Topological order in quantum matterqpt.physics.harvard.edu/talks/stanford17a.pdf · 2018. 3. 25. · that cannot be explained without using quantum mechanics. To ﬁnd the ground

Topological order in

quantum matterStanford University

Subir SachdevNovember 30, 2017

HARVARD

Talk online: sachdev.physics.harvard.edu

Mathias Scheurer

Shubhayu Chatterjee

Wei Wu

Michel Ferrero Antoine Georges

arXiv:1711.09925

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





ZXY =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,…..

T

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 ⌘ ei✓i)

lim|ri�rj |!1

⌦ i

⇤j

↵= | 0|2 6= 0 .

We break the symmetry by choosing an overall phase so that

h ii = 0 6= 0

Tc

h ii = 0 6= 0 h ii = 0

LRO

Figure 1: Schematic picture of ferro- and antiferromagnets. The chequerboard pat-

tern in the antiferromagnet is called a Néel state.

the role of symmetry in physics. Using new experimental techniques, hiddenpatterns of symmetry were discovered. For example, there are magnetic mate-rials where the moments form a chequerboard pattern where the neighbouringmoments are anti-parallel, see Fig. 1. In spite of not having any net magneti-zation, such antiferromagnets are nevertheless ordered states, and the patternof microscopic spins can be revealed by neutron scattering. The antiferro-magnetic order can again be understood in terms of the associated symmetrybreaking.

In a mathematical description of ferromagnetism, the important variable isthe magnetization, ~mi = µ ~Si, where µ is the magnetic moment and ~Si the spinon site i. In an ordered phase, the average value of all the spins is different fromzero, h~mii 6= 0. The magnetization is an example of an order parameter, whichis a quantity that has a non-zero average in the ordered phase. In a crystal itis natural to think of the sites as just the atomic positions, but more generallyone can define “block spins” which are averages of spins on many neighbouringatoms. The “renormalization group” techniques used to understand the theoryof such aggregate spins are crucial for understanding phase transitions, andresulted in a Nobel Prize for Ken Wilson in 1982.

It is instructive to consider a simple model, introduced by Heisenberg, thatdescribes both ferro- and antiferromagnets. The Hamiltonian is

HF = �JX

hiji

~Si · ~Sj � µX

i

~B · ~Si (1)

2

SRO

⌦ i

⇤j

↵⇠ exp(�|ri � rj |/⇠)

Wilson-Fisher theory(Nobel Prize, 1982)

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 = TKT ,

where the nature of the correlations changes

lim|ri�rj |!1

⌦ i

⇤j

↵⇠

8<

:

|ri � rj |�↵, for T < TKT , (QLRO)

exp(�|ri � rj |/⇠), for T > TKT , (SRO)

.

The low T phase also has topological order associated

with the suppression of vortices.

KT theory(Nobel Prize, 2016)

Kosterlitz-Thouless theory in d=2

T TKT

QLROTopological order SRO

Figure 3: To the left a single vortex configuration, and to the right a vortex-

antivortex pair. The angle ✓ is shown as the direction of the arrows, and the cores of

the vortex and antivortex are shaded in red and blue respectively. Note how the arrows

rotate as you follow a contour around a vortex. (Figure by Thomas Kvorning.)

by the Hamiltonian,

HXY = �JX

hiji

cos(✓i � ✓j) (3)

where hiji again denotes nearest neighbours and the angular variables, 0 ✓i < 2⇡ can denote either the direction of an XY-spin or the phase of asuperfluid. We shall discuss this model in some detail below.

Although the GL and BCS theories were very successful in describing manyaspects of superconductors, as were the theories developed by Lev Landau(Nobel Prize 1962), Nikolay Bogoliubov, Richard Feynman, Lars Onsager andothers for the Bose superfluids, not everything fit neatly into the Landauparadigm of order parameters and spontaneous symmetry breaking. Problemsoccur in low-dimensional systems, such as thin films or thin wires. Here, thethermal fluctuations become much more important and often prevent orderingeven at zero temperature [39]. The exact result of interest here is due toWegner, who showed that there cannot be any spontaneous symmetry breakingin the XY-model at finite temperature [53].

So far we have discussed phenomena that can be understood using classicalconcepts, at least as long as one accepts that superfluids are characterisedby a complex phase. There are however important macroscopic phenomenathat cannot be explained without using quantum mechanics. To find theground state of a quantum many-body problem is usually very difficult, butthere are some important examples where solutions to simplified problems givedeep physical insights. Electromagnetic response in crystalline materials is an

6

Vortices suppressed

Vortices proliferate

Kosterlitz-Thouless theory in d=2In 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 = TKT ,

where the nature of the correlations changes

lim|ri�rj |!1

⌦ i

⇤j

↵⇠

8<

:

|ri � rj |�↵, for T < TKT , (QLRO)

exp(�|ri � rj |/⇠), for T > TKT , (SRO)

.

