Topological (chiral) spin liquids with PEPS ?

Didier PoilblancLaboratoire de Physique Théorique, Toulouse• Spin liquids: trivial versus topologicalspin liquids, analogs of the Fractional• ChiralQuantum Hall states: key features
exemples / simple models hosting SU(2) /• Some
SU(N) CSL
no-go theorem: no real “obstruction” to• PEPS
describe CSL with PEPSExotic «topological liquids» beyond the«order parameter» paradigm* no spontaneous broken symmetry* no local order but...
* Topological order

EX. G. Wen, International Journal of Modern Physics B4, pp. 239-271 (1990)

Excitations are fractionalized anyons

Degeneracy from«topological order»{TWO TYPES OF SPIN
LIQUIDS:
spin-S AKLT

spin-1/2 RVB

Bi3 Mn4 O12 (NO3 ) material

J. Lavoie et al., Nat. Phys. 6, 850 (2010)
M. Matsuda et al., Phys. Rev. Lett. 105, 187201 (2010)

P. Fazekas and P.W. Anderson
Philosophical Magazine 30, 423-440 (1974)

S = z/2

Equal-weightsuperpositionof NN singlet
coverings

S=0«Trivial» liquid

Topological liquid
Hasting-Oshikawa-LSM theorem

Topological sectorsscylinder geometry«even»
Gv = +1s̄«odd»
Gv =1

Chiral topological spin liquids(chiral) states are genuine in the eld of the• Topological
Fractional Quantum Hall effect• Spin analogs on the lattice ?
Chiral SL (CSL) analogs of FQH states
1. Abelian CSL analog of Laughlin FQH state
2. Non-Abelian CSL analogs of non-Abelian FQHstates (Moore-Read, Read-Rezayi, etc…)
1⌫ = FQHS on a lattice (Kalmeyer-Laughlin, 1987):
2(N=Nsites/2)

fi

Paradigmatic “Abelian chiral spin liquid”

if T & P are broken :chiral spin liquidsanalogs of FQH states
Protected edge modes
described by Wess-Zumino-Witten(WZW) SU(N)k CFTs

“Long rangeentanglement”

Chiral spin-1/2 frustrated Heisenberg modelon the square lattice
A. Nielsen, G. Sierra, J.I. Cirac, Nature Com. 4, 2864 (2013)

CFT correlator SU(2)1 CFT

0.8

Parent Hamiltonien (long-range)

Chiral spin-1/2 Heisenberg model

hc/J1
0.6

J1-hc model

truncation

0.4

Kalmeyer-Laughlinchiral spin liquid

0.2

J1-J2-hc model00fi

Can be realized with synthetic gauge eld

0.2

0.4

J2/J10.6

Schematic phase diagram based on their results

Abelian chiral SL in spin-1/2 chiral AFM (I)B. Bauer, L. Cincio, B. P. Keller, M. Dol , G. Vidal, S. Trebst, A. W. W. Ludwig
Nature Communications 5, 5137 (2014)

Kagome lattice

0Chiral SL(Critical) SLJ

T-invariant H:
Yin-Chen He, D. N. Sheng, Yan Chen
Phys. Rev. Lett. 112, 137202 (2014)
Shou-Shu Gong, Wei Zhu, D. N. Sheng
Scienti c Reports 4, 6317 (2014)

ffiEdge modes : chiral c=1 CFT (SU(2) 1 WZW )More Abelian CSL in spin-1/2 AFM…

• Triangular lattice
Shou-shu Gong et al., , Phys. Rev. B 96, xx (2017)
A. Wietek, A.M. Lauchli, Phys. Rev. B 95, 035141 (2017)• Spontaneous symmetry breaking in J -J -J kagome AFM
1 2 3

S. Gong, W. Zhu, D.N. Sheng, Nat. Sci. Rep. 4, 6317 (2014)
Yin-Chen He et al., PRL 112 (2014)
A. Wietek, A. Sterdyniak, A.M. Lauchli, Phys. Rev. B 92, 125122 (2015)

