Table Of ContentEmergent Quantum Coherence from Instabilities of a Perturbed Fractionalized Spin
Liquid
S. Acharya1,∗ M. S. Laad2,† and A. Taraphder1,3‡
1Department of Physics, Indian Institute of Technology, Kharagpur, Kharagpur 721302, India.
2Institute of Mathematical Sciences, Taramani, Chennai 600113, India and
3Centre for Theoretical Studies, Indian Institute of Technology Kharagpur, Kharagpur 721302, India.
Effects of perturbations on topologically ordered (TO) states with fractionalized excitations is
very intriguing. While small perturbations are not expected to affect a robust TO, sizable pertur-
bationsmustleadtoonsetofnovelroutestoorder(s). Observabilityofsuchemergentordersinreal
condensedmattersystemscontinuestobeanopenissueofparamountinterest. Here,wedetailhow
a featureless Kitaev spin liquid with rigorous TO gives way to partial topological ordered state(s)
coexistingwithawidestaircaseofnovel‘excitonic’ordersoffractionalizedentitiesinaZeemanfield.
5
1 Weprovidequasi-exactandnovelrealizationforarangeofstrangefeatures,fromnodalspin-metals,
0 fractionalizationinselective-Mottstates,toLuttingersurfacesandhiddencoherence,allcharacter-
2 istics of the influential resonating- valence-bond as well as fractionalized-Fermi-Liquid views in the
context of strange metals. As a novel application, we show how Josephson-junction array may
l
u exhibit fractional Josephson oscillations upon flux tuning as a particularly novel manifestation of
J emergent quantum coherence.
8
PACSnumbers: 74.70.-b,74.25.Ha,76.60.-k,74.20.Rp
]
l
e
Topological excitations have fascinated physicists for Hence, they offer an attractive avenue to investigate
-
r many decades. Remarkably, recent advances in con- the above issues in a realistic context. Further, models
t
s densed matter have reached a stage where the ability thatarevariationsoftheKMandcompassmodels[5]are
.
t to observe and even manipulate such exotic excitations believed to apply even to certain transition-metal oxides
a
m seemtobewithintherealmofpossibility. Cold-atomset- (TMO) [6], though, in this context, additional perturba-
ups, topological insulators and appropriately engineered tion(s) must be included [7, 8]. In this context, fascinat-
-
d Josephson junction arrays (JJA) are poised to offer re- ingissuesrelatetowhetherafractionalizedorbitalliquid
n newed insight into topological excitations arising from can emerge, and whether analogues of orbital-Kondo de-
o
ground states with topological order (TO). In many in- stroying QCPs exist in TMOs.
c
stances, however, ensuring their stability is a potential
[ A Simple Perturbed Kitaev Model
hurdle. Another, more exacting hurdle is to unearth
3
specific examples of systems capable of exhibiting stable ConsidertheKitaevmodelinanexternalZeemanmag-
v
(symmetry protected) TO. netic field:
3
8
3 ThebirthoftheexactlysolubleKitaevmodel(KM)[1]
(cid:88) (cid:88) (cid:88)
2 led to a spurt of proposals to ‘simulate’ its TO ground H =− Jα SiαSjα−hz Siz (1)
0 state in various contexts [2, 3]. In contrast, a micro- α <i,j> i
.
1 scopic understanding of effects of various perturbations
UsingtheD =2Jordan-Wignermapping[9],H(seesup-
0 is lacking: while mostly studied numerically [4], a co-
5 herent picture of their effects in terms of changes in plementary informations[? ]) can be brought to the fol-
1 lowing instructive form (this also bares the exact emer-
the spectrum of topological (elementary) excitations of
: gent low-dimensional gauge symmetries, as pointed out
v the TO state has remained elusive. Given that the KM
Xi hosts a fractionalized spin liquid phase, novel, unantici- by Fradkin et al. [10]): H =HK +Hz, where
patedorderedstates(partialorcompletedeconfinement-
r
a confinement transition(s)), depending upon specific na- H =(cid:88)[(cid:15) c†c + i∆q(c†c† +h.c)]
ture of perturbations, are expected theoretically. Classi- K q q q 2 q −q
fyingsuchnovelordersaidsthequestfortheirobservabil- q
ity in natural or engineered condensed matter systems, + Jz (cid:88)(2n −1)(2c†c −1) (2)
apart from the attractive motivation of application(s) to 4 i,α i i
i
fault-tolerant quantum computing. Here, we are moti-
vated by recent proposals in perturbed Kitaev models and H = 2h (cid:80) (c†α + h.c). This is just a spinless
z z i i i
constructed by suitably engineered JJA [3]. Moreover, Falicov-Kimball model with a local hybridization h in
z
for the realistic range of physical parameters, such JJA d=2, but with an additional ‘spin’-triplet pairing term,
(see Fig.(1) of supplementary informations) are always whence topologically ordered ground state of H now
K
perturbed by ‘magnetic fields’ along x and/or z bonds. appearsasaspin-tripletBCSsuperfluidofJWfermions.
