Holographic Topological Semimetals - uni-wuerzburg.de

Holographic Topological Semimetals

Yan Liu Beihang University

“Gauge/gravity duality 2018”

Based on collaborations with Karl Landsteiner, Ya-Wen Sun, Jun-Kun ZhaoarXiv: 1505.04772, 1511.05505, 1604.01346, 1801.09357, 1808.xxxxx and work in progress

Topological Semimetal (TSM) Weyl semimetal Nodal line semimetal

Beyond the Landau-Ginzburg paradigm Macroscopic effect of quantum anomaly (chiral anomaly,

mixed gauge gravitational anomaly)

Most known TSM: based on weak coupling

Motivation for Holographic TSM Topological semimetals (TSM) with strong interactions：How does TSM work in strongly coupled case?

without quasiparticle no notion of band structure, Berry phase (Weyl points)

Topological semimetals (TSM) with strong interactions：How does TSM work in strongly coupled case?

without quasiparticle no notion of band structure, Berry phase (Weyl points)

Holography: strong-weak duality

A holography model for TSM can teach us qualitative lessons!

New entry in the holographic dictionary: topological states of matter;

New predictions from holography for transport properties

Motivation for Holographic TSM

OutlineHolographic Weyl Semimetal [Landsteiner, YL, PLB, 2015; Landsteiner, YL, Y. W. Sun, PRL, 2016; YL, J.-K. Zhao, in progress] [Landsteiner’s talk] [Fernandez-Pendas & Padhi’s posters]

Holographic Topological Nodal Line Semimetal [YL, Y. -W. Sun, 1801.09357; YL, Y. -W. Sun, to appear]

Summary and open questions

QFT of WSM[Grushin; Jackiw; Burkov, Balents; Kostolecky et al.]

2beff eF

Topological phase transition

2Meff

Anomalous Hall Effect (AHE) [Haldane, 1987]

J =e2

2⇡2be↵ ⇥E

QFT of WSM[Grushin; Jackiw; Burkov, Balents; Kostolecky et al.]

Topological phase transition

Holographic model

Ward identity

Holographic WSM

⇣A5 ^R ^R+

Order parameter of topological WSM: AHE

0.0 0.5 1.0 1.5 2.0 2.5 3.0

0.0

0.2

0.4

0.6

0.8

1.0

M

b

ΣAHE

8Αb

QFT result

Holographic WSM

(partially gapped)

Holographic WSM

Transports in holographic WSM Conductivities, viscosities have peak/dip behaviour in the QC regime

Temperature scaling behaviours of viscosities and conductivities in the QC regime: emergent Lifshitz-like symmetry in the IR at the transition point

⌘k/s / T 2�2� , ⌘Hk / T 4�� ,

⌘H? / T 2+� , �k / T 2�� ,

�? / T � , �AHE / T � ,

Odd viscosity is due to the presence of the mixed gauge-gravitational anomaly

analytic nontrivial relation

⌘k⌘?

=2⌘Hk

⌘H?

=�k

�?=

gxxgzz

��r=r0

T/b=0.05T/b=0.04T/b=0.03

��

��

��

��

��

��

��

η�⊥�ζ��

⌘H? = 8⇣q2�2Azgxx��r=r0

mixed axial-gravitational anomaly

Transports in holographic WSM

Holographic Nodal Line Semimetal [YL, Y.-W. Sun, 1801.09357]

Weakly coupled field theory model for NLSM

L = i ��µ@µ �m� �µ⌫bµ⌫

�

m2 < 4b2xy m2 > 4b2xy

�µ⌫ =i

2[�µ, �⌫ ]

