dl/DEQiFEqbLHmtFHJ/poonUvCWxntW39b60Cr6wFyQTfXm6i5/sFl1luULIBcZQLMClUnwRCrpAZ0bMJZYpbxcC61L4IY7+SRzTafyZj0y2IwVNzwkPLW7TqK1yW1qeRGfD/ZLtZb7TqzXfNufWXQ8emnUfOY2fRaTirzrrzxtlwthzmfnI/u1/db7UvtR+1n7Vfg1Z3anjnoTMRtd9/AQxIZdQ=</latexit>

We have exactly transformed the Hubbard model

to the “spin-fermion” model with electronic Hamil-

tonian described by electrons ci↵ on the square

or triangular lattice with dispersion

Hc = �

X

i,⇢

t⇢⇣c†i,↵ci+v⇢,↵

+ c†i+v⇢,↵ci,↵

⌘

�µX

i

c†i,↵ci,↵ +Hint

are coupled to a magnetic moment order pa-

rameter �p(i), p = x, y, z

Hint = ��X

i

�p(i)c†i,↵�

p↵�ci,� + V�

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Gauge theory of fluctuating antiferromagnetismFor fluctuating antiferromagnetism

(spin density waves (SDW)), we transform to a ro-

tating reference frame using the SU(2) rotation Ri

✓ci"ci#

◆= Ri

✓ i,+

i,�

◆,

in terms of fermionic “chargons” s and a Higgs

field Ha(i)

�p�

p(i) = Ri �

aH

a(i)R

†i

The Higgs field is the SDW order in the

rotating reference frame.

We will see later that the s are ✏ particles of the

Z2 spin liquid.<latexit sha1_base64="384hPzZYh1sgyoe7z2aUuJ/MCck=">AAAE63icdVTBbttGEGUstXXUpLXTYy6DWkEkxBEkBUkNFAGCNgh8dJs4CmraynI5pCZe7rK7yygCwa/orei1f9RLvqYdUjRqJfZeuPtm9s3Mmx1GuSLnx+OPN7Y63S++/Gr7Zu/rW7e/+XZn985rZwor8VgaZeybSDhUpPHYk1f4JrcoskjhLDr/ubbP3qN1ZPQrv8rxNBOppoSk8AzNd7f+CbUhHaP2vRfGQqIK6Qs26hSE9pSgtaa+g55cFoYwcDlpYH9HfgVL8R4dDF4+nw2H+7BE8FZolxibgTcgoAybFEuLcQXW+DWxRaZFLbGXWJFhBYWrYb9AeHk8mA5bT6Oh/+uc+r0wwpR0GZPLlVhlwi+qXqgw8QNoTcJasapKWYGclxQWOQNmWQEn3ACxWeoLCHXc+kNoKV34ITwFjgPXUYa5IybZf9DwXZweXkm132vAjVRZL482c2AS4MIzLowkvH0rF8KmRrv796HfsLo+ax43skUJHFKaOkgIFWvXPzwTMKDhNWI4SjNxlkN4tCD+sONFTfvQGgW0FDXGprMwFmmK9oqEN7r2irtyKRUgt27U8xkYG6OFurwFsjKf9xea/o6qMNzknCEsSSlwiKAEi8MMwje8/yth6wPySRndB+6oJ6mwUXHtWCcbReVv1Xzah+ZZKvq9oHhUzXf2xqNxs+DS5uDxwXj8BCYtshe062i+8y8/EVlkPAZSCedOJuPcn5ZtTBa4cJgLeS5SPOGt5qrcadlUVME9RmJI6ukx2kODXr5Risy5VRaxZ52y+9TWiH6F7aTwycFpSTovPKu5DpQUqh6tepYhJovSqxVvhLTEuUL9poRkRTejRFn9SJB/Dzr1izL0+MEvKea45aPRhHSt2IUscP3m9XQ0eTSa/jLde/ZTq912cDf4PhgEk+CH4FlwGBwFx4Hs/NgRnXed827W/aP7Z/evtevWjfbOd8HG6v79H1RTpBk=</latexit>

Gauge theory of fluctuating antiferromagnetism

The SU(2) rotation Ri obeys R†iR = 1 and

so we write

R =

✓z" �z⇤#z# z⇤"

◆

The z↵ are spin S = 1/2 bosonic spinons.We will see later that the z↵ will becomethe e particles of the Z2 spin liquid.

<latexit sha1_base64="j1L8C4yiPWq/bO1fczyKiT172fw=">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</latexit>

S. Sachdev, M. A. Metlitski, Y. Qi, and C. Xu, PRB 80, 155129 (2009)

For (fluctuating) SDW SRO, we transform to arotating reference frame using the SU(2) rotation Ri

✓ci"ci#

◆= Ri

✓ i,+

i,�

◆,

in terms of fermionic “chargons” s and a Higgs field Ha(i)

�`�`(i) = Ri �

aH

a(i)R†i

The Higgs field is the SDW order in the rotating reference frame.Note that this representation is ambiguous up to aSU(2) gauge transformation, Vi

✓ i,+

i,�

◆! Vi

✓ i,+

i,�

◆

Ri ! RiV†i

�aH

a(i) ! Vi �bH

b(i)V †i .

Gauge theory of fluctuating antiferromagnetismField Symbol Statistics SU(2)gauge SU(2)spin U(1)e.m.charge

Electron c fermion 1 2 -1

AF order � boson 1 3 0

Chargon fermion 2 1 -1

Spinon R or z boson 2 2 0

Higgs H boson 3 1 0

TABLE I. Quantum numbers of the matter fields in L and Lg. The transformations under the SU(2)’s

are labelled by the dimension of the SU(2) representation, while those under the electromagnetic U(1)

are labeled by the U(1) charge. The antiferromagnetic spin correlations are characterized by � in (5.3).

The Higgs field determines local spin correlations via (5.12).

A summary of the charges carried by the fields in the resulting SU(2) gauge theory, Lg, is

in Table I. This rotating reference frame perspective was used in the early work by Shraiman

and Siggia on lightly-doped antiferromagnets [89, 90], although their attention was restricted to

phases with antiferromagnetic order. The importance of the gauge structure in phases without

antiferromagnetic order was clarified in Ref. [85].

Given the SU(2) gauge invariance associated with (5.6), when we express L in terms of we

naturally obtain a SU(2) gauge theory with an emergent gauge field Aa

µ= (Aa

⌧,Aa), with a = 1, 2, 3.

We write the Lagrangian of the resulting gauge theory as [85–87]

Lg = L + LY + LR + LH . (5.9)

The first term for the fermions descends directly from the Lc for the electrons

L =X

i

†i,s

✓@

@⌧� µ

◆�ss0 + iA

a

⌧�a

ss0

� i,s0 +

X

i,j

tij †i,s

ei�

aAa·(ri�rj)

�

ss0 j,s0 , (5.10)

and uses the same hopping terms for as those for c, along with a minimal coupling to the SU(2)

gauge field. Inserting (5.6) into Lcn, we find that the resulting expression involves 2 complex Higgs

fields, Ha

xand H

a

y, which are SU(2) adjoints; these are defined by

Ha

x⌘

1

2�x`Tr[�

`R�

aR

†], (5.11)

and similarly for Ha

y. Let us also note the inverse of (5.11)

�x` =1

2H

a

xTr[�`R�a

R†], (5.12)

20


Gauge theory of fluctuating antiferromagnetismThe simplest e↵ective Hamiltonian for the fermionic chargons is the

same as that for the electrons, with the magnetic order replaced

by the Higgs field.

H = �

X

i,⇢

t⇢

⇣ †i,s i+v⇢,s

+ †i+v⇢,s

i,s

⌘� µ

X

i

†i,s i,s

+Hint

Hint = ��

X

i

Ha(i)

†i,s�a

ss0 i,s0 + VH

IF we can transform to a rotating reference frame in whichHa(i) =

a constant independent of time, THEN the fermions in the

presence of fluctuating magnetism will inherit the Fermi surfaces

(if present) of the electrons in the presence of static magnetism.

For insulating spin liquids, we consider the case where the chargons

are fully gapped, and there are no Fermi surfaces.<latexit sha1_base64="MdR+oKSoNmtUFy0yVJo4LZlN6nM=">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</latexit>

Field Symbol Statistics SU(2)gauge SU(2)spin U(1)e.m.charge





Higgs H boson 3 1 0












µ= (Aa

⌧,Aa), with a = 1, 2, 3.


Lg = L + LY + LR + LH . (5.9)


L =X

i

†i,s

✓@

@⌧� µ

◆�ss0 + iA

a

⌧�a

ss0

� i,s0 +

X

i,j

tij †i,s

ei�

aAa·(ri�rj)

�

ss0 j,s0 , (5.10)



fields, Ha

xand H

a


Ha

x⌘

1

2�x`Tr[�

`R�

aR

†], (5.11)



�x` =1

2H

a

xTr[�`R�a

R†], (5.12)

20

The Higgs phases of the SU(2) gauge theory can realize states with topological order. The topological order depends upon the

structure of the Higgs condensate.


