Journal ArticleDOI

# Electronic nature of chiral charge order in the kagome superconductor Cs V 3 Sb 5

15 Jul 2021-Physical Review B (American Physical Society)-Vol. 104, Iss: 3, pp 035131

Abstract: Author(s): Wang, Z; Jiang, YX; Yin, JX; Li, Y; Wang, GY; Huang, HL; Shao, S; Liu, J; Zhu, P; Shumiya, N; Hossain, MS; Liu, H; Shi, Y; Duan, J; Li, X; Chang, G; Dai, P; Ye, Z; Xu, G; Wang, Y; Zheng, H; Jia, J; Hasan, MZ; Yao, Y | Abstract: Kagome superconductors with TC up to 7 K have been discovered for over 40 y. Recently, unconventional chiral charge order has been reported in kagome superconductor KV3Sb5, with an ordering temperature of one order of magnitude higher than the TC. However, the chirality of the charge order has not been reported in the cousin kagome superconductor CsV3Sb5, and the electronic nature of the chirality remains elusive. In this paper, we report the observation of electronic chiral charge order in CsV3Sb5 via scanning tunneling microscopy (STM). We observe a 2 × 2 charge modulation and a 1 × 4 superlattice in both topographic data and tunneling spectroscopy. 2 × 2 charge modulation is highly anticipated as a charge order by fundamental kagome lattice models at van Hove filling, and is shown to exhibit intrinsic chirality. We find that the 1 × 4 superlattices form various small domain walls, and can be a surface effect as supported by our first-principles calculations. Crucially, we find that the amplitude of the energy gap opened by the charge order exhibits real-space modulations, and features 2 × 2 wave vectors with chirality, highlighting the electronic nature of the chiral charge order. STM study at 0.4 K reveals a superconducting energy gap with a gap size 2Δ=0.85meV, which estimates a moderate superconductivity coupling strength with 2Δ/kBTC=3.9. When further applying a c-axis magnetic field, vortex core bound states are observed within this gap, indicative of clean-limit superconductivity.
Topics: Charge (physics) (52%), Superconductivity (50%)