The low T phase also has topological order associated

with the suppression of vortices.

KT theory(Nobel Prize, 2016)





J Jc

h ii = 0 6= 0

LRO






HF = �JX

hiji

~Si · ~Sj � µX

i

~B · ~Si (1)

2

Can we modify the XY model Hamiltonian to obtain a phase with

“topological order” in d=3 ?

SRO

h ii = 0⌦ i

⇤j

↵⇠ exp(�|ri � rj |/⇠)

J Jc

h ii = 0 6= 0

LRO






HF = �JX

hiji

~Si · ~Sj � µX

i

~B · ~Si (1)

2

Kh ii = 0

SRONo topological

order

SROTopological

order

h ii = 0⌦ i

⇤j

↵⇠ exp(�|ri � rj |/⇠)

⌦ i

⇤j

↵⇠ exp(�|ri � rj |/⇠)

SRONo topological

order

J Jc

h ii = 0 6= 0

LRO






HF = �JX

hiji

~Si · ~Sj � µX

i

~B · ~Si (1)

2

K

SRONo topological

order

h ii = 0 SROTopological

order

h ii = 0

⌦ i

⇤j

↵⇠ exp(�|ri � rj |/⇠)

⌦ i

⇤j

↵⇠ exp(�|ri � rj |/⇠)

Only even (±4⇡, ±8⇡ . . .)vortices proliferate

All (±2⇡, ±4⇡ . . .)vortices proliferate

Add terms which suppress single but not double vortices…..

eZXY =Y

i

Z 2⇡

0

d✓i

2⇡exp

⇣� eH/T

⌘

eH = �J

X

hiji

cos(✓i � ✓j)

+X

ijk`

Jijk` cos(✓i + ✓j � ✓k � ✓`) + . . . . . .

eZXY =X

{�ij}=±1

Y

i

Z 2⇡

0

d✓i

2⇡exp

⇣� eH/T

⌘

eH = �J

X

hiji

�ij cos [(✓i � ✓j)/2]�K

X

⇤

Y

(ij)2⇤�ij

�

�

�

� K

Lattice gauge theory precursors (without

symmetry broken phases):

F.Wegner, J.Math. Phys. 12, 2259 (1971).E. Fradkin and S. H. Shenker, PRD 19,3682 (1979).

S. Sachdev and N. Read, Int. J. Mod.

Phys. B, 5, 219 (1991).

R. Jalabert and S. Sachdev, PRB 44, 686(1991).

S. Sachdev and M. Vojta, J. Phys. Soc.

Jpn 69 Supp. B, 1 (2000).

P. E. Lammert, D. S. Rokhsar, and J. Toner,

PRL 70, 1650 (1993).

T. Senthil and M. P. A. Fisher, PRB 62,7850 (2000).

R. D. Sedgewick, D. J. Scalapino, R. L. Sugar,

PRB 65, 54508 (2002).

K. Park and S. Sachdev, PRB 65, 220405(2002).

eZXY =X

{�ij}=±1

Y

i

Z 2⇡

0

d✓i

2⇡exp

⇣� eH/T

⌘

eH = �J

X

hiji

�ij cos [(✓i � ✓j)/2]�K

X

⇤

Y

(ij)2⇤�ij

• At small K, we can explicitly sum over �ij , order-by-order in K, and the theory reduces to an ordinaryXY model with multi-site interactions. The resultinge↵ective action of the XY model is periodic in ✓i !✓i + 2⇡ (for any site i), and preserves the symmetry✓i ! ✓i + c (for all sites i).

• The theory has a Z2 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 = ei✓i is gauge invari-

ant, as are all physical observables. So this is an

XY model with a modified Hamiltonian, and no ad-

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eZXY =X

{�ij}=±1

Y

i

Z 2⇡

0

d✓i

2⇡exp

⇣� eH/T

⌘

eH = �J

X

hiji

�ij cos [(✓i � ✓j)/2]�K

X

⇤

Y

(ij)2⇤�ij

• A single (odd) 2⇡ vortex in ✓i hasQ(ij)2⇤ cos [(✓i � ✓j)/2] < 0.

• So for J > 0, such a vortex will preferQ

(ij)2⇤ �ij = �1,i.e. a 2⇡ vortex has Z2 flux = �1 in its core.

• So a large K > 0 will suppress (odd) 2⇡ vortices.