We obtain di↵erent numbers of adjoint Higgs scalars, Nh,

depending upon the spatial dependence of the local spin

correlations:

Neel correlations (un- and electron-doped cuprates):

Nh = 1,

K = (⇡,⇡),

Ha(i) = H

a1 (r)e

iK·ri

Coplanar spin correlations on the triangular lattice :

Nh = 2,

K = (4⇡/3, 4⇡/p3),

Ha(i) = Re

�[H

a1 (r) + iH

a2 (r)] e

iKx·ri

Bidirectional incommensurate correlations (hole doped cuprates):

Nh = 4,

Ky = (⇡,⇡ � �), Kx = (⇡ � �,⇡),

Ha(i) = Re

�[H

a1 (r) + iH

a2 (r)] e

iKx·ri + [Ha3 (r) + iH

a4 (r)] e

iKy·ri

<latexit sha1_base64="dkYSdyyQpz9dWk4b6ilNqxGn0v8=">AAAGBXicvVTbbhMxEN1CwiXcCjzyMqJBSkVacilQISGh9qUSUgWIUqQ6RF7vJDH12ovthUarfQPxNbwhXvkOfoJfgMlmq6Y34AlL2Z2dM2OfOeNMmCjpfKv1Y+7M2Ur13PkLF2uXLl+5em3++o1XzqRW4JYwytjXIXeopMYtL73C14lFHocKt8Pd9Qm+/R6tk0a/9OMEezEfajmQgnty9a+f/cm0kTpC7WvbCCb0XGqI5GCAlnyg0zikdDAD4NFbivSwIYdDB05wxa1rQn2zP6o3IcIEdST1ENLEaPAjBJfQIVyVEGqBk20miDKUTTgdJYy1qAo27lGNvXe7MoHWcuvhfQIPuGWsqDWzGOWwiagOJUIj1UvAdQSoUHhr9FJkEoxApInlHt3iI8YKpvAY2vW8SV9Qz1gYw9OcXA2WyCbQY5EqYaxW33jDG3KREDL6bWgUoTZfBHyTSdhPZCIyHkqsL/P6Af/O6ezXTaK45vZ4/VAK563kepiSvkCIlyTcDP/OCfxXiPu9bhOKN3PvrM+6+bFigMWh2cteIFFXOPAsm753yjJn6rwLsnB2Zp3MyuHI9w6J0N87JkMZx/5RjzUZSUtdIwXoUkgtTByjdumkcUe6PDIK4fTOrhxRpj+e7e0Si1B5PlGlVj9gP42AJZjC+9eAhIMD5f6TcHdnt+2etO3KX7cd/6EfjPXnF+jPVSyYMVbvr7ZaD6BdehaCcj3rz/9ikREpdcQLxZ3babcS38u4pWupMK+x1GHCxS4f4g6ZmsfoelnR3hzukCeCgbH0o8lReGczMh47N45Dioy5H7mj2MR5EraT+sFqL5M6ST3NlelBg1SBNzCZcjC9UWpMBhdWElcQI2658DTMDp0SxlSDQxqceuhHGfO45z/IiM7Nuss0g3JSbF8WON141Vlud5c7zzsLT9ZK7S4Et4LbQSNoBw+DJ8FG8CzYCkRlq5JVPlY+VT9Xv1S/Vr9NQ8/MlTk3g0Or+v031cnxfw==</latexit>


Spin liquid on the triangular lattice

SU(2) gauge theory

For the triangular lattice, Nh = 2, we define

the complex Higgs field

Ha= H

a1 + iH

a2 .

The SU(2) gauge theory is

L =1

2

��@µHa� ✏abcA

bµH

c��2 + 1

4g2F

aµ⌫F

aµ⌫ + V (H

a)

<latexit sha1_base64="kCE+qJ5zDUfOizWpkNRewymccB0=">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</latexit>

F aµ⌫ = @µA

a⌫ � @⌫A

aµ � ✏abcA

bµA

c⌫

V (Ha) = s (Ha⇤H

a) + u0 (Ha⇤H

a)2 + u1 |HaH

a|2

+ u3 (HaH

a)3 + c.c. .<latexit sha1_base64="jIJLOCIEvFdUbng6qLTVyceDysI=">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</latexit>

• For u1 > 0, we obtain a Higgs phase hHai / (1, i, 0)

• This corresponds to fluctuating coplanar spin configurations of the

triangular lattice.

• Although hHai 6= 0, spin rotation symmetry is preserved. The gauge-

invariant observable HaH

acorresponds to a charge-density-wave at

wavevector 2K, but hHaH

ai = 0.

• The Higgs condensate breaks the SU(2) gauge symmetry down to a Z2

gauge symmetry (this is di↵erent from the Weinberg-Salam model):

the condensate is only invariant a gauge transformation �aHa!

V �bHbV †with V =

✓�1 0

0 �1

◆.

• The Higgs phase of the SU(2) gauge theory has finite energy Z2 vor-

tex defects (visons!) associated with ⇡1(SO(3)) = Z2. These are

analogous to Z Abrikosov vortices in the Ginzburg-Landau theory of

superconductivity<latexit sha1_base64="Cm/lwVLYCtShvyxorq+/iC1rUgQ=">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</latexit>


Spin liquid on the triangular lattice• For u1 > 0, we obtain a Higgs phase hH

ai / (1, i, 0)

• This corresponds to fluctuating coplanar spin configurations of the

triangular lattice.

• Although hHai 6= 0, spin rotation symmetry is preserved. The gauge-

invariant observable HaH

acorresponds to a charge-density-wave at

wavevector 2K, but hHaH

ai = 0.

• The Higgs condensate breaks the SU(2) gauge symmetry down to a Z2

gauge symmetry (this is di↵erent from the Weinberg-Salam model):

the condensate is only invariant a gauge transformation �aHa!

V �bHbV †with V =

✓�1 0

0 �1

◆.

• The Higgs phase of the SU(2) gauge theory has finite energy Z2 vor-

tex defects (visons!) associated with ⇡1(SO(3)) = Z2. These are

analogous to Z Abrikosov vortices in the Ginzburg-Landau theory of

superconductivity<latexit sha1_base64="Cm/lwVLYCtShvyxorq+/iC1rUgQ=">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</latexit>

VISON


�aH

a = V (✓)�bH

b0 V

†(✓)

V (✓) = exp (ina�a✓/2)

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✓<latexit sha1_base64="Lsj802UANc6bE0Eh3H9yNBmimPA=">AAACHHicbVDLSsNAFJ34rPVVdekmWAVXIWl97kQ3LivYWjBFJpObdnAyCTM3agn9CbeKX+NO3Ap+jU7bLLR64MLhnHs5lxOkgmt03U9ranpmdm6+tFBeXFpeWa2srbd0kikGTZaIRLUDqkFwCU3kKKCdKqBxIOAquD0b+ld3oDRP5CX2U+jEtCt5xBlFI7W3fewB0u2bStV13BFs16kfuIfHQ1IoXkGqpEDjpvLlhwnLYpDIBNX62nNT7ORUIWcCBmU/05BSdku7cG2opDHoTj76d2DvGCW0o0SZkWiP1J8XOY217seB2Ywp9vSkNxT/9YJ4Ihmjo07OZZohSDYOjjJhY2IPy7BDroCh6BtCmeLmd5v1qKIMTWVlX4PpU3axl/sID3jPQ5Ob7zn7XA5MY95kP39Jq+Z4dad2UauenBbdlcgm2SK7xCOH5ISckwZpEkYEeSRP5Nl6sV6tN+t9vDplFTcb5Besj2+qbKK8</latexit>



VISON

ssc




A.V. Chubukov, T. Senthil and S. Sachdev, Physical Review Letters 72, 2089 (1994)

Higgs condensate hHai / (1, i, 0)

Spinons R gapped hRi = 0<latexit sha1_base64="n6VeYryfJ6b8fJAtDxdXcoiJS8w=">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</latexit>

Higgs condensate hHai / (1, i, 0)

Spinons R condensed hRi 6= 0<latexit sha1_base64="o+8dd+R27RO3NT6iIpWJOZ56QOg=">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</latexit>

O(4)⇤CFT3

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Field Symbol Statistics SU(2)gauge SU(2)spin U(1)e.m.charge





Higgs H boson 3 1 0












µ= (Aa

⌧,Aa), with a = 1, 2, 3.


