nphys3517-s1.pdf

SUPPLEMENTARY INFORMATION

DOI: 10.1038/NPHYS3517

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1

Supplementary

Information for

Evidence of a

n odd

-parity hidden o

rder

in a strongly

spin- orbit

coupled correlated iridate

Contents:

S1

. RA

-SHG data for S

in

-P

out

and S

in

-S

out

geometries above and below

T

Ω

S2

. Mathematical expressions used for fitting RA-SHG patterns

S3. Visualizing the 2/

m

1



magnetic point group symmetry of the

Néel

phase

S4

. Loop-current order in cuprates versus iridates

S5. Behavior of hidden order domains under repeated thermal cycles

S6. Survey of RA-SHG patterns of Sr

2

IrO

4

across spatial regions

S7

. Functional form of the SHG temperature dependence

S8

.

Landau free energy expansion

for Né

el and hidden

order in Sr

2

IrO

4

S9

. Survey of

T

Ω

in

Sr

2

Ir

1-x

Rh

x

O

4

across spatial regions and across samples

Evidence of an odd-parity hidden order in a spin–orbit

coupled correlated iridate

2

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2

S1

. RA

-SHG data

for S

in

-P

out

and S

in

-S

out

geometries above and below

T

Ω

RA

-SHG data collected under two other polarization geometries S

in

-P

out

and S

in

-S

out

, which are

complementary to those shown in Figs 1

c & d

of the main text, are shown for completeness in

Fig. S1. The lowering of rotational symmetry from C

4

to C

1

upon cooling below

T



is visible in

both S

in

-P

out

and S

in

-S

out

geometries. The effect is present but less pronounced in S

in

-P

out

geometry owing to the different set of

susceptibility tensor elements probed

(Section S2). The

high temperature data in Fig.

S1

are

very well described by

bulk

EQ

-induced SHG from a

crystallographic 4/

m

point group while the low temperature data are very well described by a

coherent sum of

an EQ contribution (4/

m

point group) and an ED

contribution (2



/

m

or

m

1



magnetic point group). These results are fully consistent with Figs 1c & d

of the main text.

Fig. S1

.

RA

-

SHG data for S

in

-

P

out

(left column) and S

in

-

S

out

(right column) polarization geometries

collected at (a)

T

= 295 K and (b)

T

= 170 K. The tetragonal crystallographic axes

a

and

b

are labeled in

the upper left plot. Lines are best fits to expressions given in

Section S2

.

3

S2

. Mathematical expressions used for fitting RA-SHG patterns

i)

Fitting

RA

-SHG data for

T

>

T



Recent n

eutron diffraction

1,2

, RA

-SHG

and optical third harmonic generation measurements

3

show that the crystal structure of Sr

2

IrO

4

belongs to the centrosymmetric tetragonal 4/

m

point

group as opposed to the previously accepted 4/

mmm

point group

4,5

. Given the presence of

inversion symmetry, t

he leading

order contribution to SHG is the non-local term of electric-

quadrupole type, which can be expressed as an effective nonlinear polarization as

     



lkj

EQ

ijk l

i

EE

P





2

where

휒휒

푖푖푖푖푖푖푖푖

퐸퐸퐸퐸

is the electric-quadrupole susceptibility tensor. By

enforcing 4/

m

point group symmetry,

휒휒

푖푖푖푖푖푖푖푖

퐸퐸퐸퐸

is reduced to having 21 non-zero independent

elements

6

:

