Transport and symmetry breaking in strongly correlated

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Transport and symmetry breaking in strongly correlated

Transcript Of Transport and symmetry breaking in strongly correlated

Transport and symmetry breaking in strongly correlated systems with topological order
a dissertation presented by
Shubhayu Chatterjee to
The Department of Physics in partial fulfillment of the requirements
for the degree of Doctor of Philosophy
in the subject of Physics
Harvard University Cambridge, Massachusetts
April 2018

© 2018 - Shubhayu Chatterjee All rights reserved.

Thesis advisor: Subir Sachdev

Author: Shubhayu Chatterjee

Transport and symmetry breaking in strongly correlated systems with topological order
Abstract
This thesis is devoted to the study of strongly correlated phases of quantum matter in two spatial dimensions. In presence of strong interactions, the electron can split up into separate spin and charge degrees of freedom. The diagnosis of such fractionalized excitations in recent experiments, as well as finding new probes for their detection, comprise the two major themes of this thesis.
The first part of the thesis, comprising chapters 2-6, discusses the relevance of fractionalization to the pseudogap phase of the underdoped high Tc cuprate superconductors. This is motivated by transport experiments that show a violation of Luttinger’s theorem, possible only in the presence of topological order that can arise naturally as a consequence of fractionalization. Chapters 2 and 3 focus on phenomenological models with bosonic charge carriers and fermionic spin carriers, and investigate transport properties as well as various confinement transitions to symmetry broken phases. Chapters 4, 5 and 6 deal with a different framework of electron fractionalization, where the charge carriers are fermionic and spin carriers are bosonic. We find that these models can better explain the thermal and electrical transport properties in chapter 5. In chapters 6 and 7, we show how such models arise naturally from quantum fluctuations of antiferromagnetism, and can simultaneously intertwine the anti-nodal spectral gap with the discrete broken symmetries that are observed in the cuprates.
The second part of the thesis, comprising chapters 7 and 8, proposes new experimental probes to study the behavior of insulating two dimensional quantum magnets. Aided by geometric frustration and strong quantum fluctuations, they may realize a spin liquid ground state which can be extremely difficult to detect via conventional probes. Chapter 7 discusses how novel spin-transport probes from the spintronics community can be used to detect fractionalized excitations in a quantum spin liquid. Chapter 8 studies how the temperature dependent anisotropy of in-plane versus out-of-plane thermal conductivities can serve as an explicit signature of fractionalization in layered two-dimensional materials.
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Contents

1 Introduction

1

1.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Topological order and Fractionalization . . . . . . . . . . . . . . . . . . . . 13

1.3 Experimental motivation: Cuprates and beyond . . . . . . . . . . . . . . . . 38

1.4 Organization of the thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

2 Superconductivity from confinement transition of a Z2 FL* metal 50 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 2.2 Mapping between bosonic and fermionic spin liquids on the rectangular lattice via symmetry fractionalization . . . . . . . . . . . . . . . . . . . . . 53 2.3 Superconducting transition of the FL* . . . . . . . . . . . . . . . . . . . . . 68 2.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

3 FL* with bosonic chargons as a candidate for the pseudogap metal 80 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80 3.2 Model of Z2-FL* with bosonic chargons . . . . . . . . . . . . . . . . . . . . 83 3.3 Confinement transitions to translation invariant superconductors . . . . . . 86 3.4 Evolution of the Hall Number . . . . . . . . . . . . . . . . . . . . . . . . . . 92 3.5 Confined phases with broken translation symmetry . . . . . . . . . . . . . . 98 3.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

4 Transport across antiferromagnetic and topological quantum tran-

sitions

106

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

4.2 Model and formalism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

4.3 Antiferromagnetic metal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116

4.4 Co-existing antiferromagnetism and superconductivity . . . . . . . . . . . . 126

4.5 Influence of doping-dependent scattering in the dirty limit . . . . . . . . . . 134

4.6 Topological order in the pseudogap phase . . . . . . . . . . . . . . . . . . . 138

4.7 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141

4.8 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143

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5 Insulators and metals with topological order and discrete symme-