Lg = L + LY + LR + LH . (5.9)


L =X

i

†i,s

✓@

@⌧� µ

◆�ss0 + iA

a

⌧�a

ss0

� i,s0 +

X

i,j

tij †i,s

ei�

aAa·(ri�rj)

�

ss0 j,s0 , (5.10)



fields, Ha

xand H

a


Ha

x⌘

1

2�x`Tr[�

`R�

aR

†], (5.11)



�x` =1

2H

a

xTr[�`R�a

R†], (5.12)

20

Z2

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RAPID COMMUNICATIONS

PHYSICAL REVIEW B 94 , 121111(R) (2016)

Symmetry fractionalization in the topological phase of the spin- 12 J1 - J2 triangular Heisenberg model

S. N. Saadatmand* and I. P. McCullochARC Centre for Engineered Quantum Systems, School of Mathematics and Physics,

The University of Queensland, St. Lucia, Queensland 4072, Australia(Received 15 July 2016; published 13 September 2016)

Using density-matrix renormalization-group calculations for infinite cylinders, we elucidate the properties ofthe spin-liquid phase of the spin- 1

2 J1-J2 Heisenberg model on the triangular lattice. We find four distinct groundstates characteristic of a nonchiral, Z2 topologically ordered state with vison and spinon excitations. We shedlight on the interplay of topological ordering and global symmetries in the model by detecting fractionalizationof time-reversal and space-group dihedral symmetries in the anyonic sectors, which leads to the coexistence ofsymmetry protected and intrinsic topological order. The anyonic sectors, and information on the particle statistics,can be characterized by degeneracy patterns and symmetries of the entanglement spectrum. We demonstrate theground states on finite-width cylinders are short-range correlated and gapped; however, some features in theentanglement spectrum suggest that the system develops gapless spinonlike edge excitations in the large-widthlimit.

DOI: 10.1103/PhysRevB.94.121111

Introduction. Topological phases [1– 3] are an intriguingform of quantum matter, which have been challenging theoristsfor the last two decades. Before then, it was believed thatLandau symmetry-breaking theory [4] can explain orderingand phase transitions of matter through (spontaneous) breakingof a Hamiltonian symmetry. However, topological phases canpreserve all symmetries and still acquire a finite energy gap.Topological phases fall into two broad categories, “intrinsictopological order” [3] on D ! 2 dimensional lattices, and“symmetry protected topological” (SPT) [5,6] order, whichcan also exist in one dimension (1D). For the former phase,there is no local unitary transformation to smoothly deformthe state into a product state without passing through a phasetransition, regardless of the existence of symmetries. Thecanonical example of an intrinsic topological order is the Z2ground state of the toric code [7]. On the other hand, SPTsare undeformable into product states only if protected by asymmetry. The best studied example is surely the Haldanephase of odd-integer spin chains [5,6], including the groundstate of the exactly solved Affleck-Kennedy-Lieb-Tasaki(AKLT) [8] model. A key breakthrough was the realization thatanyonic statistics associated with intrinsic topological ordercorresponds to fractionalization of symmetry. Therefore, whenintrinsic topological order is coupled with lattice symmetries,the symmetries themselves fractionalize and lead to SPTordering [9– 11], which is readily detectable in many numericalmethods.

In 1973, Anderson [12] conjectured that the spin- 12

triangular Heisenberg model (THM) with antiferromagneticnearest-neighbor (NN) bonds should stabilize a resonating-valence-bond (RVB) ground state. The failure of analytic andnumerical studies [13– 16] to find such a state motivates thesearch for a minimal extension that increases the frustrationwith a next-nearest-neighbor (NNN) term. The Hamiltonian is

*[email protected]

defined as

H = J1

!

⟨i,j⟩Si · Sj + J2

!

⟨⟨i,j⟩⟩Si · Sj , (1)

where ⟨i,j ⟩ (⟨⟨i,j ⟩⟩) indicates the sum over all NN (NNN)bonds. We set J1 = 1 as the unit of the energy henceforth.Previous numerical studies using a range of techniques[15,17– 23] have suggested a spin-liquid [1,3] (SL) region,with phase boundaries in the range of J low

2 ≈ 0.05 [18] up toJ

high2 ≈ 0.19 [17]. Employing finite-size density-matrix renor-

malization group (DMRG) [24,25] and using fixed aspect-ratio scaling of magnetic order parameters, we find phaseboundaries of 0.101(4) " J2 " 0.136(4), the calculation ofwhich will be described in more depth in a future work [26].In this Rapid Communication, we focus on the propertiesof the SL phase itself. For classical spins, the model hasa phase transition at J2 = 0.125 between two magneticallyordered phases [14]. This point roughly coincides with thecenter of the spin-liquid region for the quantum model, andin this work we focus on J2 = 0.125. While there is nothingforbidding the coexistence of spontaneous symmetry breakingand topological order, the Hastings-Oshikawa-Lieb-Schultz-Mattis theorem [27] in two dimensions (2D) states that theabsence of symmetry breaking in a spin- 1

2 system on even-width cylinders implies that the ground state is a SL with eithergapless (algebraic) excitations, or gapped with degenerateground states and anyonic excitations. Thus the absence ofsymmetry breaking is a sufficient (but not necessary) conditionfor a SL. Previous DMRG studies [20,21] have argued fora gapped Z2 toric-code SL, and have obtained two possibleground states by the presence (absence) of free spins nearthe boundaries of finite cylinders. However, the propertiesof these states are unclear, since, depending on the sectorchosen, the state may develop chiral order [21], or breakingof C6 rotational symmetry [20,21,28], leading to a nematicSL. Recent studies [11,29,30] focused on the kagome latticeshow that the time-reversal symmetric Z2 SL can be fullycharacterized by the symmetry properties of lattices on tori

2469-9950/2016/94(12)/121111(6) 121111-1 ©2016 American Physical Society


PHYSICAL REVIEW B 94 , 121111(R) (2016)

Symmetry fractionalization in the topological phase of the spin- 12 J1 - J2 triangular Heisenberg model

S. N. Saadatmand* and I. P. McCullochARC Centre for Engineered Quantum Systems, School of Mathematics and Physics,

The University of Queensland, St. Lucia, Queensland 4072, Australia(Received 15 July 2016; published 13 September 2016)

Using density-matrix renormalization-group calculations for infinite cylinders, we elucidate the properties ofthe spin-liquid phase of the spin- 1

2 J1-J2 Heisenberg model on the triangular lattice. We find four distinct groundstates characteristic of a nonchiral, Z2 topologically ordered state with vison and spinon excitations. We shedlight on the interplay of topological ordering and global symmetries in the model by detecting fractionalizationof time-reversal and space-group dihedral symmetries in the anyonic sectors, which leads to the coexistence ofsymmetry protected and intrinsic topological order. The anyonic sectors, and information on the particle statistics,can be characterized by degeneracy patterns and symmetries of the entanglement spectrum. We demonstrate theground states on finite-width cylinders are short-range correlated and gapped; however, some features in theentanglement spectrum suggest that the system develops gapless spinonlike edge excitations in the large-widthlimit.

DOI: 10.1103/PhysRevB.94.121111

Introduction. Topological phases [1– 3] are an intriguingform of quantum matter, which have been challenging theoristsfor the last two decades. Before then, it was believed thatLandau symmetry-breaking theory [4] can explain orderingand phase transitions of matter through (spontaneous) breakingof a Hamiltonian symmetry. However, topological phases canpreserve all symmetries and still acquire a finite energy gap.Topological phases fall into two broad categories, “intrinsictopological order” [3] on D ! 2 dimensional lattices, and“symmetry protected topological” (SPT) [5,6] order, whichcan also exist in one dimension (1D). For the former phase,there is no local unitary transformation to smoothly deformthe state into a product state without passing through a phasetransition, regardless of the existence of symmetries. Thecanonical example of an intrinsic topological order is the Z2ground state of the toric code [7]. On the other hand, SPTsare undeformable into product states only if protected by asymmetry. The best studied example is surely the Haldanephase of odd-integer spin chains [5,6], including the groundstate of the exactly solved Affleck-Kennedy-Lieb-Tasaki(AKLT) [8] model. A key breakthrough was the realization thatanyonic statistics associated with intrinsic topological ordercorresponds to fractionalization of symmetry. Therefore, whenintrinsic topological order is coupled with lattice symmetries,the symmetries themselves fractionalize and lead to SPTordering [9– 11], which is readily detectable in many numericalmethods.

In 1973, Anderson [12] conjectured that the spin- 12

triangular Heisenberg model (THM) with antiferromagneticnearest-neighbor (NN) bonds should stabilize a resonating-valence-bond (RVB) ground state. The failure of analytic andnumerical studies [13– 16] to find such a state motivates thesearch for a minimal extension that increases the frustrationwith a next-nearest-neighbor (NNN) term. The Hamiltonian is

*[email protected]

defined as

H = J1

!

⟨i,j⟩Si · Sj + J2

!