{

푥푥푥푥푥푥푥푥

=

푦푦푦푦푦푦푦푦

,

푧푧푧푧푧푧푧푧

,

푧푧푧푧푥푥푥푥

=

푧푧푧푧푦푦푦푦

,

푥푥푦푦푧푧푧푧

=

−푦푦푥푥푧푧푧푧

,

푥푥푥푥푦푦푦푦

=

푦푦푦푦푥푥푥푥

,

푥푥푥푥푥푥푦푦

=

−푦푦푦푦푦푦푥푥

,

푥푥푥푥푧푧푧푧

=

푦푦푦푦푧푧푧푧

,

푧푧푧푧푥푥푦푦

=

−푧푧푧푧푦푦푥푥

,

푥푥푦푦푥푥 푦푦

=

푦푦푥푥푦푦푥푥

,

푥푥푥푥푦푦푥푥

=

−푦푦푦푦푥푥푦푦

,

푧푧푥푥푧푧푥푥

=

푧푧푦푦 푧푧푦푦

,

푥푥푧푧푦푦푧푧

=

−푦푦푧푧푥푥푧푧

,

푥푥푦푦푦푦푥푥

=

푦푦푥푥푥푥 푦푦

,

푥푥푦푦 푥푥푥푥

=

−푦푦푥푥푦푦푦푦

,

푥푥푧푧 푥푥푧푧

=

푦푦푧푧 푦푦푧푧

,

푧푧푥푥푧푧푦푦

=

−푧푧푦푦 푧푧푥푥

,

푦푦푥푥푥푥푥푥

=

−푥푥푦푦푦푦푦푦

,

푧푧푥푥푥푥푧푧

=

푧푧푦푦푦푦푧푧

,

푧푧푥푥푦푦푧푧

=

−푧푧 푦푦푥푥푧푧

,

푥푥푧푧푧푧푥푥

=

푦푦푧푧푧푧푦푦

,

푥푥푧푧푧푧푦푦

=

−푦푦푧푧푧푧푥푥

}

With the four additional constraints from degen

erate SHG

{

푧푧푧푧푥푥푥푥

=

푧푧푥푥푥푥푧푧

,

푧푧푧푧푦푦푥푥

=

푧푧푥푥푦푦푧푧

,

푥푥푥푥푦푦푦푦

=

푥푥푦푦푦푦푥푥

,

푥푥푥푥푥푥푦푦

=

푥푥푦푦푥푥푥푥

}

, the number of non-zero independent tensor elements is further reduced to

17. Th

e rotation of the crystal by an angle

φ

about the

c

-axis is carried out mathematically by

applying a basis transformation on the reduced tensor from the original (primed) to rotated (un-

primed) reference frame using

휒휒

푖푖푖푖푖푖푖푖

퐸퐸퐸퐸

(

휑휑

)

=

푅푅

푖푖푖푖

′

(

휑휑

)

푅푅

푖푖푖푖

′

(

휑휑

)

푅푅

푖푖푖푖

′

(

휑휑

)

푅푅

푖푖푖푖

′

(

휑휑

)

휒휒

푖푖

′

푖푖

′

푖푖

′

푖푖

′

퐸퐸퐸퐸

where

푅푅

푖푖푖푖

(

휑휑

)

is the rotation matrix about the

c

-axis

. Finally, the expression that is used to fit the

RA

-SHG data

at

T

>

T

Ω

is given by

퐼퐼

(

2

휔휔

,

휑휑

)

=

|퐴퐴푒푒̂

푖푖

(

2

휔휔

)

휒휒

푖푖푖푖푖푖푖푖

퐸퐸퐸퐸

(

휑휑

)

푒푒̂

푖푖

(

휔휔

)

휕휕

푖푖

푒푒̂

푖푖

(

휔휔

)

|

2

퐼퐼

(

휔휔

)

2

, where

A

is a constant determined by the experimental geometry,

퐼퐼

(

휔휔

)

is the intensity of the incident

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3

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DOI: 10.1038/NPHYS3517

2

S1

. RA

-SHG data

for S

in

-P

out

and S

in

-S

out

geometries above and below

T

Ω

RA

-SHG data collected under two other polarization geometries S

in

-P

out

and S

in

-S

out

, which are

complementary to those shown in Figs 1

c & d

of the main text, are shown for completeness in

Fig. S1. The lowering of rotational symmetry from C

4

to C

1

upon cooling below

T



is visible in

both S

in

-P

out

and S

in

-S

out

geometries. The effect is present but less pronounced in S

in

-P

out

geometry owing to the different set of

susceptibility tensor elements probed

(Section S2). The

high temperature data in Fig.

S1

are

very well described by

bulk

EQ

-induced SHG from a

crystallographic 4/

m

point group while the low temperature data are very well described by a

coherent sum of

an EQ contribution (4/

m

point group) and an ED

contribution (2



/

m

or

m

1



magnetic point group). These results are fully consistent with Figs 1c & d

of the main text.

Fig. S1

.

RA

-

SHG data for S

in

-

P

out

(left column) and S

in

-

S

out

(right column) polarization geometries

collected at (a)

T

= 295 K and (b)

T

= 170 K. The tetragonal crystallographic axes

a

and

b

are labeled in

the upper left plot. Lines are best fits to expressions given in

Section S2

.