try breaking

145

5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145

5.2 O(3) non-linear sigma and CP1 models . . . . . . . . . . . . . . . . . . . . . 150

5.3 SU(2) lattice gauge theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153

5.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169

6 Intertwining topological order and broken symmetry via fluctuat-

ing SDW

172

6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172

6.2 Magnetic order . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173

6.3 CP1 theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178

6.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181

7 Probing excitations in insulators via injection of spin currents

182

7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182

7.2 Formalism to evaluate spin current . . . . . . . . . . . . . . . . . . . . . . . 183

7.3 Spin current for ordered antiferromagnets . . . . . . . . . . . . . . . . . . . 189

7.4 Spin current for systems with no magnetic order . . . . . . . . . . . . . . . 192

7.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201

8 Signatures of fractionalization from interlayer thermal transport 203 8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203 8.2 Clean Z2 quantum spin liquid . . . . . . . . . . . . . . . . . . . . . . . . . . 205 8.3 Disordered Z2 quantum spin liquid . . . . . . . . . . . . . . . . . . . . . . . 210 8.4 U(1) quantum spin liquid . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214 8.5 Experimental considerations . . . . . . . . . . . . . . . . . . . . . . . . . . . 217 8.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 220

Appendices

221

A Appendices to Chapter 2

222

A.1 Derivation of the bosonic PSG . . . . . . . . . . . . . . . . . . . . . . . . . 222

A.2 PSG corresponding to the nematic bosonic ansatz . . . . . . . . . . . . . . 224

A.3 Alternate derivation of the vison PSG . . . . . . . . . . . . . . . . . . . . . 226

A.4 Derivation of the fermionic PSG . . . . . . . . . . . . . . . . . . . . . . . . 231

A.5 Trivial and non-trivial fusion rules . . . . . . . . . . . . . . . . . . . . . . . 233

A.6 Solution for the fermionic ansatz . . . . . . . . . . . . . . . . . . . . . . . . 235

A.7 Alternative derivation of the specific fermionic PSG . . . . . . . . . . . . . 238

A.8 PSG for the site bosons and constraints on HB . . . . . . . . . . . . . . . . 240

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B Appendix to Chapter 3

242

B.1 Overestimation of carrier densities by nH for an elliptical pocket . . . . . . 242

C Appendices to Chapter 4

244

C.1 Néel ordered d-wave superconductor and the phenomenon of nodal collision 244

C.2 Derivation of thermal conductivity for co-existing Néel order and supercon-

ductivity in the clean limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . 246

C.3 Particle-particle bubble in spiral antiferromagnet or algebraic charge liquid 249

D Appendices to Chapter 5

251

D.1 Momentum space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251

D.2 Relation between loop currents and Higgs condensate . . . . . . . . . . . . . 252

D.3 Real space perturbation theory for current in presence of large Higgs field . 253

E Appendices to Chapter 6

257

E.1 O(3) non-linear sigma model . . . . . . . . . . . . . . . . . . . . . . . . . . 257

E.2 Spin density wave theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 259

E.3 SU(2) gauge theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262

E.4 Derivation of CP1 theory from SU(2) gauge theory . . . . . . . . . . . . . . 268

F Appendices to Chapter 7

272

F.1 Details of spin current calculations . . . . . . . . . . . . . . . . . . . . . . . 272

F.2 Dynamic spin structure factor for an antiferromagnetic interface . . . . . . 275

G Appendices to Chapter 8

278

G.1 Layered Kitaev honeycomb model . . . . . . . . . . . . . . . . . . . . . . . 278

G.2 Evaluation of the integral in Eq. (8.4) . . . . . . . . . . . . . . . . . . . . . 284

G.3 Disordered Z2 QSL . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285

G.4 Kubo formula for the thermal conductivity . . . . . . . . . . . . . . . . . . 290

G.5 U(1) Quantum Spin Liquid . . . . . . . . . . . . . . . . . . . . . . . . . . . 299

