Quantum simulation of exotic PT-invariant topological nodal loop bands with ultracold atoms in an optical lattice

0.76 MB·

by  Dan-Wei Zhang

Quantum Simulation of Exotic PT-invariant Topological Nodal Loop Bands with
Ultracold Atoms in an Optical Lattice
Dan-Wei Zhang,1,2,∗ Y. X. Zhao,3,2,† Rui-Bin Liu,1 Zheng-Yuan Xue,1 Shi-Liang Zhu,4,5,‡ and Z. D. Wang2,§
1Guangdong Provincial Key Laboratory of Quantum Engineering and Quantum Materials,
SPTE, South China Normal University, Guangzhou 510006, China
2Department of Physics and Center of Theoretical and Computational Physics,
The University of Hong Kong, Pokfulam Road, Hong Kong, China
3Max-Planck-Institute for Solid State Research, D-70569 Stuttgart, Germany
4National Laboratory of Solid State Microstructures and School of Physics, Nanjing University, Nanjing 210093, China
6 5Synergetic Innovation Center of Quantum Information and Quantum Physics,
1
University of Science and Technology of China, Hefei 230026, China
0
(Dated: April 21, 2016)
2
Since the well-known PT symmetry has its fundamental significance and implication in physics,
r
p where PT denotes a joint operation of space-inversion P and time-reversal T, it is important and
A intriguingtoexploreexoticPT-invarianttopologicalmetalsandtophysicallyrealizethem. Herewe
developatheoryforanewtypeoftopologicalmetalsthataredescribedbyatwo-bandmodelofPT-

0 invariant topological nodal loop states in a three-dimensional Brillouin zone, with the topological
2 stabilitybeingrevealedthroughthePT-symmetry-protectednontrivialZ topologicalchargeevenin
2
theabsenceofbothPandTsymmetries. Moreover,thegaplessboundarymodesaredemonstrated
]
tobeoriginatedfromthenontrivialtopologicalchargeofthebulknodalloop. Basedontheseexact
s
a results,weproposeanexperimentalschemetorealizeandtodetecttunablePT-invarianttopological
g nodalloopstateswithultracoldatomsinanopticallattice,inwhichatomswithtwohyperfinespin
- statesareloadedinaspin-dependentthree-dimensionalopticallatticeandtwopairsofRamanlasers
t
n are used to create out-of-plane spin-flip hopping with site-dependent phase. It is shown that such
a a realistic cold-atom setup can yield topological nodal loop states, having a tunable band-touching
u ring with the two-fold degeneracy in the bulk spectrum and non-trivial surface states. The nodal
q loopstatesareactuallyprotectedbythecombinedPTsymmetryandarecharacterizedbyaZ -type
2
t. invariant (or topological charge), i.e., a quantized Berry phase. Remarkably, we demonstrate with
a numericalsimulationsthat(i)thecharacteristicnodalringcanbedetectedbymeasuringtheatomic
m
transferfractionsinaBloch-Zeneroscillation;(ii)thetopologicalinvariantmaybemeasuredbased
- onthetime-of-flightimaging;and(iii)thesurfacestatesmaybeprobedthroughBraggspectroscopy.
d The present proposal for realizing topological nodal loop states in cold atom systems may provide
n
a unique experimental platform for exploring exotic PT-invariant topological physics.
o
c
PACSnumbers: 37.75.Ss,03.65.Vf,37.10.Jk,03.67.Ac,
[

2
v I. INTRODUCTION ones, which have the two-fold and four-fold degenerate
1 Fermi points, respectively, which are topologically pro-
7 tectedbyatopologicalinvariant(suchastheChernnum-
3 Sincethediscoveryoftopologicalinsulators,thestudy ber for Weyl semimetals) and give rise to exotic Fermi-
0
of band topology of insulating and semimetallic mate- arc surface states [17–20]. While an NL semimetal has
0
. rials has attracted a broad interest [1–16]. A signif- the valence and conduction bands crossing along closed
1 icant theoretical advance has been made for classifica- lines instead of isolated points, which may topologically
0 tion of various kinds of gapped and gapless topological be protected by certain discrete symmetry and may give
6
1 bandsystems[3–11]. Veryrecently,agreaterattentionof rise to nearly flat surface bands [21, 22]. Weyl and Dirac
: exploring symmetry protected topological phases seems semimetals have been theoretically and experimentally
v tomovefromgappedinsulators/superconductorstogap- explorednotonlyinmaterials[9,17–20,23–28], butalso
i
X less metals/semimetals. For three-dimensional systems, in some artificial systems, such as photonic and acoustic
r two kinds of topological semimetals, which respectively crystals [29–31]. However, topological NL bands are yet
a consist of the nodal point and nodal loop (NL), have to be experimentally observed or realized, even though
been addressed [17–22]. Weyl semimetals [18–20] and some theoretical proposals have been suggested for their
certain Dirac semimetals [9, 17] belong to the former realization in real materials very recently [32–37]. In ad-
dition, the so-called PT symmetry may actually be un-
derstood as a generalized inversion symmetry with re-
gard to the space-time dimension, which is fundamen-
∗Electronicaddress: [email protected] tal in physics, and thus a realistic model for describing
†Electronicaddress: [email protected] exotic PT-invariant topological NLs is highly desirable.
‡Electronicaddress: [email protected] Notably, a purely combined PT-invariant topological NL
§Electronicaddress: [email protected]
2
statethathasneitherPsymmetrynorTone, canhardly the surface states may be probed through Bragg spec-
berealizedinrealcondensedmaterials,andthusitsphys- troscopy. The realization of the combined PT topologi-
ical implementation presents a great challenge to physi- cal NL states in cold atom systems would definitely be
cists both theoretically and experimentally. ofgreatimportanceincontributingtotopologicalmatter
research across disciplines.
On the other hand, fortunately, ultracold atoms in
The paper is organized as follows. Section II describes
optical lattices [38] with synthetic electromagnetic field
the topological classification of NLs in systems with the
and spin-orbit coupling [13–15] provide a clean and tun-
combined PT symmetry, interprets physical meanings of
able platform for exploring exotic topological quantum
the Z topological invariant in a lattice model, and dis-
phases [39–46]. Remarkably, the Zak phase in topologi- 2
cussesthetopologicalstabilityagainstperturbationspre-
cally nontrivial Bloch bands realized in one-dimensional
serving the PT symmetry but breaking both P and T
double-well optical lattices has been measured [39]. By
symmetries. In Sec. III, we propose an experimentally
engineering the atomic hopping configurations in opti-
feasibleschemeforrealizingthePT-invarianttopological
cal lattices, the celebrated Harper-Hofstadter model [47]
NL states with ultracold atoms in an optical lattice. We
andHaldanemodel[48]havebeenrealized[40–43],where
first introduce the proposed system and lattice model,
the Chern number characterizing the topological bands
and then study the tunable NLs and their experimen-
has also been measured. The experimental observation
tal detection. In Sec. IV, we elaborate the topological
of chiral edge states in one-dimensional optical lattices
properties of the simulated NL states by calculating the
subjected to a synthetic magnetic field and an artificial
quantized Berry phase and the surface states, and also
dimension has also been reported [45, 46]. An impor-
present practical methods for their experimental detec-
tant question then is whether the other predicted topo-
tioninthecoldatomsystemwithnumericalsimulations.
logical phases that are rare in sold-state materials can
Finally, a short conclusion is given in Sec. V.
be realized in these cold atom systems. Several schemes
have been proposed to realize Z topological insulators
2
[49,50], chiraltopologicalstates[51–53], andtopological
II. CLASSIFICATION OF PT-INVARIANT
nodal point semimatals [54–59] using ultracold atoms in
TOPOLOGICAL METALS/SEMIMETALS AND
optical lattices. However, an experimentally feasible and
THE TOPOLOGICAL STABILITY OF NODAL
tunableschemeforrealizingthecombinedPTtopological
LOOPS
NL states and their detection is still badly awaited.