3

S2

. Mathematical expressions used for fitting RA-SHG patterns

i)

Fitting

RA

-SHG data for

T

>

T



Recent n

eutron diffraction

1,2

, RA

-SHG

and optical third harmonic generation measurements

3

show that the crystal structure of Sr

2

IrO

4

belongs to the centrosymmetric tetragonal 4/

m

point

group as opposed to the previously accepted 4/

mmm

point group

4,5

. Given the presence of

inversion symmetry, t

he leading

order contribution to SHG is the non-local term of electric-

quadrupole type, which can be expressed as an effective nonlinear polarization as

     



lkj

EQ

ijk l

i

EE

P





2

where

휒휒

푖푖푖푖푖푖푖푖

퐸퐸퐸퐸

is the electric-quadrupole susceptibility tensor. By

enforcing 4/

m

point group symmetry,

휒휒

푖푖푖푖푖푖푖푖

퐸퐸퐸퐸

is reduced to having 21 non-zero independent

elements

6

:

{

푥푥푥푥푥푥푥푥

=

푦푦푦푦푦푦푦푦

,

푧푧푧푧푧푧푧푧

,

푧푧푧푧

푥푥푥푥

=

푧푧푧푧

푦푦푦푦

,

푥푥

푦푦푧푧푧푧

=

−푦푦

푥푥푧푧푧푧

,

푥푥푥푥

푦푦푦푦

=

푦푦푦푦

푥푥푥푥

,

푥푥푥푥푥푥

푦푦

=

−푦푦푦푦푦푦

푥푥

,

푥푥푥푥

푧푧푧푧

=

푦푦푦푦

푧푧푧푧

,

푧푧푧푧

푥푥푦푦

=

−푧푧푧푧

푦푦푥푥

,

푥푥

푦푦

푥푥

푦푦

=

푦푦

푥푥

푦푦

푥푥

,

푥푥푥푥

푦푦

푥푥

=

−푦푦푦푦

푥푥

푦푦

,

푧푧

푥푥

푧푧

푥푥

=

푧푧

푦푦

푧푧

푦푦

,

푥푥

푧푧푦푦푧푧

=

−푦푦

푧푧푥푥푧푧

,

푥푥

푦푦푦푦

푥푥

=

푦푦

푥푥푥푥

푦푦

,

푥푥

푦푦

푥푥푥푥

=

−푦푦

푥푥

푦푦푦푦

,

푥푥

푧푧

푥푥

푧푧

=

푦푦

푧푧

푦푦

푧푧

,

푧푧

푥푥

푧푧

푦푦

=

−푧푧

푦푦

푧푧

푥푥

,

푦푦

푥푥푥푥푥푥

=

−푥푥

푦푦푦푦푦푦

,

푧푧

푥푥

푧푧

=

푧푧

푦푦푦푦

푧푧

,

푧푧

푥푥푦푦

푧푧

=

−푧푧

푦푦푥푥

푧푧

,

푥푥

푧푧푧푧

푥푥

=

푦푦

푧푧푧푧

푦푦

,

푥푥

푧푧푧푧

푦푦

=

−푦푦

푧푧푧푧푥푥

}

With the four additional constraints from degen

erate SHG

{

푧푧푧푧

푥푥푥푥

=

푧푧

푥푥

푧푧

,

푧푧푧푧

푦푦푥푥

=

푧푧푥푥

푦푦푧푧

,

푥푥푥푥

푦푦푦푦

=

푥푥

푦푦푦푦

푥푥

,

푥푥푥푥푥푥

푦푦

=

푥푥

푦푦

푥푥푥푥

}

, the number of non-zero independent tensor elements is further reduced to

17. Th

e rotation of the crystal by an angle

φ

about the

c

-axis is carried out mathematically by

applying a basis transformation on the reduced tensor from the original (primed) to rotated (un-

primed) reference frame using

휒휒

푖푖푖푖푖푖푖푖

퐸퐸퐸퐸

(

휑휑

)

=

푅푅

푖푖푖푖

′

(

휑휑

)

푅푅

푖푖푖푖

′

(

휑휑

)

푅푅

푖푖푖푖

′

(

휑휑

)

푅푅

푖푖푖푖

′

(

휑휑

)

휒휒

푖푖

′

푖푖

′

푖푖

′

푖푖

′

퐸퐸퐸퐸

where

푅푅

푖푖푖푖

(

휑휑

)

is the rotation matrix about the

c

-axis

. Finally, the expression that is used to fit the

RA

-SHG data

at

T

>

T

Ω

is given by

퐼퐼

(

2

휔휔

,

휑휑

)

=

|퐴퐴푒푒̂

푖푖

(

2

휔휔

)

휒휒

푖푖푖푖푖푖푖푖

퐸퐸퐸퐸

(

휑휑

)

푒푒̂

푖푖

(

휔휔

)

휕휕

푖푖

푒푒̂

푖푖

(

휔휔

)

|

2

퐼퐼

(

휔휔

)