References

302

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Citations to Previously Published Work
Most of this thesis has appeared in print elsewhere. Details for particular chapters are given below.
Chapter 2:
• Shubhayu Chatterjee, Yang Qi, Subir Sachdev and Julia Steinberg, “Superconductivity from a confinement transition out of a fractionalized Fermi liquid with Z2 topological and Ising-nematic orders”, Physical Review B 94, 024502 (2016), arXiv:1603.03041
Chapter 3:
• Shubhayu Chatterjee and Subir Sachdev, “Fractionalized Fermi liquid with bosonic chargons as a candidate for the pseudogap metal”, Physical Review B 94, 205117 (2016), 165113 (2015), arXiv:1607.05727
Chapter 4:
• Shubhayu Chatterjee, Subir Sachdev and Andreas Eberlein, “Thermal and electrical transport in metals and superconductors across antiferromagnetic and topological quantum transitions”, Physical Review B 96, 075103 (2017), arXiv:1704.02329
Chapter 5:
• Shubhayu Chatterjee and Subir Sachdev, “Insulators and metals with topological order and discrete symmetry breaking”, Physical Review B 95, 205133 (2017), arXiv:1703.00014
Chapter 6:
• Shubhayu Chatterjee, Subir Sachdev and Mathias S. Scheurer, “Intertwining topological order and broken symmetry in a theory of fluctuating spin density waves”, Physical Review Letters 119, 227002 (2017), arXiv:1705.06289
Chapter 7:
• Shubhayu Chatterjee and Subir Sachdev, “Probing excitations in insulators via injection of spin currents”, Physical Review B 92, 165113 (2015), arXiv:1506.04740
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Chapter 8: • Yochai Werman, Shubhayu Chatterjee, Siddhardh C. Morampudi and Erez Berg, “Signatures of fractionalization in spin liquids from interlayer thermal transport ”, arXiv:1708.02584
Some related work on theoretical models of doped Mott insulators and comparison with numerics on the 2d Hubbard model, which is not included in this thesis, has appeared in the following papers.
• Subir Sachdev, Erez Berg, Shubhayu Chatterjee and Yoni Schattner, “Spin density wave order, topological order, and Fermi surface reconstruction”, Physical Review B 94, 115147 (2016), arXiv:1606.07813
• Wei Wu, Mathias S. Scheurer, Shubhayu Chatterjee, Subir Sachdev, Antoine Georges and Michel Ferrero, “Pseudogap and Fermi surface topology in the two-dimensional Hubbard model”, to appear in Physical Review X, arXiv:1707.06602
• Mathias S. Scheurer, Shubhayu Chatterjee, Wei Wu, Michel Ferrero, Antoine Georges and Subir Sachdev, “Topological order in the pseudogap metal”, Proceedings of the National Academy of Sciences 115, E3665 (2018), arXiv:1711.09925
Electronic preprints (shown in typewriter font) are available on the Internet at the following URL:
http://arXiv.org
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Listing of figures
1.1 (a) Symmetry broken ground state of the quantum Ising model in one spatial dimension. (b) Single spin flip excitation (local) costs energy 4J. (c) Nonlocal domain wall excitation costs energy 2J. . . . . . . . . . . . . . . . . . 3
1.2 Phase diagram of the classical XY model in three spatial dimensions. There is a finite temperature symmetry breaking phase transition at Tc to a longrange ordered phase. Adapted from Ref. [252]. . . . . . . . . . . . . . . . . 5
1.3 Phase diagram of the classical XY model in two spatial dimensions. There is a finite temperature phase transition at TBKT associated with vortex proliferation, but there is no symmetry breaking at this transition. Adapted from Ref. [252]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.4 Pictorial depiction of the Fermi Hubbard model. An electron can hop to a neighboring site with amplitude t, and two electrons have an on-site repulsion of U . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.5 Ordered Néel antiferromagnet on the square lattice. The spins on different sublattices are indicated by crimson and blue colors. . . . . . . . . . . . . . 9
1.6 Fermi surface with gapless excitations in a metal in two spatial dimensions. 11 1.7 Reconstruction of a large Fermi surface into electron and hole pockets due
to the onset of long raneg Néel order. Adapted from Ref. [252]. . . . . . . . 12 1.8 The plaquette operator (in red) that measures the Z2 flux, and the star
operator Gi (in blue) that generates gauge transformations at site ri. . . . . 14 1.9 Schematic depiction of a Wegner-Wilson loop. Such a loop can be used to