Holographic Nodal Line Semimetal

Holographic Nodal Line SemimetalConservation equation

@µJµ = 0 ,

@µJµ5 = im �5 + 2ibµ⌫ �

µ⌫�5 ,

Operator Field

Axially charged Complex scalar filed

Axially charged complex two form field

, �5 <latexit sha1_base64="6jb/LI6WnYGvg9eniICrdY1aRIA=">AAACEXicbVC7TsMwFHXKq5RXgJElokLqUFUJAsFYwcJYJPqQmlDduE5r1U4i20GqovIJLPwKCwMIsbKx8Tc4bYbSciRbx+fcq+t7/JhRqWz7xyisrK6tbxQ3S1vbO7t75v5BS0aJwKSJIxaJjg+SMBqSpqKKkU4sCHCfkbY/us789gMRkkbhnRrHxOMwCGlAMSgt9cyK64NI3VjSSXa51erjnDIAzuH+PHv0zLJds6ewlomTkzLK0eiZ324/wgknocIMpOw6dqy8FISimJFJyU0kiQGPYEC6mobAifTS6UYT60QrfSuIhD6hsqbqfEcKXMox93UlBzWUi14m/ud1ExVceikN40SREM8GBQmzVGRl8Vh9KghWbKwJYEH1Xy08BAFY6RBLOgRnceVl0jqtOXbNuT0r16/yOIroCB2jCnLQBaqjG9RATYTRE3pBb+jdeDZejQ/jc1ZaMPKeQ/QHxtcviHCeEQ==</latexit><latexit sha1_base64="6jb/LI6WnYGvg9eniICrdY1aRIA=">AAACEXicbVC7TsMwFHXKq5RXgJElokLqUFUJAsFYwcJYJPqQmlDduE5r1U4i20GqovIJLPwKCwMIsbKx8Tc4bYbSciRbx+fcq+t7/JhRqWz7xyisrK6tbxQ3S1vbO7t75v5BS0aJwKSJIxaJjg+SMBqSpqKKkU4sCHCfkbY/us789gMRkkbhnRrHxOMwCGlAMSgt9cyK64NI3VjSSXa51erjnDIAzuH+PHv0zLJds6ewlomTkzLK0eiZ324/wgknocIMpOw6dqy8FISimJFJyU0kiQGPYEC6mobAifTS6UYT60QrfSuIhD6hsqbqfEcKXMox93UlBzWUi14m/ud1ExVceikN40SREM8GBQmzVGRl8Vh9KghWbKwJYEH1Xy08BAFY6RBLOgRnceVl0jqtOXbNuT0r16/yOIroCB2jCnLQBaqjG9RATYTRE3pBb+jdeDZejQ/jc1ZaMPKeQ/QHxtcviHCeEQ==</latexit><latexit sha1_base64="6jb/LI6WnYGvg9eniICrdY1aRIA=">AAACEXicbVC7TsMwFHXKq5RXgJElokLqUFUJAsFYwcJYJPqQmlDduE5r1U4i20GqovIJLPwKCwMIsbKx8Tc4bYbSciRbx+fcq+t7/JhRqWz7xyisrK6tbxQ3S1vbO7t75v5BS0aJwKSJIxaJjg+SMBqSpqKKkU4sCHCfkbY/us789gMRkkbhnRrHxOMwCGlAMSgt9cyK64NI3VjSSXa51erjnDIAzuH+PHv0zLJds6ewlomTkzLK0eiZ324/wgknocIMpOw6dqy8FISimJFJyU0kiQGPYEC6mobAifTS6UYT60QrfSuIhD6hsqbqfEcKXMox93UlBzWUi14m/ud1ExVceikN40SREM8GBQmzVGRl8Vh9KghWbKwJYEH1Xy08BAFY6RBLOgRnceVl0jqtOXbNuT0r16/yOIroCB2jCnLQBaqjG9RATYTRE3pBb+jdeDZejQ/jc1ZaMPKeQ/QHxtcviHCeEQ==</latexit><latexit sha1_base64="6jb/LI6WnYGvg9eniICrdY1aRIA=">AAACEXicbVC7TsMwFHXKq5RXgJElokLqUFUJAsFYwcJYJPqQmlDduE5r1U4i20GqovIJLPwKCwMIsbKx8Tc4bYbSciRbx+fcq+t7/JhRqWz7xyisrK6tbxQ3S1vbO7t75v5BS0aJwKSJIxaJjg+SMBqSpqKKkU4sCHCfkbY/us789gMRkkbhnRrHxOMwCGlAMSgt9cyK64NI3VjSSXa51erjnDIAzuH+PHv0zLJds6ewlomTkzLK0eiZ324/wgknocIMpOw6dqy8FISimJFJyU0kiQGPYEC6mobAifTS6UYT60QrfSuIhD6hsqbqfEcKXMox93UlBzWUi14m/ud1ExVceikN40SREM8GBQmzVGRl8Vh9KghWbKwJYEH1Xy08BAFY6RBLOgRnceVl0jqtOXbNuT0r16/yOIroCB2jCnLQBaqjG9RATYTRE3pBb+jdeDZejQ/jc1ZaMPKeQ/QHxtcviHCeEQ==</latexit>