Inthisarticle,wefirstdepicttheclassificationoftopo- In this section, we first describe a mathematically
logical NLs in systems with the pure PT symmetry. In rigorous classification of topological metals/semimetals
three-dimensional momentum space, only PT-invariant protected by the PT symmetry using the KO-theory.
NLs are topologically protected with a Z2 charge. We ThenweconstructaphysicallyrealisticPT-invariantNL
thenconstructatheoreticalmodeltorealizethetopolog- model,whichhasanontrivialZ topologicalcharge. The
2
ical nontrivial NL, and demonstrate its topological sta- stability of the NL is investigated in detail, making con-
bility under PT invariant perturbations that break both crete implications of its nontrivial topological charge.
T and P symmetries. It is also shown that the nontriv- Other physical meanings of the topological charge are
ial topological charge of the bulk NL directly determines also discussed.
the certain boundary gapless modes. Based on these
theoretical results, we propose an experimental scheme
to realize and to detect tunable PT-invariant topolog- A. Classification of topological metals/semimetals
ical NL states with ultracold atoms in an optical lat- with PT symmetry
tice. In our proposal, fermionic (or bosonic) atoms with
two hyperfine spin states are loaded in a spin-dependent Let us consider a non-interacting fermionic system,
three-dimensionalopticallattice,andtwopairsofRaman which is described by the Hamiltonian H(k) in the mo-
lasers are used to create out-of-plane spin-flip hopping mentum space. For such a system, the time-reversal T
with site-dependent phase. We show that such a realis- and inversion P symmetries are represented by Tˆ and Pˆ
tic cold-atom setup can yield topological NL states hav- as, respectively
ing tunable ring-shaped band-touching lines with two-
folddegeneracyinthebulkspectrumandnon-trivialsur- TˆH(k)Tˆ−1 =H(−k), PˆH(k)Pˆ−1 =H(−k). (1)
facestates. TheNLstatesareactuallytopologicallypro- Tˆ is anti-unitary while Pˆ is unitary,
tected by the combined PT symmetry even in the pres-
ence of P and T breaking perturbations, and are charac- TˆiTˆ−1 =−i, PˆiPˆ−1 =i. (2)
terizedbyaquantizedBerryphase(aZ -typeinvariant).
2 Wehereareinterestedmainlyinthecombinedsymme-
Moreover, with numerical simulations, we demonstrate
try A=PT, namely only PT is required to be preserved,
that (i) the characteristic nodal ring can be detected by
whileTandPmaybebrokenindividually. Theoperation
measuringtheatomictransferfractionsinaBloch-Zener
of the anti-unitary symmetry A is given by
oscillation; (ii) the topological invariant (charge) can be
measured based on the time-of-flight imaging; and (iii) AˆH(k)Aˆ−1 =H(k), AˆiAˆ−1 =−i. (3)
3
It is clear that A operates pointwisely in the momen-
tum space as an anti-unitary operator, and therefore the
corresponding Berry bundle generated by H(k) in the
gapped region has a real relation on each fiber, since ev-
ery state |α,k(cid:105) and its complex conjugate |α,k(cid:105)∗ are re-
lated by an unitary tranformation, namely U |α,k(cid:105)∗ =
A
|α,k(cid:105). The topological classification of fiber bundles
with such a pointwise real relation (and dimension high
enough) is given by an abelian group in the framework
FIG. 1: (Color online) An NL in the three-dimensional mo-
oftheKOtheory,whereeachgroupelementcorresponds
mentum space, enclosed by a tiny circle.
to a topological class [62]. The Clifford algebra may be
used as a powerful tool in the computation of the KO
groups [61], as being illustrated below [60]. as the complex-conjugate operator and Pˆ = σ , the PT
3
To find the topological space of the gapped Hamilto- symmetry with Aˆ=σ K, requires
3
nians under the restriction of the symmetry A, we re-
combine the relevant operators as Aˆ, iAˆ and iH, which σ H∗(k)σ =H(k), (7)
3 3
anti-commutewitheachotherformingaCliffordalgebra,
whichsimplymeanstheabsenceofσ terminH(k). The
1
{Aˆ,iAˆ}=0, {Aˆ,iH}=0, {iH,iAˆ}=0. (4) time-reversal symmetry implies
For any gapped k, we adiabatically flatten H(k) to be H∗(k)=H(−k), (8)
H˜(k), which has the unital normalization H˜2(k) = 1
N
leading to that the coefficients of σ , σ , and σ are even
with 1 being the N ×N identity matrix. It is suffi- 0 1 3
N
cient, for topological purpose, to consider H˜. Thus for functions of k, while that of σ2 is odd. Note that the
conservation of TP and T implies also the conservation
the gapped region of the momentum space, we assume,
of P. From the above points, we start with a simple
without loss of generality, the following normalizations,
continuum model in three dimensions as
(iH)2 =−1, Aˆ2 =(iAˆ)2 =±1, (5)
H (k)=[R2−B(k2+k2)−B k2]σ +C k σ (9)
0 x y z z 3 z z 2
where Aˆ2 = (iAˆ)2 comes from the anti-unitarity of Aˆ in
withk=(k ,k ,k )andB >0,whichhasbothPandT
x y z
Eq.(3).
symmetries, and therefore the PT symmetry. Note that
For the case of Aˆ2 =1, it is found that Aˆand iAˆform we abandon σ term for clarity, since it has nothing to
0
theCliffordalgebraC0,2,whichisextendedbytheiH to do with the spectrum gap. We also keep the rotation
be C1,2. Since C0,2 ⊂ C1,2 ≈ C0,0 ⊂ C0,1 [61, 62], it is symmetry of the k -k plane for simplicity. It is found
x y
found that the space of the gapped Hamiltonians H(k) that the gapless points form an NL on the k -k plane
x y
is given by the classifying space R0 up to homotopy. For with kz = 0, which may be enclosed by a loop from the
an NL in a three-dimensional momentum space, we may gapped region, for instance a tiny circle on the k -k
y z
choose a circle S1 to enclose the NL from its gapped plane as shown in Fig. 1. The circle is parametrized
√
transverse dimensions, and H|S1(k) gives a map from S1 as (0,R/ B+ρcosφ,ρsinφ), where ρ is the radius and
to R0, which is classified by the KO theory as [61, 62] φ the angle. If ρ is sufficiently small, the Hamiltonian
restricted on the circle is expanded as
K(cid:103)O(S1)∼=π1(R0)∼=π0(R1)∼=Z2. (6) √
h(φ)=−2 BRρcosφσ +C ρsinφσ +O(ρ2). (10)
3 z 2
This result implies that there exists a topological non-
trivial class of NLs with Z charge, whose topological It is a well-known result that the Berry phase of the
2
stability depends solely on the combined TP symmetry, occupied state wave function of such a Hamiltonian is
regardless of individual T and P symmetries. equivalent to one modulo 2, namely
1 (cid:90)
γ = a(φ)dφ≡1 mod 2, (11)
S1 π
B. A two-band nodal loop model and the meanings
of its topological charge
wherea(φ)=(cid:104)φ|i∂ |φ(cid:105)with|φ(cid:105)beingtheoccupiedstate
φ
of h(φ). The unit topological charge γ means the NL is
We now construct a simple two-band model, which in the nontrivial class of the Z classification, which was
2
has PT symmetry with (PˆTˆ)2 =1, to illustrate our the- obtained above.
ory. A two-band model may be written in a unified form According to our classification theory, the topological
H(k) = fµ(k)σµ, where fµ are real functions of k and protection of the stability of the gapless modes requires
σµ = (σ0,σj) are Pauli matrices. Choosing the time- onlythecombinedPTsymmetry,ratherthanbothPand
reversal and inversion symmetry as Tˆ = K with the K T symmetries. In other words, the gapless modes still
4
existfortopologicalreasonunderperturbationsbreaking
both P and T symmetries but preserving PT symmetry.