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PHYSICAL REVIEW B 104, 035131 (2021)
Intrinsic nature of chiral charge order in the kagome superconductor RbV
3
Sb
5
Nana Shumiya,
1,*
Md. Shafayat Hossain ,
1,*
Jia-Xin Yin ,
1,*,
Yu-Xiao Jiang,
1,*
Brenden R. Ortiz,
2
Hongxiong Liu,
3,4
Youguo Shi,
3,4
Qiangwei Yin,
5
Hechang Lei,
5
Songtian S. Zhang ,
1
Guoqing Chang ,
6
Qi Zhang,
1
Tyler A. Cochran,
1
Daniel Multer,
1
Maksim Litskevich ,
1
Zi-Jia Cheng,
1
Xian P. Yang,
1
Zurab Guguchia,
7
Stephen D. Wilson,
2
and M. Zahid Hasan
1,8,9,10,
1
Laboratory for Topological Quantum Matter and Advanced Spectroscopy (B7), Department of Physics, Princeton University,
Princeton, New Jersey 08544, USA
2
Materials Department and California Nanosystems Institute, University of California Santa Barbara, Santa Barbara, California 93106, USA
3
Beijing National Laboratory for Condensed Matter Physics and Institute of Physics, Chinese Academy of Sciences, Beijing 100190, China
4
University of Chinese Academy of Sciences, Beijing 100049, China
5
Department of Physics and Beijing Key Laboratory of Opto-electronic Functional Materials & Micro-nano Devices, Renmin University of
China, Beijing 100872, China
6
Division of Physics and Applied Physics, School of Physical and Mathematical Sciences, Nanyang Technological University,
Singapore 637371, Singapore
7
Laboratory for Muon Spin Spectroscopy, Paul Scherrer Institute, CH-5232 Villigen PSI, Switzerland
8
Princeton Institute for the Science and Technology of Materials, Princeton University, Princeton, New Jersey 08544, USA
9
Materials Science Division, Lawrence Berkeley National Laboratory, Berkeley, California 94720, USA
10
Quantum Science Center, Oak Ridge, Tennessee 37831, USA
(Received 2 May 2021; accepted 6 July 2021; published 15 July 2021)
Superconductors with kagome lattices have been identiﬁed for over 40 years, with a superconducting transition
temperature T
c
up to 7 K. Recently, certain kagome superconductors have been found to exhibit an exotic charge
order, which intertwines with superconductivity and persists to a temperature being one order of magnitude
higher than T
c
. In this work, we use scanning tunneling microscopy to study the charge order in kagome
superconductor RbV
3
Sb
5
. We observe both a 2 × 2 chiral charge order and nematic surface superlattices
(predominantly 1 × 4). We ﬁnd that the 2 × 2 charge order exhibits intrinsic chirality with magnetic ﬁeld
tunability. Defects can scatter electrons to introduce standing waves, which couple with the charge order to
cause extrinsic effects. While the chiral charge order resembles that discovered in KV
3
Sb
5
, it further interacts
with the nematic surface superlattices that are absent in KV
3
Sb
5
but exist in CsV
3
Sb
5
.
DOI: 10.1103/PhysRevB.104.035131
Kagome lattices [1], made of corner sharing triangles,
are tantalizing quantum platforms for studying the interplay
between geometry, topology, and correlation. For instance, in-
sulating kagome magnets have been investigated for decades
in the hopes of realizing quantum spin liquids [2]. Re-
cently, focused STM research on correlated kagome magnets
has revealed many topological and many-body phenomena
[3], including Chern gapped phases [4,5], tunable electronic
nematicity [4], orbital magnetism [68], and many-body in-
terplay [9,10]. These observations are all closely related to the
emergent physics arising from the fundamental kagome band
structure, which includes Dirac cones, ﬂat bands, and Van
Hove singularities. Notably, the many-body fermion-boson
interplay [10] observed in certain kagome paramagnets leads
us to conjecture from the spectroscopic point of view that
there can be superconductivity instability that competes with
magnetism. Then, we realize that kagome superconductors
*
These authors contributed equally to this work.
Corresponding author: jiaxiny@princeton.edu
Corresponding author: mzhasan@princeton.edu
with competing magnetism have been identiﬁed for at least
over 40 years [11], such as LaRu
3
Si
2
with T
c
of 7 K and a
fundamental kagome band structure [12]. Recently, another
layered kagome superconductor, AV
3
Sb
5
(A = K, Rb, Cs),
was discovered [1315], providing research opportunities,
particularly for STM studies [1621]. While RbV
3
Sb
5
has
been studied by several experimental techniques [15,22,23], it
has not yet been studied with scanning tunneling microscopy
(STM). In our earlier studies [16], we have reported the chiral
2 × 2 charge order in KV
3
Sb
5
, which displays robust chirality
with magnetic ﬁeld tunability on the defect-free region. Now,
we ﬁnd that RbV
3
Sb
5
also features 2 × 2 charge order with
additional nematic superlattices. It is crucial to reconﬁrm the
chiral charge order in this material and test its robustness
against the surface superlattices.
RbV
3
Sb
5
has a layered structure with the stacking of Rb
1
hexagonal lattice, Sb
2
honeycomb lattice, V
3
Sb
1
kagome lat-
tice, and Sb
2
honeycomb lattice shown in Figs. 1(a)1(c).
Owing to the bonding length and geometry, the V and Sb
layers have a stronger chemical bonding, and the material
tends to cleave between Rb and Sb layers. The Sb surface
is most interesting, as it is strongly bonded to the V kagome
2469-9950/2021/104(3)/035131(6) 035131-1 ©2021 American Physical Society