�µ⌫ , �µ⌫�5 <latexit sha1_base64="jSNmIM8xmk1DeZylMRYqimnB6as=">AAACMHicdVDLSgMxFM34tr5GXboJFsFFKTOi6LLoQpcKtgqdsdxJM20wyQxJRihD/SM3fopuFBRx61eYaWfh80Dg5JxzSe6JUs608bxnZ2Jyanpmdm6+srC4tLzirq61dJIpQpsk4Ym6jEBTziRtGmY4vUwVBRFxehFdHxX+xQ1VmiXy3AxSGgroSRYzAsZKHfc4iEDlQarZMOiBEHCVByILZDYstKBWu/03gMv7XmF23KpX90bAv4lfkioqcdpxH4JuQjJBpSEctG77XmrCHJRhhNNhJcg0TYFcQ4+2LZUgqA7z0cJDvGWVLo4TZY80eKR+nchBaD0QkU0KMH390yvEv7x2ZuKDMGcyzQyVZPxQnHFsEly0h7tMUWL4wBIgitm/YtIHBcTYjiu2BP/nyr9Ja6fue3X/bLfaOCzrmEMbaBNtIx/towY6QaeoiQi6Q4/oBb06986T8+a8j6MTTjmzjr7B+fgEwAyryw==</latexit><latexit sha1_base64="jSNmIM8xmk1DeZylMRYqimnB6as=">AAACMHicdVDLSgMxFM34tr5GXboJFsFFKTOi6LLoQpcKtgqdsdxJM20wyQxJRihD/SM3fopuFBRx61eYaWfh80Dg5JxzSe6JUs608bxnZ2Jyanpmdm6+srC4tLzirq61dJIpQpsk4Ym6jEBTziRtGmY4vUwVBRFxehFdHxX+xQ1VmiXy3AxSGgroSRYzAsZKHfc4iEDlQarZMOiBEHCVByILZDYstKBWu/03gMv7XmF23KpX90bAv4lfkioqcdpxH4JuQjJBpSEctG77XmrCHJRhhNNhJcg0TYFcQ4+2LZUgqA7z0cJDvGWVLo4TZY80eKR+nchBaD0QkU0KMH390yvEv7x2ZuKDMGcyzQyVZPxQnHFsEly0h7tMUWL4wBIgitm/YtIHBcTYjiu2BP/nyr9Ja6fue3X/bLfaOCzrmEMbaBNtIx/towY6QaeoiQi6Q4/oBb06986T8+a8j6MTTjmzjr7B+fgEwAyryw==</latexit><latexit sha1_base64="jSNmIM8xmk1DeZylMRYqimnB6as=">AAACMHicdVDLSgMxFM34tr5GXboJFsFFKTOi6LLoQpcKtgqdsdxJM20wyQxJRihD/SM3fopuFBRx61eYaWfh80Dg5JxzSe6JUs608bxnZ2Jyanpmdm6+srC4tLzirq61dJIpQpsk4Ym6jEBTziRtGmY4vUwVBRFxehFdHxX+xQ1VmiXy3AxSGgroSRYzAsZKHfc4iEDlQarZMOiBEHCVByILZDYstKBWu/03gMv7XmF23KpX90bAv4lfkioqcdpxH4JuQjJBpSEctG77XmrCHJRhhNNhJcg0TYFcQ4+2LZUgqA7z0cJDvGWVLo4TZY80eKR+nchBaD0QkU0KMH390yvEv7x2ZuKDMGcyzQyVZPxQnHFsEly0h7tMUWL4wBIgitm/YtIHBcTYjiu2BP/nyr9Ja6fue3X/bLfaOCzrmEMbaBNtIx/towY6QaeoiQi6Q4/oBb06986T8+a8j6MTTjmzjr7B+fgEwAyryw==</latexit><latexit sha1_base64="jSNmIM8xmk1DeZylMRYqimnB6as=">AAACMHicdVDLSgMxFM34tr5GXboJFsFFKTOi6LLoQpcKtgqdsdxJM20wyQxJRihD/SM3fopuFBRx61eYaWfh80Dg5JxzSe6JUs608bxnZ2Jyanpmdm6+srC4tLzirq61dJIpQpsk4Ym6jEBTziRtGmY4vUwVBRFxehFdHxX+xQ1VmiXy3AxSGgroSRYzAsZKHfc4iEDlQarZMOiBEHCVByILZDYstKBWu/03gMv7XmF23KpX90bAv4lfkioqcdpxH4JuQjJBpSEctG77XmrCHJRhhNNhJcg0TYFcQ4+2LZUgqA7z0cJDvGWVLo4TZY80eKR+nchBaD0QkU0KMH390yvEv7x2ZuKDMGcyzQyVZPxQnHFsEly0h7tMUWL4wBIgitm/YtIHBcTYjiu2BP/nyr9Ja6fue3X/bLfaOCzrmEMbaBNtIx/towY6QaeoiQi6Q4/oBb06986T8+a8j6MTTjmzjr7B+fgEwAyryw==</latexit>