2
Here,wedefineα(cid:48) =(2n −1)=iB B =±1,whichis
i iα i1 i2
now a dynamic gauge field variable on the center of each
zz-links of the honeycomb Kitaev model (when n =
i,α
0,1 for all i. This transforms the original spin model on
a honeycomb lattice to a two- “orbital” spinless model
on an effective square lattice. While [n ,H] = 0 for
iα
all i (reminiscent of the spinless FK model) ensures the
exactsolvabilityoftheKMwhenh =0,afiniteZeeman
z (a) (b) (c)
field immediately spoils the beautiful integrability of H.
Interestingly, however, finite h simply translates to a
z
local hybridization between the c,α JW fermions. These
observations are crucial for an ‘almost exact’ analysis of
the KM in a Zeeman field as detailed below.
Alternatively, we motivate the instabilities of the TO
state of H as follows. In Kitaev’s language, a par-
K
tial local Z symmetry associated with [baba,H ] = 0
2 i j K
for all (ij) and a = x,y still holds when h (cid:54)= 0 (only
z
[bzbz,H ](cid:54)=0alongzz-links,sinceH =i(cid:80) bzc mixes
i j K z i i i
bz,coneachsite). Thisdirectlyimpliesadimensionalre-
duction and associated lower-D(= d < D) (here, d = 1)
gauge-likesymmetries(GLS)naturallyemerge. Theonly
order that survives the action of these GLS is a direc-
tional spin-nematic order [10], whose order parameter,
(cid:104)n(cid:105) = (cid:104)SxSx −SySy (cid:105) is the only one that remains
i i+ex i i+ey
invariant under the emergent d = 1 GLS: thus, the TO
of H is only partially “melted” by a Zeeman field. The
K
ultra-short-ranged spin-liquid phase known for h = 0
z (d)
thus immediately becomes unstable to a deconfined crit-
ical liquid inherently unstable to incipient spin-nematic
FIG. 1. Upper panel: Magnetization and static susceptibil-
order (on xx,yy zig-zag links) co-existing with a (with fi- ity as a function of externally applied field for (a) J = 0,
z
nite flux on zz-links) field-induced zz-spin order, for any (b) J =0.2J , and (c) J =0.4J . Lower panel: Plateaus and
z x z x
h (cid:54)= 0 at T = 0! It also turns out that the zz-spin jumps in the anisotropic limit of the KM, with J = 0.6J
z z x
correlation function, χzz(i−j) (cid:39) (i−j)−4 (not shown, (d). IntheinsettheηisshownasafunctionofJz (Jx =0.5).
The axes labels for figures (a), (b) and (c) are same as (d).
proof essentially identical to Tikhonov et al. [11]) using
a simple bubble composed of the G (i−j),G (i−j)
bzbz cc
propagators for H in Majorana language. Remarkably,
namicalspinfluctuationspectruminaD =2model,and
field-induced magnetization now appears as an excitonic
(iii) the fact that spin correlations on all length scales
condensate of JW-fermions.
alongxx,yy bondshavebeenexactlysubsumedintoone-
In JW-fermion language, for special values of h , the
z fermion hopping and p-wave BCS pairing terms obviates
now dynamical Z flux can, by itself, self-organize into
2 theusualdifficultyofignoringlong-rangespatialcorrela-
stripe-ordered patterns, in analogy with crystal states
tions, wherein any mean-field theory, including DMFT,
found in the FK model [12] (here, n =(1+α )/2 play
iα i is usually critiqued in D = 2. Moreover, H also has a
the role of immobile fermions in the FK-like model for z
local excitonic form and the zz-couplings appear as lo-
h =0). Inthepresentcase,suchcrystalsoccurringwith
z cal interaction. Thus, these novelties, again specific for
alargerangeofperiodicstructureshaveanoveloriginas
the KM, make the DMFT almost exact and allows for
condensation of topological kink dipoles (a condensation
exhaustive analysis.