• There is no analogous suppression of (even) 4⇡ vortices.

eZXY =X

{�ij}=±1

Y

i

Z 2⇡

0

d✓i

2⇡exp

⇣� eH/T

⌘

eH = �J

X

hiji

�ij cos [(✓i � ✓j)/2]�K

X

⇤

Y

(ij)2⇤�ij

J Jc

h ii = 0 6= 0

h ii = 0

LRO






HF = �JX

hiji

~Si · ~Sj � µX

i

~B · ~Si (1)

2

Kh ii = 0

SRONo topological

order

SROTopological order

eZXY =X

{�ij}=±1

Y

i

Z 2⇡

0

d✓i

2⇡exp

⇣� eH/T

⌘

eH = �J

X

hiji

�ij cos [(✓i � ✓j)/2]�K

X

⇤

Y

(ij)2⇤�ij



J Jc

h ii = 0 6= 0

h ii = 0






HF = �JX

hiji

~Si · ~Sj � µX

i

~B · ~Si (1)

2

Kh ii = 0

Deconfined phase of Z2 gauge theory.Z2 flux is expelled

Confined phase of Z2 gauge theory.Z2 flux fluctuates

Higgs phase of Z2 gauge theory

eZXY =X

{�ij}=±1

Y

i

Z 2⇡

0

d✓i

2⇡exp

⇣� eH/T

⌘

eH = �J

X

hiji

�ij cos [(✓i � ✓j)/2]�K

X

⇤

Y

(ij)2⇤�ij





K �z

�z

�z

�z

A quantum Hamiltonian in 2+1 dimensionseH = �J

X

hiji

�zij cos [(✓i � ✓j)/2]�K

X

⇤

Y

(ij)2⇤�zij

+U

X

i

(ni)2 � g

X

hiji

�xij ; [✓i, nj ] = i�ij

-1

• In the topological phase,the suppressed Z2 fluxesof -1 become well-definedgapped quasiparticleexcitations (‘visons’) abovethe ground state.


X

hiji

�zij cos [(✓i � ✓j)/2]�K

X

⇤

Y

(ij)2⇤�zij

+U

X

i

(ni)2 � g

X

hiji

�xij ; [✓i, nj ] = i�ij

• In the topological phase, on atorus, inserting the Z2 flux of-1 into one of the cycles of thetorus leads to an orthogonal statewhose energy cost vanishes ex-ponentially in the size of thetorus: there are 4 degenerateground states on a large torus.


X

hiji

�zij cos [(✓i � ✓j)/2]�K

X

⇤

Y

(ij)2⇤�zij

+U

X

i

(ni)2 � g

X

hiji

�xij ; [✓i, nj ] = i�ij

J Jc

h ii = 0 6= 0

h ii = 0

LRO






HF = �JX

hiji

~Si · ~Sj � µX

i

~B · ~Si (1)

2

Kh ii = 0

SRONo topological

order


eH = �J

X

hiji

�zij cos [(✓i � ✓j)/2]�K

X

⇤

Y

(ij)2⇤�zij

+U

X

i

(ni)2 � g

X

hiji

�xij ; [✓i, nj ] = i�ij

A quantum Hamiltonian in 2+1 dimensions



J Jc

h ii = 0 6= 0

h ii = 0

LRO






HF = �JX

hiji

~Si · ~Sj � µX

i

~B · ~Si (1)

2

Kh ii = 0

SRONo topological

order


eH = �J

X

hiji

�zij cos [(✓i � ✓j)/2]�K

X

⇤

Y

(ij)2⇤�zij

+U

X

i

(ni)2 � g

X

hiji

�xij ; [✓i, nj ] = i�ij

A quantum Hamiltonian in 2+1 dimensionsThe topological order is the same

as that of the ‘toric code’,or of the U(1)⇥U(1) Chern-Simons theory

Lcs =1

⇡✏µ⌫�aµ@⌫b�







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

Will study on the square lattice

U/t

h~�i 6= 0

AF Metal with “small” Fermi surface

Increasing SDW order

Fermi surface+antiferromagnetism

Mean-field theory with an antiferromagnetic

order parameter ~�i = (�1)ix+iyD~Si

E

h~�i = 0

Metal with “large” Fermi surface

We can (exactly) transform the Hubbard model to the “spin-fermion”

model: electrons ci↵ on the square lattice with dispersion

Hc = �

X

i,⇢

t⇢⇣c†i,↵ci+v⇢,↵

+ c†i+v⇢,↵ci,↵

⌘� µ

X

i

c†i,↵ci,↵ +Hint

are coupled to an antiferromagnetic order parameter �`(i),

` = x, y, z

Hint = ��X

i

⌘i�`(i)c†i,↵�

`↵�ci,� + V�

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 long-

range AF order, which transforms the Fermi surfaces from large to

small.

We can (exactly) transform the Hubbard model to the “spin-fermion”

model: electrons ci↵ on the square lattice with dispersion

Hc = �

X

i,⇢

t⇢⇣c†i,↵ci+v⇢,↵

+ c†i+v⇢,↵ci,↵

⌘� µ

X

i

c†i,↵ci,↵ +Hint

are coupled to an antiferromagnetic order parameter �`(i),

` = x, y, z

Hint = ��X

i

⌘i�`(i)c†i,↵�

`↵�ci,� + V�

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 long-

range AF order, which transforms the Fermi surfaces from large to

small.


