Chiral spin liquid phases in SU(N) quantum magnets - IPAM, 4/23/2021

IPAM, 4/23/2021

Chiral spin liquid phases in
 SU(N) quantum magnets
 Sylvain Capponi
 Toulouse Univ.

Topological phases of matter
• Robustness of topological states

 Topological quantum computation (no error correction needed !)

 How to engineer topological phases ?

 How to detect them: ground-state degeneracy
 excitations
 entanglement properties

 Use topology !

Topological phases of matter
• Robustness of topological states

 Topological quantum computation (no error correction needed !)

• Quasiparticles are anyons (fractional statistics) i.e. not necessarily bosons
 or fermions (spin statistics theorem breaks down in 2+1D)

• Excitations can be abelian or not

 topological quantum field theory
 (e.g. Chern-Simons),
 braid group, fusion rules…

Chiral topological spin liquids
Chiral topological phase is found in the fractional quantum
Hall (FQH) effect

 topological phases, exotic excitations
 (abelian or not)
 unconventional superconductor when doped

 Is it possible to reach the same physics without Landau levels, on a lattice ?

 See e.g. talk by
 Cécile Repellin

 Mimic an effective magnetic field, Look for lattice models with similar
 flat bands etc. wavefunctions
 Fractional Chern insulators

OUTLINE
 Introduction: chiral spin liquids in a nutshell

 Combined numerical methods of a family of SU(N) models

 Conclusions and outlook
Collaborators/Refs:
 • Ji-Yao Chen, L. Vanderstraeten, S. Capponi, D. Poilblanc, Phys. Rev. B 98, 184409 (2018)
 • Ji-Yao Chen, S. Capponi, A. Wietek, M. Mambrini, N. Schuch, D. Poilblanc, Phys. Rev. Lett. 125, 017201
 (2020)
 • Ji-Yao Chen, Jheng-Wei Li, Pierre Nataf, Sylvain Capponi, Matthieu Mambrini, Keisuke Totsuka, Hong-
 Hao Tu, Andreas Weichselbaum, Jan von Delft, Didier Poilblanc, in preparation

he 1980s it was which is separated by the insulating bulk in wise edge channels, whose direction was

 n April 18, 2021
 rriers are con- between, and whenever quantization occurs determined by the spin orientation (either up
 tem (or sheet), in the transverse resistance, the longitudi- or down) of the occupying electrons, forced

 Hall
 (1879)

 Quantum Hall
 (1980)
 Chiral spin liquids (CSL)
 Spin Hall
 (2004)

 Quantum spin Hall
 (2007)
 Anomalous Hall
 (1881)

 Quantum anomalous Hall
 (2013)

 H
 = lattice analogue of FQH states M

 Low-energy physics described by 2+1 Chern-Simons theory
 Quantum Hall Quantum spin Hall Quantum anomalous Hall

 1
 Quantum Hall trio. Numbers in parentheses indicate the years of each discovery. H is the external magnetic field, and M is

 ⌫=
 the magnetization. For all three quantum Hall effects, electrons flow through the lossless edge channels, with the rest of the
 FQH state
 system insulating. When there is a net forward flow of electrons for Hall resistance measurement, (left) those extra electrons lattice spin S=1/2 model
 2
 occupy only the left edge channels in the quantum Hall system regardless of their spins, (center) opposite-spin electrons occupy
 opposite sides in the quantum spin Hall system, and (right) only spin-down electrons flow through the left edge in the quantum
 anomalous Hall system. The locking schemes between spin and flow direction, and the number of edge channels depend on the
 material details, and only the simplest cases are illustrated here.

 incompressible (gapped) in the bulk
 www.sciencemag.org SCIENCE VOL 340 12 APRIL 2013
 same
 153
 Published by AAAS

 charged e/2 fractional excitation neutral s=1/2 fractional excitation

 robust gapless chiral edge states same

 SU(2)1 CFT
 triangular lattice: Kalmeyer-Laughlin, 1987

chiral spin liquids = lattice analogs of FQH states

These states break time-reversal symmetry (T) and parity (P)

 Protected edge modes
 described by SU(2)1 CFT

 “Long range Xiao-Gang Wen
 entanglement”

 Tensor networks formalism well suited
 Dubail-Read 2015
 No-go theorem for a gaussian PEPS to have a bulk gap

chiral Jχ interactions. Fo
 clear.
 I. INTRODUCTION Jχ ∼ #t 3 /U 2 where J1 (resp. Jχ ) is the nearest-neighborthe same chirality direc
 The emergence of In this spin
 quantum article,
 liquids inwe study Heisenberg
 frustrated the spin-1/2 1 -J
 (resp., scalarJchirality)
 2 Heisenberg
 coupling.
 Another open question in frustrated magnetism of thetriangular lattice. This o
 Abelian CSL in spin-1/2 SU(2) models on frustrated lattices quantum magnetism is an exciting phenomenon in contem-
 porary condensed model
 exhibit fascinatingsymmetry
 on [1].
 matter physics theThese
 (TRS)
 properties such as
 triangular
 long-rangebreaking
 lattice
 states of matter
 phase
 withlattice
 triangular
 diagram
 additional
 of
 is the nature oftime-reversal
 the
 ground-state chiral interaction Jχ using DMRGS = 1/2
 the intermediate phase in the
 Heisenberg model with added toward the corners of a te breaks T
 entanglement [2,3] or anyonic braiding statistics of quasipar- next-nearest-neighbor couplings around J2 /J1 ≈ 1/8. Severalmodel with growing J2 a
 simulations.
 ticle excitations, relevant The
 for a potential model Hamiltonian
 implementation of is given
 authors [20,33,34] found aas spin disordered state. Recently
 order, J1 -J2 spin liquid (
 topological quantum computation [4]. Only very recently have several numerical studies [35– 40] proposed that a topological
 PHYSICAL REVIEW B 96,
 such 075116
 phases been(2017) ! in realistic local spin!spin
 found to be stabilized liquid state of some ! kind might be realized in this regime.liquid (CSL), and tetrahe
 S=1/2 on triangular lattice models [5– 19]. H = J1 S⃗i · ⃗
 S j +
 Triangular lattice Heisenberg models are a paradigm of
 J 2
 The⃗exact⃗nature of this phase yet
 we S · S
 advocate
 i j +
 the J
 presence
 χ of
 ◦
 ( ⃗
 S
 a i
 remains
 ×
 O(4) ∗ ⃗
 S unclear.
 )
 quantum
 j · ⃗
 S In this paper
 ,
 critical
 k pointlines) are obtained by m
 frustrated magnetism. Although the ⟨i,jHeisenberg
 ⟩ model with [41–
 ⟨⟨i,j ⟩⟩ 44] separating the 120
 △/▽ Néel order from a putative Z2spin correlation function
 eemingly only nearest-neighbor interaction is known to stabilize a spin liquid. The diverging correlation length at this quantum
 critical point and the neighboring first-order phase transitionboundary between the 12
netheless, regular 120◦ Néel order [20– 23], adding further interac-
eatures in PHYSICAL REVIEW B 96 , 075116 (2017) tion terms may increase where J and
 frustration
 1 andJinduce
 2 denote magneticthe into NN the and the
 stripy collinear NNN magnetic interactions,
 ordered phase render the
 J2 disorder to the system. Experimentally, several materials unambiguous
 PHYSICAL identification
 REVIEW B 95, 035141 of the(2017) intermediate spin liquid
tor shows Global phase 1 with triangular respectively.
 lattice geometry do notThe
 exhibit scalar
 any sign chiral
 of phaseinteraction
 challenging, however. Jχ has the same
 J1 diagram and quantum spin liquids in a spin- triangular antiferromagnet
 liquid and quantum criticality in extended S = 12 Heisenberg models
 with the J
 2
 magnetic orderingmagnitude down to lowestfor all Chiral[24–
 temperatures the spin27].
 up (△) and down on the
 (▽)
 triangular
 triangles,
 II. lattice
 MODEL
 and the topological nature in t
 Shou-Shu Gong, W. Zhu, J.-X. Zhu, D. N. Sheng, and KunThese Yang include, for example, the organic Mott insulators like
 developed. A possible
 1 2 2,3 4 5

 e [65,66], National 1
 J
 High Magnetic Field Laboratory, Florida State University, Tallahassee, Florida 32310, USA2 Cu
 κ-(BEDT-TTF) three
 2 (CN)3 sites or EtMe3for
 [24,25] i,j,k Sb[Pd(dmit)Jχ follow 2 ]2 Wethe clockwise
 investigate * the order
 Heisenberg
 Alexander Wietek and Andreas M. Läuchli
 in
 model all
 with the
 nearest- and
 Austriauniformby increasing Jχ is ne
 [26,27]
 Mexicoand are USA
 thus candidates realizing spin liquid physics.
 ctor. The
 2
 Theoretical Division, T-4 and CNLS, Los Alamos National Laboratory, Los Alamos, New
 Center for Integrated Nanotechnologies, Los Alamos National Laboratory, Los Alamos, NewHistorically
 3
 87545,
 Mexico 87545,Kalmeyer
 triangles
 USA
 as shown
 and Laughlin [28] introduced
 in Fig.
 (Received the
 Institut 1(a). Physically,
 fürnext-nearest-neighbor
 Theoretische Physik, Universität the
 interactions
 Innsbruck, scalar
 with an
 A-6020 chiral
 additional
 Innsbruck,
 10 June 2016; revised manuscript received 21 December 2016; published 24 January 2017)