⟨⟨i,j⟩⟩Si · Sj , (1)

where ⟨i,j ⟩ (⟨⟨i,j ⟩⟩) indicates the sum over all NN (NNN)bonds. We set J1 = 1 as the unit of the energy henceforth.Previous numerical studies using a range of techniques[15,17– 23] have suggested a spin-liquid [1,3] (SL) region,with phase boundaries in the range of J low

2 ≈ 0.05 [18] up toJ

high2 ≈ 0.19 [17]. Employing finite-size density-matrix renor-

malization group (DMRG) [24,25] and using fixed aspect-ratio scaling of magnetic order parameters, we find phaseboundaries of 0.101(4) " J2 " 0.136(4), the calculation ofwhich will be described in more depth in a future work [26].In this Rapid Communication, we focus on the propertiesof the SL phase itself. For classical spins, the model hasa phase transition at J2 = 0.125 between two magneticallyordered phases [14]. This point roughly coincides with thecenter of the spin-liquid region for the quantum model, andin this work we focus on J2 = 0.125. While there is nothingforbidding the coexistence of spontaneous symmetry breakingand topological order, the Hastings-Oshikawa-Lieb-Schultz-Mattis theorem [27] in two dimensions (2D) states that theabsence of symmetry breaking in a spin- 1

2 system on even-width cylinders implies that the ground state is a SL with eithergapless (algebraic) excitations, or gapped with degenerateground states and anyonic excitations. Thus the absence ofsymmetry breaking is a sufficient (but not necessary) conditionfor a SL. Previous DMRG studies [20,21] have argued fora gapped Z2 toric-code SL, and have obtained two possibleground states by the presence (absence) of free spins nearthe boundaries of finite cylinders. However, the propertiesof these states are unclear, since, depending on the sectorchosen, the state may develop chiral order [21], or breakingof C6 rotational symmetry [20,21,28], leading to a nematicSL. Recent studies [11,29,30] focused on the kagome latticeshow that the time-reversal symmetric Z2 SL can be fullycharacterized by the symmetry properties of lattices on tori

2469-9950/2016/94(12)/121111(6) 121111-1 ©2016 American Physical Society


S. N. SAADATMAND AND I. P. MCCULLOCH PHYSICAL REVIEW B 94, 121111(R) (2016)

FIG. 1. Visualization of the triangular lattice on an infinite YCstructure. The size and color of the spheres indicate the long-rangecorrelation with the principal (gray) site. The color of the bondsindicates the strength of the NN correlations. The average of NNcorrelation is subtracted from each bond to highlight the anisotropypattern.

or infinite cylinders via the projective symmetry group (PSG)classifications [1,11,29,31,32].

Method. We consider a triangular lattice structure that iswrapped around an infinite cylinder. We employ the infinitematrix product states (iMPS) [25,33,34] ansatz, via theinfinite DMRG (iDMRG) algorithm [25,33] with single-siteoptimization [35] and utilizing SU (2) symmetry to obtaintranslationally invariant variational ground states on an infinitecylinder. We keep up to m = 5000 states, approximatelyequivalent to 15 000 states of a U (1)-symmetric basis. We usethe so-called YC structure, where the infinite-length cylinderhas a circumference equal to the number of sites in the Ydirection, L y . The mapping of the MPS chain on the cylinderis set to minimize the one-dimensional range of NN and NNNinteractions. The setup is shown in Fig. 1, indicating alsothe correlations of a typical ground state. Bipartite quantities,e.g., reduced density matrix (ρr ) and entanglement entropy[28], are measured by defining a Y -direction cut through thecylinder without crossing any vertical bond. The frameworkof iDMRG is a natural candidate for calculating symmetryproperties, since excitations can be introduced at cylinderedges by manipulating the symmetries of the wave function.Unlike the case for finite systems, the “edges” are effectivelyat infinity, so they do not affect the translation symmetryof the wave function. The Z2 SL of the RVB type carriesvison excitations, and bosonic and fermionic spinons [36].We control the even/odd parity of spinon flux in the groundstate by setting SU (2) quantum numbers (global spins, S)to be either integers (even sector—no spinon) or odd halfintegers (odd sector—with spinons) at the unit cell boundary.We cannot directly control the vison flux through the cylinder,so we can only obtain two ground states for each cylindergeometry. However, for finite-L y cylinders the degeneracy isexpected to be lifted, and fortunately we find that the groundstates for different width cylinders also give the vison andnonvison sectors, allowing us to obtain all four combinationsof even/odd spinons and the presence/absence of a vison flux.We note that Metlitski and Grover [37] and Kolley et al. [38]established the observation of a tower of states (TOS) in thelow-lying part of the entanglement spectrum (ES) [38,39] asa “smoking gun” evidence for the existence of magneticallyordered states (carrying Nambu-Goldstone excitations). Weconfirm the nonmagnetic nature of the phase by the absence

of TOS in the ES, regardless of the anyon sector (see belowand also Ref. [26]).

We obtain a structure of four anyon sectors of the Z2 toric-code-type topological order [40] that comprises the identityi anyon (carries no spinon or vison flux), a bosonic spinon banyon (carries a S = 1

2 spin), a v anyon (carries a vison andhas a π flux threading the cylinder, equivalent to possessingantiperiodic boundary conditions in the Y direction), andfinally a fermionic spinon f anyon (a composite excitation,which carries both a S = 1

2 spin and a π flux). In thisRapid Communication, we work in a minimally entangledstates (MES) [41,42] basis introduced in Refs. [11,30] for thefour-dimensional ground-state manifold and preserves SPTordering. For even-L y cylinders, each unique MES state [11]corresponds to threading an anyonic flux in the long directionand creating a particle/antiparticle pair of a at infinity, namely,|Ua/a

L y⟩ (also denoted as the YCL y-a sector). Given a particular

MES, the action of a global symmetry group (g) member #g

on the state can be considered as two independent actionson each anyon, i.e., #g|Ua/a

L y⟩ = ϒg|Ua

L y⟩ ⊗ ϒg|Ua

L y⟩, where

ϒg’s are unitary operators acting on a single anyon |UaL y

⟩.Anyons can fractionalize [11,29,30,32] the symmetry g byfactorizing an identity member of the group (square root of g).g is always a linear representation (it is describing a physicalsymmetry), but ϒg’s can now form a nontrivial PSG, whichis a central extension of the original group [1]. In the MPSrepresentation of the ground state, the ϒg can be expressedas operators acting on the “auxiliary” basis, i.e., the basisof the entanglement Hamiltonian − ln(ρr ) on a bipartite cut.Thus the existence of a PSG through measurements of ϒg’simplies 1D SPT ordering [11], by considering rings as single“supersites” (global symmetries along the Y direction are nowinternal symmetries when viewed as a 1D chain), which isstraightforward to detect using iMPS techniques [43].

Ground-state energies. We present ground-state energiesof anyonic sectors in Fig. 2. Energies are extrapolated to thethermodynamic limit of basis size m → ∞, using a linear fitagainst the energy variance per site (see Ref. [28] for details).We suggest that fitting against variances is the most accuratemethod for extrapolating energies in DMRG (more reliablethan extrapolation with respect to the DMRG truncation error).The different topological sectors are expected to acquireslightly different energies on finite-width cylinders. Dependingon L y , we find that the actual ground state in the even/oddsectors varies as to whether or not it contains a π visonflux. In some cases, especially for smaller widths, we havebeen able to construct variational wave functions in theother sectors by manipulating the wave function (i.e., toforce a particular symmetry state), but the resulting statesare rather unstable and have considerably higher energies.However, the overall behavior of energies indicates that thedifference between energy of even and odd sectors is rapidlydecreasing with increasing L y . This is consistent with having adegenerate ground state in the thermodynamic limit L y → ∞.Interestingly, there is an energy crossover between even/oddsectors already for YC10, which makes it unreliable to estimatean energy for the L y → ∞ limit. We note that our energiesper site for larger system widths are somewhat lower thanpreviously published results.

121111-2

YBa2Cu3O6+x

High temperature superconductors

CuO2 plane

Cu

O

on, inelastic neutron scattering demonstrate that there is no long-range antiferromagnetic order for the reduced samples in the dop-ing range from x ¼ 0:13–0.18 but that the large magnetic fluctua-tions provide some indications of its role for a possible pairingmechanism. Although similar studies of antiferromagnetism andsuperconductivity are made difficult by the severe difficultiesencountered in their growth, a similar temperature-doping phasediagram for the IL Sr1"xLaxCuO2 polycrystalline samples has beengathered by Ishawata et al. [19] from a few rare data in the

literature. The phase diagram already presented in Fig. 3(a) showsTNðxÞ behaving very similarly to the T0 phase with a very similarsharp transition between the antiferromagnetic and the supercon-ducting samples.

In Fig. 5, we compare the phase diagrams of the T0 phase withthat of hole-doped La2"xSrxCuO4 underlining in particular theextent of the doping range for the antiferromagnetic phase onthe electron-doped side and the very narrow range for ceriumcontents with superconductivity. One can also observe the pseu-dogap line which will be described in Section 4.1 and a zone (inlight blue) where magnetic correlations are strong but long-rangeorder does not set in as will be discussed in Section 4.5. Despitean apparent asymmetry in these phase diagrams, the mostimportant message of this schematic is the presence ofantiferromagnetism and superconductivity on either side, sug-gesting that the mechanism modifying the electronic propertiesand leading to superconductivity is most probably the same forboth types of doping. Of course, there are many differences inthese phase diagrams: for example, the pseudogap signatures inthe physical properties are quite different in electron- and hole-doped cuprates. There are solid evidence of two-band-like beha-viours in the transport properties of electron-doped cuprateswhile only hole-type carriers are observed for hole-doped cup-rates (except at very large doping [30]). Moreover, hole-dopedcuprates present a wealth of electronic phases like stripe order,charge-density waves, . . .[1] that have not yet been confirmedin electron-doped cuprates and may be absent (and perhapsirrelevant for the mechanism of superconductivity).