U/t

h~�i 6= 0



Fermi surface+antiferromagnetism

Mean-field theory with an antiferromagnetic

order parameter ~�i = (�1)ix+iyD~Si

E

h~�i = 0


U/t

Fermi surface+antiferromagnetism+topological order

h~�i 6= 0



h~�i = 0

Increasing SDW orderMetal with “small” Fermi surface

and topological order?

h~�i = 0


For fluctuating antiferromagnetism, we transform to a

rotating 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 AFM order in the rotating reference frame.

Note that this representation is ambiguous up to a

SU(2) gauge transformation, Vi

✓ i,+

i,�

◆! Vi

✓ i,+

i,�

◆

Ri ! RiV†i

�aH

a(i) ! Vi �

bH

b(i)V

†i .

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

For fluctuating antiferromagnetism, we transform to a

rotating 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 AFM order in the rotating reference frame.

Note that this representation is ambiguous up to a

SU(2) gauge transformation, Vi

✓ i,+

i,�

◆! Vi

✓ i,+

i,�

◆

Ri ! RiV†i

�aH

a(i) ! Vi �

bH

b(i)V

†i .


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

⌘� µ

X

i

†i,s i,s

+Hint

Hint = ��

X

i

⌘iHa(i)

†i,s�a

ss0 i,s0 + VH

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

a constant independent of i and time, THEN the fermions in the

presence of fluctuating AFM will inherit the small Fermi surfaces

of the electrons in the presence of static AFM.

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

⌘� µ

X

i

†i,s i,s

+Hint

Hint = ��

X

i

⌘iHa(i)

†i,s�a

ss0 i,s0 + VH

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

a constant independent of i and time, THEN the fermions in the

presence of fluctuating AFM will inherit the small Fermi surfaces

of the electrons in the presence of static AFM.

Fluctuating antiferromagnetism


Fluctuating antiferromagnetismWe cannot always find a single-valued SU(2) rotation Ri to make

the Higgs field Ha(i) a constant !

n-fold vortex in

AFM order

(assume easy-plane AFM for

simplicity)

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 antiferromagnetismWe cannot always find a single-valued SU(2) rotation Ri to make


n-fold vortex in

AFM order

R

(�1)nR


simplicity)

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)

We cannot always find a single-valued SU(2) rotation Ri to make


n-fold vortex in

AFM order

R

(�1)nR

Topological order

Vortices with n odd must be suppressed: such a metal with

“fluctuating antiferromagnetism” has BULK Z2

TOPOLOGICAL ORDER and fermions which inherit the

small Fermi surfaces of the antiferromagnetic metal.


simplicity)

U/t


h~�i 6= 0 h~�i = 0AF Metal with “small”

Fermi surface


h~�i = 0

Increasing SDW orderMetal with “small” Fermi surface

and topological order?



U/t



Increasing SDW orderMetal with “small” Fermi surface;

Higgs phase of a SU(2) gauge theorywith Z2 or U(1) topological order (with suppressed Z2 vortices and

hedgehogs respectively)

S. Sachdev and D. Chowdhury, Prog. Theor. Exp. Phys. 12C102 (2016)

h ~Hi 6= 0

hRi = 0

h ~Hi 6= 0

hRi 6= 0AF Metal with “small”

Fermi surface

Confining phase of SU(2) gauge theory.

h ~Hi = 0

hRi 6= 0

Topological order in the pseudogap metal

Mathias S. Scheurer,1 Shubhayu Chatterjee,1 Wei Wu,2, 3 MichelFerrero,2, 3 Antoine Georges,2, 4, 3, 5 and Subir Sachdev1, 6, 7

1Department of Physics, Harvard University, Cambridge MA 02138, USA2Centre de Physique Theorique, Ecole Polytechnique,

CNRS, Universite Paris-Saclay, 91128 Palaiseau, France3College de France, 11 place Marcelin Berthelot, 75005 Paris, France

4Center for Computational Quantum Physics, Flatiron Institute, 162 Fifth Avenue, New York, NY 10010, USA5DQMP, Universite de Geneve, 24 quai Ernest Ansermet, CH-1211 Geneve, Suisse6Perimeter Institute for Theoretical Physics, Waterloo, Ontario, Canada N2L 2Y5

7Department of Physics, Stanford University, Stanford CA 94305, USA(Dated: November 11, 2017)

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 comparedwith cluster extensions of dynamical mean field theory, and quantum Monte Carlo calculations, onthe pseudogap phase of the strongly interacting hole-doped Hubbard model. Good agreement isfound in the momentum, frequency, hopping, and doping dependencies of the spectral function andelectronic self-energy. We show that (approximate) lines of zeros of the zero-frequency electronicGreen’s function are signs of the underlying topological order of the gauge theory, and describehow 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.