Extension to SU(N) Heisenberg models
on the square latticeN-dim fundamental irrep on every site :iijkJ.-Y. Chen, J.-W. Li, P. Nataf, S. Capponi, M. Mambrini, K. Totsuka, H.-H. Tu,A. Weichselbaum, J. von Delft & DP, Phys. Rev. B104, 235104 (2021)

See also e.g. P. Nataf et al. PRL 2016 for triangular latticeED on torus geometry

gap
Topo deg(N-fold){ED on open systems

Chiral edge mode

WZW SU(4)1 CFTKeisuke Totsuka

angular momentum

PEPS tensor networks as variational ansatz
|

↵=X

↵

Ci1 ,i2 ,...,iN |i1 , i2 , ..., iNs
i = {1, · · · , dphys }I. Cirac
F. Verstraete
G.Vidaldimension of auxilary(or virtual) space↵,= {1, . . . , D}fi

in nite-PEPS or “iPEPS”
in thermodynamic limit

PEPS with continuous U(1) or SU(N)symmetries
First, introduced for SU(2)-invariant PEPS
M. Mambrini, R. Orus & DP, Phys. Rev. B 94, 205124 (2016)* virtual space : V = S1
<latexit sha1_base64="HKaUFo9NzK8erjB3jXMjRffK4Kk=">AAACFHicbZBNS8MwGMfT+TbnW9Wjl+AQBGG0Q9CLMPTicTL3AmspaZpuYWlTklQYZR/Ci1/FiwdFvHrw5rcx7XrQzQcCv/yflzz5+wmjUlnWt1FZWV1b36hu1ra2d3b3zP2DnuSpwKSLOeNi4CNJGI1JV1HFyCARBEU+I31/cpPn+w9ESMrjezVNiBuhUUxDipHSkmeeOcWMTJBgBntXHc92eMJSCTtec04ODrjK74ln1q2GVQRcBruEOiij7ZlfTsBxGpFYYYakHNpWotwMCUUxI7Oak0qSIDxBIzLUGKOISDcrFprBE60EMORCn1jBQv3dkaFIymnk68oIqbFczOXif7lhqsJLN6NxkioS4/lDYcqg4jB3CAZUEKzYVAPCgupdIR4jgbDSPta0Cfbil5eh12zYmu/O663r0o4qOALH4BTY4AK0wC1ogy7A4BE8g1fwZjwZL8a78TEvrRhlzyH4E8bnD4i9nno=</latexit>· · · SpS2Can be adapted to SU(N) (Ji-Yao Chen)

Add discrete point group symmetry
Chiral PEPS ansatz: A = AR + iAI<latexit sha1_base64="yaudnxbEBEBJOsXXqaXhxFtjNk0=">AAAB8XicbVBNS8NAEJ3Ur1q/qh69LBZBEEoiRb0IrV70VsV+YBvCZrtpl242YXcjlNB/4cWDIl79N978N27bHLT1wcDjvRlm5vkxZ0rb9reVW1peWV3Lrxc2Nre2d4q7e00VJZLQBol4JNs+VpQzQRuaaU7bsaQ49Dlt+cPrid96olKxSDzoUUzdEPcFCxjB2kiPtcuad3/Cat6tVyzZZXsKtEicjJQgQ90rfnV7EUlCKjThWKmOY8faTbHUjHA6LnQTRWNMhrhPO4YKHFLlptOLx+jIKD0URNKU0Giq/p5IcajUKPRNZ4j1QM17E/E/r5Po4MJNmYgTTQWZLQoSjnSEJu+jHpOUaD4yBBPJzK2IDLDERJuQCiYEZ/7lRdI8LTtn5cpdpVS9yuLIwwEcwjE4cA5VuIE6NICAgGd4hTdLWS/Wu/Uxa81Z2cw+/IH1+QP7po/U</latexit>

AR =X↵(A1 )
A↵ ↵

AI =X

A

(A2 )Different irreps !