We should underline that Sr in SrCuO2 can be substituted by iso-valent Ca as shown first by Siegrist et al. [31]. In fact, it was the firstreport on the growth of the IL structure in the Sr1"yCayCuO2 systemas the addition of Ca allows one to stabilize the IL crystal structureand favour the growth of small Ca0.86Sr0.14CuO2 single crystals. Weshould underline also that CaCuO2 do exists in the IL structure andhas been inserted successfully in (CaCuO2)n/(SrTiO3)m and(SrCuO2)n/(BaCuO2)m heterostructures with surprising results likesuperconductivity up to 70 K [32,33].

3. Growth and reduction

The growth of T0 and IL electron-doped cuprates is generallycomplicated by the stability of other related oxide phases thatcompete in the temperature-composition phase diagram.3 Theseoxide phases can even be intercalated in the bulk as an epitaxiallayer and are difficult to eliminate completely. In fact, in someinstances, they may appear during the post-annealing process.Unfortunately, these competing phases can lead to many erroneousconclusions on the physical properties and extra care has been takenin recent years to improve the quality of the materials. Moreover, thepost-annealing required for the reduction is tricky as it involvesremoving a tiny amount of oxygen in conditions approaching thedecomposition line of the phase. In here, we will simply enumerateseparately the typical route used to grow high quality single crys-talline samples for the T0 and IL compounds, namely single crystalsand epitaxial thin films. We will then discuss the reduction condi-tions and the resulting structural impacts.

3.1. T0 single crystal growth

Single crystals of the T0 phase have mostly been grown success-fully by two different methods: in-flux solidification and travelling-solvent floating zone (TSFZ). The first group of techniques involvesthe directional growth of millimeter size high-quality single crystals

Fig. 4. (a) Temperature-doping phase diagram of Nd2"xCexCuO4 polycrystallinesamples obtained by muon spin resonance [2] showing the Néel temperature TN

and the superconducting transition Tc as a function of doping. (b) Temperature-doping phase diagram for as-grown (solid circles) and reduced (open symbols)Nd2"xCexCuO4 single crystals. The solid grey circles result from a shift assuming thatoxygen removal provides additional carriers (electrons). Adapted from Ref. [29].

Fig. 5. Schematic of the phase diagrams comparing hole-doped La2"xSrxCuO4 andelectron-doped Ln2"xCexCuO4. The diagram shows the doping range for supercon-ductivity (in red), antiferromagnetism (in blue) and T%ðxÞ as the pseudogap line(Section 4.1). In the light blue region above the superconducting dome, strongmagnetic correlations without long-range order persist (Section 4.5). Adapted fromRef. [6]. (For interpretation of the references to colour in this figure legend, thereader is referred to the web version of this article.)

3 In the case of thin films, the growth phase diagram involves also oxygen pressure.

318 P. Fournier / Physica C 514 (2015) 314–338

p














Insulating Antiferromagnet














p

K. Fujita, Chung Koo Kim, Inhee Lee, Jinho Lee, M. H. Hamidian, I. A. Firmo, S. Mukhopadhyay, H. Eisaki, S. Uchida, M. J. Lawler, E.-A. Kim, J. C. Davis, Science 344, 612 (2014)

Yang He, Yi Yin, M. Zech, A. Soumyanarayanan, I. Zeljkovic, M. M. Yee, M. C. Boyer, K. Chatterjee, W. D. Wise, Takeshi Kondo, T. Takeuchi, H. Ikuta, P. Mistark, R. S. Markiewicz, A. Bansil, S. Sachdev, E. W. Hudson, and J. E. Hoffman, Science 344, 608 (2014)

0.4

0.3

0.2

0.1

0

Intensity

0.300.250.200.150.100.050

doping

Smoothed Sx Sy

Hole doped cuprates














p

We have exactly transformed the Hubbard model

to the “spin-fermion” model with electronic Hamil-

tonian described by electrons ci↵ on the square

or triangular lattice with dispersion

Hc = �

X

i,⇢

t⇢⇣c†i,↵ci+v⇢,↵

+ c†i+v⇢,↵ci,↵

⌘

�µX

i

c†i,↵ci,↵ +Hint

are coupled to a magnetic moment order pa-

rameter �p(i), p = x, y, z

Hint = ��X

i

�p(i)c†i,↵�

p↵�ci,� + V�

<latexit sha1_base64="QaEafyESiH/HHcLplD2kMuhT0qg=">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</latexit>

�

Hole states

occupied

Electron states

occupied

�

+

Fermi surface+antiferromagnetism

The electron spin polariza-tion obeys

D~�(r, ⌧)

E= ~N eiK·r

where K = (⇡,⇡) is the or-dering wavevector.

<latexit sha1_base64="utEO16IOHYnrNwep7KKB6mYWu6I=">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</latexit>


In momentum space, the coupling between ~N and the electronstakes the form

Hint = �X

k,q,↵,�

~Nq · c†k+q,↵~�↵�ck+K,�

where ~� are the Pauli matrices, the boson momentum q is small,while the fermion momenum k extends over the entire Brillouinzone. In the antiferromagnetically ordered state, we may take~N / (0, 0, 1) , and the electron dispersions obtained by diago-nalizing Hc +Hint are

Ek± ="k + "k+K

2±

s✓"k � "k+K

2

◆2

+ �2| ~N|2

This leads to the Fermi surfaces shown in the following slidesas a function of increasing | ~N|.

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Metal with “large” Fermi surface


Fermi surfaces translated by K = (�,�).


“Hot” spots



Electron and hole pockets inantiferromagnetic phase with h~�i 6= 0

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Square lattice Hubbard model with hole doping

Metal with “large” Fermi

surface

s

Increasing SDW order

Metal with electron and hole pockets

Metal with hole pockets

Increasing SDW orderIncreasing SDW orderand smalland large

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Square lattice Hubbard model with electron doping


surface

s


Metal with electron pockets

and smalland largeh~�i 6= 0

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VOLUME 88, NUMBER 25 P H Y S I C A L R E V I E W L E T T E R S 24 JUNE 2002

00.4 0 0.4 00.400.51.01.5

0.0

0.04

0.10

0.15

00.51.01.5

Binding Energy (eV)

E

E

x=0.15x=0.10x=0.04(a) (b) (c) (d) (e)

FIG. 2 (color). (a) EDCs in-tegrated in a region near the!p"2, p"2# position. (b) EDCsintegrated in a region near the!p , 0.3p# position. (c), (d), and(e) EDCs from around the puta-tive LDA Fermi surface for x !0.04, 0.10, and 0.15 samples, re-spectively.

spectral weight that develops along the zone diagonal forthe x ! 0.04 sample is gapped by $150 meV. This isin contrast to near !p, 0.3p# [Fig. 2(b)], where there isspectral weight at EF for x ! 0.04 doping levels. For x !0.10, the zone diagonal spectral weight has moved closerto EF and even stronger EF intensity has formed near!p, 0.3p#. A weak dispersion is evident along the zone di-agonal for the x ! 0.10 sample (not shown). At x ! 0.15there is large near-EF weight and a strong dispersionthroughout the zone as detailed in our previous work[14,15]. The small chemical potential shift !$150 meV#seen by x-ray photoemission spectroscopy in NCCO byHarima et al. [17] in this doping range is consistent withour result where it appears that the band at !p, 0# moveson order of this amount when doping from x ! 0 tox ! 0.15.

Following our previous analysis of the optimally dopedcompound [14], we construct Fermi surfaces by integratingEDCs in a small window about EF !240 meV, 120 meV#and plotting this quantity as a function of "k (Fig. 3). Con-sistent with the above observation that spectral weightalong the zone diagonal is gapped for x ! 0.04, one cansee that it is only the states at !p, 0# that can contributeto low-energy properties. Here, a Fermi “patch” indicatesthat there is an extremely low-energy shallow band in thispart of the Brillouin zone. At x ! 0.10, weight at EF withlow intensity begins to appear near the zone diagonal. Near!p, 0#, the “band” becomes deeper and the Fermi patchbecomes a Fermi surface. At x ! 0.15, the zone diago-nal region has become intense. The full Fermi surface hasformed and only in the intensity-suppressed regions near!0.65p, 0.3p# (and its symmetry-related points) at the in-tersection of the Fermi surface with the antiferromagneticBrillouin zone boundary does the underlying Fermi surfaceretain its anomalous properties [14].