of (cid:104)iB B (cid:105) on each zz-bond). On the other hand, field-
i1 i2
induced metamagnetic transitions, now associated with The impurity model of DMFT is solved by multi-
‘excitonicsolid’states,arealsogenericallyexpectedfrom orbital iterated perturbation theory as an impurity
a Falicov-Kimball model with hybridization. solver: though not “exact”, its proven quantitative
We analyze H within dynamical mean-field theory accuracy vis-a-vis exact QMC results for the spinful
(DMFT). Our choice is motivated by features special Anderson lattice model [15] ensures its efficacy in this
for H : (i) when h = 0, the KM possesses ultra- case. We focus on the c,α-fermion spectral functions
K z
short-range spin correlations [13] rigorously uncorre- and field-induced magnetization, now simply equal to
latedbeyondnearestneighbors, (ii)aDMFT-relatedap- a local excitonic average m(h ) = (cid:104)(c†α + h.c)(cid:105). To
z i i
proach [14] thus gives the first instance of an exact dy- facilitate discussion, we show the evolution of m(h )
z
3
tary informations), and go hand-in-hand with jumps in
(cid:104)n (h (cid:105) at the same h values. These topological crystal
α z z
phases occur (Fig. 2) for a range of odd-denominator
valued magnetizations, as well as at certain even de-
nominators (relative to saturation, at m(h )/m =
z sat
1/16,1/12,1/11,1/9,1/8,1/7,2/13,1/6,2/11,1/5,2/9,
1/4,2/7,1/3,3/8,3/7,2/5,4/7,5/8,3/4,...). Odddenom-
inator plateaus in the off-diagonal conductivity are well-
known in the fractional quantum Hall effect (FQHE),
which is also the only other example of a real system
rigorously exhibiting topological order. In spin systems,
such exotica have also been found and widely discussed
in frustrated Shastry-Sutherland models [18, 19], where
they emerge due to competition between frustration and
field-induced magnetism (even-denominator plateaus
are also visible there). So our plateaus are not related
to FQHE-like physics, even though (in contrast to
earlier work) they arise in a perturbed (field-induced)
partially TO phase. Nevertheless, in Fig. 2, we exhibit
m(h ) and χ(h ) = (d/dh )m(h ) together to illustrate
z z z z
FIG. 2. Plateaus and jumps in the ‘magnetization’ in the “analogies” with FQHE. We interpret the oscillations
isotropic limit of the KM. The corresponding susceptibilities
in χ(h ) as “de Haas van-Alphen (dHvA)” quantum
are also shown. z
oscillations of topological JW fermions in a partially
magnetized spin liquid phase. Such exotica may show
up [20] in magnetic torque studies. This reflects “hid-
vs. h in Fig. 1. When J << J = J , m(h ) rises
z z x y z den” coherence in a spin liquid: interestingly, hidden
monotonically with h , as expected of a field-polarized
z off-diagonal long-range order (ODLRO) was conjectured
paramagneticphase. However,itsunderlyingtopological
by early on by Anderson precisely in this context[21].
origin is underscored by observing that m(h ) (cid:39) hη
z z Here, h (cid:54)=0 partially polarizes the spinons reflecting in
with 1.0 > η > 0.85 for 0 ≤ J ≤ 0.25(= J /4): for z
z x magnetization plateaus, and these ordered phases “ ride
larger J , η(J ) reverses its slope (Fig. 1 lower panel
z z on top” of a partially topological ordered state. This
(inset)), increasing monotonically toward 1.0. The
component has in-built coherence, thanks to the p-wave
special character of J /J = 1/4 is related to its being
z x BCS term in H , manifesting as coherent “quantum
precisely twice the (mean-field) excitation energy of K
oscillations” we unearth. Thus, in stark contrast to
the three flat bands that arise in a mean-field theory
conventional paramagnets, the TO phase of the Kitaev
(related to the original resonating valence bond (RVB)
modelundergoesanintricatesequenceofpartialordering
description in the cuprate context [16]) distinct to
“solid” instabilities, co-existing with the remnant of the
ours [17]. It marks the point where the α-fermion level
TOfractionalizedspinliquidinanexternalZeemanfield
and concomitant excitonic tendency begins to play a
before reaching full saturation.
role for J /J ≥ 1/4. That m(h ) starts developing
z x z
The DMFT spectral functions reveal deeper insights
plateaus (or incompressible JW-excitonic solid phases)
(See supplementary informations). Clear “orbital” se-
as a function of h only for J /J > 1/4 also supports
z z x
lectivity is seen in c and α-fermion spectral functions,
this view.
and this is a direct fall-out of the in-built differentiation
Remarkably, for J /J > 1/4, m(h ) as a function between these states in the JW-fermionized H itself.
z x z K
of h shows an extensive sequence of magnetization While hybridization and local Hubbard correlations con-
z
plateaus, now associated with incompressible excitonic spiretoproduceaJW“Kondo”insulatorforα-fermions,
solid phases, separated by excitonic “fluid” phases. theyproducelarge-scalespectralchangesinthec-fermion
These are intimately linked to the kink-dipole “con- sector. Focussingonthevariationsinρ (ω),ρ (ω)across
c α
densates” (crystals of the “flux”, (cid:104)iB B (cid:105): see the representative samples where magnetization steps ap-
i1 i2
n (h ) vs h in Suppl. Info). These are thus first- pear, it is clear that large-scale reshuffling of dynamical
α z z
order quantum phase transitions between sequences of spectral weight across energy scales of order the band-
“topological crystal” states, each sandwiched between width (equal to 2J here) accompany emergence of each
x
topological Bose-Einstein condensates of excitonic kink plateau
dipoles. These are nothing but zeeman-field induced Thus, the plateau structure has its origin in the ubiq-
metamagnetic transitions with accompanying hysteresis uitous competition between Mott localization (Hubbard
at each magnetization jump (see Fig.(3b) of supplemen- term) and itinerance (hybridization and hopping) in H,
4
poles, of G (k,ω): this is evident from the structure of
cc
G (k,ω)itself, whichfeaturesananomalous“BCS”-like
cc
self-energy structure given by Σan(k,ω) = ω+(cid:15)−∆k−2kh2/ω,
so that the poles of Σ appear as zeros of G (k,ω).