DP, J. Ignacio Cirac & Norbert Schuch, Phys. Rev. B 91, 224431 (2015)

SU(2) CSL in frustrated models
with iPEPS
Symmetric PEPS

DP, Phys. Rev. B 96, 121118 (2017)
J. Hasik, M. Van Damme, DP, L. Vanderstraeten,Phys. Rev. Lett. 129, 1772001 (2022).
Non-symmetric PEPS

nteraaM

No-go theorem for chiral PEPS
Chiral TNS of free fermions (Gaussian PEPS) haveno gapped local parent Hamiltonians
J. Dubail, N. ReadPhys. Rev. B 92, 205307 (2015)
T.B. Wahl, H.-H. Tu, N. Schuch, J.I. Cirac
Phys. Rev. Lett. 111, 236805 (2013)

For interacting spins,
likely such a no-go theorem applies(see argument next)

Hand-waiving argument from PEPS bulk-edge correspondencebulkBAedge⇢A = TrB |<latexit sha1_base64="MnbiwLY3IvhsEpd04wp3M235s9Y=">AAACFnicbVDLTgIxFO3gC/E16tJNIzHBBWTGEHUhBnHDEhNeCTOZdEqBhs4jbcdIBr7Cjb/ixoXGuDXu/BsLzELBk9ybk3PuTXuPGzIqpGF8a6mV1bX1jfRmZmt7Z3dP3z9oiiDimDRwwALedpEgjPqkIalkpB1ygjyXkZY7vJ36rXvCBQ38uhyFxPZQ36c9ipFUkqPnLT4InJtSbHEP1vnEqcCxVRP0+mrax7AELfIQxrl81XFPJ46eNQrGDHCZmAnJggQ1R/+yugGOPOJLzJAQHdMIpR0jLilmZJKxIkFChIeoTzqK+sgjwo5nZ03giVK6sBdwVb6EM/X3Row8IUaeqyY9JAdi0ZuK/3mdSPYu7Zj6YSSJj+cP9SIGZQCnGcEu5QRLNlIEYU7VXyEeII6wVElmVAjm4snLpHlWMM8LxbtitlxJ4kiDI3AMcsAEF6AMqqAGGgCDR/AMXsGb9qS9aO/ax3w0pSU7h+APtM8f8M6d+Q==</latexit><latexit sha1_base64="fPubFoo3BVhkCwx8WltGnCaOQ9E=">AAACBnicbZBPS8MwGMbfzn9z/qt6FCE4BE+jFUGPQy8eJ7hNWEtJ03QLS9qSpOIoO3nxq3jxoIhXP4M3v43Z1oNuPhD48bzvm+R9wowzpR3n26osLa+srlXXaxubW9s79u5eR6W5JLRNUp7KuxAryllC25ppTu8ySbEIOe2Gw6tJvXtPpWJpcqtHGfUF7icsZgRrYwX2offAgsKTAoU5H46Rp5hAHjcXRBihwK47DWcqtAhuCXUo1QrsLy9KSS5oognHSvVcJ9N+gaVmhNNxzcsVzTAZ4j7tGUywoMovpmuM0bFxIhSn0pxEo6n7e6LAQqmRCE2nwHqg5msT879aL9fxhV+wJMs1TcjsoTjnSKdokgmKmKRE85EBTCQzf0VkgCUm2iRXMyG48ysvQue04Rq+Oas3L8s4qnAAR3ACLpxDE66hBW0g8AjP8Apv1pP1Yr1bH7PWilXO7MMfWZ8/O0SYTA==</latexit>⇠bulk ⇠><| = exp ( Hb )range of boundary H

J. Ignacio Cirac, DP, Norbert Schuch, Frank Verstraete, Phys. Rev. B 83, 245134 (2011)

If chiral edge mode (with discontinuous dispersion)
Boundary H long-range
bulk-edge correspondence

Bulk correlation length (strictly speaking) in nitefiNevertheless, is there a real obstruction to construct“Physically relevant” chiral topological PEPS ?

iPEPS method=constructed by renormalization• Environmentof the corner transfer matrix (CTM)⟨ΨPEPS | H□ | ΨPEPS⟩

T. Nishino & K. Okunichi, J. Phys. Soc. J. 65, 891 (1996)
R. Orus & G.Vidal, Phys. Rev. B 80, 094403 (2009)

optimisation scheme based on a• Variational
conjugate gradient method

χ

L.Vanderstraeten, J. Haegeman, P. Corboz, F.Verstraete,Phys. Rev. B 94, 155123 (2016)used for non-chiral AFM:
DP & M. Mambrini, Phys. Rev. B 96, 014414 (2017)used for spin-1/2 chiral AFM:

DP, Phys. Rev. B 96, 121118 (2017)non-symmetric PEPS: use automatic• Ifdifferenciation
J. Hasik (2019)

CTM Renormalization Group algorithm

fifi• Speci c features for symmetric PEPS providing better stability :• The CTM is Hermitian -> SVD replaced by ED (better stability)• All C (corner) and T (edges) matrices are the same
• SU(2) symmetry preserved in the CTM xed point.