We can gain more insight by looking at plots of theEDCs around the putative LDA Fermi surface, as shown inFigs. 2(c)–2(e). In Fig. 2(c), for x ! 0.04, a large broadfeature is gapped by $150 meV near the zone-diagonal re-gion. As one proceeds around the ostensible LDA Fermisurface, the high-energy feature loses spectral weight and

may disappear, while another feature pushes up at EF. It isthis second component that contacts EF near !p, 0.3p# toform the small Fermi surface (FS) for the x ! 0.04 sample.Similar behavior is seen in the x ! 0.10 and 0.15 plots;the lowest-energy features become progressively sharper,closer to and upon entering the metallic state. The factthat there are two components supports our conjecturethat at low dopings the material can be characterized by asmall FS or electron pocket around !p, 0# (volume consis-tent with x ! 0.04) with doping-induced spectral weightat higher energy elsewhere in the zone. As the carrier con-centration is increased, the !p, 0# FS deforms and a newFS segment emerges. It derives from the diagonal fea-ture progressively moving to EF, as seen by comparingthe bottom EDCs of Figs. 2(c)–2(e). These two segmentsconnect to form the LDA-like Fermi surface with volume1.12 6 0.05.

Γ

(π,π)

(π,0)

(π,π)

(π,0) Γ

(π,π)

(π,0)

(a) (b) (c)x=0.10 x=0.15

Bin

ding

Ene

rgy

(eV

)

Γ Γ Γ Γπ,π π,π π,π π,0π,0π,0π,π

x=0 x=0.04 x=0.10 x=0.15

x=0.04

-0.5

-1.0

-1.5

0

(d)

FIG. 3 (color). Fermi surface plots: (a) x ! 0.04, (b) x !0.10, and (c) x ! 0.15. EDCs integrated in a 60 meV window!240 meV, 120 meV# plotted as a function of "k. Data weretypically taken in the displayed upper octant and symmetrizedacross the zone diagonal. (d) Panels showing doping dependent“band structure.” Features are plotted for the doping levels andmomentum space regions where they can be resolved. We donot include the slight low-energy shoulder of the x ! 0 sample,as this is probably reflective of a small intrinsic doping level.

257001-3 257001-3

N. P. Armitage, F. Ronning, D. H. Lu, C. Kim, A. Damascelli, K. M. Shen, D. L. Feng, H. Eisaki, Z.-X. Shen, P. K. Mang, N. Kaneko, M. Greven, Y. Onose, Y. Taguchi, and Y. TokuraPhys. Rev. Lett. 88, 257001 (2002)

Doping Dependence of an n-Type Cuprate Superconductor Investigated by

Angle-Resolved Photoemission Spectroscopy

Nd2�xCexCuO4±�<latexit sha1_base64="vAOEVOlTX9QAEWqRHpJE5qFGURo=">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</latexit><latexit sha1_base64="vAOEVOlTX9QAEWqRHpJE5qFGURo=">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</latexit><latexit sha1_base64="vAOEVOlTX9QAEWqRHpJE5qFGURo=">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</latexit><latexit sha1_base64="vAOEVOlTX9QAEWqRHpJE5qFGURo=">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</latexit>

Electron doped cuprates

PNAS 116, 3449 (2019) Fermi surface reconstruction in electron-doped cuprates without antiferromagnetic long-range orderJunfeng He, C. R. Rotundu, M. S. Scheurer, Y. He, M. Hashimoto, K. Xu, Y. Wang, E. W. Huang, T. Jia, S.-D. Chen, B. Moritz, D.-H. Lu, Y. S. Lee, T. P. Devereaux and Z.-X. Shen

• New photoemission measurements at zero magneticfield show Fermi surfaces in quantitative agreementwith quantum oscillation measurements.

• The energy gap between the electron and hole pock-ets collapses near x = 0.17 like an order parameter.

• “The totality of the data points to a mysterious or-der between x = 0.14 and x = 0.17, whose appear-ance favors the FS reconstruction and disappearancedefines the quantum critical doping. A recent topo-logical proposal provides an ansatz for its origin.”

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surface

s



and smalland large

�<latexit sha1_base64="rrJ5pnU8WKd738lOqlEZLPHRWg4=">AAAB7XicdVDLSgMxFM3UV62vqks3wSK4GjIy7XRZdKHLCvYB7VAyaaaNJpkhyQhl6D+4caGIW//HnX9j+hBU9MCFwzn3cu89UcqZNgh9OIWV1bX1jeJmaWt7Z3evvH/Q1kmmCG2RhCeqG2FNOZO0ZZjhtJsqikXEaSe6u5j5nXuqNEvkjZmkNBR4JFnMCDZWavcvsRB4UK4g1w9QUPcgclHNr1Z9S6qoFtQD6LlojgpYojkov/eHCckElYZwrHXPQ6kJc6wMI5xOS/1M0xSTOzyiPUslFlSH+fzaKTyxyhDGibIlDZyr3ydyLLSeiMh2CmzG+rc3E//yepmJ62HOZJoZKsliUZxxaBI4ex0OmaLE8IklmChmb4VkjBUmxgZUsiF8fQr/J+0z10Oud+1XGufLOIrgCByDU+CBADTAFWiCFiDgFjyAJ/DsJM6j8+K8LloLznLmEPyA8/YJzy+PSQ==</latexit><latexit sha1_base64="rrJ5pnU8WKd738lOqlEZLPHRWg4=">AAAB7XicdVDLSgMxFM3UV62vqks3wSK4GjIy7XRZdKHLCvYB7VAyaaaNJpkhyQhl6D+4caGIW//HnX9j+hBU9MCFwzn3cu89UcqZNgh9OIWV1bX1jeJmaWt7Z3evvH/Q1kmmCG2RhCeqG2FNOZO0ZZjhtJsqikXEaSe6u5j5nXuqNEvkjZmkNBR4JFnMCDZWavcvsRB4UK4g1w9QUPcgclHNr1Z9S6qoFtQD6LlojgpYojkov/eHCckElYZwrHXPQ6kJc6wMI5xOS/1M0xSTOzyiPUslFlSH+fzaKTyxyhDGibIlDZyr3ydyLLSeiMh2CmzG+rc3E//yepmJ62HOZJoZKsliUZxxaBI4ex0OmaLE8IklmChmb4VkjBUmxgZUsiF8fQr/J+0z10Oud+1XGufLOIrgCByDU+CBADTAFWiCFiDgFjyAJ/DsJM6j8+K8LloLznLmEPyA8/YJzy+PSQ==</latexit><latexit sha1_base64="rrJ5pnU8WKd738lOqlEZLPHRWg4=">AAAB7XicdVDLSgMxFM3UV62vqks3wSK4GjIy7XRZdKHLCvYB7VAyaaaNJpkhyQhl6D+4caGIW//HnX9j+hBU9MCFwzn3cu89UcqZNgh9OIWV1bX1jeJmaWt7Z3evvH/Q1kmmCG2RhCeqG2FNOZO0ZZjhtJsqikXEaSe6u5j5nXuqNEvkjZmkNBR4JFnMCDZWavcvsRB4UK4g1w9QUPcgclHNr1Z9S6qoFtQD6LlojgpYojkov/eHCckElYZwrHXPQ6kJc6wMI5xOS/1M0xSTOzyiPUslFlSH+fzaKTyxyhDGibIlDZyr3ydyLLSeiMh2CmzG+rc3E//yepmJ62HOZJoZKsliUZxxaBI4ex0OmaLE8IklmChmb4VkjBUmxgZUsiF8fQr/J+0z10Oud+1XGufLOIrgCByDU+CBADTAFWiCFiDgFjyAJ/DsJM6j8+K8LloLznLmEPyA8/YJzy+PSQ==</latexit><latexit sha1_base64="rrJ5pnU8WKd738lOqlEZLPHRWg4=">AAAB7XicdVDLSgMxFM3UV62vqks3wSK4GjIy7XRZdKHLCvYB7VAyaaaNJpkhyQhl6D+4caGIW//HnX9j+hBU9MCFwzn3cu89UcqZNgh9OIWV1bX1jeJmaWt7Z3evvH/Q1kmmCG2RhCeqG2FNOZO0ZZjhtJsqikXEaSe6u5j5nXuqNEvkjZmkNBR4JFnMCDZWavcvsRB4UK4g1w9QUPcgclHNr1Z9S6qoFtQD6LlojgpYojkov/eHCckElYZwrHXPQ6kJc6wMI5xOS/1M0xSTOzyiPUslFlSH+fzaKTyxyhDGibIlDZyr3ydyLLSeiMh2CmzG+rc3E//yepmJ62HOZJoZKsliUZxxaBI4ex0OmaLE8IklmChmb4VkjBUmxgZUsiF8fQr/J+0z10Oud+1XGufLOIrgCByDU+CBADTAFWiCFiDgFjyAJ/DsJM6j8+K8LloLznLmEPyA8/YJzy+PSQ==</latexit> �

<latexit sha1_base64="rrJ5pnU8WKd738lOqlEZLPHRWg4=">AAAB7XicdVDLSgMxFM3UV62vqks3wSK4GjIy7XRZdKHLCvYB7VAyaaaNJpkhyQhl6D+4caGIW//HnX9j+hBU9MCFwzn3cu89UcqZNgh9OIWV1bX1jeJmaWt7Z3evvH/Q1kmmCG2RhCeqG2FNOZO0ZZjhtJsqikXEaSe6u5j5nXuqNEvkjZmkNBR4JFnMCDZWavcvsRB4UK4g1w9QUPcgclHNr1Z9S6qoFtQD6LlojgpYojkov/eHCckElYZwrHXPQ6kJc6wMI5xOS/1M0xSTOzyiPUslFlSH+fzaKTyxyhDGibIlDZyr3ydyLLSeiMh2CmzG+rc3E//yepmJ62HOZJoZKsliUZxxaBI4ex0OmaLE8IklmChmb4VkjBUmxgZUsiF8fQr/J+0z10Oud+1XGufLOIrgCByDU+CBADTAFWiCFiDgFjyAJ/DsJM6j8+K8LloLznLmEPyA8/YJzy+PSQ==</latexit><latexit sha1_base64="rrJ5pnU8WKd738lOqlEZLPHRWg4=">AAAB7XicdVDLSgMxFM3UV62vqks3wSK4GjIy7XRZdKHLCvYB7VAyaaaNJpkhyQhl6D+4caGIW//HnX9j+hBU9MCFwzn3cu89UcqZNgh9OIWV1bX1jeJmaWt7Z3evvH/Q1kmmCG2RhCeqG2FNOZO0ZZjhtJsqikXEaSe6u5j5nXuqNEvkjZmkNBR4JFnMCDZWavcvsRB4UK4g1w9QUPcgclHNr1Z9S6qoFtQD6LlojgpYojkov/eHCckElYZwrHXPQ6kJc6wMI5xOS/1M0xSTOzyiPUslFlSH+fzaKTyxyhDGibIlDZyr3ydyLLSeiMh2CmzG+rc3E//yepmJ62HOZJoZKsliUZxxaBI4ex0OmaLE8IklmChmb4VkjBUmxgZUsiF8fQr/J+0z10Oud+1XGufLOIrgCByDU+CBADTAFWiCFiDgFjyAJ/DsJM6j8+K8LloLznLmEPyA8/YJzy+PSQ==</latexit><latexit sha1_base64="rrJ5pnU8WKd738lOqlEZLPHRWg4=">AAAB7XicdVDLSgMxFM3UV62vqks3wSK4GjIy7XRZdKHLCvYB7VAyaaaNJpkhyQhl6D+4caGIW//HnX9j+hBU9MCFwzn3cu89UcqZNgh9OIWV1bX1jeJmaWt7Z3evvH/Q1kmmCG2RhCeqG2FNOZO0ZZjhtJsqikXEaSe6u5j5nXuqNEvkjZmkNBR4JFnMCDZWavcvsRB4UK4g1w9QUPcgclHNr1Z9S6qoFtQD6LlojgpYojkov/eHCckElYZwrHXPQ6kJc6wMI5xOS/1M0xSTOzyiPUslFlSH+fzaKTyxyhDGibIlDZyr3ydyLLSeiMh2CmzG+rc3E//yepmJ62HOZJoZKsliUZxxaBI4ex0OmaLE8IklmChmb4VkjBUmxgZUsiF8fQr/J+0z10Oud+1XGufLOIrgCByDU+CBADTAFWiCFiDgFjyAJ/DsJM6j8+K8LloLznLmEPyA8/YJzy+PSQ==</latexit><latexit sha1_base64="rrJ5pnU8WKd738lOqlEZLPHRWg4=">AAAB7XicdVDLSgMxFM3UV62vqks3wSK4GjIy7XRZdKHLCvYB7VAyaaaNJpkhyQhl6D+4caGIW//HnX9j+hBU9MCFwzn3cu89UcqZNgh9OIWV1bX1jeJmaWt7Z3evvH/Q1kmmCG2RhCeqG2FNOZO0ZZjhtJsqikXEaSe6u5j5nXuqNEvkjZmkNBR4JFnMCDZWavcvsRB4UK4g1w9QUPcgclHNr1Z9S6qoFtQD6LlojgpYojkov/eHCckElYZwrHXPQ6kJc6wMI5xOS/1M0xSTOzyiPUslFlSH+fzaKTyxyhDGibIlDZyr3ydyLLSeiMh2CmzG+rc3E//yepmJ62HOZJoZKsliUZxxaBI4ex0OmaLE8IklmChmb4VkjBUmxgZUsiF8fQr/J+0z10Oud+1XGufLOIrgCByDU+CBADTAFWiCFiDgFjyAJ/DsJM6j8+K8LloLznLmEPyA8/YJzy+PSQ==</latexit>

�<latexit sha1_base64="rrJ5pnU8WKd738lOqlEZLPHRWg4=">AAAB7XicdVDLSgMxFM3UV62vqks3wSK4GjIy7XRZdKHLCvYB7VAyaaaNJpkhyQhl6D+4caGIW//HnX9j+hBU9MCFwzn3cu89UcqZNgh9OIWV1bX1jeJmaWt7Z3evvH/Q1kmmCG2RhCeqG2FNOZO0ZZjhtJsqikXEaSe6u5j5nXuqNEvkjZmkNBR4JFnMCDZWavcvsRB4UK4g1w9QUPcgclHNr1Z9S6qoFtQD6LlojgpYojkov/eHCckElYZwrHXPQ6kJc6wMI5xOS/1M0xSTOzyiPUslFlSH+fzaKTyxyhDGibIlDZyr3ydyLLSeiMh2CmzG+rc3E//yepmJ62HOZJoZKsliUZxxaBI4ex0OmaLE8IklmChmb4VkjBUmxgZUsiF8fQr/J+0z10Oud+1XGufLOIrgCByDU+CBADTAFWiCFiDgFjyAJ/DsJM6j8+K8LloLznLmEPyA8/YJzy+PSQ==</latexit><latexit sha1_base64="rrJ5pnU8WKd738lOqlEZLPHRWg4=">AAAB7XicdVDLSgMxFM3UV62vqks3wSK4GjIy7XRZdKHLCvYB7VAyaaaNJpkhyQhl6D+4caGIW//HnX9j+hBU9MCFwzn3cu89UcqZNgh9OIWV1bX1jeJmaWt7Z3evvH/Q1kmmCG2RhCeqG2FNOZO0ZZjhtJsqikXEaSe6u5j5nXuqNEvkjZmkNBR4JFnMCDZWavcvsRB4UK4g1w9QUPcgclHNr1Z9S6qoFtQD6LlojgpYojkov/eHCckElYZwrHXPQ6kJc6wMI5xOS/1M0xSTOzyiPUslFlSH+fzaKTyxyhDGibIlDZyr3ydyLLSeiMh2CmzG+rc3E//yepmJ62HOZJoZKsliUZxxaBI4ex0OmaLE8IklmChmb4VkjBUmxgZUsiF8fQr/J+0z10Oud+1XGufLOIrgCByDU+CBADTAFWiCFiDgFjyAJ/DsJM6j8+K8LloLznLmEPyA8/YJzy+PSQ==</latexit><latexit sha1_base64="rrJ5pnU8WKd738lOqlEZLPHRWg4=">AAAB7XicdVDLSgMxFM3UV62vqks3wSK4GjIy7XRZdKHLCvYB7VAyaaaNJpkhyQhl6D+4caGIW//HnX9j+hBU9MCFwzn3cu89UcqZNgh9OIWV1bX1jeJmaWt7Z3evvH/Q1kmmCG2RhCeqG2FNOZO0ZZjhtJsqikXEaSe6u5j5nXuqNEvkjZmkNBR4JFnMCDZWavcvsRB4UK4g1w9QUPcgclHNr1Z9S6qoFtQD6LlojgpYojkov/eHCckElYZwrHXPQ6kJc6wMI5xOS/1M0xSTOzyiPUslFlSH+fzaKTyxyhDGibIlDZyr3ydyLLSeiMh2CmzG+rc3E//yepmJ62HOZJoZKsliUZxxaBI4ex0OmaLE8IklmChmb4VkjBUmxgZUsiF8fQr/J+0z10Oud+1XGufLOIrgCByDU+CBADTAFWiCFiDgFjyAJ/DsJM6j8+K8LloLznLmEPyA8/YJzy+PSQ==</latexit><latexit sha1_base64="rrJ5pnU8WKd738lOqlEZLPHRWg4=">AAAB7XicdVDLSgMxFM3UV62vqks3wSK4GjIy7XRZdKHLCvYB7VAyaaaNJpkhyQhl6D+4caGIW//HnX9j+hBU9MCFwzn3cu89UcqZNgh9OIWV1bX1jeJmaWt7Z3evvH/Q1kmmCG2RhCeqG2FNOZO0ZZjhtJsqikXEaSe6u5j5nXuqNEvkjZmkNBR4JFnMCDZWavcvsRB4UK4g1w9QUPcgclHNr1Z9S6qoFtQD6LlojgpYojkov/eHCckElYZwrHXPQ6kJc6wMI5xOS/1M0xSTOzyiPUslFlSH+fzaKTyxyhDGibIlDZyr3ydyLLSeiMh2CmzG+rc3E//yepmJ62HOZJoZKsliUZxxaBI4ex0OmaLE8IklmChmb4VkjBUmxgZUsiF8fQr/J+0z10Oud+1XGufLOIrgCByDU+CBADTAFWiCFiDgFjyAJ/DsJM6j8+K8LloLznLmEPyA8/YJzy+PSQ==</latexit>

h�ai 6= 0<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>

h�ai = 0<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>


�a ) Antiferromagnetism at (⇡,⇡)<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>



surface

s


and large








Higgs phasewith

Topological order?


x = 0.175<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>

x = 0.14<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>



Gauge theory of fluctuating antiferromagnetismThe simplest e↵ective Hamiltonian for the fermionic chargons is the

same as that for the electrons, with the magnetic order replaced

by the Higgs field.