diagnose phase transitions in the pure gauge theory, but fails in the presence of dynamical matter fields. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 1.10 The confining and deconfining phases of the pure Z2 gauge theory. (i) g > gc describes the confining phase. The energy to separate test electric charges (denoted by black dots) grows linearly with their separation L. (ii) g < gc describes the deconfining phase. This phase has gapped fluxes or visons (indicated by green π-flux plaquettes) that can be created locally only in pairs and then moved by the action of σℓx (shown in blue). A plaquette with an even number of blue lines on its edges carries no gauge flux, so the string is invisible. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 1.11 Examples of non-simply connected surfaces in two dimension with different genus, from Wikipedia [https://en.wikipedia.org/wiki/Genus_ (mathematics)]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 1.12 X creates a vison through the hole, while WC measures the Aharonov Bohm phase due to the vison by taking a Z2 gauge charge around the cylinder. Adapted from Ref. [222]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
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1.13 Schmatic phase diagram for the Z2 gauge theory with dynamical matter fields, at g = 1/K. Self-duality under λ ↔ g implies reflection symmetry about the diagonal. The dotted line indicates a first order phase transition, whereas solid lines indicate continuous phase transitions. Adapted from Ref. [41]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
1.14 Cartoon of spin liquid. An electron splits into a charge and spin degree of freedom. The charge remains attached to the positive ion core while the spin-half emergent quasiparticle is mobile. (Figure courtesy of T. Senthil) . 25
1.15 Insertion of flux Φ adiabatically through one hole of the torus. . . . . . . . 28 1.16 Schematic figure showing a vison threading the hole of a cylinder deep in
the deconfined phase. The dark blue (black) bonds correspond to σℓz = 1 (σℓz = 1). (b) The translation operator along the x direction moves the dark (σℓz = 1) bonds by one lattice spacing. Equivalently, once can act with the gauge transformation operator Gi (which changes the sign of σℓz on all bonds ≪ emanating from ri) on every indicated by a black dot (The light blue bonds represent the previous location of the operator X). Adapted from Ref. [221]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 1.17 (a) Hole doped spin liquid state. (b) Short-range interactions bind spinons and holons, producing electron-like dimers in a FL∗. Adapted from Ref. [230]. 34 1.18 Phase diagram adapted from Refs. [45, 47, 244]. The x and y axes are parameters controlling the condensates of H and R respectively. There is long-range antiferromagnetic order only in phase A, where both R and H condensates are present. Phase C describes an ACL with topological order. Phase B is a Fermi liquid. . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 1.19 (a) The fundamental building blocks of all cuprates, the CuO2 planes. (b) A schematic phase diagram of YBCO as a function of hole-doping p and temperature T . The phases (discussed in the main text) are abbreviated as follows: AF - antiferromagnet, PG - pseudogap, dSC - d-wave superconductor, DW - (bond) density waves, SM - strange metal, FL -Fermi liquid. The line T ∗ marks the appearance of the anti-nodal spectral gap. Adapted from Refs. [104, 244]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 1.20 (a) An antiferromagnet with hole density p. Relative to the filled band (2 electrons per site) there are 1 + p holes per site. (b) Large Fermi surface on the overdoped side, with size 1 + p. Adapted from Refs. [225, 244] . . . . . 40 1.21 The spectral function of the cuprates in the pseudogap phase at (nearly) zero energy. The spectral gap is clearly visible along (0, π) and symmetry related directions. Adapted from Ref. [277]. . . . . . . . . . . . . . . . . . . 41 1.22 Doping dependence of the Hall number nH = V /(eRH ) in hole doped YBCO in magnetic fields upto 88T . The drop in nH from 1 + p to p happens before CDW sets in. Adapted from Ref. [12]. . . . . . . . . . . . . . . . . . . . . . 42 1.23 The in-plane thermal conductivity of dmit-131, which has a linear in T behavior at low T , compared to κ-et and dmit-221 (which orders at low T). Adapted from Ref. [319]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 1.24 Dynamic spin structure factor S(q, ω) measured by inelastic neutron scattering at (a) ω = 6 meV, (b) ω = 2 meV, and (c) ω = 0.75 meV, all at T = 1.6K. It is quite diffuse, and no sharp peaks corresponding to magnetic ordering is seen. Adapted from Ref. [106]. . . . . . . . . . . . . . . . . . . . 45
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SymmetrySubir SachdevThesisShubhayu ChatterjeeFractionalization