an cc
Thus, this is a generalized Luttinger surface, rather
than a conventional Fermi surface. This is analogous to
what happens in correlation-driven selective-Mott tran-
sitions in f-electron systems in the context of the FL∗
phase, or in the YRZ ansatz [23]. In our case, clear
changesintopologyoftheLuttingersurfaceoccurateach
plateau, and are thus intimately linked to correlation-
driven Lifshitz-like transitions accompanying each field-
(a)
induced plateau. Moreover, given that the c-fermion
spectral function is that of p-wave nodal Bogoliubov
quasiparticles,thisthisstateisap-wavenodalspinsemi-
metal. These novelties are signatures of the underlying
partial TO surviving in the partially magnetized phases.
AZeemanfieldresultsinthetransferofdynamicalspec-
tralweightfromthefractionalizedspinliquidcomponent
totheexcitonicsolid,butitisindeedremarkable,thanks
totheundisturbedemergentd=1-GLS,thatitpartially
preserves the original TO.
Finally, the large-J limit of H is known to be noth-
z K
(b) ingbuttheKitaevToricCode(TC)model[1]. Here,this
implies a Mott-like insulating phase of the JW fermions.
The no double-occupancy constraint of a c,α fermion
on each site i is naturally implemented by a Gutzwiller
projector, P = Π (1 − n n ) acting on the p-wave
G i ic iα
BCS ground state. Explicitly, |Ψ(cid:105) = P Π (u +
TC G k k
v c†c† )|vac(cid:105) where |vac(cid:105) is the c-fermion vacuum. But
k k −k
this is precisely the Gutzwiller-projected (p-wave) BCS
state, longknownasoneoftheoptimalstatesdescribing
a (d-wave) RVB spin liquid state in the context of the
famed RVB description of the cuprates [16]. Thus, the
ground state of the TC model is just the p-wave RVB
(c) (d) state! Thissuggeststheenticingpossibilityofinvestigat-
ing TC physics and its non-Abelian excitations in future
FIG. 3. The renormalized dispersions ((a) and (b)), corre- in terms hitherto developed for quantum spin liquids in
sponding Luttinger surfaces ((c) and (d)) for dispersive JW quantum Heisenberg models [24].
Fermions at plateaus 1/3 and 2/3 respectively. The bound-
ary of yellow and blue regions corresponds to the contour of We now detail a specific quasi-realistic scenario where
ω(k ,k )=0andhasbeenmarkedsimilarlyinsidethefigures
x y our findings could aid finding of novel orders in a real-
in the lower panel. k and k are in units of π/a, where a in
x y world set-up:
lattice constant.
1. Kitaev Josephson Junction Arrays As men-
tioned before (see also supplementary informations), a
novel implementation of the perturbed KM using JJA
andthefieldsimplytunesthebalancebetweentheseten-
has been proposed [3]. Our results can now be used to
dencies, now by favoring the tendency to exciton forma-
unearth novelties in such engineered systems.
tion. This is reminiscent of the state of affairs character-
izing the FL∗ view proposed in the context of f-electron
We now see that varying the flux through a qubit
quantum criticality [22].
leads to a generation of a whole host of ordered “JW-
More unanticipated features show up in DMFT “band excitoniccooper-pair”crystals,withasuccessionofperi-
structures”. In Fig. S1, we exhibit the c-fermion dis- odssetbythecompetitionbetweenh (flux)andthema-
z
persions and “Fermi” surfaces obtained as usual from terial parameters of each JJ (these enter the Kitaev cou-
the DMFT propagators. Remarkably, we find that plings [3] in J ,J = J,J ). These are now interpreted
x y z
the “Fermi surface” corresponds to zeros, rather than as kink-dipole crystals, and arise as an explicit mani-
5
festation of the fact that strong geometric frustration- tional nature. Suitably engineered JJAs as proposed [3]
induced topological liquid state of the KM undergoes an could unearth the structures we propose by flux tun-
intricate sequence of “partially ordered” solid phases be- ing, with novel applications to novel quantum interfer-
fore achieving a full solid at saturation. In view of the ence phenomena, plasmonics and TQC. Finally, if ex-
co-existing and remnant topological critical spin-liquid amples of such perturbed KMs to real transition-metal
arising from emergent “d = 1” GLS in this state, one oxides could be found, manifestations of the underlying
may also refer to each of these states as critical topo- TO emerging as lower-d(=1) gauge-like symmetries, to-
logical supersolids: these turn out to be unstable to ne- getherwiththeirintimatelinktounconventionalnematic
matic order ∀J (cid:54)=J (see supplementary informations). order, could reveal themselves under appropriate exter-
x y
A direct manifestation of this insight is that, in anal- nal perturbations. This remains an enticing perspective
ogy with excitonic Josephson effects in electron-hole bi- for future work.