H = �

X

i,⇢

t⇢

⇣ †i,s i+v⇢,s

+ †i+v⇢,s

i,s

⌘� µ

X

i

†i,s i,s

+Hint

Hint = ��

X

i

Ha(i)

†i,s�a

ss0 i,s0 + VH

IF we can transform to a rotating reference frame in whichHa(i) =

a constant independent of time, THEN the fermions in the

presence of fluctuating magnetism will inherit the Fermi surfaces

(if present) of the electrons in the presence of static magnetism.

For the electron-doped cuprates, the chargons acquire the small,

reconstructed Fermi surfaces of the doped antiferromagnet.<latexit sha1_base64="PzAX/SNq0TtEWPw5oVLFFSKG5W0=">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 OPX72xyKlvvVToRvi64QUuPiwtzna+H20+W5z5dUfHXM3ol+jh9FatBG9iF5Fk+httBvx3pfe371/et+uf+sv9+/3H8xDr17pcu5FZ77+yn/h3jNE</latexit>

We obtain di↵erent numbers of adjoint Higgs scalars, Nh,

depending upon the spatial dependence of the local spin

correlations:

Neel correlations (un- and electron-doped cuprates):

Nh = 1,

K = (⇡,⇡),

Ha(i) = H

a1 (r)e

iK·ri

Coplanar spin correlations on the triangular lattice :

Nh = 2,

K = (4⇡/3, 4⇡/p3),

Ha(i) = Re

�[H

a1 (r) + iH

a2 (r)] e

iKx·ri

Bidirectional incommensurate correlations (hole doped cuprates):

Nh = 4,

Ky = (⇡,⇡ � �), Kx = (⇡ � �,⇡),

Ha(i) = Re

�[H

a1 (r) + iH

a2 (r)] e

iKx·ri + [Ha3 (r) + iH

a4 (r)] e

iKy·ri

<latexit sha1_base64="dkYSdyyQpz9dWk4b6ilNqxGn0v8=">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</latexit>


SU(2) gauge symmetry

broken down to U(1)

9

237

Figure 1 | Fermi surface reconstruction in optimal-doped NCCO. a, Schematic diagram of a 238 reconstructed Fermi surface with electron-like pockets near antinode and hole-like pockets near 239 node. The dashed lines indicate the antiferromagnetic Brillouin zone. b, Schematic band dispersion 240 along a momentum cut on the electron-like pocket (near hotspot), marked by the red arrow in (a). 241 The original dispersion is split into conduction and valence bands by an AFM energy gap. The 242 reconstructed bands bend back at the AFMZB. c, Schematic band dispersion along a momentum cut 243 on the hole-like pocket (nodal cut), marked by the blue arrow in (a). The AFM energy gap is slightly 244 above EF, but the folded band (back-bent hole band) disperses below EF. The gray (dashed) line in 245 (b,c) represents the original (folded) band. d-f, the same as a-c, but for the original Fermi surface 246 without reconstruction. g-i, Photoemission intensity plot (g), second derivative image with respect 247 to energy (h) and raw energy distribution curves (EDCs) (i) for optimal-doped NCCO, measured along 248 a momentum cut on the electron-like pocket (near hotspot, labelled by the red arrow in the inset of 249 h). Conduction and valence bands extracted from the EDCs (blue triangles in i) are also presented in 250 g (black circles) and h (white circles). The EDC at the AFMZB is shown in red (i). The main band and 251 folded band are marked by “MB” and “FB”, respectively. j-l, the same as g-i, but for the nodal cut 252 (labelled by the blue arrow in the inset of h). 253

Junfeng He, C. R. Rotundu, M. S. Scheurer, Y. He, M. Hashimoto, K. Xu, Y. Wang, E. W. Huang, T. Jia, S.-D. Chen, B. Moritz, D.-H. Lu, Y. S. Lee, T. P. Devereaux and Z.-X. Shen PNAS 116, 3449 (2019)

11

Spectral function of SU(2) gauge theory with topological order

Gap opening and back bending of the bands can be obtained in the absence of long-range order

and translational symmetry breaking if the system exhibits topological order1,2. Fig. 7 shows an

example of the calculated spectral function of an SU(2) gauge theory of fluctuating

antiferromagnetism within the formalism of Ref. 2 for the parameters relevant to our experiments

on NCCO.

The gap opening, band folding and back bending in the simulated spectral function are similar to

those observed by ARPES, but there are also additional weak features inside the gap which are not

visible in the experimental data.

Figure 7 | Spectrum of SU(2) gauge theory with topological order. Spectral function 𝐴𝜔(𝑘) (b)

along a momentum cut through the electron-like pocket (a). The parameters used in the simulation

are: |𝐻0| = 0.06𝑡0 , 𝑇 = 0.006𝑡0 , Δ = 0.002𝑡0 , t0′ = −0.45𝑡0 , 𝐽 = 0.1𝑡0 , 𝜂 = 0.003 𝑡0 , 𝑡 = 𝑍𝑡0 =

380meV and doping level of 𝑥 = 0.15. Please refer to Ref. 2 for the SU(2) gauge theory and the

physical meaning of the parameters.

Reference:

1. Sachdev, S., Topological order and Fermi surface reconstruction. arXiv: 1801.01125v3 (2018).

2. Scheurer, M. S. et al. Topological order in the pseudogap metal. Proc. Nat. Acad. Sci. 115,

E3665-E3672 (2018).

9

237

Figure 1 | Fermi surface reconstruction in optimal-doped NCCO. a, Schematic diagram of a 238 reconstructed Fermi surface with electron-like pockets near antinode and hole-like pockets near 239 node. The dashed lines indicate the antiferromagnetic Brillouin zone. b, Schematic band dispersion 240 along a momentum cut on the electron-like pocket (near hotspot), marked by the red arrow in (a). 241 The original dispersion is split into conduction and valence bands by an AFM energy gap. The 242 reconstructed bands bend back at the AFMZB. c, Schematic band dispersion along a momentum cut 243 on the hole-like pocket (nodal cut), marked by the blue arrow in (a). The AFM energy gap is slightly 244 above EF, but the folded band (back-bent hole band) disperses below EF. The gray (dashed) line in 245 (b,c) represents the original (folded) band. d-f, the same as a-c, but for the original Fermi surface 246 without reconstruction. g-i, Photoemission intensity plot (g), second derivative image with respect 247 to energy (h) and raw energy distribution curves (EDCs) (i) for optimal-doped NCCO, measured along 248 a momentum cut on the electron-like pocket (near hotspot, labelled by the red arrow in the inset of 249 h). Conduction and valence bands extracted from the EDCs (blue triangles in i) are also presented in 250 g (black circles) and h (white circles). The EDC at the AFMZB is shown in red (i). The main band and 251 folded band are marked by “MB” and “FB”, respectively. j-l, the same as g-i, but for the nodal cut 252 (labelled by the blue arrow in the inset of h). 253

S. Sachdev, Topological order and Fermi surface reconstruction, Reports on Progress in Physics 82, 014001 (2019)

M. S. Scheurer, S. Chatterjee, Wei Wu, M. Ferrero, A. Georges, and S. Sachdev, Proceedings of the National Academy of Sciences 115, E3665 (2018)

Junfeng He, C. R. Rotundu, M. S. Scheurer, Y. He, M. Hashimoto, K. Xu, Y. Wang, E. W. Huang, T. Jia, S.-D. Chen, B. Moritz, D.-H. Lu, Y. S. Lee, T. P. Devereaux and Z.-X. Shen PNAS 116, 3449 (2019)

Mathias Scheurer



surface

s


and large








Higgs phasewith

Topological order?


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Emergent gauge fields are obtained by transformations to a “rotating reference frame”.

SU(2) gauge theory Higgsed down to Z2 yields quantum phases with Z2 topological order

Theory of fluctuating antiferromagnetism in the electron-doped cuprates. Found a metallic state with topological order, reconstructed Fermi surfaces, and violation of the Luttinger theorem.This phase can explain recent photoemission experiments near optimal doping.

spin liquids in insulators and metalsqpt.physics.harvard.edu/talks/samarkand19.pdfsamarkand, may 14,...

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