# Magnetic field-induced intermediate quantum spin liquid with a spinon Fermi surface

## Abstract

The Kitaev model with an applied magnetic field in the $H\parallel \left[111\right]$ direction shows two transitions: from a nonabelian gapped quantum spin liquid (QSL) to a gapless QSL at ${H}_{c1}\simeq 0.2K$ and a second transition at a higher field ${H}_{c2}\simeq 0.35K$ to a gapped partially polarized phase, where K is the strength of the Kitaev exchange interaction. We identify the intermediate phase to be a gapless U(1) QSL and determine the spin structure function $S\left(\mathbf{k}\right)$ and the Fermi surface ${\u03f5}_{F}^{S}\left(\mathbf{k}\right)$ of the gapless spinons using the density matrix renormalization group (DMRG) method for large honeycomb clusters. Further calculations of static spin-spin correlations, magnetization, spin susceptibility, and finite temperature-specific heat and entropy corroborate the gapped and gapless nature of the different field-dependent phases. In the intermediate phase, the spin-spin correlations decay as a power law with distance, indicative of a gapless phase.

- Authors:

- Publication Date:

- Research Org.:
- The Ohio State Univ., Columbus, OH (United States)

- Sponsoring Org.:
- USDOE Office of Science (SC)

- OSTI Identifier:
- 1524092

- Alternate Identifier(s):
- OSTI ID: 1609873

- Grant/Contract Number:
- FG02-07ER46423

- Resource Type:
- Published Article

- Journal Name:
- Proceedings of the National Academy of Sciences of the United States of America

- Additional Journal Information:
- Journal Name: Proceedings of the National Academy of Sciences of the United States of America Journal Volume: 116 Journal Issue: 25; Journal ID: ISSN 0027-8424

- Publisher:
- National Academy of Sciences

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS; Science & Technology

### Citation Formats

```
Patel, Niravkumar D., and Trivedi, Nandini. Magnetic field-induced intermediate quantum spin liquid with a spinon Fermi surface. United States: N. p., 2019.
Web. doi:10.1073/pnas.1821406116.
```

```
Patel, Niravkumar D., & Trivedi, Nandini. Magnetic field-induced intermediate quantum spin liquid with a spinon Fermi surface. United States. doi:https://doi.org/10.1073/pnas.1821406116
```

```
Patel, Niravkumar D., and Trivedi, Nandini. Fri .
"Magnetic field-induced intermediate quantum spin liquid with a spinon Fermi surface". United States. doi:https://doi.org/10.1073/pnas.1821406116.
```

```
@article{osti_1524092,
```

title = {Magnetic field-induced intermediate quantum spin liquid with a spinon Fermi surface},

author = {Patel, Niravkumar D. and Trivedi, Nandini},

abstractNote = {The Kitaev model with an applied magnetic field in the H ∥ [ 111 ] direction shows two transitions: from a nonabelian gapped quantum spin liquid (QSL) to a gapless QSL at H c 1 ≃ 0.2 K and a second transition at a higher field H c 2 ≃ 0.35 K to a gapped partially polarized phase, where K is the strength of the Kitaev exchange interaction. We identify the intermediate phase to be a gapless U(1) QSL and determine the spin structure function S ( k ) and the Fermi surface ϵ F S ( k ) of the gapless spinons using the density matrix renormalization group (DMRG) method for large honeycomb clusters. Further calculations of static spin-spin correlations, magnetization, spin susceptibility, and finite temperature-specific heat and entropy corroborate the gapped and gapless nature of the different field-dependent phases. In the intermediate phase, the spin-spin correlations decay as a power law with distance, indicative of a gapless phase.},

doi = {10.1073/pnas.1821406116},

journal = {Proceedings of the National Academy of Sciences of the United States of America},

number = 25,

volume = 116,

place = {United States},

year = {2019},

month = {5}

}

DOI: https://doi.org/10.1073/pnas.1821406116

*Citation information provided by*

Web of Science

Web of Science

Works referenced in this record:

##
Field-induced quantum criticality in the Kitaev system $\alpha -{\mathrm{RuCl}}_{3}$

journal, July 2017

- Wolter, A. U. B.; Corredor, L. T.; Janssen, L.
- Physical Review B, Vol. 96, Issue 4

##
The density-matrix renormalization group in the age of matrix product states

journal, January 2011

- Schollwöck, Ulrich
- Annals of Physics, Vol. 326, Issue 1

##
Rare-Earth Chalcogenides: A Large Family of Triangular Lattice Spin Liquid Candidates

journal, October 2018

- Liu, Weiwei; Zhang, Zheng; Ji, Jianting
- Chinese Physics Letters, Vol. 35, Issue 11

##
Anyons in an exactly solved model and beyond

journal, January 2006

- Kitaev, Alexei
- Annals of Physics, Vol. 321, Issue 1

##
Dynamical correlation functions using the density matrix renormalization group

journal, July 1999

- Kühner, Till D.; White, Steven R.
- Physical Review B, Vol. 60, Issue 1

##
Two-dimensional Fermi surfaces in Kondo insulator SmB _{6}

journal, December 2014

- Li, G.; Xiang, Z.; Yu, F.
- Science, Vol. 346, Issue 6214

##
Impact of off-diagonal exchange interactions on the Kitaev spin-liquid state of $\alpha \text{\u2212}{\mathrm{RuCl}}_{3}$

journal, June 2019

- Takikawa, Daichi; Fujimoto, Satoshi
- Physical Review B, Vol. 99, Issue 22

##
Emergence of a field-driven U(1) spin liquid in the Kitaev honeycomb model

journal, January 2019

- Hickey, Ciarán; Trebst, Simon
- Nature Communications, Vol. 10, Issue 1

##
Observation of two types of fractional excitation in the Kitaev honeycomb magnet

journal, May 2018

- Janša, Nejc; Zorko, Andrej; Gomilšek, Matjaž
- Nature Physics, Vol. 14, Issue 8

##
Variational study of triangular lattice spin- $1\u22152$ model with ring exchanges and spin liquid state in $\kappa \text{\u2212}{\left(\mathrm{ET}\right)}_{2}{\mathrm{Cu}}_{2}{\left(\mathrm{CN}\right)}_{3}$

journal, July 2005

- Motrunich, Olexei I.
- Physical Review B, Vol. 72, Issue 4

##
Physics of the Kitaev Model: Fractionalization, Dynamic Correlations, and Material Connections

journal, March 2018

- Hermanns, M.; Kimchi, I.; Knolle, J.
- Annual Review of Condensed Matter Physics, Vol. 9, Issue 1

##
Neutron scattering in the proximate quantum spin liquid α-RuCl _{3}

journal, June 2017

- Banerjee, Arnab; Yan, Jiaqiang; Knolle, Johannes
- Science, Vol. 356, Issue 6342

##
Quantum spin liquids: a review

journal, November 2016

- Savary, Lucile; Balents, Leon
- Reports on Progress in Physics, Vol. 80, Issue 1

##
Cooper pairing in non-Fermi liquids

journal, March 2015

- Metlitski, Max A.; Mross, David F.; Sachdev, Subir
- Physical Review B, Vol. 91, Issue 11

##
The density-matrix renormalization group

journal, April 2005

- Schollwöck, U.
- Reviews of Modern Physics, Vol. 77, Issue 1

##
Excitons in topological Kondo insulators: Theory of thermodynamic and transport anomalies in ${\mathrm{SmB}}_{6}$

journal, March 2017

- Knolle, Johannes; Cooper, Nigel R.
- Physical Review Letters, Vol. 118, Issue 9

##
Quantum spin liquid states

journal, April 2017

- Zhou, Yi; Kanoda, Kazushi; Ng, Tai-Kai
- Reviews of Modern Physics, Vol. 89, Issue 2