In general, a σ term with even functions of k and a
2
σ term with odd functions of k break both P and T.
3
From our discussion of P and T symmetries above, the
perturbations below, for instance, satisfy the symmetry
conditions.
H(cid:48)(k)=[D(k +k )+D k ]σ
x y z z 3
+[µ+(cid:15) k2+(cid:15)(k2+k2)]σ , (12)
z z x y 2 FIG.2: (Coloronline)AnNLinathree-dimensionalBrillouin
zone, enclosed by different loops.
under which gapless modes still survive, although the
shape of the NL has been distorted. For instance when (cid:15)
and (cid:15) vanish, the gapless region is given by the solution
z III. PT-INVARIANT NODAL LOOP STATES IN
of the equation COLD ATOM SYSTEMS
B(k2+k2)−D(k +k ) Although the two-band Hamiltonian of PT-invariant
x y x y
+[B (µ/C )2+D µ/C −κ2]=0, (13) NL is simple, the direct implementation of this ideal
z z z z
model in electronic materials is difficult due to the spin-
orbit coupling or complex lattice structure therein [32–
for which it is seen that the radius of the NL is changed
37]. In this section, we turn to propose an experimen-
and the center is moved by the perturbations.
tal scheme to realize and detect tunable PT-invariant
We now consider the realistic situation that the NL
topological NL states with ultracold atoms in a three-
exists in a lattice model, where the momentum coordi-
dimensional optical lattice. We first describe the pro-
natesareperiodic,formingaBrillouinzoneasillustrated
posed system and the lattice model, which may have
in Fig. 2. The NL in Fig. 2 as the collection of gapless
both P and T symmetries or generically only preserves
points is still denoted by the red circle as in Fig. 1. But
the combined PT symmetry while breaks both of them.
inalatticemodeltheperiodicityofthemomentumcoor-
Then we study the tunable NLs and their experimental
dinates allows large circles, such as L and L in Fig. 1,
1 2 detection in this cold atom system.
since the two ends of such a line segment are identified.
Thus for a lattice model, the tiny circle in Fig. 1 enclos-
ingtheNLcanbecontinuouslydeformedtobetheS1 in
A. Proposed cold atom system and lattice model
the gapped region, which may further be divided as two
large circles L and L , namely
1 2
Our proposed system is based on a fermionic gas (or
bosonicgas)ofnoninteractingatomswithtwochosenhy-
S1 ≈L −L (14)
1 2 perfine spin states |↑(cid:105) and |↓(cid:105) in a spin-dependent cubic
optical lattice, as shown in Fig. 3(a). The lattice poten-
with the sign indicating the direction of the line. The
tial takes the form
topological charges satisfy the relation
V (r)=−V [cos2(k x)+cos2(k y)+cos2(k z)], (16)
σ 0,σ 0 0 0
γ =γ −γ mod 2, (15)
S1 L1 L2
where σ =↑,↓, V denote the potential strengths for
0,σ
where actually L and L can be any large circles in- the spin state |σ(cid:105), and k is the wave number with the
1 2 0
side and outside the NL, respectively, as in Fig. 2. For lattice constant a = π/k . As shown in Fig. 3(a), we
0
a topologically nontrivial NL, if the topological charge consideratwo-photonRamantransitionbetween|↑(cid:105) and
of any outside line is trivial, then every large circle go- |↓(cid:105) along the z axis, which is achieved by coupling the
ing inside the NL has nontrivial topological charge. For twogroundhyperfinestatestoanexcitedstate|e(cid:105) witha
straight lines inside, each of them may be regarded as largesingle-photondetuning∆ . Inaddition,thetwohy-
d
corresponding to a one-dimensional gapped system that perfine states differ in the magnetic quantum number by
is of topologically nontrivial band structure, leading to one and thus the atomic addressing is achieved through
certain gapless boundary modes. polarization selection. This configuration has been used
The arguments above are applicable to general cases to create the equal-Rashba-Dresselhaus spin-orbit inter-
in the classification. We may note an essential differ- action [13–15]. The Rabi frequencies of the correspond-
encebetweenZandZ2 topologicalcharges,namelyforZ ing laser fields are chosen as Ω1(z) = Ω0cos(k0z/2)eik0(cid:48)z
topological charge, reversing the direction also reverses and Ω (z)=Ω sin(k z/2), respectively, where Ω is the
2 0 0 0
the topological charge, while Z topological charge is in- strength constant controlled by the laser intensities of
2
sensitive to the direction since 1≡−1 mod 2 [12]. the Raman fields and k(cid:48) denotes a small deviation of
0
5
(a) (b) of the spin state |σ(cid:105) at the lattice site r. A straightfor-
ward calculation yields the tight-binding Hamiltonian
Raman field Ω (z)
eff
Ω1,+ ↑ Ω1,− Hˆ =− (cid:88) tσaˆ†r,σaˆr+ηˆ,σ+H.c.
r,σ,η
tso Ω2,+z ↓ Raman transitΩio2n,− +(cid:88)trs,r±ˆzaˆ†r,↑aˆr±ˆz,↓+H.c. (19)
r,±
-t so ∆ e +(cid:88)m (aˆ† aˆ −aˆ† aˆ ).
- t d z r,↑ r,↑ r,↓ r,↓
t↑ z↓ Ω1 Ω2 r
t Here the spin-conserved hopping along the η axis (η =
so t
t↓↑ x y ↑ δ ↓ xV,σy]w,zσ)(ri(cid:48)s−dre−rivηˆe)d≈a(s4/t√σπ)=V03,−/σ4E(cid:82)R1d/34re(cid:48)−w2σ∗√(rV(cid:48)0,−σ/rE)R[2pimn2ath+e
FIG. 3: (Color online) The proposed cold atom system and harmonic approximation (by using Gaussian wavefunc-
laserconfiguration. (a)Thecubicopticallatticeandtheeffec- tions of the ground state centered in each lattice well
tive atomic hopping configuration between nearest-neighbor for the Wannier functions) with E = (cid:126)2k2/2m be-
R 0 a
lattice sites. It contains the spin-conserved hopping t↑ along ing the recoil energy [64]. The spin-flip hopping terms
each axis, t↓ in the xy plane but −t↓ out of the plane, the induced by the Raman field take the form tr,r±ˆz =
spin-fliphoppingfrom|↓(cid:105) to|↑(cid:105) whichis±tsostaggeredalong (cid:82) d3r(cid:48)w∗(r(cid:48)−r)(cid:126)Ω (r(cid:48))w (r(cid:48)−r∓ˆz). s
the z axis, and its hermitian conjugation progress. (b) The ↑ eff ↓
In this lattice system, the strength of the Raman
spin-dependentlatticestructureandlaserconfigurationofthe
field proportional to sin(k z) and the lowest band Wan-
Raman transition between spin states |↑(cid:105) and |↓(cid:105) along the 0
nier functions are antisymmetric and symmetric with
z axis . The effective Raman field strength proportional to
respect to the center of each lattice site, respectively.