layers [25], one expects the Josephson critical current, Acknowledgement
J , to depend on the excitonic correlations. Explicitly,
c MSL wishes to thank R. Moessner for a helpful dis-
J (cid:39) h2(cid:104)c†α ;c†α (cid:105) (cid:39) h2(cid:104)c†α (cid:105)(cid:104)c†α (cid:105). As a result of the
c z i i j j z i i j j cussion and MPIPKS, Dresden support when this work
p-wave “BCS” term in the fermionized H , there will
K was conceived. He also thanks G. Baskaran for helpful
also be a direct “fermionic” current, now caused by the
discussions. ATandSAwouldliketoacknowledgeIMSc,
p-wave JW-fermions. Thus, remarkably, we expect the
wherethisworkstarted,forhospitality. SAwouldliketo
critical “Josephson” current to exhibit fractional oscilla-
thank UGC (India) for a research fellowship.
tionsasthefluxisslowlyrampedup. FractionalShapiro
steps are also directly predicted as a consequence of the
underlyingsuperstructureofkink-dipoles. Inducingtran-
sitions between successive steps with a pulsed electro-
magnetic field can result in emission of coherent electro- ∗ [email protected]
magneticradiationwhoseenergy(cid:126)Ωcanbetunedinthis † [email protected]
‡ [email protected]
setup. With realistic JJ parameters, Ω can be tuned to
[1] A. Kitaev, Annals of Physics 303, 2 (2003), ISSN 0003-
lie in the GHz or sub-GHz region: thus, this setup can
4916.
achieve flux-tuned control of EM radiation. Excitations
[2] A. Buhler et al., Nature Communications 5 (2014).
of such ordered states as those found here will be slid- [3] J. Q. You et al., Phys. Rev. B 81, 014505 (2010).
ingcollectivemodesoftheexcitonicsolidakintophason [4] J. Reuther, R. Thomale, and S. Trebst, Phys. Rev. B 84,
modes in the CDW context, and these are precisely the 100406 (2011).
“Josephson plasmons”, suggesting novel applications to [5] Zohar Nussinov and Jeroen van den Brink,
arXiv:1303.5922v1 (2013).
“topologicalplasmonics”. Oursisthusafirstproposalto
[6] G.JackeliandG.Khaliullin,Phys.Rev.Lett.102,017205
exploitanalmostexactlysolvablemodeloffractionalized
(2009).
JW-fermions in D =2 to detect topological analogues of
[7] Subhro Bhattacharjee et al., New Journal of Physics 14
famed quantum interference effects. Other subtle man- (2012).
ifestations of ODLRO, such as topological analogues of [8] Y. Yamaji et al., Phys. Rev. Lett. 113, 107201 (2014).
Little-Parks experiment, etc, could also be imagined in [9] H.-D. Chen and J. Hu, Phys. Rev. B 76, 193101 (2007).
theJJAcontext. Further,totheextentthatitisdifficult [10] Z. Nussinov and E. Fradkin, Phys. Rev. B 71, 195120
(2005).
to generate single JW fermions by splitting up JW exci-
[11] K. S. Tikhonov et al., Phys. Rev. Lett. 106, 067203
tons, the states we find have a much more efficient topo-
(2011).
logical protection (the topological liquid phases are in-
[12] T. Kennedy, Reviews in Mathematical Physics 06, 901
herently symmetry-protected topological (SPT) phases), (1994).
these findings can be exploited for possible topological [13] G. Baskaran, S. Mandal, and R. Shankar, Phys. Rev.
quantum computation (TQC) engineering, along lines Lett. 98, 247201 (2007).
proposed by You et al. [3]. [14] J. Knolle et al., Phys. Rev. Lett. 112, 207203 (2014).
[15] N. Dasari et al., arXiv:1504.04097v1 (2015): excellent
Conclusion
agreementbetweenMO-IPTandCTQMCisseentherefor
To conclude, we have presented strong evidence link-
both one- and multi-orbital Hubbard models, especially
ing a large sequence of “excitonic ordered solid” phases with finite crystal-field terms.
of the Zeeman-field driven KM to novel instabilities of a [16] P. W. Anderson, The theory of superconductivity in
rigorously known TO phase. These could be touted as the high-tc cuprate superconductors, (Princeton Univer-
topological analogues of the famous ”Barkhausen steps” sity Press, 1997).