##
Models and materials for generalized Kitaev magnetism

journal, November 2017

- Winter, Stephen M.; Tsirlin, Alexander A.; Daghofer, Maria
- Journal of Physics: Condensed Matter, Vol. 29, Issue 49

##
Majorana quantization and half-integer thermal quantum Hall effect in a Kitaev spin liquid

journal, July 2018

- Kasahara, Y.; Ohnishi, T.; Mizukami, Y.
- Nature, Vol. 559, Issue 7713

##
Density-matrix algorithms for quantum renormalization groups

journal, October 1993

- White, Steven R.
- Physical Review B, Vol. 48, Issue 14

##
Field-induced quantum spin liquid in the Kitaev-Heisenberg model and its relation to $\alpha \text{\u2212}{\mathrm{RuCl}}_{3}$

journal, October 2019

- Jiang, Yi-Fan; Devereaux, Thomas P.; Jiang, Hong-Chen
- Physical Review B, Vol. 100, Issue 16

##
Spin Waves and Revised Crystal Structure of Honeycomb Iridate ${\mathrm{Na}}_{2}{\mathrm{IrO}}_{3}$

journal, March 2012

- Choi, S. K.; Coldea, R.; Kolmogorov, A. N.
- Physical Review Letters, Vol. 108, Issue 12

##
Spectral functions with the density matrix renormalization group: Krylov-space approach for correction vectors

journal, November 2016

- Nocera, A.; Alvarez, G.
- Physical Review E, Vol. 94, Issue 5

##
The density matrix renormalization group for strongly correlated electron systems: A generic implementation

journal, September 2009

- Alvarez, G.
- Computer Physics Communications, Vol. 180, Issue 9

##
Robust non-Abelian spin liquid and a possible intermediate phase in the antiferromagnetic Kitaev model with magnetic field

journal, June 2018

- Zhu, Zheng; Kimchi, Itamar; Sheng, D. N.
- Physical Review B, Vol. 97, Issue 24

##
Dynamical and topological properties of the Kitaev model in a [111] magnetic field

journal, July 2018

- Gohlke, Matthias; Moessner, Roderich; Pollmann, Frank
- Physical Review B, Vol. 98, Issue 1

##
Signatures of magnetic-field-driven quantum phase transitions in the entanglement entropy and spin dynamics of the Kitaev honeycomb model

journal, April 2019

- Ronquillo, David C.; Vengal, Adu; Trivedi, Nandini
- Physical Review B, Vol. 99, Issue 14

##
A Field Guide to Spin Liquids

journal, March 2019

- Knolle, J.; Moessner, R.
- Annual Review of Condensed Matter Physics, Vol. 10, Issue 1

##
Dynamics and Transport of the ${Z}_{2}$ Spin Liquid: Application to $\kappa \mathrm{\text{\u2212}}(\mathrm{ET}{)}_{2}{\mathrm{Cu}}_{2}(\mathrm{CN}{)}_{3}$

journal, April 2009

- Qi, Yang; Xu, Cenke; Sachdev, Subir
- Physical Review Letters, Vol. 102, Issue 17

##
Quantum spin liquid in the spin- $1\u22152$ triangular antiferromagnet $\mathrm{Et}{\mathrm{Me}}_{3}\mathrm{Sb}{\left[\mathrm{Pd}{\left(\text{dmit}\right)}_{2}\right]}_{2}$

journal, March 2008

- Itou, T.; Oyamada, A.; Maegawa, S.
- Physical Review B, Vol. 77, Issue 10

##
Spin Gaps in a Frustrated Heisenberg Model for ${\mathrm{CaV}}_{4}{O}_{9}$

journal, October 1996

- White, Steven R.
- Physical Review Letters, Vol. 77, Issue 17

##
Density matrix formulation for quantum renormalization groups

journal, November 1992

- White, Steven R.
- Physical Review Letters, Vol. 69, Issue 19

##
Direct evidence of a zigzag spin-chain structure in the honeycomb lattice: A neutron and x-ray diffraction investigation of single-crystal Na ${}_{2}$ IrO ${}_{3}$