 r shows a National HighDepartment
 4
 of Physics and Astronomy, California State University, Northridge, California chiral 91330, USA interaction
 spin liquid (CSL) state on the J χ term
 triangular
 Wecan
 lattice. be
 This induced in the Hubbard model with
 investigate the J -J Heisenberg model on the triangular lattice with an additional scalar chirality term
 to the neighboring ph
 G
 5
 Magnetic Field Laboratory and Department of Physics, Florida State University, state Tallahassee, Florida 32306, USA
 is closely related to the celebrated Laughlinand wave function 1 2

mplication (Received 8 May 2017; published 9 August 2017) (d) of the fractional quantum large Hallin
 U a magnetic field
 been a[76,77]. Starting from the Hubbard
 show that a chiral spin liquid is stabilized in a sizable region of the phase diagram. This topological phase is
 triplet gap in the eve
 t DMRG M R 0.5 Tetrahedral phase
 We study the spin-1/2 Heisenberg model on the triangular lattice with the nearest-neighbor shown to be J > the
 0, ground
 model,
 the
 effect
 state of several extended
 a t/U (t E D
 and has situated
 recently in
 Heisenberg
 and
 discuss
 between
 U
 coplanar
 are
 120
 ◦

 the
 Néel ordered
 hopping
 and a noncoplanar
 and
 1 2
 tetrahedrally ordered
 interaction,
 phase.
 the nature of the spin-disordered intermediate phase in the J -J model. We compare the ground states
 Furthermore we
 could be consistent w
 D
 1
 models
 next-nearest-neighobr J > 0 Heisenberg interactions, and the additional scalar chiral interaction J (S × S ) · S
 2 on
 ⃗ the⃗ kagomé
 ⃗ lattice [5– 7,9]. The question
 from exact arises
 diagonalization with a Dirac spin liquid wave function and propose a scenario where this wave function
 in DMRG calculation
 χ i j k

either the for the three spins in all0.4
 the triangles using large-scale density matrix renormalization groupwhether
 geometry. With increasing J (J /J ! 0.3) and J (J /J ! 1.0) interactions, we establish
 calculationaonCSLcylinder respectively)
 can indeed be realized expansion
 on describes
 the triangular to the second ◦
 order at half-filling
 the quantum critical point between the 120 magnetically ordered phase and a putative Z spin liquid.
 2

 latticea quantum phase proposed. In a recent study
 as originally [10] this
 ⃗i × S⃗j ) · S⃗k with gapless spin liquid [67
 2 2 1 χ χ 1 DOI: 10.1103/PhysRevB.95.035141
 oped due 0.3
 diagram with the magnetically ordered 120 , stripe, and noncoplanar tetrahedral phase. Inwas
 ◦
 between these magnetic gives the for effective
 N ! 3. In this paper chiral we interaction Jχ (S
 Jχ

 shown for SU(N) models
 order phases, we find a chiral spin liquid (CSL) phase, which is identified as a ν = 1/2 bosonic fractional quantumevidence that indeed the CSL is stabilized
 ations, or provide
 Hall state with possible spontaneous rotational symmetry breaking. By switching on the chiral interaction, we
 0.2 in a
 conclusive
 spin-1/2 J
 Heisenberg χ ∼ #t
 model
 3
 /U
 upon
 2
 ,
 adding where
 a further
 I. # is
 INTRODUCTION
 scalar the magnetic flux χ J enclosed
 3
 ∼ #t /U whereby
 2
 1 the
 J (resp.
 χ We study the sys
 J ) is the nearest-neighbor
 find that the previously identified spin liquid in theCSL Heisenberg (resp., scalar chirality) coupling.
 in frustrated DMRG [78] with spi
 J -J triangular model (0.08 " J /J " 0.15) shows a phase
nal study, 1 2
 transition to the CSL phase at very small J . We also compute the spin triplet gap in bothchirality
 χ
 2 1
 spin liquidterm Sitriangle.
 ⃗
 Jχ and
 phases,
 ⃗ ⃗ We
 · (Sj × Sk ), similar
 The emergence of quantum spin liquids in frustrated
 take
 to
 quantumRefs. J18,10].
 [6–
 magnetism =is an1.0
 Such as phenomenon
 exciting the energy in contem- scale. Using
 Another DMRG
 open question magnetism of the
 a term gap caninbetherealized as a lowest orderporaryeffective
 condensedHeisenberg triangular lattice is the nature of the intermediate phase in the
 S = 1/2 Heisenbergchoose two geometries
 0.1 a large gap in the odd topological sector but a small or vanishing
s the best our finite-size results suggest o
 120 phase
 sector. We discuss the implications of our results on the nature of the spin liquidstripe Hamiltonian of thesimulation,
 phase
 phases.
 even
 Hubbard model upon weadding
 exhibit obtain
 # flux
 fascinating
 matter physics [1]. These states of matter
 a
 through
 properties quantum
 such phase
 as long-range ground-state diagram
 phase diagram as of theshown model with added
 J1-JJ2 SL next-nearest-neighbor couplings around J /J ≈ 1/8. Several
d remains 0
 0
 DOI: 10.1103/PhysRevB.96.075116 0.05 0.1 0.15 0.2 0.25or by introducing in Fig. 1(d).
 the elementary plaquettes [6,29,30], entanglement
 ticle Besides
 excitations, relevantthe
 [2,3] or anyonic
 either via a magnetic
 for 120
 field braiding
 a
 ◦ FIG.
 potential Néel phase,
 statistics of quasipar-
 1. Approximate
 implementation of T = the
 authors
 0 phase stripe
 [20,33,34] ofphase,
 diagram found
 1 the Ja -J
 2 χ
 2 1
 modeleither
 spin-Jdisordered the x axis (XC) o
 state. Recently
 artificial gauge fields in possible cold-atom
uid phase, J 2 experiments [31,32]. andThethe time-reversal
 coupling constants thenfound invariant
 topological quantum computation [4].
 relate tostabilized
 onOnly
 the very
 spin
 triangular
 liquid
 recently have cf. several
 lattice,
 in
 Eq. (1).numerical
 the
 The extent
 J -J
 studies
 model
 [35– 40]
 of phases is proposed
 inferred that a topological
 andin this
 1(b).
 regime.These cylind
vestigated
 I. INTRODUCTION
 Quantum spin liquid (QSL) is a kind of a long-range
 the phase boundaries of several ordered phases, suggesting
 the Hubbard
 possible strong competition of the different physical
 model
 (here
 parameters
 we
 mecha-
 t
 such
 and
 denote
 models
 phases been
 U as
 [5– J
 it
 19].
 1 ∼
 as
 t 2
 to
 /U and
 J 1
 be
 -J S=1/2 on frustrated square lattice
 2
 from in excitation
 realistic spectra
 local spin fromspin liquid
 ED on state
 1
 a of2some
 periodic kind might
 36-site
 simulation cluster; see main text for details. Orange: S = 1 K.A1L and L are the num
 SL), we find a welarge
 advocate the regime
 presence of of∗
 be realized
 triangular
 The exact nature of this phase yet remains unclear. In this paper
 a O(4) quantum y critical pointx RAP
 lculations FIG. 1. Model Hamiltonian and nismsquantum
 in the kagomephase spin liquiddiagram of thea fully frustrated magnetism. Although the Heisenberg model with [41– 44] separating the 120 Néel order from a putative Z
 regime. In particular, Triangular lattice Heisenberg models
 (120◦ are a
 Néel); paradigm
 light blue:of S = 0 $.E2b (CSL); ◦ green: S = 0 $.E2a,