sin(k z) (green dotted line) consists two pairs of plane waves
0
denoted by their Rabi frequencies Ω (gives Ω laser field) Due to this spatial configuration, the periodic field
1,± 1
and Ω (gives Ω laser field). Two spin states are coupled does not couple the intra-site spins and the spin-flip
2,± 2
viaatwo-photonRamantransitionwithalargedetuning∆ hopping terms satisfy tr,r±ˆz = ±(−1)z/ae±iϕt [51],
d s so
from an excited state |e(cid:105) and a two-photon detuning δ. (cid:82)whdezrwe↑∗(tszo)ei=k0(cid:48)z2(cid:126)s∆Ωin20d((cid:82)k0dzx)ww↓↑∗((zx)−w↓a()xa)n×d ϕ(cid:82) d=ywk0↑(cid:48)∗a(y.)Twh↓(eyl)as×t
the wave numbers (k0(cid:48) (cid:28) k0) that can be tuned from term mz = (V0,↑ −V0,↓ −(cid:126)δ)/2 is equivalent to a Zee-
zero to finite value via the laser beams. Here Ω and Ω manfieldalongthez axisandcanbepreciselytunedvia
1 2
can be realized, respectively, with a pair of laser beams the laser frequencies of the Raman beams with acoustic-
Ω1,± = 21Ω0e±i(k0±k0(cid:48))z and Ω2,± = 21Ω0e±i(k0z+π/2). opticmodulatororthroughthelatticepotentialsforfixed
Generally,thereisatwo-photondetuningδintheRaman laser frequencies. By redefining the spin-down operator
transition, as shown in Fig. 3(b). aˆr,↓ →eiπz/aaˆr,↓, Hamiltonian (19) can be rewritten as
In the presence of large detuning |∆|(cid:29)Ω0,|δ| for the Hˆ =−(cid:88)t (aˆ† aˆ +aˆ† aˆ )+H.c.
Ramantransition,wecaneliminatetheexcitedstateand σ r,σ r+xˆ,σ r,σ r+yˆ,σ
consider the atomic motions in the ground state mani- r,σ
(cid:88)
fold. Under this condition, the effective single-particle − (t aˆ† aˆ −t aˆ† aˆ )+H.c. (20)
↑ r,↑ r+ˆz,↑ ↓ r,↓ r+ˆz,↓
Hamiltonian in the spin basis {|↑(cid:105),|↓(cid:105)} is given by
r
(cid:88)
p2 (cid:18) V (r) (cid:126)Ω (z) (cid:19) + tso(eiϕaˆ†r,↑aˆr+ˆz,↓−e−iϕaˆ†r,↑aˆr−ˆz,↓)+H.c.
H0 = 2ma + (cid:126)Ω↑∗eff(z) V↓(re)ff+(cid:126)δ , (17) +(cid:88)r m (aˆ† aˆ −aˆ† aˆ ).
z r,↑ r,↑ r,↓ r,↓
where p and ma respectively denote the atomic momen- r
tduemnotaensdthmearsess,ualtnindgΩReaffm(za)n=coΩu∆∗1pΩdli2ng=inΩt20h2e∆−eidktw0(cid:48)zos-ipnh(okt0ozn) Ffigiguurraeti3o(na)foarlstoheshϕow=s0th(iis.ee.ff,ekc(cid:48)ti=ve0a)tcoamseicihnotphpeinogptciocna-l
0
transition. We consider all the atoms in the lowest band lattice.
of the optical lattice, and then the Hamiltonian in the In the three-dimensional Brillouin zone, the resultant
second quantization formalism takes the form Bloch Hamiltonian is given by
(cid:90) H =f (k)σ −2t sin(k a+ϕ)σ −f (k)σ , (21)
Hˆ = d3r(cid:48)Ψ†(r(cid:48))H Ψ(r(cid:48)). (18) B z 3 so z 2 0 0
0
wheref (k)=m −α [cos(k a)+cos(k a)]−α cos(k a)
z z − x y + z
and f (k)=α [cos(k a)+cos(k a)]+α cos(k a), with
Here Ψ=(Ψ ,Ψ )T is the two-component field operator 0 + x y − z
with Ψ (r(cid:48)) =↑ (cid:80)↓ aˆ w (r(cid:48)−r), which is expanded in α± ≡ t↑ ±t↓ are also tunable parameters. The Bloch
σ r r,σ σ Hamiltonian can be rewritten as
termsoftheWannierfunctionw (r(cid:48)−r)andaˆ repre-
σ r,σ
senting the annihilation operator of the fermionic mode H =H˜ +H , (22)
B 0 P
6
where H˜ = f (k)σ − 2t cosϕsin(k a)σ − f (k)σ Zeeman field and the hopping amplitudes can be tuned
0 z 3 so z 2 0 0
and the perturbation part H = −2t sinϕcos(k a)σ . independently, we can define two ratio parameters β =
P so z 2
Here H˜0 preserves both P and T symmetries but HP mz/α− and τ =α+/α− in this system.
(which vanishes in the case ϕ = 0) breaks the two sym- We then numerically solve the Eqs. (23) and (24) for
metries. However, the whole Hamiltonian HB satisfies the existence and the shape of the NLs, and the resul-
the combined PT symmetry. As analysized in Sec. II, tant phase diagram is shown in Fig. 4(a), where the
this guarantees the existence and the topological stabil- phases III, II.a, II.b and I, respectively, correspond to
ityofsymmetry-protectedNLsinthree-dimensionalBril- trivial insulator, an NL on the k = 0 plane, an NL on
z
louinzone. TheperturbationpartHP inthissystemonly the kz = π/a plane and two coexisting NLs on the two
shifts the center of the NLs by replacing kza→kza+ϕ, planes. The two NLs belong to the same nontrivial Z2
and do not modified their shape and topological proper- caseasthatbeingdiscussedintheprevioussection. Dur-
ties. Therefore, without loss of generality, we take ϕ=0 ing the increase of β (denoted by the black solid arrows
case in the following sections to study the NL states and in Fig. 4), the NLs have the same evolution procession
theirdetectioninthissystem. Notethattheproposedop- as illustrated in Fig. 4(b). First, a singular point is cre-
tical lattice system and the Raman coupling scheme are ated at the corner of the sub-Brillouin zone with k =0
z
also applicable to the bosonic atoms [13–15, 40–42]. As or π/a, then spread to be a circle centered at the cor-
for other different types of NL states, a four-band model ner. The circle expands bigger and bigger, going across
allowing Dirac or Weyl rings was also recently suggested the whole sub-Brillouin zone (the red large circle), then
to be simulated with cold atoms [63]. becomes a circle centered at the origin, and finally dis-
appears after decaying as a singular point at the origin.