[17] F. J. Burnell and C. Nayak, Phys. Rev. B 84, 125125
encountered in the magnetization process of a conven-
(2011).
tional field-driven magnet. Here, however, they corre-
[18] Introduction to frustrated magnetism: Models, experi-
spond to partially magnetically ordered complex unit-
ment, theory, eds. Lacroix. C, Mendels. P, and Mila, F.,
cell patterns emerging as partial ordered states of a frac- Springer Series in Solid State Sciences (2011).
tionalized spin liquid, betraying their highly unconven- [19] O. A. Starykh, H. Katsura, and L. Balents, Phys. Rev.
6
B 82, 014421 (2010).
[20] D. Watanabe et al., Nature Communications 3 (2012).
[21] P. W. Anderson et al., Journal of Physics: Condensed
Matter 16 (2004).
[22] T. Senthil et al., Phys. Rev. Lett. 90, 216403 (2003).
[23] Kai-Yu Yang et al., Europhysics Letters 86 (2009).
[24] S. Yunoki and S. Sorella, Phys. Rev. Lett. 92, 157003
(2004).
[25] Y. N. Joglekar et al., Phys. Rev. B 72, 205313 (2005).
1
Supplementary Informations (cid:90) dν (cid:90) d2k iν 1
Π(0)(iω)=−
2π (2π)2(iν)2−(vk)2iν−iω+J /2
z
D =2 Duality Transformations (S3)
Kitaev’sfamedsolutionreliesonmappingspin-1/2sto behaves as Π(0)(iω)(cid:39)(2πv2)−1[(iω+Jz/2)ln((Jz/2−
bilinears of four (bx,by,bz,c) Majorana fermions along iω)/W)−W], where W (cid:39) Jx is a cut-off of order the
the three (xx,yy,zz) bonds (Fig. S1). We start with spinon bandwidth. The log-singularity is now cut-off by
an alternative but equivalent approach due to Chen and the factor (iω+Jz/2), which itself arises from the linear
Hu [S2] and Nussinov [S3] that exploits a duality map- DOSofthenon-interactingc-fermions. Thus,thereisno
ping of the spins via a D = 2 Jordan-Wigner (JW) infra-red singularity in this case, and, as expected from
fermionization of H. Remarkably, since this affords an this line of argument, a RPA calculation using the bare
exact mapping between topological and classical orders propagators indeed captures the essential features of the
in D =2 spin systems, studying TO(s), their excitations exact solution to a high accuracy [S5].
andinstabilitiesistransmutedintothebetter-knownlan- In the TO phase (hz = 0), the c-fermions thus expe-
guage of well-known models hosting classical order(s) rience a strongly fluctuating (Z2) potential, which ‘sud-
characterized by symmetry-breaking. Thus, denly’ switches between 0,Jz during every hop of the
c-fermions. If the pairing term were absent, this would
S+ =2[Π S ][Π Sz ]c† (S1) constitute a complete reconstruction of the entire Fermi
ij j(cid:48)<j,i i(cid:48)j(cid:48)z i(cid:48)<i i(cid:48)j ij
seaduetotheAnderson-Nozieres-de-Dominicis“orthogo-
and nalitycatastrophe”(OC).Alongwiththerigorousultra-
short spin correlations in H , this is what makes the
K
Sizj =(2c†ijcij −1) (S2) ‘local’ approximation exact here. But with hz = 0, (cid:15)q
has a Dirac-like form which cuts off the infra-red singu-
anddefiningMajoranafermionsonthetwo(“black”and larbehaviorandtheOC,sincetheDOSvanisheslinearly
“white”) sublattice sites of the honeycomb or brick-wall atω =0(=(cid:15)F)asshownabove;thiswillnotbetrueany-
lattice as Aw = (c−c†)w/i,Bw = (c+c†)w and Ab = more when hz is finite and Fermi pockets, corresponding
(c+c†) ,B = (c−c†) /i, followed by the introduction to finite DOS at ω = 0, are expected to appear in the
b b b
of fermions c=(A +iA )/2,c† =(A −iA )/2. Using c-fermion spectrum. However, this is still not enough, as
w b w b
these transformations, we get the fermionized form of the local hybridization also produces finite recoil of the
H =H +H usedinthemaintext. InH asthusdefined α-fermion: this must cut-off the X-ray-edge singularities
K z
in the main text, αi(cid:48) =iBi1Bi2 is a bilinear of Majorana in spectral functions. But since any finite hz also leads
fermions. Since [α(cid:48),H] = 0 for all i, it takes eigenvalues to an excitonic condensate at T = 0, this provides an
i
±1. Thus, we define α(cid:48) = (2n −1), where n = 0,1 additional route to removal of infra-red singularities by
i i,α i,α
and [n ,H]=0 for all i as well, preserving the local Z a (partial in our case) coherence-restoring instability to
i,α 2
symmetry . Using this representation, the zeeman term field-induced order. This is precisely what we find (see
maps onto the excitonic term in JW-fermion language. main text).