0.8SU(2) case

Chiral spin-1/2 Heisenberg modelhc/J1Energetics

0.6

J1-hc model
0.4

Kalmeyer-Laughlinchiral spin liquid

0.2

J1-J2-hc modeliPEPSMPS

Faster convergence than state-of-the-art
MPS on cylinders !

000.2

0.4

J2/J10.6

Entanglement spectrum(in nitely-long edge)chie
modralpstee

mode

D=5, particle ansatz methodfiPerfectly chiral when<latexit sha1_base64="zsdSIFpb2qlZ5BsByrNuIUa0ddU=">AAAB/nicbVBNS8NAEN3Ur1q/ouLJS7AInkoiRT0WvXisYD+gCWWz3TRLN7thd6KUUPCvePGgiFd/hzf/jds2B219MPB4b4aZeWHKmQbX/bZKK6tr6xvlzcrW9s7unr1/0NYyU4S2iORSdUOsKWeCtoABp91UUZyEnHbC0c3U7zxQpZkU9zBOaZDgoWARIxiM1LePfBIzX7FhDFgp+egzEcG4b1fdmjuDs0y8glRRgWbf/vIHkmQJFUA41rrnuSkEOVbACKeTip9pmmIywkPaM1TghOogn50/cU6NMnAiqUwJcGbq74kcJ1qPk9B0JhhivehNxf+8XgbRVZAzkWZABZkvijLugHSmWTgDpigBPjYEE8XMrQ6JscIETGIVE4K3+PIyaZ/XvIta/a5ebVwXcZTRMTpBZ8hDl6iBblETtRBBOXpGr+jNerJerHfrY95asoqZQ/QH1ucPIKuWQQ==</latexit>!1

CTMRG environment
dimensionVery fast decay ofspin-spin correlations

D=5 iPEPS

Consistent with bulk gap

But… diverging correlation length !
Transfer matrix

ξn = − 1/ln(λn /λmax)

Closer look at spin-spin correlations…

Very small weight“Gossamer” taildevelops at large rC(d) = Cbulk(d) + Ctail(d)Ctail(d) =∑i>itailw(ξi) exp(−d/ξi)Discussion(Ia)
features in higher-S SU(2) and SU(N) CSL• Similar(square lattice) :D=15
D=6Spin-1 SU(2)2 non-Abelian CSL
Ji-Yao Chen, Laurens Vanderstraeten, Sylvain Capponi & DP
Phys. Rev. B 98, 184409 (2018)

SU(4)1 Abelian CSL
Ji-Yao Chen, Jheng-Wei Li, Pierre Nataf et al.
Phys. Rev. B 104, 235104 (2021)

Discussion(Ib)• Similar features in other lattices :Kagome spin-1/2 SU(2)1 CSL:

D=8Sen Niu, Juraj Hasik, Ji-Yao Chen & DP
arXiv:2209.04485 (2022)

Discussion(II)
PEPS possess all features of top CSL. Topo order from Z• Chiralgauge symmetry.

N

long-range correlation “tail” is an artifact of chiral PEPS
• The
needed to comply with PEPS bulk-edge correspondence• NOT a practical limitation in PEPS descriptions of chiral SL• TN techniques are variational approaches becoming as
successful in 2D (and 3D ?) as DMRG in 1D.

• Can provide classi cation of topological phases of quantum
spin systems

fi!! Acknowledgements:

Many thanks to :I.Cirac

M. Mambrini

N. Schuch

S. Capponi

R. OrusJi-Yao Chen

L. Vanderstraeten

K. Totsuka

Juraj Hasik

Sen Niu