After such a Z nontrivial small nodal circle is created
2
III II.b I II.a III at the corner of the sub-Brillouin zone, the topological
number of any one-dimensional subsystem parametrized
- 2-τ -τ τ 2+τ β byk insidethesmallcircleisincreasedbyone,whileany
z
I: Double NLs 1 1 oneoutsidehasitstopologicalnumberunchanged,which
IIII..ab:: AA NNLL aatt kkzz==π0 /a πk(/a)y0 πk (/a)y0 cecaroinnntgibneauosfaeucestndtfherfoaomtrmthtaheteicormnesaotdfiooelnrdotihfsectuhoseusectdisricdilneeFroenisgeu.sl.t2sHcinoanovsniindlyg-
III: Trivial insulator
-1 -1 this in mind we can infer the topological number of any
-1 0 1 -1 0 1
k(π/a) k (π/a) one-dimensional system with fixed in-plane momentum
x x
k ≡ (k ,k ) in any region of the phase diagram. For
(cid:107) x y
FIG.4: (Coloronline)PhasediagramandtunableNLstates. instance, the distribution of topological number γ in the
(a) Phase diagram. III, II.a,II.b and I correspond to trivial k -k plane along the white arrow in Fig. 4(a) is shown
x y
insulator,anNLonthekz =0plane,anNLonthekz =π/a in Fig. 4(c), which has also been confirmed by the nu-
planeandtwocoexistingNLsonthetwoplanes. (b)Theevo- mericalsimulation. Itisnotedthatinthefifthsub-figure
lutionofanNLduringtheincreaseofβ denotedbytheblack with β = 0, ones at the corner of the sub-Brillouin zone
arrowsin(a). (c)Thedistributionofthetopologicalnumbers
have the topological number γ = 1+1 ≡ 0 mod 2. In
of one-dimensional subsystems on k -k plane in difference
x y addition, the shape of the NL can changes from a circle
phasesduringtheincreaseofβ markedbythewhitearrowin
to square as shown in Fig. 4(b) due to the fact that the
(a).
k2 terms in the continuum Hamiltonian (9) are replaced
η
by the cos(k a) terms in this lattice system with Hamil-
η
tonian(21). Thus,intheproposedopticallatticesystem,
one can realize tunable NL states by simply varying the
B. Tunable nodal loop states
ratio parameter β via the laser fields.
In the following, we focus on the single NL cases and
Now we proceed to study the properties and the de-
furtherstudythepropertiesoftheNLstate. InFigs. 5(a)
tection method of the NL states in the proposed model
and5(b),weplotthebulkenergygapE (k)−E (k)for
system. For ϕ = 0, the bulk spectrum is E (k) = + −
±
(cid:113) typical parameters, which clearly show the gap-closing
f0(k) ± 4t2sosin2(kza)+fz(k)2. The bulk bands are points forming the NLs on the kz = 0 and kz = π/a
fully gapped except the points in momentum positions planes, respectively. On the two planes, the upper and
that satisfy the following conditions: lower energy surfaces touch along a ring with a con-
stant energy ε in phase II.a and ε in phase II.b. The
a b
cos(kxa)+cos(kya)=(mz−α+)/α− for kz =0,(23) constant energies εa = −α+(α+ − mz)/α− − α− and
cos(kxa)+cos(kya)=(mz+α+)/α− for kz =π/a,(24) εb = −α+(α+ +mz)/α− +α− come from the f0(k)σ0
terminHamiltonian(21),respectively. Thistermbreaks
which can give rise to NLs with two-fold degeneracy in the chiral symmetry but only shifts the nodal ring from
the three-dimensional Brillouin zone. Without loss of zero energy without modifying its shape. Actually, it
generality, we assume t ≥ t > 0. Since the effective does not affect the topological stability of the NLs but
↑ ↓
7
cycle, we can obtain the momentum distribution of the
(a) (b)
Bulk gap transfer fraction to the upper band from time-of-flight
imaging[65,66]. Inthiscase,thetransferfractionξ (k )
z (cid:107)
is given by
ξz(k(cid:107))=PLZ(k(cid:107))=e−π∆2z(k(cid:107))/4vzF, (26)
whereP istheLandau-Zenertransitionprobabilityand
LZ
∆ =(E −E )| denotes the energy gap for the
z + − kz=0,π/a
transition along the k direction. Figures 5(c) and (d)
z
(c) (d)
show the typical momentum distribution ξ (k ), where
ξ z (cid:107)
z eachpointofmaximumtransferinthekx-ky planecorre-
sponds to a nodal point and all the points form a nodal
loop, corresponding to the cases in Figs. 5(a) and 5(b),
respectively. The peaks in ξ (k ) are sharp as the tran-
z (cid:107)
sition probability in a Landau-Zener event increases ex-
ponentially as the energy gap decreases, such that the
momentum distribution of the atomic transfer fraction
with its maximum positions can well reveal the shape of
an NL.
FIG. 5: (Color online) The bulk gap with the gap-closing
pointsforminganodalringanditsdetection. (a)Anodalring
onthekz =0planewiththeparametermz =2.0. (b)Anodal IV. TOPOLOGICAL PROPERTIES OF THE
ring on the kz = π/a plane with the parameter mz = −1.8. SIMULATED NODAL LOOP STATES
(c,d)Themomentumdistributionoftheatomictransferfrac-
tion ξ (k ,k ) with the parameter F = 0.2. The maximum
z x y In this section, we precede to investigate the topolog-
positions form a curve that reveals the nodal ring in (a) and
ical properties of the simulated NL states and present
(b),respectively. Otherparametersin(a-d)aret =1asthe
↑
practicalmethodsfortheirexperimentaldetectioninthis
energy unit, t =t =0.3.
↓ so
cold atom system. We first show the quantized Berry
phase characterizing the NL states in the bulk and then
givesthesurfacestatesadispersion[3,21]. InphaseII.a, consider the related non-trivial surface states.
the low-energy effective Hamiltonian near K = (0,0,0)
r
is given by
A. Quantized Berry phase: Z topological invariant
2
H (q)=[b (q2+q2)+b q2−λ ]σ −v q σ
eff 1 x y 0 z 1 3 z z 2
+[b (q2+q2)+b q2−λ ]σ , (25) As analysed in the previous section, Hamiltonian (21)
0 x y 1 z 0 0 preserves the combined PT symmetry. This guarantees
the symmetry-protected topological stability of the PT-
where q=k−K , b = a2α /2, b = a2α /2, λ =
r 0 + 1 − 1
invariant NLs in three-dimensional Brillouin zone and a
2α +α −m , v = 2at , and λ = 2α +α . In
− + z z so 0 + −
quantized Berry phase γ in units of π that characterizes
phase II.b, the low-energy effective Hamiltonian near
its topological protection, even in the presence of P and
K(cid:48) = (π/a,π/a,π/a) is H(cid:48) (q) = −H (q) with rede-
r eff eff T breaking perturbation H . If a closed loop in the mo-
finedq=k−K(cid:48) inthiscase. TheeffectiveHamiltonian P
r mentum space is pierced by the NL, one has γ =1 (i.e.,
(25) takes the form of the two-band model (9), and cap-
π Berry phase), otherwise γ =0, which represents a Z -
turestheessentialfeaturesofthenodalringstatesonthe 2
(cid:112) type invariant. The same invariant can be written if the
q =0 plane of radius λ /b when its size is small (see
z 1 1
loop is chosen parallel to k , i.e., with fixed k , since it
Fig. 4). z (cid:107)
is closed at k = ±π/a due to periodic boundary con-
Atthisstage,weturntoelaboratethattheNLscanbe z
ditions. Thus the topological invariant of the NL states
detectedusingtheBloch-Zener-oscillationmethod,which
may be evaluated as
hasbeenexperimentallydemonstratedtoprobetheDirac
points in a honeycomb optical lattice [65, 66]. A basic
i (cid:88) (cid:90) π/a
idea lies in that the band touching points can be moni- γ(k )=− (cid:104)u (k)|∂ |u (k)(cid:105)dk , (27)
tored from the atomic fraction tunnelling to the excited (cid:107) π En<EF −π/a n kz n z
band in Bloch oscillations. We can use noninteracting
fermionic atoms or an incoherent distribution of bosonic wherethesumisoverthefilledBlocheigenstates|u (k)(cid:105)
n
atomswiththepopulationbeinghomogeneousinthemo- of Hamiltonian (21) with the Fermi energy E . Equa-
F
mentum space initially prepared in the lower band [42]. tion (27) indicates that the topological properties of the
AconstantforceF isappliedalongthez axisandpushes systemcanbeparameterizedbyk asthenontrivialZak
(cid:107)
the atoms moving along the k direction. After a Bloch phase for an effective gapped one-dimensional system.