NANA SHUMIYA et al. PHYSICAL REVIEW B 104, 035131 (2021)
FIG. 1. (a)–(c) Crystal structure of kagome superconductor
RbV
3
Sb
5
from three-dimensional view, top view, and side view,
respectively. (d) Topographic image of a clean Sb surface (V =
100 mV, I = 0.5nA). (e) Fourier transform of the topography
showing Bragg peaks and charge ordering vector peaks. The 2 × 2
charge order vector peaks are highlighted by the shaded red ring,
which includes pairs of Q
1
, Q
2
,andQ
3
.
lattice. Previous STM studies have unambiguously resolved
Sb honeycomb surfaces in KV
3
Sb
5
and CsV
3
Sb
5
[1621], and
studied the charge order and surface superlattices. We study
RbV
3
Sb
5
with STM at 4.2 K. Through cryogenic cleaving,
we have also obtained large clean Sb surfaces in RbV
3
Sb
5
,
as shown in Fig. 1(d). The Fourier transform of the to-
pography reveals a 2 × 2 charge order as marked by the
shaded red region in Fig. 1(e). Such 2 × 2 charge order has
been consistently observed in KV
3
Sb
5
and CsV
3
Sb
5
by both
STM [1519,21] and bulk [13,16] x-ray measurements. In
addition, there are also nematic superlattice modulations (pre-
dominantly 1 × 4) along the Q
1
direction, and other weaker
superlattice signals along this direction. A similar superlat-
tice signal is also observed in the Sb surface in CsV
3
Sb
5
[1719,21]. However, such a signal has not been detected in
K/Cs/Rb surfaces (or in the bulk x-ray data), while a bulk
modulation will project and appear on all surfaces. Therefore,
while the nematicity may be a bulk phenomenon, the speciﬁc
1 × 4 modulation is more likely to be a surface phenomenon.
KV
3
Sb
5
does not exhibit a 1 × 4 superlattice for Sb surface
[16], and because CsV
3
Sb
5
has a factor of 3 higher T
c
than
that of KV
3
Sb
5
, previous STM observations in CsV
3
Sb
5
con-
jectured a close relationship between the 1 × 4 superlattices
and higher T
c
[19]. Our observation in RbV
3
Sb
5
, which has a
T
c
similar to KV
3
Sb
5
, makes such a scenario unlikely.
Now we focus on the intrinsic anisotropy of the 2 × 2
charge order on a large defect-free region in Fig. 2. We per-
form spectroscopic dI/dV maps at the same region with a
magnetic ﬁeld perpendicular to the surface. The maps taken at
FIG. 2. (a)–(c) dI/dV maps taken at the same clean Sb surface with B = 0T,3T,+3 T, respectively. The magnetic ﬁeld is applied along
the c axis. The maps are all taken at E = 30 meV with V =−100 mV and I = 0.5 nA. (d)–(f) Spectroscopic 2 × 2 vector peaks taken at
B = 0T, 3T,+3 T, respectively. The images are Fourier transforms of spectroscopic maps. A circular region of the full Fourier-transformed
image is shown for clarity, highlighting the six 2 × 2 vector peaks. The height of the three pairs of vector peaks is marked with arbitrary units
for each data. The chirality can be deﬁned as the counting direction (clockwise or anticlockwise) from the lowest to highest pair vector peaks
as marked by the rotating arrows.
035131-2