The pseudogap metal is a novel state of electronic mat-ter found in the hole-doped, cuprate high temperaturesuperconductors [1]. It exhibits clear evidence of electri-cal transport with the temperature and frequency depen-dence of a conventional metal obeying Fermi liquid the-ory [2, 3]. However, a long-standing mystery in the studyof the cuprates is that photoemission experiments do notshow the ‘large’ Fermi surface that is expected from theLuttinger theorem of Fermi liquid theory [4]. (Brokensquare lattice translational symmetry can allow ‘small’Fermi surfaces, but there is no sign of it over a widerange of temperature and doping over which the pseudo-gap state is present [1, 5], and we will not discuss stateswith broken symmetry here.) There are non-perturbativearguments [6–9] that deviations from the Luttinger vol-ume are only possible in quantum states with topologicalorder. But independent evidence for the presence of topo-logical order in the pseudogap has so far been lacking.

In this paper we employ a SU(2) gauge theory of fluc-tuating antiferromagnetism (AF) in metals [10, 11] todescribe the pseudogap metal. Such a gauge theory de-scribes fluctuations in the orientation of the AF order,while preserving a local, non-zero magnitude. An alter-native, semiclassical treatment of fluctuations of the AForder parameter has been used to describe the electron-doped cuprates [12], but this remains valid at low tem-peratures (T ) only if the AF correlation length ⇠AF di-verges as T ! 0. We are interested in the case where ⇠AF

remains finite at T = 0, and then a gauge theory formu-lation is required to keep proper track of the fermionicdegrees of freedom in the background of the fluctuatingAF order. Such a gauge theory can formally be derivedfrom a lattice Hubbard model, as we will outline in thenext section. The SU(2) gauge theory yields a pseudogapmetal with only ‘small’ Fermi surfaces when the gauge

group is ‘Higgsed’ down to a smaller group. We will de-scribe examples of Higgsing down to U(1) and Z2, andthese will yield metallic states with U(1) and Z2 topolog-ical order. See Appendix A for a definition of topologicalorder in gapless systems; for the U(1) case we primarilyconsider, the topological order is associated [13, 14] withthe suppression of ‘hedgehog’ defects in the spacetimeconfiguration of the fluctuating AF order.

We will present a mean-field computation of the elec-tronic Green’s function across the entire Brillouin zonein the U(1) Higgs phase of the SU(2) gauge theory. Suchresults allow for a direct comparison with numerical com-putations on the Hubbard model. One of our main resultswill be that for a reasonable range of parameters in theSU(2) gauge theory, both the real and imaginary parts ofthe electron Green’s function of the gauge theory withtopological order closely resemble those obtained fromdynamical cluster approximation (DCA), a cluster ex-tension of dynamical mean field theory (DMFT). WhileDCA allows us to study the regime of strong correlationsdown to low temperature, it has limited momentum-space resolution. For this reason, we have also performeddeterminant quantumMonte Carlo (DQMC) calculationsand find self-energies that, in the numerically accessibletemperature range, agree well with the gauge theory com-putations. Additional results on the comparison betweenthe SU(2) gauge theory and the DCA and DQMC com-putations, as a function of doping and second-neighborhopping, appear in a companion paper [15].

In several discussions in the literature [16–22], viola-tions of the Luttinger theorem have been linked to thepresence of lines of zeros (in two spatial dimensions) inthe electron Green’s function on a “Luttinger surface”.The conventional perturbative proof of the Luttinger the-orem yields an additional contribution to the volume en-

DRAFT

Topological order in the pseudogap metalMathias S. Scheurera,*, Shubhayu Chatterjeea, Wei Wub,c, Michel Ferrerob,c, Antoine Georgesb,c,d,e, and Subir Sachdeva,f,g

aDepartment of Physics, Harvard University, Cambridge MA 02138, USA; bCentre de Physique Théorique, École Polytechnique, CNRS, Université Paris-Saclay, 91128Palaiseau, France; cCollège de France, 11 place Marcelin Berthelot, 75005 Paris, France; dCenter for Computational Quantum Physics, Flatiron Institute, 162 Fifth Avenue,New York, NY 10010, USA; eDQMP, Université de Genève, 24 quai Ernest Ansermet, CH-1211 Genève, Suisse; fPerimeter Institute for Theoretical Physics, Waterloo, Ontario,Canada N2L 2Y5; gDepartment of Physics, Stanford University, Stanford CA 94305, USA

This manuscript was compiled on November 11, 2017

We compute the electronic Green’s function of the topologically or-dered Higgs phase of a SU(2) gauge theory of fluctuating antiferro-magnetism on the square lattice. The results are compared with clus-ter extensions of dynamical mean field theory, and quantum MonteCarlo calculations, on the pseudogap phase of the strongly interact-ing hole-doped Hubbard model. Good agreement is found in the mo-mentum, frequency, hopping, and doping dependencies of the spec-tral function and electronic self-energy. We show that lines of (ap-proximate) zeros of the zero-frequency electronic Green’s functionare signs of the underlying topological order of the gauge theory,and describe how these lines of zeros appear in our theory of theHubbard model. We also derive a modified, non-perturbative versionof the Luttinger theorem that holds in the Higgs phase.