Non-singular Spin Fluctuation Spectrum Non-Trivial Topological Order with hz (cid:54)=0
Here, we point out how the “Dirac” spectrum for c- Here, we show how the emergent D = 1 GLS men-
fermions cuts-off the infra-red (X-ray edge) singularities tioned in the text are linked to a novel “hidden” topo-
in the Kitaev model with hz = 0. In a normal Fermi logical order even when hz (cid:54)=0.
liquid,theinfra-redsingularnatureofthelocalizedprop- Noticing that a Zeeman field along zz-bonds implies
agator (as well as of the “excitonic” propagator) can be [ibaibaj,HK +Hz] = 0 for all (ij)||xx,yy implies persis-
tracedbacktotheappearanceoflogarithmicdivergences tence of partial local Z2 gauge symmetry and emergent
in the particle-hole polarization function. This necessi- low (d = 1) gauge-like symmetries, we show that these
tates summing of infinite number of Feynman diagrams hide a novel kind of topological quantum critical point
for any arbitrary coupling strength [S4], or “summing (TQCP), separating two topological phases. Since topo-
the logs”, leading to a power-law singularity in dynam- logical order can be described by conventional (Landau-
ical response functions. In the Kitaev model, however, like) order when expressed in terms of dual (Jordan-
two features mitigate against this: first, the immobile- Wigner) variables, we take this approach. Remarkably,
fermion level lies at −Jz/2 rather than at ω = 0, and, it will turn out that the special point Jx =Jy(=J) sep-
second, the “Dirac”-like DOS for the c-fermions (thanks arates two Ising-like nematic phases at T =0.
tothep-waveBCSterminHK)cutsofftheinfra-redsin- In the single chain case, HK is trivially diagonal-
gularities. Explicitly, writing G = 1 and ized to give two quasiparticle bands with dispersions
αα|(0)(iω) iω+Jz/2 (cid:113)
Gcc(k,iω)= (iω)2i−ωv2k2 Ek± = ± Jx2+Jy2+2JxJycos(k) (with one band fully
near the ‘Dirac” points, we find that the one-loop po- occupied and the other empty). The second derivative
larization function, of the ground state energy diverges at J = J , where
x y
2
ImplementationofthePerturbedKitaevModel
in Engineered Josephson Junction Arrays
Here, we briefly recapitulate the proposal for simulat-
ing the perturbed Kitaev model we have studied by us-
ingaspecificengineeredJosephsonjunctionarrayonthe
honeycomb lattice.
Our results are directly applicable to the engineered
JJA of You et al. [S1] as long as we identify the Kitaev
spins as pseudospin-1/2s associated with a pair charge
qubit: defining|↑(cid:105) ,|↓(cid:105) astwochargestateswithcooper
i i
pair number n = 1,0, the S = 1/2 representation per-
i
mits a rigorous mapping, n =(1−Sz)/2, cosφ =Sx/2
i i i i
and sinφ = −Sy/2 with φ being the pair phase at i,
i i i
conjugate to the number as implied by n =−id/dφ . If
i i
each charge qubit is placed at an “optimal” point where
n = 1/2, and J (= J ) < J is considered, one ends
g x y z
up with the anisotropic Kitaev model in an “external
field” h(cid:80) Sx (not hSz). However, this is of no con-
i i i
sequence, as we obtain our model simply by relabelling
Sx → Sz,Sz → −Sx. Further, we then have J < J (=
i i i i z x
J), which is precisely the limit we are concerned with.
FIG. S1. (Color online) JJA on a honeycomb lattice( [S1]). Remarkably, since h = E (Φ ) = 2E cos(πΦ /Φ ) with
Ji i J i 0
Φ = π(cid:126)/e the flux quantum in the JJA, changing the
0
Josephson coupling energy offers a subtle knob to tune
the excitation gap vanishes. This quantum transition at the “Zeeman” field in the KM, and, in fact, in the JJA
T = 0 does not involve any symmetry change, rather, it context, a real external magnetic field or adding non-
is a change in topological order. This is made explicit by magnetic impurities can change E (Φ) [S8].
Ji
the transformation [S6, S7]
Sx =τx τx,Sy =Π2Nτy (S4)
i i−1 i i l=i l
which recasts H as the D = 1 quantum Ising model
K
(QIM) on the dual lattice. When J > J the dual spin-
x y
correlation function shows long-range order:
Lim (cid:104)τxτx(cid:105)(cid:39)[1−(J /J )2]1/4 (S5)
i→∞ 0 2i y x
In the original spin-representation, this is a string order
with “order” parameter ∆ (i)=(cid:104)Π2i Sx(cid:105).
x l=1 l
For J > J , we exploit the well-known Kramers-
y x
Wannier duality of the D =1 QIM to conclude that the FIG. S2. (Color online) The DOS of the interacting c and α
orderisthatofstringsoftheSy,i.e,theorderparameter (left and right respectively) fermions at plateaus 1 and 2 .