J1 -J2 spin
 S=1/2 on kagome lattice
 entangled state with fractionalized
 ARTICLEquasiparticles [1]. Since
 spin-1/2Received
 the proposal by P. W. Anderson,
 1 -J
 the -J2014
 χ Heisenberg
 concept
 Junderstanding
 212Jan | Accepted 4 Sep 2014model
 of QSL has been
 gapped chiral spin liquid (CSL) [44,45] is found
 on smallonfurther-neighbor
 the triangular [15,16]lattice.
 or chiral (a)
 the noncoplanar
 by switching tetrahedral
 and [14] regular 120 Néel order [20– 23],$.E2a,
 interactions only nearest-neighbor interaction order
 $.E2b
 is known tofor
 degenerate large
 (Dirac/Z
 stabilize a
 $.E2b degenerate (stripy
 2 spinJ
 spin
 χ , whose
 liquid);
 liquid.
 PHYSICAL
 magnetic
 dark blue:
 The diverging
 REVIEW
 order);
 spin S = 0 $.A1,
 correlation
 B 96,
 dark red / light
 Toat (2017)
 length
 121118(R)
 study
 2
 this quantumthe phase di
 red: phase transition
 magnetic orderedwe phaseperform calculatio
 correlated | Published 10 Oct 2014 critical point and the neighboring first-order
 configuration is shown in Fig. 1(c). Below into the tetrahedral
 DOI: 10.1038/ncomms6137
 playing an important role for strongly ◦
 adding further interac-
 on the NN kagome model. alexander.wietek@uibk.ac.at
 S
 *
 materials and(b)
 e kagome are the schematic figures of the
 unconventional superconductors [2]. Although 120promising
 ◦
 and the stripe magnetic tion terms may increase frustration S = 1
 and M.A
 induce/ S = 0
 magnetic$.E2a the stripy
 (tetrahedral collinear
 magnetic order). render the
 Chiral spin liquid and emergent
 Another spinanyons
 liquid candidate is the antifer-
 phase for disorder to the system. Experimentally, severalInvestigation
 ! 0.25, we identify P a CSL as the
 of the chiral
 unambiguous
 1/2
 antiferromagnetic
 identification
 bosonic
 of the intermediate Heisenberg
 10. We keep
 spin liquid model
 up to 400
 um phase order
 R G
 QSLs have been pursued for more than two decades [3– 9],
 on the XC and YC
 only recently such novel states have been found in realistic spin
 in a Kagome
 cylinders.
 lattice
 romagnet
 Mott
 The on the triangular
 edge-sharing model
 is insulator
 triangular has lattice.
 2469-9950/2017/95(3)/035141(6) the
 Although J 2
 with triangular lattice geometry
 E
 035141-1 do not exhibit any
 materials
 sign of
 ν
 phase = challenging,
 using however.entangled pair states
 projected
 ©2017 American Physical Society

o far from M nearest-neighbor J 1 , next-nearest-neighbor
 models [10– 23], in which geometric frustration and competing
 frustration
 it turns out
 present in the spin-1/2 NN triangular model,
 to J
 still
 2 , and
 exhibit three-spin
 a 120 ◦ fractional
 scalar
 antiferromagnetic order quantum
 These include, for
 Hall
 example, P
 state
 the organic
 by
 magnetic ordering down to lowest temperatures [24– 27].
 observing
 Mott insulators like
 the gapless chiral
 II. MODEL
 with the truncation err
 states. D
 interactions play important roles1 for developing
 chiral Jχ interactions. For all the triangles,
 B. Bauer , L. Cincio
 spin liquid
 2, B.P. Keller3, M. Dolfi4[46,47].

 One of the most promising spin liquid candidates is
 , G. VidalIn recent
 2, S.
 sulators such the
 Trebstexperiments
 5 & A.W.W. on
 chiral
 as κ-(ET) Cu (CN) 2
 triangular
 interactions
 Ludwig 3
 edge
 have mode.
 organic
 and EtMe Sb[Pd(dmit)
 2 3
 Mott
 3
 in-
 ] 2 2
 The
 κ-(BEDT-TTF)
 strong Cu 2 (CN)
 2
 nematic
 3 [24,25] or
 order
 EtMe
 [26,27] and are thus candidates realizing spin liquid physics.
 3
 of
 Sb[Pd(dmit)2 2
 Laboratoire bond
 ]
 de Physiqueenergy
 We Théorique,
 Didier Poilblanc
 investigatesuggests
 the Heisenberg
 CNRS modeldewith
 and Université
 next-nearest-neighbor interactions with an additional uniform
 nearest-
 Toulouse, and Toulouse, France
 31062

Abelian CSL in SU(N) models
 Enlarging SU(2) to SU(N) is known to destabilize magnetic order
 This talk: ⇤

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NATURE PHYSICS DOI: 10.1038/NPHYS2878

 LETTERS
 PUBLISHED ONLINE: 2 FEBRUARY 2014 | DOI: 10.1038/NPHYS2878

A one-dimensional liquid of fermions with
tunable spin
Guido Pagano1,2, Marco Mancini1,3, Giacomo Cappellini1, Pietro Lombardi1,3, Florian Schäfer1,
Hui Hu4, Xia-Ji Liu4, Jacopo Catani1,5, Carlo Sias1,5, Massimo Inguscio1,3,5 and Leonardo Fallani1,3,5*

Correlations in systems with spin degree of freedom are at the 1D systems with a high degree of complexity, including spin–orbit-
heart of fundamental phenomena, ranging from magnetism to coupled materials22 or SU(N ) Heisenberg and Hubbard chains23,24 .
superconductivity. The e�ects of correlations depend strongly Moreover, the investigation of these multi-component fermions
on dimensionality, a striking example being one-dimensional is relevant for the simulation of field theories with extended
(1D) electronic systems, extensively studied theoretically over SU(N ) symmetries25 .

X a chiral PEPS X
 the fundamentalWe representation of SU(N
 open strips provide complementary results. Finaly, for N = 4,will move
 then start by generalizi
 ansatz is also constructed providing a good vari-PEPS
 to a description of this SU
 In the following, we will investigate model (1) using
 whose ED
 plementary physical
 and DMRG spintechniques
 degrees providing
 (N
 of freedom
 ) 1 CSL
 overwh tr
 = J1 Pational
 ij + Jenergy
 2
 hi,ji bulk anyonic
 we
 kl
 consider
 andPenabling
 correlations.
 hhk,lii
 the
 Models and definitionsmost general SU(N)-sym
 to connect the edge spectra to thecording
 defined on a square lattice by to the of
 ing evidence fundamental irrep CSL
 a stable topological of SU(N
 phase. ),
 ical nature of this CSL phase is precisely established fro
 The here
 top

 + JR (P ijk +onPaijk
 defined
 1 three-site interaction:
 X We then start by generalizing
 ) + lattice
 square
 X the Hamiltonian of (1)
 iJI by (P ijk onPevery
 placing,
 Ref. [28]spin-
 N-dimensional spin degree o
 1
 the Nchiral
 ijk ),lattice site, aand ters)
 N =
 -fold GSPEPS.
 and3, (ii)we
 the shall
 Following
 degeneracy
 existence
 on a torusthe
 focusof chiral
 on the edge
 prescription
 geometry
 Nmodes
 (periodic
 = 4–case obse
 Model
 4ijk N-dimensional spin degree
 defined in terms
 4ijk Xthe fundamental
 of X representatio
 of freedom, which transforms as
 permutation
 the fundamental representation of SU(N ). As for N = 3,
 both
 features on open
 observed systems
 for and
 these in the
 three entanglement
 values
 finite cylinders – whose content follows exactly the pr
 of Nspectra
 allow o

 we (SU(N)
 consider the most H=J
 symmetry)
 general P + J
 SU(N)-symmetric
 the first (second) term corresponds to two-site permu- P
 short-range heuristic
 tion of rules
 the SU and
 (N )conclusions
 WZW CFT for
 theory. general
 In a N
 second . step
 we consider the most genera
 1 ij 2 kl
 1
 three-site interaction: will move to a description of this SU (N ) CSL phase u
 s over all (next-)nearest-neighbor bonds, and the third 1
 X X J + iJ

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Numerical methods for SU(N)
 ... an introduction
 Analytics: large-N, mean-field, parton wavefunctions
g tableau: α PRL ; 2 ;127204
 ¼ ½3113, 2 %; (2014) PHYSICAL REVIEW LETTERS
 week ending

erator ofExact diagonalization (U(1)+lattice symmetries or SU(N) symmetry)
 • 19 SEPTEMBER 2014
 the dimension
 les of standard tableaux
ence. (e) Normal product week ending

 using standard Young tableaux PRL 113, 127204 (2014) PHYSICAL REVIEW LETTERS 19 SEPTEMBER 2014

 Exact Diagonalization of Heisenberg SUðNÞ Models
 Pierre Nataf and Frédéric Mila

 to the main diagonal,
 Institute of Theoretical Physics, École Polytechnique Fédérale de Lausanne (EPFL), CH-1015 Lausanne, Switzerland
 (Received 23 May 2014; published 18 September 2014)

for a box above (below)
 Building on advanced results on permutations, we show that it is possible to construct, for each
 irreducible representation of SUðNÞ, an orthonormal basis labeled by the set of standard Young tableaux
 in which the matrix of the Heisenberg SUðNÞ model (the quantum permutation of N-color objects) takes

 Hilbert space
 FIG. can
 1. (a)be Example of a Young tableau: α ¼ ½3 ; 2 ; 2 %;
 an explicit and extremely simple form. Since the relative dimension of the full Hilbert space to that of
 the singlet space on n sites increases very fast with N, this formulation allows us to extend exact
 α (b) Integers di;N that enter the numerator of the dimension
 ere V is the of α;Hilbert
 (c) Hook lengths li ; (d) Examples of standard tableaux
 diagonalizations of finite clusters to much larger values of N than accessible so far. Using this method,
 we show that, on the square lattice, there is long-range color order for SU(5), spontaneous dimerization for
 SU(8), and evidence in favor of a quantum liquid for SU(10).
 α α ranked according to the last letter sequence. (e) Normal product
 dN > 1, V can ½3itself;2 ;2 % DOI: 10.1103/PhysRevLett.113.127204 PACS numbers: 75.10.Jm, 02.70.-c, 67.85.-d, 71.10.Fd
 state jΦ1α i ¼ jAAABBCCi.
ent subsectors V i as There is currently considerable experimental activity on decreases when the dimension of the local Hilbert space
 α α α distance from the ith box to the main diagonal,
 dimðV Þ ¼ f dN and
 algebraic ultracold multicomponent fermions [1–3]. When loaded
 in an optical lattice, these systems are expected to be, for
 increases, and EDs are severely limited by the size of the
 available clusters. Alternatives are clearly called for.
 counted positively (resp. negatively) for a box above (below) integer number of particles per site and sufficiently large In this Letter, we introduce a simple method to perform
 the diagonal (see Fig. 1). The full Hilbert space can be on-site repulsion, in a Mott insulating phase described by EDs of any quantum permutation Hamiltonian separately in
 ⊗n the SUðNÞ Heisenberg model [4–7]. This effective model each irreducible representation (irrep) of SUðNÞ. Since the
ty is that, since it has ⊕ α α
 decomposed as □ ¼ α V , where V is the Hilbert is a generalization of the familiar SU(2) model, and in the dimension of the irreps relevant at low energy (for instance