pseudogap metal | topological order | Hubbard model | DMFT | QMC

The pseudogap metal is a novel state of electronic mat-ter found in the hole-doped, cuprate high temperature

superconductors [1]. It exhibits clear evidence of electricaltransport with the temperature and frequency dependence ofa conventional metal obeying Fermi liquid theory [2, 3]. How-ever, a long-standing mystery in the study of the cuprates isthat photoemission experiments do not show the ‘large’ Fermisurface that is expected from the Luttinger theorem of Fermiliquid theory [4]. (Broken square lattice translational symme-try can allow ‘small’ Fermi surfaces, but there is no sign of itover a wide range of temperature and doping over which thepseudogap state is present [1, 5], and we will not discuss stateswith broken symmetry here.) There are non-perturbative ar-guments [6–9] that deviations from the Luttinger volume areonly possible in quantum states with topological order. Butindependent evidence for the presence of topological order inthe pseudogap has so far been lacking.

In this paper we employ a SU(2) gauge theory of fluctuatingantiferromagnetism (AF) in metals [10, 11] to describe thepseudogap metal. Such a gauge theory describes fluctuationsin the orientation of the AF order, while preserving a local,non-zero magnitude. An alternative, semiclassical treatmentof fluctuations of the AF order parameter has been used todescribe the electron-doped cuprates [12], but this remainsvalid at low temperatures (T ) only if the AF correlation length›AF diverges as T æ 0. We are interested in the case where›AF remains finite at T = 0, and then a gauge theory formula-tion is required to keep proper track of the fermionic degreesof freedom in the background of the fluctuating AF order.Such a gauge theory can formally be derived from a latticeHubbard model, as we will outline in the next section. TheSU(2) gauge theory yields a pseudogap metal with only ‘small’Fermi surfaces when the gauge group is ‘Higgsed’ down to asmaller group. We will describe examples of Higgsing downto U(1) and Z2, and these will yield metallic states with U(1)

and Z2 topological order. See SI Appendix A for a definitionof topological order in gapless systems; for the U(1) case weprimarily consider, the topological order is associated [13, 14]with the suppression of ‘hedgehog’ defects in the spacetimeconfiguration of the fluctuating AF order.

We will present a mean-field computation of the electronicGreen’s function across the entire Brillouin zone in the U(1)Higgs phase of the SU(2) gauge theory. Such results allowfor a direct comparison with numerical computations on theHubbard model. One of our main results will be that for areasonable range of parameters in the SU(2) gauge theory, boththe real and imaginary parts of the electron Green’s function ofthe gauge theory with topological order closely resemble thoseobtained from dynamical cluster approximation (DCA), a clus-ter extension of dynamical mean field theory (DMFT). WhileDCA allows us to study the regime of strong correlations downto low temperature, it has limited momentum-space resolution.For this reason, we have also performed determinant quan-tum Monte Carlo (DQMC) calculations and find self-energiesthat, in the numerically accessible temperature range, agreewell with the gauge theory computations. Additional resultson the comparison between the SU(2) gauge theory and theDCA and DQMC computations, as a function of doping andsecond-neighbor hopping, appear in a companion paper [15].

In several discussions in the literature [16–22], violationsof the Luttinger theorem have been linked to the presenceof lines of zeros (in two spatial dimensions) in the electronGreen’s function on a “Luttinger surface”. The conventionalperturbative proof of the Luttinger theorem yields an addi-tional contribution to the volume enclosed by the Fermi surfacewhen the electron Green’s function has lines of zeros: it was

Significance Statement

The copper oxide based high temperature superconductors dis-play a mysterious ‘pseudogap’ metal phase at temperatures justabove the critical temperature in a regime of low hole density.Extensive experimental and numerical studies have yieldedmuch information on the nature of the electron corrections,but a fundamental theoretical understanding has been lacking.We show that a theory of a metal with topological order andemergent gauge fields can model much of the numerical data.Our study opens up a route to a deeper understanding of thelong-range quantum entanglement in these superconductors,and to the direct detection of the topological characteristics ofthe many-body quantum state.

Author contributions.

The authors declare no conflict of interest.