INTRINSIC NATURE OF CHIRAL CHARGE ORDER IN THE PHYSICAL REVIEW B 104, 035131 (2021)
FIG. 3. (a),(b) Real space image of the 2 × 2 chiral charge or-
der taken at B =−3T and B =+3T, respectively. The data are
produced by the inverse Fourier transform of the 2 × 2 vector peaks
in Fig. 2(e) and 2(f), respectively. (c),(d) Line-cut proﬁle for along
three directions marked in (a) and (b), respectively. The modulations
along the three directions are different in both cases, deﬁning a
chirality. Magnetic ﬁeld switch induces a switch of the strengths of
two stronger modulations.
30 meV with B = 0T, 3T,+3 T are displayed in Figs. 2(a)
2(c), respectively. We ﬁnd that the +3 T map is different
from the others. To better visualize the difference, we perform
Fourier transform analysis of these maps. Particularly, we
extract the six 2 × 2 vector peaks as shown in Figs. 2(d)
2(f), which reveals pronounced intensity anisotropy along
with different directions for all cases. This corresponds to
the fact that the amplitudes of the 2 × 2 modulation in real
space along three directions are different from each other. The
observed anisotropy can be due to a chiral charge order as
initially discussed in certain transition-metal dichalcogenides
and high-temperature superconductors [24,25]. The chirality
can be deﬁned as the counting direction (clockwise or anti-
clockwise) from the lowest to highest vector peaks. We ﬁnd
the chirality at the same atomic area can be switched by the
magnetic ﬁeld applied along the c axis for opposite directions.
A real space elaboration of the chirality switch is further
shown in Fig. 3, demonstrating that the strength of 2 × 2
modulation is switched by the magnetic ﬁeld. Figure 4 further
shows the energy-resolved vector peak intensity for different
magnetic ﬁelds. The vector peaks have weak intensity for
negative energies, hindering the identiﬁcation of chirality. For
higher positive energies where the intensities are strong, we
observe strong anisotropy. The intensity of Q
1
is always the
strongest, and we note that this direction is the same as that of
the nematic superlattices. Moreover, the intensities between
Q
2
and Q
3
are different, from which we can determine chiral-
ity. The reversal of their intensities between 3 T and +3T
then demonstrates a chirality switch.
FIG. 4. Magnetic ﬁeld tunability of the chiral charge order at a
defect-free region. Comparison of intensities of three 2 × 2 vector
peaks as a function of energy for the same defect-free region for
B = 0T, 3T,+3 T, respectively. The vector peaks have weak in-
tensities at negative energies, and the magnetic ﬁeld induced chirality
switching effects are primarily observed at higher positive energies.
As a comparison, we also perform experiments around the
defect-rich region in Fig. 5. Defects can backscatter electrons
to induce standing waves. Figure 4(a) shows rich standing
waves in the dI/dV map of this region. The Fourier trans-
form of this map shows clear ringlike signals just within
the 2 × 2 charge order vector peaks. A detailed plot of the
energy-resolved vector peak intensity at B = 0T, 3T,+3T
is displayed in Fig. 4(b). Different from the defect-free case
in Figs. 24, Q
2
and Q
3
basically have similar intensities over
all measured energies, suggesting a diminishing of chirality.
Moreover, there is no strong magnetic ﬁeld response. All these
observations are again consistent with our reports for KV
3
Sb
5
[16]. We believe, because the standing wave signals in the q
space are close to the 2 × 2 charge order peaks, there exists
a defect-pinning effect [26], which is an extrinsic property of
the charge order. The interplay between the charge order and
defects can be studied by the Bogoliubov–de Gennes method
in future.
Now we discuss the implications of our experiments. The
observations not only reconﬁrm the ubiquitous chiral charge
order in AV
3
Sb
5
, but also suggest that the chirality and ﬁeld
switching are both robust against nematic superlattices. The
2 × 2 charge order has been proposed by pioneering theories
of kagome lattices [2729] at Van Hove singularity ﬁlling. Re-
cently, several theoretical works focused on AV
3
Sb
5
[16,30
36] have conﬁrmed 2 × 2 charge order with unconventional
features, including time-reversal symmetry breaking, chiral-
ity, nematicity, and topology. The unconventional features
arise from the interferences of three kagome sublattices with
extended Coulomb interactions, and they can further interact
with the topologically nontrivial band structure in these mate-
rials. While the nematicity of the charge order observed here
can be consistent with the surface manifestation of the 2 ×
2 × 2 charge order [37], the chirality ubiquitously observed in
KV
3
Sb
5
[16], RbV
3
Sb
5
(this work), and CsV
3
Sb
5
[38] cannot
be explained by the conventional 2 × 2 × 2 charge order. As
the chirality can be switched by a magnetic ﬁeld that explicitly
breaks time-reversal symmetry, it implies a complex set of
order parameters of the charge order, which contain relative
035131-3