 FIG.
 α
 using
 2 (color online). α SU(N)
 Real-space U(1)
 space associated to irrep α, and, if dαN > 1, V α can itself (N-1hPCartan)
 case of one particle per site, it takes the general form of a
 correlations i −1=N for
 quantum permutation Hamiltonian:
 the singlet, to which the ground state belongs) is much
 smaller than that of the sector used in traditional ED, this

energy, correlati
 Monte Carlo methods in many cases.
 Tensor network representation.—The method proposed The key observ

 Numerical methods for SU(N) here can be applied to any spin, bosonic, or fermionic
 systems, but we shall use spin-1=2 lattice models as
 illustrations [see Fig. 1(a)]. The lattice sites are labeled
 by j ∈ ½1; N" and the spin operators are Saj (a ¼ x, y, z).
 single-particle o
 bond dimension

 !
 The Abrikosov fermion representation is Saj ¼ d†m ¼
 P † a
 ... an introduction 1
 2 αβ cjα τ αβ c jβ , where c†
 jα (cjα ) are fermionic creation
 (annihilation) operators at j, α ¼ ↑; ↓ is the spin index, and
 One dummy col
 τa are Pauli matrices. This is an overcomplete representa-
 ensure that all M
 tion with unphysical states (empty and doubly occupied) the dummy vect
 that need to be removed by the single-occupancy constraint
 recover a usual M
Analytics: large-N, mean-field, parton wavefunctions P †
 α cjα cjα ¼ 1. The Schwinger boson representation is
 very similar, where the fermionic operators are replaced
 then straightforw
 tion of Eq. (1)
 by their bosonic counterparts. MPOs correspo
• Exact diagonalization (U(1)+lattice symmetries or SU(N) symmetry) One popular class of trial wave functions for spin models
 is the projected Fermi sea
 (2) apply the pro
 Pj acting on tw
 YN tensor network
 jΨi ¼ PG d†m j0i; ð1Þ bosonic paired
 m¼1 create fermionic
• DMRG (U(1) or SU(N) symmetry) + parton wavefunction where j0i is the vacuum, †
 QN the dm are single-particle orbitals
 Compressing
 derived above
 of the partons, PG ¼ j¼1 Pj is a product of projectors that computed simpl
 contraction of
 closed loops is
 it imperative to
Projected Fermi sea would enable ac
 sequentially act
 (a) (b) vacuum as the
has a tensor network representation However, the
 exponentially w
 to carry out the p
MPO: D=2 end, we need to
 PHYSICAL REVIEW LETTERS 124, 246401 (2020) that its bond dim
 The simplest t
 FIG. 1. (a) Schematics of parton construction for spin-1=2 decomposition,
 latticeREVIEW
 PHYSICAL LETTERS
 models. (b) Schematics 124,network
 of the tensor 246401 (2020) so-called mixed
 representa-
using MPO-MPS compression -> MPS tion of the projected Fermi sea in Eq. (1). gular values [3
 Tensor Network Representations of Parton Wave Functions
 246401-2
 4,*
 Ying-Hai Wu ,1 Lei Wang,2,3 and Hong-Hao Tu
 1
 School of Physics and WuhanNetwork
 Tensor Representations
 National High Magnetic Field Center, Huazhong Wave
 of Parton Functions
 University of Science and Tec
 Wuhan 430074, China

Numerical methods for SU(N)
 ... an introduction
Analytics: large-N, mean-field, parton wavefunctions
• Exact diagonalization (U(1)+lattice symmetries or SU(N) symmetry)

• DMRG (U(1) or SU(N) symmetry) + parton wavefunction

 SciPost Phys. Le
• PEPS using SU(N) symmetric tensors + lattice point-group symmetry

 Both DMRG and PEPS can use
SU(N) symmetry, e.g. QSpace library ,

 Andreas Weichselbaum

Chiral spin liquid with PEPS PHYSICAL REVIEW B 94, 205124 (2016)

Using a classification of SU(2)-invariant PEPS Systematic construction of spin liquids on the square lattice from tensor
 networks with SU(2) symmetry

 Matthieu Mambrini,1 Román Orús,2 and Didier Poilblanc1
 1
 Laboratoire de Physique Théorique, C.N.R.S. and Université de Toulouse, 31062 Toulouse, France
 2
 Institute of Physics, Johannes Gutenberg University, 55099 Mainz, Germany
 (Received 29 August 2016; published 14 November 2016)

 * virtual space : V = S1 S2 · · · Sp
 We elaborate a simple classification scheme of all rank-5 SU(2) spin rotational symmetric tensors accordin
 (i) the onsite physical spin S, (ii) the local Hilbert space V ⊗4 of the four virtual (composite) spins attached to e

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Exact Diagonalization on torus
 Predictions:
 • If Ns=k*N: singlet ground-state degeneracy on a torus = N

 we start by examining, for larger N
 obtained on Ns -site periodic cluster
 about clusters used). We first consid
 an integer multiple of N so that, in
 particle excitations would be popula
 gap above
 of the the Nspectra
 singlet energy quasi-for fixe
 sus
 deg, are shown in Fig. 1 for N ran
 ground-states
 For all the values of N studied here
 values, a clear gap is observed betwe
 and quasi-degenerate states and the
 trum. Interestingly, we note that =
 a pure imaginary E 3-site cyclic permu
 within the gapped phase (except perh
In 2d: generalization
 FIG. 2. Zoom of Hastings-Oshikawa-Lieb-Mattis
 of the singlet low energy spectra at ✓ = ⇡/4 and = ⇡/4 instead was chosen in Ref.
 theorem
 = ⇡/2, for N ranging from 2 to 10, and the same cluster sizes as is also stable within a significant rang
 forbids a non-degenerate gapped state
 in Fig. 1 (for N = 2, Ns = 20). The GS energy is subtracted off for yond ✓ = ⇡/4, e.g. also at ✓ = ⇡/6.
 gaplessbetter
 orcomparison
 discrete symmetry
 between breaking
 the various spectra. The exact or we shall mostly report results obtain
 topological
 degeneracy

5 5
 5 5

 Exact Diagonalization on torus

 excitation energy
 4 M 4 7 3
 4 4
 0 6 2 Σ0
 3
 Γ X Γ 5 Σ2
 3 2 3 1 0
 Γ
 Predictions: 1 8 4
 0 3 1
 Σ1
 3

 2
 • If Ns=k*N: singlet ground-state degeneracy on a torus = N
 2
 2 2

 1 1 1 1

 • Lattice momenta can be obtained from a generalized Pauli principle
 0 0 0 0

 Haldane, Bernevig, Regnault,….

 M.Eb

 Γ.A

 Γ.A
 Γ.B

 Γ.B
 M.Ea

 Γ.A
 Γ.B
 Γ.Eb
 Γ.Ea
 M.A
 M.B

 X.A
 X.B
 Γ.A
 Γ.B

 0
 1
 2
 3
 4
 5
 6
 7
 8

 0

 1

 0
 1
 2
 0
 1
 2
 3

 Σ0

 Σ1

 Σ2
 6 6
 M 6 X 6
 6 ∆ 6 3 6 2 6
 Σ Z 0 4
 5 0
 5 Γ 0
 Γ
 0 24 Γ 3
 5 Z1 5 5 5 5 5
 4 5
 1 1

 excitation energy
 1
 excitation energy

 we start
 4 by examining, for larger N , theM low-energy
 4 spectra 7 3
 4 obtained on Ns -site periodic4 clusters (see Table I for details 4
 6 2
 4 4 4
 0 Σ0
 3
 about clusters used). We first consider
 Γ X the case of Ns being Γ 5 Σ2
 3
 an integer multiple of N so that, in2 a CSL 3phase, no quasi- 3 8 4
 1
 Γ
 0
 3 3 1 3 3 1 3
 particle excitations would be populating the GS. A selection 0
 of the singlet energy spectra for fixed ✓ = ⇡/4, plotted ver- Σ1
 2
 2 sus 2, are shown in Fig. 1 for 2 N ranging N = 4 to N = 10. 2 2 2 2
 For all the values of N studied here, in a broad interval of
 1
 values,
 1 a clear gap is observed1
 between a group 1 of degenerate 1 0 quasi-hole 1
 1 1
 and quasi-degenerate states and the rest of the singlet spec-
 0 quasi-hole
 trum. Interestingly, we note that = ⇡/2 – corresponding to 7
 0 a pure0 imaginary 3-site cyclic0 permutation –0 is alway located
 within the gapped phase (except perhaps for N = 3 for which
 4 0
 0
 0
 0