*To whom correspondence should be addressed. E-mail: [email protected]

www.pnas.org/cgi/doi/10.1073/pnas.XXXXXXXXXX PNAS | November 11, 2017 | vol. XXX | no. XX | 1–7

arXiv:1711.09925

M. S. Scheurer, S. Chatterjee, Wei Wu, M. Ferrero, A. Georges, and S. Sachdev

function A!k ,!=0"=−1/" ImG!k ,0" for a 2D Hubbardmodel with U=8t at zero temperature for two values of dop-ing. For n =0.78 !left panels" we have a large electron-typeFermi surface #blue/dark gray line in the r!k" panel$ separat-ing the occupied region of the Brillouin zone !green/gray",defined by r!k"# 0 from the unoccupied region !yellow/lightgray" defined by r!k"$ 0. The Fermi surface can be alsotraced in the A!k" panel as the maximum of the spectralfunction. On the other hand, for n =0.92 a qualitatively dif-ferent picture emerges. The Fermi surface !blue/dark grayline" is now represented by a hole pocket and, in addition, wehave a line of zeros of the Green function !red dashed line"close to the !" ,"" region of the Brillouin zone. Furthermore,there is no one-to-one correspondence between the Fermisurface and the maximum of the spectral function. This be-havior has two origins, !1" the proximity of a zero line sup-presses the weight of the quasiparticle on the far side of thepocket, and !2" for k points corresponding to r!k"!0 thequasiparticles are pushed away from !=0 and a pseudogapopens at the Fermi level. We show this explicitly in Fig. 5 bycomparing the low frequency dependence of the spectralfunction in three different points of the Brillouin zone,marked by A, B and C in Fig. 4. Notice the suppression ofthe zero-frequency peak at point B and the frequency shift%=−0.05t of the peak at point C. The cumulant approachprovides a simple interpretation of this effect, observed inphotoemission experiments,12 in terms of the emergence ofinfinite self-energy lines or equivalently Luttinger lines !linesof zeros of the Green function".

IV. CONCLUSIONS

In conclusion, our strong coupling CDMFT study of theHubbard model shows that the lightly doped system is char-acterized by a small, closed Fermi line that appears in thezero-frequency spectral function as an arc due to the pres-ence of a line of zeros of the Green function near the “darkside” of the Fermi surface. These lines of poles of the self-energy appear near the Mott insulator and have the importantconsequence of violating the Luttinger relation between thenumber of particles and the volume of the Fermi surface asdetermined by the poles of the Green function.13 The vanish-ing of both the real and imaginary parts of the Green func-tion at specific locations in the Brillouin zone is an appealingscenario that is consistent with the growth of the real andimaginary parts of the self-energy as the temperature is re-duced. This is the hallmark of the Mott transition inCDMFT,11 and should be contrasted with the weak couplingscenario where the real part of the self-energy is regular, andonly the imaginary part exhibits singularities. The divergenceof the self-energy in certain points of the Brillouin zone isobserved in cluster DMFT calculation, using both realspace14 !CDMFT" and momentum space15 cluster schemes.This behavior seems at odds with the very spirit of DMFTand shows that the self-energy is not the appropriate quantityto describe Mott physics governed by short-range correla-tions. We argue that the irreducible quantity that should beused to describe this physics is the two-point cumulant. Inparticular for the Hubbard model, a precursor of the self-energy divergence can be observed even for values of theon-site interaction smaller than the bandwidth.15 A criticalreevaluation of the data for this regime from the cumulantperspective would be extremely useful.

Max

0

n = 0.78

A(k

)

n = 0.92

r(k)

(π,π)

(π,π)(π,π)

(π,0)

(π,0)

(0,π)(0,0) (0,0)

(0,0)(0,0)

(0,π)

(0,π)(0,π)

(π,π)

(π,0)

(π,0)

B

C

A

FIG. 4. !Color online" Renormalized energy r!k" !upper panels"and spectral function A!k" !lower panels" for the 2D Hubbardmodel with U=8t and T=0. The color code for the upper panels isgreen/gray !r# 0", blue/dark gray line !r=0", yellow/light gray !r$ 0", red dashed line !r→ & ". The frequency dependence of thespectral function for the points marked by A, B, and C is shown inFig. 5.

-0.25 0 0.25 0.5ω0

0.1

0.2

0.3

0.4

0.5

0.6

A(k

,ω)

ABC

FIG. 5. !Color online" Frequency dependence of the spectralfunction for three points in the Brillouin zone marked by A, B, andC in Fig. 4. Point A !blue line with triangles" is on the Fermisurface, close to !" /2 ," /2"; point B !green squares" is on the “darkside” of the Fermi surface, in the vicinity of the zero line; and pointC !red circles" is in the pseudogap region on the line and corre-sponding to the maxima of the spectral function !see Fig. 4". Noticethat the leading edge gap is quantitatively much smaller than thedistance between the peaks at positive and negative energy.

FERMI ARCS AND HIDDEN ZEROS OF THE GREEN… PHYSICAL REVIEW B 74, 125110 !2006"

125110-5

T.D. Stanescu and G. Kotliar,PRB 74, 125110 (2006)

Common features of many cluster-DMFT computations of pseudogapmetal:

• Momentum-space di↵erentia-tion: electron self-energy isenhanced at low frequenciesin the anti-nodal region, andvanishes in the nodal region.