NANA SHUMIYA et al. PHYSICAL REVIEW B 104, 035131 (2021)
FIG. 5. Absence of chirality and magnetic tunability at standing-
waves-rich region. (a) dI/dV map data taken at a defect-rich Sb
surface. The maps are all taken at E = 0meV with V =−100mV
and I = 0.5nA. This region hosts numerous defect-induced standing
waves. The inset shows the Fourier transform of the map data, which
exhibits additional ringlike signals within the 2 × 2 vector peaks.
(b) Comparison of intensities of three 2 × 2 vector peaks for this
defect-rich region as a function of energy for B = 0T, 3T,+3T,
respectively. In this region, there is no apparent chirality of the charge
order, and we do not observe a strong magnetic ﬁeld response of the
vector peaks.
phase differences. The phase difference of three sets of the
2 × 2 order parameter, if not 0 or π , breaks time-reversal
symmetry. Recently, more direct evidence of the time-reversal
symmetry breaking comes from muon spin spectroscopy by
observation of a concurrent emergence of an internal magnetic
ﬁeld with the charge order phase transition [39]. Theoreti-
cally, a broken time-reversal symmetry charge order is also
suggested to be energetically favorable in the kagome lattice
at Van Hove ﬁlling and with extended Coulomb interactions
[16,30,32,33,36], which features orbital currents running in
the kagome lattice. Originally, charge order with broken time-
reversal symmetry was proposed as the Haldane model for
achieving quantum anomalous Hall effect [40] and orbital
currents [41,42] for modeling pseudogap phase of cuprates. Its
tantalizing visualization in kagome superconductors comes as
an experimental surprise. Since there has not yet been anoma-
lous Hall measurements for RbV
3
Sb
5
, whether our observed
intrinsic and extrinsic behavior of the chiral charge order can
be related with the intrinsic and extrinsic anomalous Hall
effects [43,44] deserves future attention. It is also crucial to
probe the magnetic ﬁeld switching effect more systematically
in the future by varying the magnetic ﬁeld strength, which can
help to determine the critical switching ﬁeld and to further
compare with anomalous Hall measurements.
Experimental and theoretical work at Princeton University
was supported by the Gordon and Betty Moore Founda-
tion [Grants No. GBMF4547 and No. GBMF9461 (M.Z.H.)].
The material characterization is supported by the United
States Department of Energy (U.S. DOE) under the Basic
Energy Sciences program (Grant No. DOE/BES DE-FG-
02-05ER46200). S.D.W. and B.R.O. acknowledge support
from the University of California Santa Barbara Quantum
Foundry, funded by the National Science Foundation (Grant
No. NSF DMR-1906325). Research reported here also made
use of shared facilities of the UCSB MRSEC (Grant No.
NSF DMR-1720256). B.R.O. also acknowledges support
from the California NanoSystems Institute through the Elings
fellowship program. T.A.C. was supported by the National
Science Foundation Graduate Research Fellowship Program
under Grant No. DGE-1656466. H.C.L. was supported by
National Natural Science Foundation of China (Grants No.
11822412 and No. 11774423), Ministry of Science and Tech-
nology of China (Grants No. 2018YFE0202600 and No.
2016YFA0300504), and Beijing Natural Science Foundation
(Grant No. Z200005). Y.S. was supported by the National
Natural Science Foundation of China (Grant No. U2032204),
and the K. C. Wong Education Foundation (Grant No. GJTD-
2018-01). G.C. would like to acknowledge the support of the
National Research Foundation, Singapore under its NRF Fel-
lowship Award (Award No. NRF-NRFF13-2021-0010) and
the Nanyang Assistant Professorship grant from Nanyang
Technological University.
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035131-5

##### Citations
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Journal ArticleDOI
Tiema Qian1, Morten Christensen2, Chaowei Hu1, Amartyajyoti Saha3  +4 moreInstitutions (3)
Abstract: The authors apply uniaxial strain on the layered kagome superconductor CsV${}_{3}$Sb${}_{5}$. They show that its charge density wave (CDW) and superconductivity (SC) transitions are dominated by the change in the $c$ axis, and the effect of the explicit rotational symmetry breaking by strain is negligible on the competition between CDW and SC. Together with theoretical studies, they propose that the trilinear coupling between the ${M}_{1}^{+}$ and ${L}_{2}^{\ensuremath{-}}$ phonon modes plays an important role in the CDW.

5 citations

Journal ArticleDOI
Ying Xiang1, Qing Li1, Yongkai Li2, Wei Xie1  +4 moreInstitutions (2)
Abstract: In transition metal compounds, due to the interplay of charge, spin, lattice and orbital degrees of freedom, many intertwined orders exist with close energies. One of the commonly observed states is the so-called nematic electron state, which breaks the in-plane rotational symmetry. This nematic state appears in cuprates, iron-based superconductor, etc. Nematicity may coexist, affect, cooperate or compete with other orders. Here we show the anisotropic in-plane electronic state and superconductivity in a recently discovered kagome metal CsV3Sb5 by measuring c-axis resistivity with the in-plane rotation of magnetic field. We observe a twofold symmetry of superconductivity in the superconducting state and a unique in-plane nematic electronic state in normal state when rotating the in-plane magnetic field. Interestingly these two orders are orthogonal to each other in terms of the field direction of the minimum resistivity. Our results shed new light in understanding non-trivial physical properties of CsV3Sb5.