 Γ.Ea
 Γ.A

 Γ.B
 ∆
 M.A

 M.B
 Γ.B 0

 Γ.Ea 1
 M.AΣ

 0

 M.EaZ1

 Γ.A

 X.A

 Γ.Eb
 Γ.B

 X.B
 Γ.A
 Γ.B

 Γ.A

 Γ.B
 0

 Γ.A 1

 2

 3

 34

 80
 1
 2
 3

 Γ.B 4
 5
 = ⇡/4 instead was chosen in Ref. [28]). The gapped phase
 M.BZ
 M.Eb

 Γ.A
 Γ.B
 Γ.Eb

 X.A
 X.B
 Γ.A

 0
 1
 2

 4
 5
 6
 7

 0

 1
 0
 1
 2
 3

 0

 Σ1

 Σ2
FIG. 2. Zoom of the singlet low energy spectra at ✓ = ⇡/4 and

 Σ
 = ⇡/2, for N ranging from 2 to 10, and the same cluster sizes as is also stable within a significant range of the parameter ✓, be-
in Fig. 1 (for N = 2, Ns = 20). The GS energy is subtracted off for yond ✓ = ⇡/4, e.g. also at ✓ = ⇡/6. Hence, in the following,

Exact Diagonalization on torus
 Predictions: 6 6
 6 6

 • If Ns=k*N: singlet ground-state degeneracy on a torus = N 5 5
 5 5

 excitation energy
 4 M 4 7 3
 4 4 X
 0 6 2 Σ0 2
 3
 Γ X Γ 5 Σ2 4
 • Lattice momenta can be obtained from a generalized Pauli principle 3 2
 1
 3
 8 4
 1
 0 3 Γ 1
 0

 Σ1
 3 Γ
 5
 3
 0

 1
 2 2
 Haldane, Bernevig, Regnault,….
 2 2

 • Quasi-hole counting: deg=Ns, 1 per momentum sector 1 1 1 1

 0 0 0 0

 4 5

 X.A
 M.Eb

 Γ.A

 Γ.A

 X.B
 Γ.B

 Γ.B
 M.Ea

 Γ.A
 Γ.B
 Γ.Eb
 Γ.Ea
 M.A
 M.B

 X.A
 X.B
 Γ.A
 Γ.B

 0
 1
 2
 3
 4
 5
 6
 7
 8

 0

 1

 0
 1
 2
 3
 4
 5
 0
 1
 2
 3

 Σ0

 Σ1

 Σ2
 6 6
 we start 6by examining, for larger N , 6the low-energy spectra
 M
 3 6 6
 X 2
 6 6 0
 ∆ 0
 Σ Z
 obtained
 5
 on Ns -site periodic clusters5
 (see Table I for details
 0
 Γ
 4 0
 Γ
 2
 Γ 2 Γ 3 1
 about clusters
 5 used). We first consider
 0 Z 5 the case of Ns being
 1 5 5 5 5
 4 5 1
 an integer multiple of N so that, 1 in a CSL phase, no quasi- 1
 excitation energy
 excitation energy
 4 M 4 7 3
 particle 4excitations would be0 populating 4 the GS. A selection6 2
 4 4 4 4 X 2
 Σ 0
 of the singlet energy spectraΓ
 3
 2
 Xfor fixed ✓ = ⇡/4, plotted Γ 5ver- Σ 4 0 2
 3 3 1 0
 sus , are 3 shown in Fig. 1 for 1 N ranging3 N = 4 to N 8=4 10. 0 3 3 Γ 1 3 3 Γ 3

 For all the values of N studied here, in a broad interval of Σ 5 1 1
 2 2
 values, a2 clear gap is observed between 2
 a group of degenerate 2
 2
 0 quasi-hole 2
 2

 and quasi-degenerate states and the rest of the singlet spec-
 1 10 quasi-hole 1 1 1 quasi-hole 1 1
 trum. Interestingly, we note that = ⇡/2 – corresponding to 1 1
 1 quasi-hole
 a pure imaginary 3-site cyclic permutation – is alway located
 0 0 0 0
 0 0
 within the gapped phase (except perhaps for N = 3 for which 0 0

 = ⇡/4 instead was chosen in Ref. [28]). The gapped phase

 Γ.Ea
 Γ.A

 Γ.B
 M.B ∆

 M.A

 M.B
 0

 3 1
 X.A Σ

 Z0

 Z1

 Γ.A

 ΣX.A

 Γ.Eb
 Γ.B

 X.B
 Γ.A
 Γ.B

 Γ.A

 Γ.B
 3 0

 1

 6 2

 3

 4

 0 0
 1
 2

 Σ0 3
 4
 5

 0

 X.A 1

 2
FIG. 2. Zoom of the singlet low energy spectra at ✓ = ⇡/4 and
 M.Eb

 Γ.A

 Γ.A

 X.B
 Γ.B

 Γ.B
 M.Ea

 Γ.A
 Γ.B
 Γ.Eb
 Γ.Ea
 M.A

 X.B
 Γ.A
 Γ.B

 0
 1
 2

 4
 5

 7
 8

 1

 0
 1
 2
 3
 4
 5
 0
 1
 2

 Σ1

 2
 = ⇡/2, for N ranging from 2 to 10, and the same cluster sizes as is also stable within a significant range of the parameter ✓, be-
in Fig. 1 (for N = 2, Ns = 20). The GS energy is subtracted off for yond FIG.
 ✓ = 3.⇡/4, e.g. also spectra
 Low-energy at ✓ = on
 M
 ⇡/6. Hence,
 periodic in theatfollowing,
 clusters 19
 fixed = ⇡/2 and for ✓ = ⇡/4 (a-d)Xor ✓ = ⇡/6 (e-h). Clusters with site numbe 13

l l0

 3 q
 1 q7/5
 0 q 2/5

 17/5
on energy excitation energyOrder excitation energy excitation energy
 l l0

 4
 2 q 12/5 2
 1
 1
 2 q
 1 q

 2
 1
 Order

 5
 5
 5
 5
 1
 1
 1
 0 q0 1 •

 5
 2
 1
 1
 1

 3 q3 2 •
 •
 24

 4
 2

 45
 45
 45
 Irreps / Multiplicities
 24
 24

 1
 1
 Irreps / Multiplicities

 2
 1

 50
 50
 75
 75

 2
 1
 1

 70
 70
 TABLE XII. SU(5)1 WZW model – Tower of states starting from
 5
 126

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DMRG Wu, Wang, Tu, PRL 124, 246401 (2020)
 8
 •Parton construction is useful to boost DMRG convergence
 Spectrum on cylinder vs ky
 TABLE II. SU(2)1 WZW model – The direct product of the confor-
 mal tower of the spin-1/2 primary (left - see Table V in Appendix B)
 Exact zero-mode edge states
 with a spin-1/2 gives a new tower (right) with a doubling of the num-
 ber of states in each Virasoro level indexed by l l0 .
 2 2 2
 l Construct
 l0 tower
 2
 N different
 towerminimally
 ⌦
 3
 1
 0 1 entangled states
 1 • 1 to target
 2 3
 1 1 different1 excitations
 • 1
 1

 2 4 ! 3 5
 1
 2 1 1 1 • 2 1
•Probe entanglement spectrum as fingerprint
 3 2
 of
 1
 topological
 2 • 3
 order
 1
 10 2 4 1 3 5 11

 2 4 1 3 5
 4 3 2 3 • 5 2

 lations [48]. For N = 2, it is known that the exact zero modes
 play an important role in constructing the MESs [44, 46].
 These exact zero modes, denoted by dL and dR , localize
 m e n t ! at the two boundaries of the cylinder. Their occurrence at
 a g re e
 p e r f 3̄
 ect 3̄
 the single-particle momentum ky = ⇡/2 requires that for
 mod(Ny , 4) = 0 (mod(Ny , 4) = 2), the parton Hamilto-
 3 3
FIG. 8. The entanglement spectra on width-6 cylinders for SU(3) CSLs. (a) Identity sector. (b) sector (⌦ ). (c) sector (⌦ ). The

Subtlety in choosing MES basis
 DMRG SU(N) subtleties
 Subtlety in choosing MES basis
 Subtlety in choosing MES basis
 = † †
 G ↑ ↓ FS

 Hong-Hao Tu et al.
 Semion sector:
 Semion sector:
 Semion sector:
 † †
 = G ↑ FS
 † † ↓
 = G ↑ ↓ FS
 Semion sector = † †
 G ↑ ↓ FS not a singlet
 SU(2)Spincase
 Not a spin-singlet!
 singlet:
 Spin singlet:
 † † † †
 − ′ = G ( ↑ ↓ − ↓ ↑ ) FS
 ′ † † † †
 − = G ( ↑ ↓ − ↓ ↑ ) FS singlet
 Entanglement spectrum:
 Not a spin-singlet!
 two copies of semion conformal
 towers ( 1/2 ⊗ semion )

 Entanglement spectrum:
 two copies of semion conformal
 towers ( 1/2 ⊗ semion )