• Gapped spectrum in the anti-nodal region

• Fermi arcs in the nodal region

• Apparent zero of Green’s func-tion on a “Luttinger surface”.

Electron Green’s function in SU(2) gauge theory higgsed down to U(1)Electron is fractionalized into bosonic spinons and fermionic chargons:

✓ci"ci#

◆= Ri

✓ i,+

i,�

◆,

The chargons, , are treated as free fermions in a Higgs background: this re-

constructs the chargon Fermi surface into “small pockets”, even though trans-

lational and spin rotation symmetries remain unbroken

H = �

X

i,⇢

t⇢

⇣ †i,s i+v⇢,s

+ †i+v⇢,s

i,s

⌘�µ

X

i

†i,s i,s��

X

i

(�1)ix+iyH

a0

†i,s�

ass0 i,s0

The diagonal chargon Green’s function is

G (!,~k) =

1

! � "~k � ⌃ (!,~k)

, ⌃ (!,~k) =

H20

! � "~k+~Q

, ~Q = (⇡,⇡) .

This has poles at the pocket Fermi surfaces, and zeros at "~k+~Q.

The electron Green’s function is computed via a convolution, and then the zeros

are smeared to approximate zeros.

Full Brillouin zone spectra of chargons ( ) and electrons (c)

Red line indicates the locusof ReG(k,! = 0) = 0

Red line indicates the locusof G(k,! = 0) = 0

Electron Green’s function in SU(2) gauge theory higgsed down to U(1)

Anti-nodal spectra compared to cluster DMFT


T = t/30 , U = 7t , p = 0.05

t0 takes di↵erent negative values

Lifshitz transition compared to cluster DMFT

✏~k = ✏~k +Re⌃~k(! = 0) = �Re⇣Gc(! = 0,~k)

⌘�1

The p-t0 dependence of the “interacting Lifshitz transition”, defined by the signchange of the renormalized quasiparticle energy ✏(⇡,0) at !peak > 0, is shown assolid blue lines calculated from the SU(2) gauge theory, part (a), and DCA, part(b). The black dashed lines show the location of the same transition for non-interacting electrons. The red lines indicate where the particle-hole asymmetryof the self-energy changes, i.e., where the peak position !peak of the anti-nodalIm(self-energy) changes sign.

(a) (b)


The imaginary part of the self-energy at the lowest Matsubara frequency !0 =

⇡T determined from DQMC on the Hubbard model (U = 7t, t0 = �0.1t, T =

0.25t, p = 0.042) and from the SU(2) gauge theory is shown in (a) and (b),

respectively. To avoid too much broadening, we have applied a slightly smaller

temperature of T = 0.15t for the gauge theory. The inset in (b) shows the gauge

theory prediction at zero frequency and low temperature (as before T = t/30).The black dashed line corresponds to the position of the Luttinger surface of

the chargons.

SM

FL

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

YBa2Cu3O6+x

SM

FL

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

YBa2Cu3O6+x

Insulating Antiferromagnet

A conventional metal:

the Fermi liquid with Fermi

surface of size 1+p

M. Plate, 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

SM

FL

SM

FL

Pseudogap metal

at low pMany 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

T. Senthil, M. Vojta and S. Sachdev, PRB 69, 035111 (2004)


U/t







h ~Hi 6= 0

hRi = 0

h ~Hi 6= 0


Fermi surface


h ~Hi = 0

hRi 6= 0


U/t







h ~Hi 6= 0

hRi = 0

h ~Hi 6= 0


Fermi surface


h ~Hi = 0

hRi 6= 0

p-dopedcuprates


U/t







h ~Hi 6= 0

hRi = 0

h ~Hi 6= 0


Fermi surface


h ~Hi = 0

hRi 6= 0

n-dopedcuprates

SM

FL

J Jc

h ii = 0 6= 0

h ii = 0

LRO






HF = �JX

hiji

~Si · ~Sj � µX

i

~B · ~Si (1)

2

Kh ii = 0

SRONo topological

order


eH = �J

X

hiji

�zij cos [(✓i � ✓j)/2]�K

X

⇤

Y

(ij)2⇤�zij

+U

X

i

(ni)2 � g

X

hiji

�xij ; [✓i, nj ] = i�ij

A quantum Hamiltonian in 2+1 dimensions




U/t






h ~Hi 6= 0

hRi = 0

h ~Hi 6= 0


Fermi surface


h ~Hi = 0

hRi 6= 0

New classes of quantum states with topological order

Can be understood as: (a) defect suppression in states with fluctuating order associated with broken symmetries (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







Mathias Scheurer

Shubhayu Chatterjee

Wei Wu

Michel Ferrero Antoine Georges

arXiv:1711.09925

Topological order in quantum matterqpt.physics.harvard.edu/talks/stanford17a.pdf · 2018. 3. 25. · that cannot be explained without using quantum mechanics. To ﬁnd the ground

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