Holographic model

Two axially charged matter fields

Ward identity

S =

Zd5x

p�g

1

22

✓R+

12

L2

◆� 1

4F2 � 1

4F 2 +

↵

3✏abcdeAa

✓3FbcFde + FbcFde

◆

� (Da�)⇤(Da�)� V1(�)�

1

3⌘

�D[aBbc]

�⇤�D[aBbc]�� V2(Bab)� �|�|2B⇤

abBab

�

@µJµ = 0 ,

@µJµ5 = lim

r!1

p�g

✓� ↵

3✏r↵�⇢�(F↵�F⇢� + F↵�F⇢�) + iq1

h�⇤(Dr�)� �(Dr�)⇤

i+

+iq2⌘

�B⇤

µ⌫D[rBµ⌫] � (D[rBµ⌫])⇤Bµ⌫

�◆+ c.t. .



QCP is stable

The phase transition is continuous

No local order parameter or transport for the phase transition


QCP is stable

No local order parameter or transport for the phase transition

The phase transition is continuous

��

-��

-��

-��

-��

-��

-��

-��

��

Ω��

Holographic Nodal Line SemimetalAPRES:

(1) Multiple while discrete Fermi nodal lines (2) Fermi nodal lines forming circles in the nodal line semimetal

phase with two bands crossing

��

��

��

��

��

��

��

��

��

distinguish different topology of the quantum wave function in the momentum space

Topological invariants for interacting systems: the topological Hamiltonian method, the zero frequency Green’s function contains all topological information [Z. Wang and S.C.Zhang, 2012, 2013]