 Entanglement spectrum=
two copies of “semion” conformal towers

 ¯ ¯

pendix D as exam-
 the
 A2 1
 Bdirect
 1
 product
 1 2
 =2
 of each
 1
 2
 2
 3
 3
 3
 3
 5 V = around
 4 N • each· site · · can be made
 >
 N following 1 heuristic(8)rules valid
 havecaseus
 = •B1 1 ··· 1 N 1 (8) for all N (which we will
 > generalize to the non-Abelian
 allofagreement
 te 2
 its primary spin with
 >
 >
 ;
 2
 (see
 2 3 3 5
 PEPS
 where the direct sum contains all N irreps defined
 in the conclusion). In
 ;
 other words, we construct the
 Matrix
 spin-
 tain the
and
 I in Appendix
 ABLE III. = 4,ofDrespec-
 N Number as exam-
 symmetric site-tensors in each class charac-
 column Young SU(N ) PEPS from site/bond tensors with virtual space,
 tableaux of 0 up to N 1 boxes, variatio
 co
 in overall agreement
 rized by the irrep of the with
 point group of the square lattice (lines) spectra
 ngle
 C
 N
 XXVI,
 d the occupation
 = 3 and
 XXVII
 Nnumbers ofand
 = 4,
 4v
 the 6,4, 4̄ andwhere
 respec- with
 the(columns).
 1 particles
 the
 direct sum contains all N irreps defined by single the larg
 N •Symmetric
 = 2 and PEPS
 N = 3 cases,
 construction V = 1 2 and
 um contains all N irreps defined column by single
 Young tableaux of 0 up to N 1 boxes, consistently 2 9 each CT
entlyTables XXVI, XXVII
 leaux of 0 up to N 1 boxes,with and 3̄.NFor
 3consistently
 the =FIG.2the
 and N
 10. PEPSN= =
 on3the caseVlattice
 4cases,
 square 2we = then
 1
 involving 2 assume
 and
 site A V>
 >
 =tensors
 3 = 1 V
 and 4 =symmet6
 bond B tensors. The bondVdimension on the black links is up QSpace
 dr1such 3 cases, Vby
 N =a multiplicity = 1 2 and
 2 taking the direct3 (withV
 3̄. For
 3 bond
 =
 product 1the N
 oftoeach dimension
 virtual=
 1350,
 4 case
 space: we
 and the vertical red
 then assume
 =
 Dsegments
 N •
 = 15). V 4 Note
 ·
 correspond
 ·
 = · 6 to > the dthat
 4 N D,
 = N the
 4̄ 1 1 of
 (8)
 site
 singu
 >
 ;
 4 casetower
=onformal we then
 with assume
 the conjugateV4 of = (with 4bond
 6 primary
 its 4̄spindimension
 1(see degreesDof =
 physical 15). Note that the
 freedom. site tensor A reps. S
4̄sion 1
 ables XXVID = 15). Note that the can
 , XXVII and XXVIII in can
 sitebe
 Appendix be seen
 seen as
 tensor
 D as as a linear map
 A a linear map (VN ) ! F, and the bond tensor
 exam- ⌦z (V N ) ⌦z
 ! F, and the
 of all b
 de

 ores).
 S
 FT asA
 ear This
 map
 shown
 brings
 (V )our
 ⌦z simulations
 ! F, andin overall
 B
 the B asas
 bond
 :
 agreement
 : inNFigs. 8 and 9 for N = 3 and N = 4, respec-(V (V
 tensor
 N ) ⌦2 with
 N )! ⌦2 •. ! where•. the direct sum contains
 The simplest adequate ansatz has the following form, all N irreps defined byhave inc
 single
 control
 column Young tableaux of 0 up to N 1 boxes, consistently
 . and a direct comparison with Tables XXVI,
 nsor
 vely,
 construction
 XVIII (see Appendix D).
 As
 As
 forXXVIIN
 for
 =and
 N
 2 and
 = with 3,
 2the we
 and
 N = classify
 2 and
 N
 3,
 X N
 R
 we =the 3 SU(4)-symmetric
 cases,N V2 = 1
 classify
 X I
 the 2 and V3 = 1
 SU(4)-
 w.r.t.
 optimiz
 it

 andCSL we
 breaksclassifyP andthe T site-tensors
 SU(4)-symmetric
 but not PT according A 3= to
 A3̄. (i)
 For
 + the
 iAthe N
 =particle
 = 4 case
 R
 A occupations
 a we+ then
 i assume
 I
 A b
 , n V =(9)=
 6 4 4̄ 1
 ction 3, 4
 R I a R b I occ variatio
 ding to (i) the particle occupations noccP {nsite-tensors
 6 , n 4 , n 4̄ ,
 = n 1 } according
 on (with
 the zbond = 4to
 dimension
 a=1
 (i)
 virtual D the=
 spaces particle
 15).b=1Note
 connected
 can be seen as a linear map (VN ) ! F, and the bondconside
 ⌦z
 that occupatio
 the site
 to tensor
 estimate
 tensor
 A
 n of
 on the chiral PEPS
 z = is4 avirtual
 Tensor used
 linear for
 spaces each
 combination
 IV. IPEPS connectedsite (
 {n6 , nof4 ,point-group n
 where↵ =
 ton4̄ , n1B}as (V
 the z)
 real and
 on (ii)
 N ) the
 elementary
 SU(N) ⌦2 the (1-dimensional)
 •.z = R4 virtual
 tensors
 !symmetric A a
 and A
 ones b
 either irreptrans-of
 spaces co
 detric
 N = 3 (see Ref. [28]).
 and (ii) the (1-dimensional) the C4v pointform
 irrep of
 P
 group of the
 according square
 to the A1 and lattice
 A2 irreps,
 I
 [61]respectively,
 (see Tableor III). ac-
 e n t s
 = ral z) PEPS
 ng the network used
 represented
 A. Symmetric
 for each
 PEPS construction
 site (
 cording
 Since the chiral spinsite-tensors n
 to As
 the
 liquid
 ↵ B =for
 1 andN z)
 only B =
 2 and
 2 and
 irreps,
 breaks (ii)
 3, we the
 respectively,
 P and classify
 T (1-dimensiona
 giving
 but the
 doesc
 rise
 o e to
 not
 c i
 SU(4)-symmetric
 ffi
 up of
 = the square
 •Optimization lattice
 is [61]
 performed (see Table
 using III).CTMRG
 two possible families A according
 and A to
 . (i)
 N the
 = particle
 16 and
 f e w N occupations
 = 17 n =
 (see Ref. [28]). the point group of the square lattice [61] (see
 occ
 irtual
 cc 3 indices on the links break Cthe4vproduct are the PT, {nthe
 numbers , n PEPS
 of, the
 n , n
 A
 complex
 } on
 elementary
 B
 the site
 R
 tensors
 z = tensor
 4 invirtual
 each A I
 should
 spaces
 class and connected to
 in liquid only breaks P and T but does notR 6 4 P
 4̄ 1
 d
 nk-2
 We
 etwork tobond
 now tensors,
 extend the
 represented z being
 construction
 PT, the PEPS complex site tensor A should
 of be
 chiralinvariant
 Since PEPS used
 the (up
 a forto
 chiral
 and a sign)
 I each
 a spin under
 site
 are arbitrary ( real n↵PT
 liquid = symmetry
 z) and (ii)
 only
 coefficients but(1-dimensional)
 the
 breaks
 of these acquires
 tensors.
 P The a irrep
 and T of
 bu
 Both

sha1_base64="kfUz+zFRFc8YJ1o5U59S8+PZFc0=">AAAB63icbVC7SgNBFL3rM8ZXVLCxGQyCVdgVRMsQGytJwDwgWcLsZDYZMjO7zMwGwpJfsLFQxNbSv/AL7Gz8FmeTFJp44MLhnHu5954g5kwb1/1yVlbX1jc2c1v57Z3dvf3CwWFDR4kitE4iHqlWgDXlTNK6YYbTVqwoFgGnzWB4k/nNEVWaRfLejGPqC9yXLGQEm0y6647y3ULRLblToGXizUmxfFz7Zu+Vj2q38NnpRSQRVBrCsdZtz42Nn2JlGOF0ku8kmsaYDHGfti2VWFDtp9NbJ+jMKj0URsqWNGiq/p5IsdB6LALbKbAZ6EUvE//z2okJr/2UyTgxVJLZojDhyEQoexz1mKLE8LElmChmb0VkgBUmxsaTheAtvrxMGhcl77Lk1mwaFZghBydwCufgwRWU4RaqUAcCA3iAJ3h2hPPovDivs9YVZz5zBH/gvP0ASXWRcA==
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PEPS: entanglement spectrum
 infinite PEPS cylinder D=15
 SU(4), Nv=4, full SU(N) symmetry
 12
 = 1350