2

, where

A

is a constant determined by the experimental geometry,

퐼퐼

(

휔휔

)

is the intensity of the incident

4

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DOI: 10.1038/NPHYS3517

4

beam

and

푒푒̂

is the polarization of the incoming or outgoing light, which we select to be either

linearly P or S polarized. We note that previous work has

already

shown that there is no evidence

of a surface electric-dipole contribution SHG

3

and that the crystallographic symmetry of the

surface remains unchanged across

T

Ω

7

.

ii)

Fitting

RA

-SHG data

for

T

<

T



The low temperature RA-SHG data are fit to a coherent sum of the electric-quadrupole term

described above and a

hidden order induced electric-dipole term. The electric-dipole contribution

is expressed as a nonlinear polarization

푃푃

푖푖

(

2

휔휔

)

∝휒휒

푖푖푖푖푖푖

퐸퐸퐸퐸

퐸퐸

푖푖

(

휔휔

)

퐸퐸

푖푖

(

휔휔

)

. By enforcing 2



/

m

magnetic

point group symmetry

,

휒휒

푖푖푖푖푖푖

퐸퐸퐸퐸

is reduced to having 14 non-zero independent elements:

{

푥푥푥푥푥푥

,

푥푥푥푥푥푥

,

푥푥푥푥푥푥

,

푥푥푥푥푥푥

,

푥푥푥푥푥푥

,

푥푥푥푥푥푥

,

푥푥푥푥푥푥

,

푥푥푥푥푥푥

,

푥푥푥푥푥푥

,

푥푥

푥푥푥푥

,

푥푥푥푥푥푥

,

푥푥푥푥푥푥

,

푥푥푥푥

푥푥

,

푥푥푥푥

푥푥

}

We only discuss the results using a 2



/

m

magnetic point group here although the same procedure

was applied to all of the magnetic point groups we surveyed. The additional constraints from

degenerate SHG

{

푥푥푥푥푥푥

=

푥푥푥푥푥푥

,

푥푥푥푥푥푥

=

푥푥푥푥푥푥

,

푥푥푥푥푥푥

=

푥푥푥푥

푥푥

,

푥푥푥푥푥푥

=

푥푥푥푥

푥푥

}

leaves 10 non-zero

independent tensor elements remaining. A basis transformation was then carried out on

휒휒

푖푖푖푖푖푖

퐸퐸퐸퐸

using

휒휒

푖푖푖푖푖푖

퐸퐸퐸퐸

(

휑휑

)

=

푅푅

푖푖푖푖 ′

푅푅

푖푖푖푖

′

푅푅

푖푖푖푖

′

휒휒

푖푖

′

푖푖

′

푖푖

′

퐸퐸퐸퐸

and the expression used to fit the RA

-SHG data at

T

<

T

Ω

is

퐼퐼

̃

(

2

휔휔

,

휑휑

)

=

|퐴퐴푒푒̂

푖푖

(

2

휔휔

)

휒휒

푖푖푖푖푖푖푖푖

퐸퐸퐸퐸

(

휑휑

)

푒푒̂

푖푖

(

휔휔

)

휕휕

푖푖

푒푒̂

푖푖

(

휔휔

)

+

퐴퐴푒푒̂

푖푖

(

2

휔휔

)

휒휒

푖푖푖푖푖푖

퐸퐸퐸퐸

(

휑휑

)

푒푒̂

푖푖

(

휔휔

)

푒푒̂

푖푖

(

휔휔

)

|

2

퐼퐼

(

휔휔

)

2

.

5

S3

. Visualizing the 2/

m

1



magnetic point group symmetry of the N

éel

phase

Fig. S2 (a) Known dipolar antiferromagnetic structure

of Sr

2

IrO

4

.

Planes through each Ir-O layer in the unit cell

are shown, with z denoting the position of the layer along the

c

-axis.

The red

and green arrows denote the

direction of the

magnetic dipole moments in the two structural

sub

-lattice

s. Resultant

magnetic structure

upon

applying the following operations contained within the 2/

m

1



point group: (b) 180



rotation about

the

c

-axis

, (c)

reflection about a mirror plane normal to the

c

-axis, (d) time

-reversal, (e) sp

atial inversion

. All structures are related

by simple lattice translation.

The antiferromagnetic structure of Sr

2

IrO

4

reported by neutron and resonant x-ray diffraction

measurements is shown in Fig. S2a.

Its magnetic point group derived from the tetragonal 4/

m

crystallographic point group of

Sr

2

IrO

4

is 2/

m

1



[Courtesy of S. Lovesey and D. Khalyavin].

Invariance of the magnetic structure under the elements of 2/

m

1



is explicitly shown in Figs S2

b-e. We note that the 2/

m

1



magnetic point group assignment does not rely on the magnitude of

the magnetic moments on the two structural sub-lattices being equal, even though experimentally

they are found to be so.