Ht(k) = �G�1(0,k)<latexit sha1_base64="RikHpaxf5xR4Q9Hspl6UlFFw2sU=">AAACEnicbVDLSsNAFJ3UV62vqEs3g0VowZZEBN0IRRd2WcE+oIlhMp20QycPZiZCCfkGN/6KGxeKuHXlzr9x0kbQ1gMDh3Pu5c45bsSokIbxpRWWlldW14rrpY3Nre0dfXevI8KYY9LGIQt5z0WCMBqQtqSSkV7ECfJdRrru+Crzu/eECxoGt3ISEdtHw4B6FCOpJEevWj6SI4xY0kwdWUks14PjtHpRu75LamZaMY5/JEcvG3VjCrhIzJyUQY6Wo39agxDHPgkkZkiIvmlE0k4QlxQzkpasWJAI4TEakr6iAfKJsJNppBQeKWUAvZCrF0g4VX9vJMgXYuK7ajILIOa9TPzP68fSO7cTGkSxJAGeHfJiBmUIs37ggHKCJZsogjCn6q8QjxBHWKoWS6oEcz7yIumc1E2jbt6clhuXeR1FcAAOQQWY4Aw0QBO0QBtg8ACewAt41R61Z+1Ne5+NFrR8Zx/8gfbxDc5onD8=</latexit><latexit sha1_base64="RikHpaxf5xR4Q9Hspl6UlFFw2sU=">AAACEnicbVDLSsNAFJ3UV62vqEs3g0VowZZEBN0IRRd2WcE+oIlhMp20QycPZiZCCfkGN/6KGxeKuHXlzr9x0kbQ1gMDh3Pu5c45bsSokIbxpRWWlldW14rrpY3Nre0dfXevI8KYY9LGIQt5z0WCMBqQtqSSkV7ECfJdRrru+Crzu/eECxoGt3ISEdtHw4B6FCOpJEevWj6SI4xY0kwdWUks14PjtHpRu75LamZaMY5/JEcvG3VjCrhIzJyUQY6Wo39agxDHPgkkZkiIvmlE0k4QlxQzkpasWJAI4TEakr6iAfKJsJNppBQeKWUAvZCrF0g4VX9vJMgXYuK7ajILIOa9TPzP68fSO7cTGkSxJAGeHfJiBmUIs37ggHKCJZsogjCn6q8QjxBHWKoWS6oEcz7yIumc1E2jbt6clhuXeR1FcAAOQQWY4Aw0QBO0QBtg8ACewAt41R61Z+1Ne5+NFrR8Zx/8gfbxDc5onD8=</latexit><latexit sha1_base64="RikHpaxf5xR4Q9Hspl6UlFFw2sU=">AAACEnicbVDLSsNAFJ3UV62vqEs3g0VowZZEBN0IRRd2WcE+oIlhMp20QycPZiZCCfkGN/6KGxeKuHXlzr9x0kbQ1gMDh3Pu5c45bsSokIbxpRWWlldW14rrpY3Nre0dfXevI8KYY9LGIQt5z0WCMBqQtqSSkV7ECfJdRrru+Crzu/eECxoGt3ISEdtHw4B6FCOpJEevWj6SI4xY0kwdWUks14PjtHpRu75LamZaMY5/JEcvG3VjCrhIzJyUQY6Wo39agxDHPgkkZkiIvmlE0k4QlxQzkpasWJAI4TEakr6iAfKJsJNppBQeKWUAvZCrF0g4VX9vJMgXYuK7ajILIOa9TPzP68fSO7cTGkSxJAGeHfJiBmUIs37ggHKCJZsogjCn6q8QjxBHWKoWS6oEcz7yIumc1E2jbt6clhuXeR1FcAAOQQWY4Aw0QBO0QBtg8ACewAt41R61Z+1Ne5+NFrR8Zx/8gfbxDc5onD8=</latexit><latexit sha1_base64="RikHpaxf5xR4Q9Hspl6UlFFw2sU=">AAACEnicbVDLSsNAFJ3UV62vqEs3g0VowZZEBN0IRRd2WcE+oIlhMp20QycPZiZCCfkGN/6KGxeKuHXlzr9x0kbQ1gMDh3Pu5c45bsSokIbxpRWWlldW14rrpY3Nre0dfXevI8KYY9LGIQt5z0WCMBqQtqSSkV7ECfJdRrru+Crzu/eECxoGt3ISEdtHw4B6FCOpJEevWj6SI4xY0kwdWUks14PjtHpRu75LamZaMY5/JEcvG3VjCrhIzJyUQY6Wo39agxDHPgkkZkiIvmlE0k4QlxQzkpasWJAI4TEakr6iAfKJsJNppBQeKWUAvZCrF0g4VX9vJMgXYuK7ajILIOa9TPzP68fSO7cTGkSxJAGeHfJiBmUIs37ggHKCJZsogjCn6q8QjxBHWKoWS6oEcz7yIumc1E2jbt6clhuXeR1FcAAOQQWY4Aw0QBO0QBtg8ACewAt41R61Z+1Ne5+NFrR8Zx/8gfbxDc5onD8=</latexit>

Topological invariants[YL, Y.-W. Sun, to appear]

For pure AdS, the topological invariants of dual Dirac nodes (@ ) are .

For holographic WSM with very small M/b, the dual Weyl nodes are at and , the topological invariants are respectively.