 sha1_base64="7DGu16HlBmlRFI/FC2R4MC9eNGU=">AAAB8HicbVDLSgNBEOyNrxhfUY9eBoPgKezG+LgIQS8eI5iHJEuYncwmQ2Zml5lZISz5Ci8eFPHq53jzb5wke9DEgoaiqpvuriDmTBvX/XZyK6tr6xv5zcLW9s7uXnH/oKmjRBHaIBGPVDvAmnImacMww2k7VhSLgNNWMLqd+q0nqjSL5IMZx9QXeCBZyAg2VnrskiG79s7O3V6x5JbdGdAy8TJSggz1XvGr249IIqg0hGOtO54bGz/FyjDC6aTQTTSNMRnhAe1YKrGg2k9nB0/QiVX6KIyULWnQTP09kWKh9VgEtlNgM9SL3lT8z+skJrzyUybjxFBJ5ovChCMToen3qM8UJYaPLcFEMXsrIkOsMDE2o4INwVt8eZk0K2Xvoly5r5ZqN1kceTiCYzgFDy6hBndQhwYQEPAMr/DmKOfFeXc+5q05J5s5hD9wPn8AUdKPcQ==
 AAAB63icbVC7SgNBFL3rM8ZXVLCxGQyCVdgVRMsQGytJwDwgWcLsZDYZMjO7zMwGwpJfsLFQxNbSv/AL7Gz8FmeTFJp44MLhnHu5954g5kwb1/1yVlbX1jc2c1v57Z3dvf3CwWFDR4kitE4iHqlWgDXlTNK6YYbTVqwoFgGnzWB4k/nNEVWaRfLejGPqC9yXLGQEm0y6647y3ULRLblToGXizUmxfFz7Zu+Vj2q38NnpRSQRVBrCsdZtz42Nn2JlGOF0ku8kmsaYDHGfti2VWFDtp9NbJ+jMKj0URsqWNGiq/p5IsdB6LALbKbAZ6EUvE//z2okJr/2UyTgxVJLZojDhyEQoexz1mKLE8LElmChmb0VkgBUmxsaTheAtvrxMGhcl77Lk1mwaFZghBydwCufgwRWU4RaqUAcCA3iAJ3h2hPPovDivs9YVZz5zBH/gvP0ASXWRcA==

matrix eigenvalues as ta ða ¼ 0; 1; …Þ with jt0 j > jt1 j >
 jt2 j > % % %, it turns out t0 is nondegenerate, suggesting nondegeneracy agrees with dimer operator bein
 PEPS: correlation lengths
 absence of long-range order in the variational wave rotation invariant. Depending on the parafermion
 ξparafermion have different values, both of which ar
 SU(3) than the spinon SU(4) correlation length. Interestingly,
 10 0 correlation
 5 lengths, except the spin(so correlation len
 that no anyons is present in
 acy is observed on small torii as

 -2
 6 no sign of saturation with increasing χ,theinGS,agreem
 surability between N and N i
 populates N quasi-de
 s
 s
 10
 our expectation that the state is not
 4 in the
 clusters following WZWZ
 pected. Finally, chiral many-bo
 CFT3c
 5
 10 -4 double phase. more stringent test of the exist
 CSL.

 -6 4 Degeneracy
 3 structure of topological chiral
 iDMRG computations by en
 systems – typically P
 infinitely-lo
 10 most valuable and complemen
 remarkable feature of our resultsGutwiller-projected is the corresp parton wa
 construct iDMRG ansatze in e
 10 -8 3
 between 2 the leading four eigenvalues to theirofSU(N the transf
 ) global singlet n
 tions carry larger entanglement
 -10 2
 and the different sectors in the ES: The
 linear Q
 combinations ¼ 0
 of MES, sex
 10 hence, show ES with more stru
 (a) (b) one branch,
 1
 while Q ¼ & 1 each have
 standing three almo
 has been fully provide
 Following the prescriptions
 10
 -12
 0 20 40 60 80 100 120 0 2 4 6 8 10
 1 erate branches.
 0 1 2 This
 3 4 is 5 in 6direct have analogy to
 constricted a familly
 and, under optimization, a goo
 th
 of c

 leading eigenvalue t0 , which hasobtained trivial spin,
 for the chiral SU(4) H
 entanglement spectra obtained
 FIG. 13. Maximum correlation lengths obtained from the transfer

Correlations in the bulk Correlation length approximate
 directly
 by D = 225. from
 matrix (in the absence ofthreefold transfer degeneracy
 numbers associatedmatrix
 gauge flux) plotted versus , normalized of
 tors , t ,
 of atinfinitely-long
 1 the chiral
 tiplicity of
 and
 cylindert
 2 modes is
 FIG. 4. Different bulk correlations in the optimized PEPS. From
 2
 The SU(4) quantum to these

 have perfectly degenerate spins 3one. Finally,
 correlation lengths are indicated.
 and growing 3̄, correlatio
 matc
 of the singlet PEPS ansatz in al
 the correlations versus distance (computed with χ¼ 392) in (a),
 Nothesaturation
 we extract so using
 correlation lengths presumably gapless
 exponential fits,
 perfect
 which are state…tions [30].degeneracy between
 We believe such a property also holds Q ¼
 for any & veryA
 (of1. smallsimilar
 mension are consistent with the
 co
 weight) in correl

 shown in (b) (using the same symbols), along with those extracted dence
 N = 4 due between
 to the large value of(approximate)
 SU(N ) CSL, although it could not be established here for
 the bond dimension D. degeneracy of
 plicitely for N = 2 [30]). We s
 tails would fade away (i.e. thei

 transfer operator and of the ES representation Dubail-Read branches 2015
 vanish) for increasing D, provi
 was
 from the transfer matrix spectrum with or without flux inserted of the GS. If corr
 theorem [67] does not practica
 No-go
 (shown as lines), withtheorem for aof free-fermion
 g the degeneracy the eigenvalue. BothPEPSfortochiralhavePEPSs a bulk withgap
 V. CONCLUSION AND OUTLOOK
 SUð2Þ1 counting, PEPS representation where
 of the topo
 We note that the SU(N ) C
 approaches agree for the spinon, vison, and dimer correlation
 Is it also true in the interacting case ? be explained
 Heisenberg as
 models on the arising
 square lattice havefrom the
 In this work, the previous pairing potential and SU(3) chiral
 been generalized symmetry of
 where the 3-site interaction theis p
 to any SU(N ) fundamental irrep as physical degrees of free- ing to = ⇡/2), mostly stud
 relevant in ultracold atom syste

Abelian CSL: spontaneous T-breaking
 Topological CSL can also be found in the absence of explicit T-breaking
 week ending
 PRL 112, 137202 (2014) PHYSICAL REVIEW LETTERS 4 APRIL 2014 PHYSICAL REVIEW B 92, 125122 (2015)

 Chiral Spin Liquid in a Frustrated Anisotropic Kagome Heisenberg Model
 WIETEK, STERDYNIAK, AND LÄUCHLI
 Nature of chiral spin liquids on the kagome lattice
 PHYSICAL REVIEW B 92, 125122 (2015)

 Yin-Chen He,1 D. N. Sheng,2 and Yan Chen1,3
 erg

 1
 Department of Physics, State Key Laboratory of Surface Physics and Laboratory of Advanced Materials, Alexander Wietek,* Antoine Sterdyniak, and Andreas M. Läuchli
 )

 Fudan University, Shanghai 200433, China
 Institut für Theoretische Physik, Universität Innsbruck, A-6020 Innsbruck, Austria
 Hu (2 eis (2

 2
 Department of Physics and Astronomy, California State University, Northridge, California 91330, USA
 (Received 16 April 2015; revised manuscript received 28 August 2015; published 14 September 2015)
 nb

 3
 Department of Physics and Center of Theoretical and Computational Physics,
 SU

 The University of Hong Kong, Pokfulam Road, Hong Kong, China
 (Received 10 January 2014; revised manuscript received 20 February 2014; published 4 April 2014) We investigate the stability and the nature of the chiral spin liquids which were recently uncovered in exte
 e

 CSL
 Kalmeyer-Laughlin (KL) chiral spin liquid (CSL) is a type of quantum spin liquid without time-reversal Heisenberg models on the kagome lattice. Using a Gutzwiller projected wave function approach, i.e., a pa
 symmetry, and it is considered as the parent state of an exotic type of superconductor—anyon PHYSICAL REVIEW X 10, 021042 (2020)
 superconductor. Such an exotic state has been sought for more than twenty years; however, it remains construction, we obtain large overlaps with ground states of these extended Heisenberg models. We fu
 H

 unclear whether it can exist in a realistic system where time-reversal symmetry is breaking (T breaking) suggest
 FIG. 1. (Color online) Sketch of thethat
 kagomethe
 lattice appearance
 and of the
 different interaction terms of the Hamiltonian (1). Heisenberg
 of the chiral spin liquid in the time-reversal invariant case is linked to a clas
 spontaneously. By using the density matrix renormalization group, we show that KL CSL existsinteractions
 in a between first, second, and third nearest neighbors are
 frustrated anisotropic kagome Heisenberg model, which has spontaneous T breaking. We find thatconsidered.
 our
 transition line between
 The third nearest neighbor Heisenberg interactions are
 two FIG. 2. (Color online) Excitation spectra from exact diagonal-
 ization.magnetically
 Different symbols and colorsordered correspondphases.
 to different
 Chiral Spin Liquid Phase of the Triangular Lattice Hubbard Model:
 )