z
8
UsingtheBlochHamiltonian(21)withgivenk , wever- 6(b)] and the sharp boundary between γ = 0 and γ = 1
(cid:107)
ifythatγ =1fork insidetheprojectedNLwhileγ =0 becomesrelativelysmooth,whichareduetothetrapping
(cid:107)
outside, as shown in Fig. 6(a). potential and the finite size effects.
For fermionic atoms in this optical lattice, one can di-
rectly probe γ(k ) by measuring the Bloch wave func-
(cid:107) (a) (b) γ
tion cnσ(k) from |un(k)(cid:105) = cn↑(k) |↑(cid:105)+cn↓(k) |↓(cid:105) with -1 -1
time-of-flightimaging[67]. Onefirstmapouttheatomic 3
momentum distribution ρ (k)=|c (k)|2 for the filled
band using the conventionnaσl time-of-nflσight imaging. One π/a) 0 π/a) 0 2
thenmaymeasurethephaseinformationofcnσ(k)byin- k(y k(y
troducingaπ/2rotationbetweenthetwospinstateswith 1
animpulsivepulselightbeforetheflightofatoms, which
√
induces the transition cn↑(k) →√ [cn↑(k) + cn↓(k)]/ 2 1-1 0 1 1-1 0 1 0
andcn↓(k)→[cn↑(k)−cn↓(k)]/ 2. Withthispulse, the kx(π/a) kx(π/a)
difference between |c↑(k)±c↓(k)|2/2 measured through (c) (d)
time-of-flight imaging gives the real part of the inter- 4 0.4
ference terms Re[c∗n↑(k)cn↓(k)]. The imaginary part 2 ky = 0 0.3 Tatw kox= skuyr=fa0ce modes
aImd[ciff∗ne↑r(ekn)tcnr↓o(tka)t]iocna.n bTehoebtmaeinaesdurbemyetnhte soafmtheewpayopwuiltah- nergy0 ensity0.2 MMooddee 12
tion and interference terms determines the Bloch wave E D0.1
-2
function up to an arbitrary overall phase c (k) →
nσ
c (k)eiχ(k),whereχ(k)ingeneraldependsonkinstead 0
nσ -4
ofthespinindex. Thearbitraryk-dependentphaseposes -1 0 1 1 16 32
k (π/a) z (units of a)
an obstacle to measure the topological invariant [67]. To x
overcomethisdifficulty,weuseagauge-invariantmethod
to calculate the Berry phase [68] FIG. 6: (Color online) (a) The Berry phase γ(kx,ky) calcu-
lated by using the Bloch Hamiltonian (21). (b) The simu-
1 N(cid:88)j−1 lated measurement of Berry phase γ(kx,ky) with the lattice
γ(k )= Arg[det(cid:104)u (k ,kz)|u (k ,kz )(cid:105)], size 32×32×32 and an additional harmonic trap with the
(cid:107) π n (cid:107) j n(cid:48) (cid:107) j+1
parameter ν = 0.001 (see the text) by using Eq. (28). The
j=0
(28) black solid line denotes the nodal loop in this case. (c) The
where k in the BZ is discretized into small N inter- energyspectrumwithrespecttokx forfixedky =0andopen
z j
vals with kz = −π/a + 2jπ/N a, the overlap phase boundary condition along the z axis, and the surface states
j j inside the gap. (d) The density distribution of two surface
Arg[•] = Imag{ln[•]}, and the determinant is that of
modes at k =k =0. Other parameters in (a-d) are t =1
a matrix formed by allowing n and n(cid:48) to run over filled x y ↑
as the energy unit, t =t =0.3, and m =2.0.
↓ so z
Bloch eigenstates.
To demonstrate that the method is feasible in a real-
istic experiment, we numerically simulate the proposed
detectionoftheBerryphaseγ(k )withfinitelatticesys-
(cid:107) B. Protected surface states
tem and an additional harmonic trap
Hˆ = 1m ω2(cid:88)d2aˆ† aˆ , (29) Accordingthethebulk-edgecorrespondence,thetopo-
trap 2 a i i,σ i,σ logical NL state with the Berry phase γ(k ) is related
(cid:107)
i,σ
to edges states at the end of the one-dimensional sys-
whereω isthetrapfrequencyandd isthedistancefrom tem with a fixed k [21, 68, 69]. Hence, for a fixed k
i (cid:107) (cid:107)
the center of the trap to the lattice site i. We can use and two surfaces (upper and lower surfaces) along the z
ν =m ω2a2/2t toparametrizetheinfluenceofthistrap- axis, two in-gap states appear at in the surface Brillouin
a ↑
ping potential. For a typical experiment with a ≈ 400 zone when γ(k ) (cid:54)= 0. This indicates that surface states
(cid:107)
nm and t ≈ 1 kHz, µ is on the order of 10−3 for 6Li or appear for all γ(k ) inside the area enclosed by the pro-
↑ (cid:107)
40Ka atoms in a trap with ω ≈2π×50 Hz. In numerical jected NL on the surface plane. With the f (k)σ term,
0 0
simulations,weperformthespinrotationandobtainthe the surface states acquire a dispersion proportional to
momentum distribution under different spin basis by di- f (k ,k ,0). In Fig. 6(c), we plot the energy spectrum
0 x y
agonalizing the real-space Hamiltonian on a finite lattice with respect to k for fixed k = 0 and open bound-
x y
and using a Fourier transformation [67]. The numeri- ary condition along the z axis (with 32 lattice sites),
cal result of γ(k ) for finite lattice 32×32×32 with a whichshowsthesurfacestatesinsidethegapdenotedby
(cid:107)
weak harmonic trap (ν = 0.001) and typical parameters red solid lines connecting the two band-touching points.