For holographic NLSM, for one set of NL the Berry phase is Pi; while for another set of NL, Berry phase is undetermined

Bulk topological structure: (1) different bulk IR solutions which are not adiabatic connected; (2) scalar field is to generate gap, another matter field is to generate FS, they behave differently in different solutions

! = k = 0 ±1

! = 0 kz = ±Az|hor±1

Topological invariants[YL, Y.-W. Sun, to appear]

Starting from the most general holographic model [Grignani, et al. 2017]

with

Z(�) = 1 ,

W (�) = �w0

h1� cosh

⇥r

2

3�⇤i

,

V (�) =9

2

h1� cosh

⇥r

2

3�⇤i

,

Y (�) = coshhr2

3�i.

S =

Zd5x

p�g

1

22

�R+ 12

�� Y (�)

4F2 � Z(�)

4F 2 +

↵

3✏abcdeAa

⇣FbcFde + 3FbcFde

⌘

� 1

2(@�)2 � W (�)

2(Aa � @a✓)

2 � V (�)

�

Holographic WSM/insulator transition[YL, J.-K. Zhao, in progress]

Three different phases, however, the QCP is unstable.

The IR geometry in the insulating phase is a gapped geometry

There is a first order quantum phase transition

0.690 0.692 0.694 0.696 0.698 0.700 0.702-2.20

-2.18

-2.16

-2.14

-2.12

-2.10

Mb

Ω

Vb4

0.0 0.2 0.4 0.6 0.8 1.00.0

0.2

0.4

0.6

0.8

1.0

Mb

σAHE8αb


0.690 0.692 0.694 0.696 0.698 0.700 0.702-2.20

-2.18

-2.16

-2.14

-2.12

-2.10

Mb

Ω

Vb4

Three different phases, however, the QCP is unstable.

The IR geometry in the insulating phase is a gapped geometry

There is a first order quantum phase transition

0.690 0.692 0.694 0.696 0.698 0.700 0.702-2.20

-2.18

-2.16

-2.14

-2.12

-2.10

Mb

Ω

Vb4

0.0 0.2 0.4 0.6 0.8 1.00.0

0.2

0.4

0.6

0.8

1.0

Mb

σAHE8αb


Summary （1）

Weakly coupled WSM vs. Holographic WSM

Weyl Semimetal Holographic WSM

SymmetryTime reversal or

invesion symmetry breaking

Time reversal breaking

Features in transport

Anomalous Hall conductivity

AHEQuantum critical physics

Mixed anomaly: Odd viscosity

Edge States Gapless surface state Fermi arc

Surface current [Ammon et al, PRL, 2017]

Topological invariants

laboratory TaAs, TaP, etc. ??

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Weakly coupled NLSM vs. Holographic NLSM

Nodal line semimetal Holographic NLSM

SymmetrySymmetry protected by mirror reflection symmetry, inversion

symmetry

Symmetry protected by inversion symmetry

Features in transport ARPES ARPES (multiple NL)

Other transport (??)

Edge States No ??

Topological invariants (1) ; (2) undetermined

laboratory PbTaSe2, ZrTe etc. ??

Summary （2）

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Open questions

Other approach (Kinetic theory, EFT) to odd viscosity?

Disorder effects (or translational invariance breaking effects) on holographic TSM?

Interesting transport physics in holographic NLSM?

Classification of strongly coupled topological semimetals (Organisation principle)

Thank You!


Ansatz for T=0

Near UV Metric: scalar field: two form field：

ds2 = u(�dt2 + dz2) +dr2

u+ f(dx2 + dy2)

� = �(r) ,

Bxy = B(r) .

r�|r!1 = M

Bxy|r!1 = br

At zero temperature: 3 distinct classes of solutions (near horizon @ IR)

M/b<0.744 (Topological phase)

Holographic WSM

u = u0r2�1 + �u r↵

�,

h = h0r��1 + �h r↵

�,

Az = r��1 + �a r↵

�,

� = �0

�1 + �� r↵

�

u = r2 ,

h = r2 ,

Az = a1 +⇡a21�

21

16re�

2a1qr ,

� =p⇡�1

⇣a1q2r

⌘3/2e�

a1qr ;

u =�1 +

3

8�

�r2 ,

h = r2 ,

Az = a1r�1 ,

� =

r3

�+ �1r

�2 ,

M/b>0.744 (Trivial phase)