3 3
is ∆ =(cid:104)Π2i+1Sy(cid:105).
y l=2 l
At J = J , both these order parameters collapse to
x y DMFT Spectral Functions
zero simultaneously. The topological excitations which
Here, we expound briefly upon our DMFT results for
destroy each of these orders are precisely proliferated
the spectral functions used to discuss spectral features
string-like excitations associated with the dual order:
and their consequences in the main text.
these are consequences of the emergent d = 1 GLS. Re-
We have used the two-orbital iterated perturbation
markably, it is easily seen that the two non-zero order
theory(IPT)asanimpuritysolverinthecontextofsolv-
parameters are nothing but the only ones invariant un-
ing H using DMFT. Though not “exact” in the sense
der the emergent GLS, i.e, they are the spin-nematic or-
of continuous-time quantum Monte Carlo (CT-QMC)
der parameters, defined by (cid:104)n(cid:105)=(cid:104)SxSx−SySy(cid:105). Thus,
i j i j or the numerical renormalization group (NRG) impurity
J = J is a TQCP separating two topological ordered
x y solvers, its numerical efficacy in the context of DMFT
phases. In dual language, it separates two Ising-nematic
for a related spinful Anderson lattice problem vis-a-vis
ordered phases.
CTQMC is well-documented, allowing a good degree of
confidence for its use in our case.
3
ergyinDMFT(notshown). Thisaspectisadirectconse-
quenceoftheimportantobservation,namely,thatafinite
h leads to emergent lower-dimensional d(= 1) < D = 2
z
gauge-likesymmetries: thus,thesespinon-likeexcitations
are topological excitations (non-local strings in the orig-
inal spin variables in H ) of this partially TO state.
K
However, this fractionalized component co-exists with a
finite“excitonic”orderparameter,whichisjustthefield-
induced magnetization (Fig. S3(a)) in original spin lan-
guage.
(a) (b) This has important consequences: clearly, our results
show that, across successive magnetization plateaus,
FIG. S3. (Color online)(a) nα(hz)/nα,sat as a function of large reshuffling of dynamical spectral weight over scales
magnetic field (hz) in the isotropic limit (nα,sat is the sat- oforderthefullc-fermionband-widthoccur. Thisclearly
uration value of n ). The jumps and plateaus in the profile
α atteststotheimportantroleofdynamicalcorrelationsin
closelycorrespondtothesamenumbersinm(h )/m inthe
z sat this interplay and generation of the plateau structures.
same limit. (b) Hysteresis in magnetization.
To the extent such dynamical correlations are crucial,
we opine that less sophisticated approximations such as
In Fig. S2, we show the c- and the α-JW fermion static Hartree-Fock may miss important aspects of the
spectral functions for the isotropic Kitaev case at values detailed structures found here.
m(h )/m = 1/3,2/3. Several notable features stand
z sat
outclearly. First,thein-built“orbital”differentiationin-
volved in the pure KM has direct consequences on their
respective evolution as a function of h . In particular,
z ∗ [email protected]
the α-fermion DOS resembles that of a Kondo insulator,
† [email protected]
owing to the fact that a combination of hz induced c−α ‡ [email protected]
hybridization and the bare energy level of the α-fermion [1] J. Q. You et al., Phys. Rev. B 81, 014505 (Jan 2010).
in the presence of a “Hubbard” (J ) interaction term [2] H-DChenetal.,J.Phys.A:MathandTheor.41,075001
z
conspire to generate such a state for the parameter val- (2008).
uessetbyH. However,thec-fermionDOSshowsdrastic [3] H-D. ChenandZ. Nussinov, Journal ofPhysics A:Math-
ematical and Theoretical 41 (2008).
spectral changes as a direct consequence of the dynami-
[4] G. D. Mahan, Many Particle Physics, Springer (2000).
cal interplay between the “Dirac” band structure of the
[5] J. Knolle et al., Phys. Rev. Lett. 112, 207203 (2014).
c-fermions,c−αhybridization,andtheHubbardinterac-
[6] E.FradkinandL.Susskind,Phys.Rev.D17,2637(1978).
tion. Notwithstanding this, however, the c-JW fermions [7] X-Y Feng et al., Phys. Rev. Lett.98, 087204 (2007).
continue to be elementary excitations of the perturbed [8] V. Gurarie and A. Tsvelik, Journal of low temperature
spin liquid, since ImΣ (ω) always vanishes at low en- physics 135, 245 (2004).
cc