is shown in Fig. 6(b). Compared with the result shown The density distribution of the two corresponding sur-
inFig. 6(a),inthiscasethek regimewithγ =1slightly face modes at k = k = 0 is shown in Fig. 6(d). The
(cid:107) x y
shrink from that of the ideal NL [black solid line in Fig. surface states for other k are similar, and they merge
y
9
intothebulkbandswhenthesystembecomeatrivialin- dimensionalmomentumspace,onlyPT-invariantNLsare
sulator. For this system in phase II.a, the surface modes topologically protected with a Z classification. Moti-
2
perpendicular to the z direction withfixed k can be de- vated by this observation, we have proposed a realistic
(cid:107)
scribedbytheeffectiveHamiltonianH (k )=iv σ ∂ + experimental scheme to realize PT-invariant topological
z (cid:107) z 2 z
g (k )σ +g (k )σ uptothelinearorderink →−i∂ , NLstateswithcoldatomsinathree-dimensionaloptical
z (cid:107) 3 0 (cid:107) 0 z z
where g (k ) = m −α −α [cos(k a)+cos(k a)] and lattice, which have tunable loop-shaped Fermi lines with
z (cid:107) z + − x y
g (k )=−α −α [cos(k a)+cos(k a)]. two-folddegeneracyinthebulkspectrumandnon-trivial
0 (cid:107) − + x y
Intheopticallattice,thesurfacestatesmaybewashed surface states. The NL states are actually protected by
outbythesmoothharmonicpotentialandbecomeundis- the combined PT symmetry even in the absence of both
tinguishable from the bulk states. To circumvent this P and T symmetries, and are characterized by a quan-
problem, one can use a steep confining potential or cut tized Berry phase (a Z -type invariant). We have also
2
the atomic hopping along the z axis by locally tuning shown that (i) the characteristic NLs can be detected by
the effective Rabi frequency of the Raman lasers. Un- measuringtheatomictransferfractionsinaBloch-Zener
der this condition, the protected surface states can be oscillation; (ii) the topological charge can be measured
probed through Bragg spectroscopy [53, 70, 71]. One based on the time-of-flight imaging; and (iii) the surface
couldshineanothertwolaserbeamsatacertainangleto states can be probed through Bragg spectroscopy. The
induceaspecificallytunedRamantransitionfromanoc- experimental realization and detection of PT-symmetry
cupiedspinstatetoanunoccupiedhyperfinelevelandfo- protectedNLstatesincoldatomsystemswillberegarded
custhemnearthesurfaceofthethree-dimensionalatomic as an important advance in the field of quantum simu-
cloud [71]. When the momentum and energy conserva- lation, paving the way for exploring exotic PT-invariant
tion conditions are satisfied, the atomic transition rate topological physics.
is peaked and can be measured. By scanning the Ra-
man frequency difference, the surface energy-momentum
dispersion relation can be mapped out from these Bragg VI. ACKNOWLEDGMENTS
signals [70, 71].
We thank M. Gong for useful discussions. This work
was supported by the NSFC (Grant No. 11474153),
V. CONCLUSION the PCSIRT (Grant No. IRT1243), the SKPBR of
China (Grant No. 2013CB921804), the SRFYTSCNU
Insummary,wehavepresentedatopologicalclassifica- (Grant No. 15KJ16), and the RGC of Hong Kong
tion of NLs in systems with the PT symmetry. In three- (HKU173051/14P and HKU173055/15P).
[1] M. Z. Hasan and C. L. Kane, Colloquium: Topological with additional symmetries from Clifford algebras, Phys.
insulators, Rev. Mod. Phys. 82, 3045 (2010). Rev. B 88, 125129 (2013).
[2] X.-L.QiandS.C.Zhang,Topological insulators and su- [9] B.-J.YangandN.Nagaosa,Classificationofstablethree-
perconductors, Rev. Mod. Phys. 83, 1057 (2011). dimensional Dirac semimetals with nontrivial topology,
[3] G. E. Volovik, The Universe in a Helium Droplet Nat.Commun.5,4898(2014);B.-J.Yang,T.Morimoto,
(Clarendon, Oxford, 2003). andN.Nagaosa,Topologicalchargesofthree-dimensional
[4] A.P.Schnyder,S.Ryu,A.Furusaki,andA.W.W.Lud- Dirac semimetals with rotation symmetry, Phys. Rev. B
wig,Classificationoftopologicalinsulatorsandsupercon- 89, 165120 (2015).
ductors in three spatial dimensions, Phys. Rev. B 78, [10] C.-K. Chiu and A. P. Schnyder, Classification of
195125 (2008). reflection-symmetry-protected topological semimetals and
[5] S. Ryu, A. P. Schnyder, A. Furusaki, and A. W. Lud- nodal superconductors, Phys. Rev. B 90, 205136 (2014).
wig, Topological insulators and superconductors: tenfold [11] C.-K. Chiu, J. C. Y. Teo, A. P. Schnyder, and S. Ryu,
wayanddimensionalhierarchy,NewJ.Phys.12,065010 Classificationoftopologicalquantummatterwithsymme-
(2010). tries, arXiv:1505.03535.
[6] Y. X. Zhao and Z. D. Wang, Topological Classification [12] Y.X.ZhaoandZ.D.Wang,Novel Z topological metals
2
and Stability of Fermi Surfaces, Phys. Rev. Lett. 110, and semimetals, Phys. Rev. Lett. 116, 016401 (2016).
240404 (2013); Y. X. Zhao and Z. D. Wang, Topological [13] J. Dalibard, F. Gerbier, G. Juzeliu¯nas, and P. O¨hberg,
connection between the stability of Fermi surfaces and Colloquium: Artificialgaugepotentialsforneutralatoms,
topological insulators and superconductors, Phys. Rev. B Rev. Mod. Phys. 83, 1523 (2011).
89, 075111 (2014); Y. X. Zhao and Z. D. Wang, Disor- [14] N. Goldman, G. Juzeliu¯nas, P. O¨hberg, and I. B. Spiel-
deredWeylsemimetalsandtheirtopologicalfamily,Phys. man,Light-inducedgaugefieldsforultracoldatoms,Rep.
Rev. Lett. 114, 206602 (2015). Prog. Phys. 77, 126401 (2014).
[7] A. Kitaev, Periodic table for topological insulators and [15] V. Galitski and I. B. Spielman, Spin-orbit coupling in
superconductors, AIP Conf. Proc. 1134, 22 (2009). quantum gases, Nature (London) 494, 49 (2013); H.
[8] T. Morimoto and A. Furusaki, Topological classification Zhai, Degenerate quantum gases with spinCorbit cou-
10
pling: a review, Rep. Prog. Phys. 78, 026001 (2015). Nat. Phys. 11, 920 (2015).
[16] L. Lu, J. D. Joannopoul, and M. Soljaˇc´ıc, Topological [32] H.Weng,Y.Liang,Q.Xu,Y.Rui,Z.Fang,X.Dai,and
photonics, Nat. Photonics 8, 821 (2014). Y. Kawazoe, Topological node-line semimetal in three-
[17] Z. Wang, Y. Sun, X.-Q. Chen, C. Franchini, G. Xu, dimensionalgraphenenetworks,Phys.Rev.B92,045108
H.Weng,X.Dai,andZ.Fang,Diracsemimetalandtopo- (2015).
logical phase transitions in A Bi (A=Na, K, Rb), Phys. [33] K.Mullen,B.Uchoa,andD.T.Glatzhofer,LineofDirac
3
Rev. B 85, 195320 (2012); Z. Wang, H. Weng, Q. Wu, nodes in hyperhoneycomb lattices, Phys. Rev. Lett. 115,
X.Dai,andZ.Fang,Three-dimensionalDiracsemimetal 026403 (2015).
and quantum transport in Cd As , Phys. Rev. B 88, [34] Y. Kim, B. J. Wieder, C. L. Kane, and A. M. Rappe,
3 2
125427 (2013). Dirac line nodes in inversion-symmetric crystals, Phys.
[18] X. Wan, A. M. Turner, A. Vishwanath, and S. Y. Rev. Lett. 115, 036806 (2015).