M/b=0.744 (Critical point)

At zero temperature: 3 distinct classes of solutions (near horizon @ IR)

M/b<0.744 (Topological phase)

Holographic WSM

u = u0r2�1 + �u r↵

�,

h = h0r��1 + �h r↵

�,

Az = r��1 + �a r↵

�,

� = �0

�1 + �� r↵

�

u = r2 ,

h = r2 ,

Az = a1 +⇡a21�

21

16re�

2a1qr ,

� =p⇡�1

⇣a1q2r

⌘3/2e�

a1qr ;

u =�1 +

3

8�

�r2 ,

h = r2 ,

Az = a1r�1 ,

� =

r3

�+ �1r

�2 ,

M/b>0.744 (Trivial phase)

M/b=0.744 (Critical point)

From free energy, we find a continuous and smooth behaviour at the critical point.


Three different IR solutions: Near horizon = leading + subleading

u =1

8(11 + 3

p13)r2

⇣1 + �ur↵1

⌘,

f =

s2p13

3� 2 b0r

↵⇣1 + �fr↵1

⌘,

� = �0r� ,

B = b0r↵⇣1 + �br↵1

⌘,

u = u0r2(1 + �ur�) ,

f = f0r↵(1 + �fr�) ,

� = �0(1 + ��r�) ,

B = b0r↵(1 + �br�) ,

u =�1 +

3

8�1

�r2 ,

f = r2 ,

� =

r3

�1+ �1r

2p

160�21+84�1+9

3+8�1�2 ,

B = b1r2p2q

3�+�13+8�1 .

M/b<critical value (Topological

NLSM phase)

M/b=critical value (Critical phase)

M/b>critical value (Trivial phase)


��-�� -�� -� � ��

��

��

��

��

��

��-�� -�� -� � ��

�

�

�

�

�

��

ϕ

Bulk profile for two-form field

Bulk profile for scalar field

M/b = 1.682

M/b = 1.702

M/b = 1.717

M/b = 1.733

M/b = 1.750

• Diagonal conductivities at T=0: • Diagonal conductivities at T>0:

Predictions of Holographic WSM: conductivity

0.0 0.5 1.0 1.5 2.0 2.5 3.00

1

2

3

4

5

6

7

M

b

Σdiag

T

�?T

=�xx

T=

�yy

T

�k

T=

�zz

T

Axisymmetric system [Landau&Lifshitz, Vol. 10]

In total: 3 shear, 2 bulk and 2 odd viscosities

Viscosity: slightly deform the metric of spacetime

Predictions of Holographic WSM: viscosity

at T=0, from field theory arguments, no substantial odd viscosity expected

Odd viscosity determined by IR properties:

Predictions of Holographic WSM: odd viscosity

• Highly suppressed in the WSM phase

• Rises steeply entering the QC region

• peaks at the critical point and drops slowly as M/b increases, finally reaching zero

T/b=0.05T/b=0.04T/b=0.03

��

�

��

��

��

��

��

η�∥

�ζ��


⌘Hk = 4⇣q2Az�2g2

xx

gzz

��r=r0

Predictions of Holographic WSM: odd viscosity

T/b=0.05T/b=0.04T/b=0.03

��

��

��

��

��

��

��

η�⊥�ζ��

• Highly suppressed in the WSM phase

• Rises steeply entering the QC region

• peaks at the critical point and drops slowly as M/b increases, finally reaching zero

Qualitatively the same as the other one


⌘H? = 8⇣q2�2Azgxx��r=r0

Transverse shear viscosity

Longitudinal shear viscosity

Predictions of Holographic WSM: shear viscosity

T/b=0.05T/b=0.04T/b=0.03

��

��

��

��

��

��

��

�πη∥�

KSS bound

⌘? = ⌘xy,xy = ⌘(xx�yy),xx�yy = gxxpgzz

��r=r0

=s

4⇡

⌘k = ⌘xz,xz = ⌘yz,yz =g2xxpgzz

��r=r0

Holographic Topological Semimetals - uni-wuerzburg.de

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