2 citations

Journal ArticleDOI

1 citations

Posted Content
Abstract: Compounds with kagome lattice usually host many exotic quantum states, including the quantum spin liquid, non-trivial topological Dirac bands and a strongly renormalized flat band, etc. Recently an interesting vanadium based kagome family $A$V$_{3}$Sb$_{5}$ ($A$ = K, Rb, or Cs) was discovered, and these materials exhibit multiple interesting properties, including unconventional saddle-point driven charge density wave (CDW) state, superconductivity, etc. Furthermore, some experiments show anomalous Hall effect which inspires that there might be some chiral flux current states. Here we report scanning tunneling measurements by using spin polarized tips. Although we have observed clearly the $2\times2$ and $1\times4$ CDW orders, the well-designed experiments with refined spin polarized tips do not reveal any trace of the chiral flux current phase in CsV$_3$Sb$_5$. Thus it remains debatable whether this state really exists in CsV$_3$Sb$_5$ and we may need additional scenario to explain the anomalous Hall effect.

1 citations

Posted Content
Abstract: The vanadium-based kagome superconductor CsV3Sb5 has attracted tremendous attention due to its unconventional anomalous Hall effect (AHE), its charge density waves (CDWs), and a pseudogap pair density wave coexisting with unconventional strong-coupling superconductivity (SC). The origins of time-reversal symmetry breaking (TRSB), unconventional SC, and their correlation with different orders in this kagome system is of great significance, but, so far, is still under debate. Doping by the chemical substitution of V atoms in the kagome layer provides the most direct way to reveal the intrinsic physics that originates from the kagome lattice, but remains unexplored. Here, we report, for the first time, the synthesis of Ti-doped CsV3Sb5 single crystals with controllable carrier doping concentration. The Ti atoms directly substitute for V in the vanadium kagome layers. Remarkably, the Ti-doped CsV3Sb5 SC phase diagram shows two distinct SC phases. The lightly-doped SC phase has a V-shaped gap pairing, coexisting with CDWs, indicating a strong-coupling unconventional SC nature. The other SC phase has a U-shaped gap pairing without CDWs, displaying a conventional SC feature. This is the first observation of the two distinct phases in superconductors, revealed through Ti doping of CsV3Sb5. These findings pave a new way to synthesise doped CsV3Sb5 and represents a new platform for tuning the superconducting pairing and multiple orders in kagome superconductors.

##### References
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Journal ArticleDOI
TL;DR: A simple derivation of a simple GGA is presented, in which all parameters (other than those in LSD) are fundamental constants, and only general features of the detailed construction underlying the Perdew-Wang 1991 (PW91) GGA are invoked.
Abstract: Generalized gradient approximations (GGA’s) for the exchange-correlation energy improve upon the local spin density (LSD) description of atoms, molecules, and solids. We present a simple derivation of a simple GGA, in which all parameters (other than those in LSD) are fundamental constants. Only general features of the detailed construction underlying the Perdew-Wang 1991 (PW91) GGA are invoked. Improvements over PW91 include an accurate description of the linear response of the uniform electron gas, correct behavior under uniform scaling, and a smoother potential. [S0031-9007(96)01479-2] PACS numbers: 71.15.Mb, 71.45.Gm Kohn-Sham density functional theory [1,2] is widely used for self-consistent-field electronic structure calculations of the ground-state properties of atoms, molecules, and solids. In this theory, only the exchange-correlation energy EXC › EX 1 EC as a functional of the electron spin densities n"srd and n#srd must be approximated. The most popular functionals have a form appropriate for slowly varying densities: the local spin density (LSD) approximation Z d 3 rn e unif