Savrasov, Topological semimetal and Fermi-arc surface [35] R. Yu, H. Weng, Z. Fang, X. Dai, and X. Hu, Topolog-
states in the electronic structure of pyrochlore iridates, ical node-line semimetal and Dirac semimetal state in
Phys. Rev. B 83, 205101 (2011). antiperovskite Cu PdN, Phys. Rev. Lett. 115, 036807
3
[19] L.Balents,Viewpoint: Weyl electrons kiss,Physics4,36 (2015).
(2011). [36] G. Bian, T.-R. Chang, R. Sankar, S.-Y. Xu, H. Zheng,
[20] G. Xu, H. Weng, Z. Wang, X. Dai, and Z. Fang, Chern T. Neupert, C.-K. Chiu, S.-M. Huang, G. Chang, I. Be-
Semimetal and the Quantized Anomalous Hall Effect in lopolski, D. S.Sanchez, M. Neupane, N. Alidoust, C.
HgCr Se , Phys. Rev. Lett. 107, 186806 (2011). Liu, B. Wang, C.-C. Lee, H.-T. Jeng, A. Bansil, F.
2 4
[21] A. A. Burkov, M. D. Hook, and L. Balents, Topological Chou, H. Lin, and M. Zahid Hasan, Topological nodal-
nodal semimetals, Phys. Rev. B 84, 235126 (2011). linefermionsinthenon-centrosymmetricsuperconductor
[22] C. Fang, Y. Chen, H.-Y. Kee, and L. Fu, Topological compound PbTaSe , arXiv: 1505.03069.
2
nodal line semimetals with and without spin-orbital cou- [37] Y.-H. Chan, C.-K. Chiu, M. Y. Chou, and A. P. Schny-
pling, Phys. Rev. B 92, 081201(R) (2015). der, Topological semi-metals with line nodes and drum-
[23] S.-Y. Xu, I. Belopolski, N. Alidoust, M. Neupane, C. head surface states, arXiv: 1510.02759.
Zhang, R. Sankar, S.-M. Huang, C.-C. Lee, G. Chang, [38] M. Lewenstein, A. Sanpera, V. Ahufinger, B. Damski,
B.Wang,G.Bian,H.Zheng,D.S.Sanchez,F.Chou,H. A. S. De, and U. Sen, Ultracold atomic gases in optical
Lin,S.Jia,andM.Z.Hasan,DiscoveryofaWeylfermion lattices: mimickingcondensedmatterphysicsandbeyond,
semimetal and topological Fermi arcs, Science 349, 613 Adv. Phys. 56, 243 (2007).
(2015). [39] M.Atala,M.Aidelsburger,J.T.Barreiro,D.Abanin,T.
[24] B.Q.Lv,H.M.Weng,B.B.Fu,X.P.Wang,H.Miao,J. Kitagawa, E. Demler, and I. Bloch, Direct measurement
Ma,P.Richard,X.C.Huang,L.X.Zhao,G.F.Chen,Z. of the Zak phase in topological Bloch bands, Nat. Phys.
Fang, X. Dai, T. Qian, H. Ding Experimental discovery 9, 795 (2013).
ofWeylsemimetalTaAs,Phys.Rev.X5,031013(2015). [40] H. Miyake, G. A. Siviloglou, C. J. Kennedy, W. C.
[25] Z.K.Liu,J.Jiang,B.Zhou,Z.J.Wang,Y.Zhang,H.M. Burton, and W. Ketterle, Realizing the Harper Hamil-
Weng, D. Prabhakaran, S-K. Mo, H. Peng, P. Dudin, T. tonian with laser-assisted tunneling in optical lattices,
Kim,M.Hoesch,Z.Fang,X.Dai,Z.X.Shen,D.L.Feng, Phys. Rev. Lett. 111, 185302 (2013).
Z. Hussain, and Y. L. Chen, A stable three-dimensional [41] M. Aidelsburger, M. Atala, M. Lohse, J. T. Barreiro,
topologicalDiracsemimetalCd As ,Nat.Mater.13,677 B. Paredes, and I. Bloch, Realization of the Hofstadter
3 2
(2014). Hamiltonian with ultracold atoms in optical lattices,
[26] Z. K. Liu, B. Zhou, Z. J. Wang, H. M. Weng, D. Prab- Phys. Rev. Lett. 111, 185301 (2013).
hakaran, S. -K. Mo, Y. Zhang, Z. X. Shen, Z. Fang, X. [42] M. Aidelsburger, M. Lohse, C. Schweizer, M. Atala, J.
Dai, Z. Hussain, and Y. L. Chen, Discovery of a three- T. Barreiro, S. Nascimb`ene, N. R. Cooper, I. Bloch, and
dimensional topological Dirac semimetal, Na Bi, Science N.Goldman,Measuring the Chern number of Hofstadter
3
343, 864 (2014). bands with ultracold bosonic atoms, Nat. Phys. 11, 162
[27] M.Neupane,S.-Y.Xu,R.Sankar,N.Alidoust,G.Bian, (2015).
C. Liu, I. Belopolski, T.-R. Chang, H.-T. Jeng, H. Lin, [43] G. Jotzu, M. Messer, R. Desbuquois, M. Lebrat, T.
A. Bansil, F. Chou, and M. Z. Hasan, Observation of Uehlinger, D. Greif, and T. Esslinger, Experimental re-
athree-dimensional topological Diracsemimetal phasein alisation of the topological Haldane model with ultracold
high-mobility Cd As , Nat. Commun. 5, 3786 (2014). fermions, Nature (London) 515, 237 (2014).
3 2
[28] S.-Y. Xu, C. Liu, S. K. Kushwaha, R. Sankar, J. W. [44] L. Duca, T. Li, M. Reitter, I. Bloch, M. Schleier-Smith,
Krizan,I.Belopolski,M.Neupane,G.Bian,N.Alidoust, and U. Schneider, An Aharonov-Bohm interferometer
T.-R. Chang, H.-T. Jeng, C.-Y Huang, W.-F. Tsai, H. for determining Bloch band topology, Science 347, 288
Lin, P. P. Shibayev, F.-C. Chou, R. J. Cava, and M. (2015).
Zahid Hasan, Observation of Fermi arc surface states in [45] M.Mancini,G.Pagano,G.Cappellini,L.Livi,M.Rider,
a topological metal, Science 347, 294 (2015). J. Catani, C. Sias, P. Zoller, M. Inguscio, M. Dalmonte,
[29] L. Lu, L. Fu, J. D. Joannopoulos, and M. Soljaˇc´ıc, Weyl andL.Fallani,Observationofchiraledgestateswithneu-
points and line nodes in gyroid photonic crystals, Nat. tralfermionsinsyntheticHallribbons,Science349,1510
Photonics 7, 294 (2013). (2015).
[30] L. Lu, Z. Wang, D. Ye, L. Ran, L. Fu, J. D. Joannopou- [46] B. K. Stuhl, H.-I. Lu, L. M. Aycock, D. Genkina, and
los, and M. Soljaˇc´ıc, Experimental observation of Weyl I. B. Spielman, Visualizing edge states with an atomic
points, Science 349, 622 (2015). Bose gas in the quantum Hall regime, Science 349, 1514
[31] M. Xiao, W.-J. Chen, W.-Y. He and C. T. Chan, Syn- (2015).
thetic gauge flux and Weyl points in acoustic systems, [47] D. R. Hofstadter, Energy levels and wave functions of