journal, May 2012

- Ye, Feng; Chi, Songxue; Cao, Huibo
- Physical Review B, Vol. 85, Issue 18

##
Kondo Breakdown and Quantum Oscillations in ${\mathrm{SmB}}_{6}$

journal, January 2016

- Erten, Onur; Ghaemi, Pouyan; Coleman, Piers
- Physical Review Letters, Vol. 116, Issue 4

##
Spin-liquid phase of the multiple-spin exchange Hamiltonian on the triangular lattice

journal, July 1999

- Misguich, G.; Lhuillier, C.; Bernu, B.
- Physical Review B, Vol. 60, Issue 2

##
Spinon Fermi Surface in a Cluster Mott Insulator Model on a Triangular Lattice and Possible Application to $1T\text{\u2212}{\mathrm{Ta}\mathrm{S}}_{2}$

journal, July 2018

- He, Wen-Yu; Xu, Xiao Yan; Chen, Gang
- Physical Review Letters, Vol. 121, Issue 4

##
Nonmagnetic Insulating States near the Mott Transitions on Lattices with Geometrical Frustration and Implications for κ-(ET) _{2} Cu _{2} (CN) _{3}

journal, September 2002

- Morita, Hidekazu; Watanabe, Shinji; Imada, Masatoshi
- Journal of the Physical Society of Japan, Vol. 71, Issue 9

##
1T-TaS _{2} as a quantum spin liquid

journal, June 2017

- Law, K. T.; Lee, Patrick A.
- Proceedings of the National Academy of Sciences, Vol. 114, Issue 27

##
Lanczos method for the calculation of finite-temperature quantities in correlated systems

journal, February 1994

- Jaklič, J.; Prelovšek, P.
- Physical Review B, Vol. 49, Issue 7

##
Dirac and Chiral Quantum Spin Liquids on the Honeycomb Lattice in a Magnetic Field

journal, May 2018

- Liu, Zheng-Xin; Normand, B.
- Physical Review Letters, Vol. 120, Issue 18

##
Theory of the field-revealed Kitaev spin liquid

journal, June 2019

- Gordon, Jacob S.; Catuneanu, Andrei; Sørensen, Erik S.
- Nature Communications, Vol. 10, Issue 1

##
Spin Bose-Metal and Valence Bond Solid Phases in a Spin- $1/2$ Model with Ring Exchanges on a Four-Leg Triangular Ladder

journal, April 2011

- Block, Matthew S.; Sheng, D. N.; Motrunich, Olexei I.
- Physical Review Letters, Vol. 106, Issue 15

##
Mott Insulators in the Strong Spin-Orbit Coupling Limit: From Heisenberg to a Quantum Compass and Kitaev Models

journal, January 2009

- Jackeli, G.; Khaliullin, G.
- Physical Review Letters, Vol. 102, Issue 1

##
Direct evidence for dominant bond-directional interactions in a honeycomb lattice iridate Na2IrO3

journal, May 2015

- Hwan Chun, Sae; Kim, Jong-Woo; Kim, Jungho
- Nature Physics, Vol. 11, Issue 6

##
Spin Dynamics of the Spin- $1/2$ Kagome Lattice Antiferromagnet ${\mathrm{ZnCu}}_{3}(\mathrm{OH}{)}_{6}{\mathrm{Cl}}_{2}$

journal, March 2007

- Helton, J. S.; Matan, K.; Shores, M. P.
- Physical Review Letters, Vol. 98, Issue 10

##
Spin Liquid State in an Organic Mott Insulator with a Triangular Lattice

journal, September 2003

- Shimizu, Y.; Miyagawa, K.; Kanoda, K.
- Physical Review Letters, Vol. 91, Issue 10