 momentum/point-group symmetry sectors. We use the cluster ge-
 rd

 only considered across the hexagons. Three-spin scalar chirality
 model has two topological degenerate ground states, which exhibit nonvanishing scalar chirality orderinteractions
 and and breaking time-reversal and parity symmetries, are ometries and notation explained in Ref. [28]. (a) Effect of the J term χ

 areQuantum
 protected by Spin Liquid gap.
 finite excitation with Emergent
 Furthermore, Chiralthis
 we identify Order
 state asin
 KLthe
 CSLTriangular-lattice
 also considered
 by the characteristic Hubbard DOI:
 on gray Model
 10.1103/PhysRevB.92.125122
 shaded triangles. A
 on the spectrum on the 30-siteDensity Matrix
 cluster. The fourfold degeneracy ofRenormalization
 PACS the
 ground state is lifted to a twofold degeneracy which corresponds to
 Group Study
 number(s): 75.10.Kt, 05.30.Pr, 75.10.Jm, 75.40
 ba
 SU

 one sign of the scalar chirality. (b) Scan across the classical transition
 edge conformal field theory from the entanglement spectrum and the quasiparticles braiding statistics 1,2,3,*
 cluster. The fourfold degeneracy 1,2 1,2,4 1,2
 Bin-Bin Chen, 1, 2
 Ziyu Chen, 1
 Shou-Shu Gong, 1, ⇤
 D. N. Sheng, 3
 Wei
 extracted from the modular matrix. We also study how this CSL phase evolves as the system approaches the Li,1, 4, †
 and Andreas
 breaking Weichselbaum
 was discovered for J
 5, 2, ‡
 χ = 0 and 0.2 ! (J 2 = J 3 )/ of
 Aaron
 the CSL is
 Szasz
 line for Jχ = 0 on the 36b sites
 only present
 ,close to J
 Johannes
 2 = J 3 (yellow
 Motruk,
 shading). (c)
 Michael P. Zaletel, and Joel E. Moore
 1 for J = J = 0.4, J = 0, and various system sizes
 Department of χPhysics, University of California, Berkeley, California 94720, USA
 b

 J1 ! 0.7. Here the ground state degeneracy is four, which can Energy spectra
 nearest-neighbor kagome Heisenberg 1 model.
 School of Physics, Beihang University, Beijing 100191, China be understood as arising from two copies of opposite chirality 2 Ns and geometries. Turquoise rectangle: (0,0) [#] momentum, even questions arise: (i) Are the two chiral sp
 I. INTRODUCTION
 2 3

 2
 Arnold Sommerfeld Center for Theoretical Physics, Center for NanoScience, of a twofold degenerate ν = 1/2 Laughlin state. Unlike several
 Materials Sciences
 under 180◦ rotation. Blue Division,
 up triangle: (0,π Lawrence
 ) [M] momentum, Berkeley
 odd National Laboratory, Berkeley, California 94720, USA
 DOI: 10.1103/PhysRevLett.112.137202 and Munich Center PACS fornumbers:
 Quantum 75.10.Kt,
 Science05.30.Pr, 75.10.Jm, 75.40.Mg
 and Technology, topological phases such as Toric code [45] and double-semion
 ◦3
 Perimeter
 under 180 rotation. Red down Institute
 triangle: for
 (0,0)Theoretical
 distinct or are they related? (ii) Is there a
 [#] momentum, Physics, Waterloo, Ontario N2L 2Y5, Canada

 Ludwig-Maximilians-Universität München, 80333 Munich, Germany The[46] quest for quantum spin liquidsDepartment
 phases that also have a fourfold ground state degeneracy, [1] is ofcurrently
 4 rotation, odd under reflection.
 even under 180 ◦
 Physics, Princeton a University, Princeton, New Jersey 08540, USA
 we will show that in this case time-reversal symmetry is (lattice-based) picture or a variational wa
 3
 Department of Physics and Astronomy, California State University, Northridge, very active endeavour in (Received
 California 91330,
 spontaneously broken.USA condensed 8analogy
 August matter
 2019;
 effect revised
 physics. manuscript This
 magnetic received
 field on the 4 describes
 December 2019;the accepted 18 March 2020; published(iii)
 22 May 20
 An SU 4(4) chiral spin liquid and quantized dipole Hall effect in moiré bilayers
 Topological order, an exoticInternational
 state of Research
 matter Institute
 that hostsof Multidisciplinary
 KL state Science, may exist Beihangin University,
 magnetic Beijingfrustrated 100191, China
 systems to the of a longitudinal chiral spin liquid? Wh
 elusive state of quantum matter comes phase, inexperimental
 variousstudies forms
 )

 5
 two degenerate ground states in a ferromagnetic Ising model
 Department of Condensed Matter Physics and Materials Science, III. ENERGY SPECTROSCOPY
 that have d’être”
 fractionalized quasiparticles with anyonic braiding sta- through spontaneously breaking time-reversal symmetry of these chiralspinspin liquids,
 phase in i.e., w
 4

 in the ordered by
 Motivated where the magnetic field immediately found signatures of a quantum liquid
 Ya-Hui
 Brookhaven
 tistics, is one of the core topics in modern condensed- Zhang ,
 National
 1
 D. N. Sheng
 Laboratory, , and
 Upton,
 2
 Ashvin
 New York
 [12,17], which are among the most Vishwanath
 11973-5000,and USA
 1
 is theoretically
 difficult
 To investigate
 systems
 the
 we studiedintensely
 persistence
 limit,for
 of
 the model for J2 =studied,
 this chiral spin liquid at the selects
 however
 one of the two ordered
 wasstates.
 difficult
 As we show later based
 (

 J3 = organic crystals whose structure is well by Jχdescribed by the two-dimensional triangular lattice, we study the
 and using theliquids
 spin stabilized for the two reported H
 thermodynamical on overlaps, the chiral spin liquid thus selected is of the
 (Dated: February 11, 2021)
 SU

 matter physics [1]. A quantum spin liquid (QSL) [2] is a theorists to study exactly. toUSApin down
 system sizesin computational studies ofto therealistic
 TRS symmetric quantum
 0.4 and Jχ = 0 up to 42 sites. The low-energy spectra for Hubbard same typemodel
 as the oneon stabilized
 this in the J1 -Jat
 lattice χ model
 half alone,
 filling infinite-system density matrix renormalization
 method. On infinite cylinders withwe come up with someanguiding principle wh
 1
 Department of Physics, Harvard University, Cambridge, thattoMA, and
 different are shown in Fig. 2(c). While the energy is connected situation in the absence
b 2021

 prominent example of topological
 The interplay order, whichspin
 between frustration andIn
 is thought this
 charge Letter,
 fluctuationwe showrise
 gives the KL state
 an exotic quantumis the ground
 state in the
 splitting between the four ground states has a nonmonotonous group of Jχ(iDMRG) finite circumference, we identify intermediate phase
 to exist in some frustrated magnets [3]. Among
 intermediate-interaction regime various state of
 of the half-filled a frustrated anisotropic
 triangular-lattice Hubbard (TLU) spin
 kagomemodel, Hamiltonians
 Heisenberg
 whilethethe model
 nature of and the fourhard to states
 characterize .
 unambiguously in interaction
 the J2 = J3 condition (in the absence of Jχ ) by fixing to stabilize CSLs on other lattices? In the f
 behavior, energy gap between lowest energy In Fig. 2(b) we investigate the effect of a deviation from
 2
 Department of Physics and Astronomy, California State University, Northridge,and theCA 91330
 fifth one increases with the system size. Moreover, the
 between observed metallic behavior at low strength and Mott insulating spin-ordered behavior
 types of QSL [3–11], the therestateisis aunder
 classdebate. Using the density (KHM)
 of time-reversal matrix renormalization
 by using group with
 the density matrix experiments
 SU(2) spin ⌦U(1)
 renormalization
 ratio ofcharge
 the energy on
 symmetries
 group quantum
 splitting to the energy gapmagnets.
 decreases with at strong
 J2 = 0.5interactions.
 and varying J3 .Chiral ordering
 One observes that from spontaneous breaking of time-reversal symmetry, a fractionally
 the fourfold
 ground state degeneracy is rapidly lifted when J3 deviates address each of these questions. In short, we
 implemented, we spin
 studyliquid
 the TLU (Dated:
 model defined March
 on the 19,
 long 2021)
 cylinder geometry
 symmetry violating QSL called chiral (CSL) (DMRG) [30], a numerical methodwith which circumference
 The
 thehas
 realized Sbeen
 system
 =
 at the
 Wtends
 proven
 size; this
 1/2
 = to4.indicate that this phase is indeed quantized
 thermodynamical Heisenberg antiferromagnet
 limit. It is also important to
 spin Hallon
 more than about 0.05–0.1 the
 response,
 from 0.5.kagome
 and characteristic level statistics in the entanglement spectrum in the
 Interestingly the line
 [12–14]. A CSL shares some A gapped
 Motivated quantum
 similar by the spin liquid,
 recent proposal
 properties with on-site
 with theof realizinginteraction
 powerful an SU(4)9 . U/t . 10.75, is identified
 Hubbardquasi-one-dimensional
 in solving model on triangular between moiré
 notice that
 the metallic
 super- sectors involved in the fourfold 0 < J = J < 1 is the classical transition line between a
 frustrated
 the momentum