117,932 citations

Journal ArticleDOI
Georg Kresse1, Jürgen Furthmüller2Institutions (2)
TL;DR: An efficient scheme for calculating the Kohn-Sham ground state of metallic systems using pseudopotentials and a plane-wave basis set is presented and the application of Pulay's DIIS method to the iterative diagonalization of large matrices will be discussed.
Abstract: We present an efficient scheme for calculating the Kohn-Sham ground state of metallic systems using pseudopotentials and a plane-wave basis set. In the first part the application of Pulay's DIIS method (direct inversion in the iterative subspace) to the iterative diagonalization of large matrices will be discussed. Our approach is stable, reliable, and minimizes the number of order ${\mathit{N}}_{\mathrm{atoms}}^{3}$ operations. In the second part, we will discuss an efficient mixing scheme also based on Pulay's scheme. A special metric'' and a special preconditioning'' optimized for a plane-wave basis set will be introduced. Scaling of the method will be discussed in detail for non-self-consistent and self-consistent calculations. It will be shown that the number of iterations required to obtain a specific precision is almost independent of the system size. Altogether an order ${\mathit{N}}_{\mathrm{atoms}}^{2}$ scaling is found for systems containing up to 1000 electrons. If we take into account that the number of k points can be decreased linearly with the system size, the overall scaling can approach ${\mathit{N}}_{\mathrm{atoms}}$. We have implemented these algorithms within a powerful package called VASP (Vienna ab initio simulation package). The program and the techniques have been used successfully for a large number of different systems (liquid and amorphous semiconductors, liquid simple and transition metals, metallic and semiconducting surfaces, phonons in simple metals, transition metals, and semiconductors) and turned out to be very reliable. \textcopyright{} 1996 The American Physical Society.

64,484 citations

Journal ArticleDOI
Frederick D. Haldane1Institutions (1)
TL;DR: A two-dimensional condensed-matter lattice model is presented which exhibits a nonzero quantization of the Hall conductance in the absence of an external magnetic field, and exhibits the so-called "parity anomaly" of (2+1)-dimensional field theories.
Abstract: A two-dimensional condensed-matter lattice model is presented which exhibits a nonzero quantization of the Hall conductance ${\ensuremath{\sigma}}^{\mathrm{xy}}$ in the absence of an external magnetic field. Massless fermions without spectral doubling occur at critical values of the model parameters, and exhibit the so-called "parity anomaly" of (2+1)-dimensional field theories.

3,679 citations

Journal ArticleDOI
Abstract: We propose that the enigmatic pseudogap phase of cuprate superconductors is characterized by a hidden broken symmetry of ${d}_{{x}^{2}\ensuremath{-}{y}^{2}}$-type. The transition to this state is rounded by disorder, but in the limit that the disorder is made sufficiently small, the pseudogap crossover should reveal itself to be such a transition. The ordered state breaks time-reversal, translational, and rotational symmetries, but it is invariant under the combination of any two. We discuss these ideas in the context of ten specific experimental properties of the cuprates, and make several predictions, including the existence of an as-yet undetected metal-metal transition under the superconducting dome.

807 citations

Journal ArticleDOI
Yi Zhou1, Kazushi Kanoda2, Tai Kai Ng3Institutions (3)
Abstract: This is an introductory review of the physics of quantum spin liquid states. Quantum magnetism is a rapidly evolving field, and recent developments reveal that the ground states and low-energy physics of frustrated spin systems may develop many exotic behaviors once we leave the regime of semiclassical approaches. The purpose of this article is to introduce these developments. The article begins by explaining how semiclassical approaches fail once quantum mechanics become important and then describe the alternative approaches for addressing the problem. Mainly spin-1/2 systems are discussed, and most of the time is spent in this article on one particular set of plausible spin liquid states in which spins are represented by fermions. These states are spin-singlet states and may be viewed as an extension of Fermi liquid states to Mott insulators, and they are usually classified in the category of so-called SU(2), U(1), or Z2 spin liquid states. A review is given of the basic theory regarding these states and the extensions of these states to include the effect of spin-orbit coupling and to higher spin (S>1/2) systems. Two other important approaches with strong influences on the understanding of spin liquid states are also introduced: (i) matrix product states and projected entangled pair states and (ii) the Kitaev honeycomb model. Experimental progress concerning spin liquid states in realistic materials, including anisotropic triangular-lattice systems [κ-(ET)2Cu2(CN)3 and EtMe3Sb[Pd(dmit)2]2], kagome-lattice system [ZnCu3(OH)6Cl2], and hyperkagome lattice system (Na4Ir3O8), is reviewed and compared against the corresponding theories.

706 citations

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