Flavour-selective localization in interacting latticefermions via SU(N) symmetry breakingDaniele Tusi

LENS - European Laboratory of Non-Linear SpectroscopyLorenzo Franchi

University of FlorenceLorenzo Livi

University of FlorenceKarla Baumann

SISSA - Scuola Internazionale Studi SuperioriDaniel Benedicto-Orenes

ICFO https://orcid.org/0000-0002-8098-012XLorenzo Del Re

Università di RomaRafael Barfknecht

CNR-INO https://orcid.org/0000-0002-3883-8437Tianwei Zhou

https://orcid.org/0000-0001-5830-6326Massimo Inguscio

LENS and Università degli Studi di FirenzeGiacomo Cappellini

LENSMassimo Capone

SISSA - Scuola Internazionale Superiore di Studi AvanzatiJacopo Catani

INO-CNR & LENS - University of FlorenceLeonardo Fallani ( fallani@lens.uni.it )

University of Florence https://orcid.org/0000-0001-5327-3254

Article

Keywords: quantum physics, superconductors, topological insulators

Posted Date: June 9th, 2021

DOI: https://doi.org/10.21203/rs.3.rs-543907/v1

License: This work is licensed under a Creative Commons Attribution 4.0 International License. Read Full License

Flavour-selective localization in interacting lattice fermions1

via SU(N) symmetry breaking2

D. Tusi1, L. Franchi2, L. F. Livi2, K. Baumann3,4, D. Benedicto Orenes5,6, L. Del Re7, R. E. Barfknecht5,3

T. Zhou2, M. Inguscio8,1,5, G. Cappellini5,1, M. Capone3,9, J. Catani5,1, L. Fallani2,1,5,104

1 LENS European Laboratory for Nonlinear Spectroscopy (Sesto Fiorentino, Italy)5

2 Department of Physics and Astronomy, University of Florence (Sesto Fiorentino, Italy)6

3 SISSA Scuola Internazionale Superiore di Studi Avanzati (Trieste, Italy)7

4 Laboratoire de Physique et Etude des Materiaux, UMR8213 CNRS/ESPCI/UPMC (Paris, France)8

5 CNR-INO Istituto Nazionale di Ottica del Consiglio Nazionale delle Ricerche (Sesto Fiorentino, Italy)9

6 ICFO - Institut de Ciencies Fotoniques, The Barcelona Institute of Science and Technology (Castelldefels, Spain)10

7 Department of Physics, Georgetown University, 37th and O Sts., NW, Washington, DC 20057, USA11

8 Department of Engineering, Campus Bio-Medico University of Rome (Roma, Italy)12

9 CNR-IOM Istituto Officina dei Materiali, Consiglio Nazionale delle Ricerche (Trieste, Italy)13

10 INFN National Institute for Nuclear Physics (Firenze, Italy)14

15

A large repulsion between particles in a quantum system can lead to their localization, as it16

happens for the electrons in Mott insulating materials. This paradigm has recently branched out17

into a new quantum state, the orbital-selective Mott insulator, where electrons in some orbitals are18

predicted to localize, while others remain itinerant. We provide a direct experimental realization19

of this phenomenon, that we extend to a more general flavour-selective localization. By using an20

atom-based quantum simulator, we engineer SU(3) Fermi-Hubbard models breaking their symmetry21

via a tunable coupling between flavours, observing an enhancement of localization and the emergence22

of flavour-dependent correlations. Our realization of flavour-selective Mott physics opens the path to23

the quantum simulation of multicomponent materials, from superconductors to topological insulators.24

Interactions shape the collective behavior of many-25

particle quantum systems, leading to rich phase diagrams26

where conventional and novel phases can be induced by a27

controlled variation of external stimuli. The most direct28

example is perhaps the Mott insulator, where the large29

repulsion amongst particles leads to an insulating state30

despite the non-interacting system being a metal or a31

superfluid1. In solid-state physics the interest in Mott32

insulators is reinforced by the observation that the proxim-33

ity to a Mott transition is a horn of plenty where a variety34

of novel and spectacular phases can be observed, high-Tc35

superconductivity2,3 being only the tip of an iceberg4.36

In recent years it has become clear that the standard37

SU(2) Fermi-Hubbard model is only a specific example, as38

many interesting materials require a description in terms39

of “multicomponent” Hubbard models, e.g. when the40

conduction electrons have an additional orbital degree41

of freedom. These systems are not merely more compli-42

cated, rather they host new phenomena, challenging the43

standard paradigm of Mott localization5. Indeed, when44

the symmetry between orbitals is broken, by some field or45

internal coupling, electrons in specific orbitals (or some46

combinations of them) can be Mott-localized while others47

remain itinerant, leading to surprising “orbital-selective48

Mott insulators”6.4950

Orbital-selective Mott physics has become a central51

concept for the description of a new class of high-Tc52

superconductors based on iron7, as it can describe the53

anomalies of the metallic state8,10 and the orbital charac-54

ter of superconductivity11 in those systems. However, a55

clean observation of selective Mott transitions is intrinsi-56

cally hard in solid-state systems, because of the limited57

FIG. 1. Sketch of the physical system. a) We consider asystem of repulsively SU(3)-interacting fermions in a lattice.The global symmetry is broken by a coherent Rabi driving Ωbetween two internal states. b) In the absence of coupling, thesystem experiences a phase transition from an SU(3) metal to aMott insulator as the hopping is reduced. c) The Rabi drivinglifts the degeneracy between the states and, in a dressed-basis picture, causes them to acquire different energies. Thecompetition with the hopping can drive a transition from ametal to an insulator already in the noninteracting case.

experimental control over the microscopic parameters and58

the orbital degree of freedom. The paradigm of selective59

2

FIG. 2. Measurement of double occupancies. a) Number of atoms in doubly-occupied sites Nd as a function of the totalatom number N for two different values of Ω and the same U = 2.6D. b) Average doublon fraction fd as a function of U/Dand Ω/D. The actual measurements are marked by the points. c,d) Subsets of the data are shown for two different crosssections of the plot in b), i.e. for Ω = 0 (c) and for U = 2.6D (d). The measurements in b,c,d) are averages over an intervalN = (5 . . . 35)× 103. Error bars in a,c,d) are obtained with a bootstrap analysis20. Dotted lines in a) are fits with a piecewisefunction (null + straight line), while color shades in c,d) are guides to the eye representing the experimental uncertainty.

Mott physics is itself far from being fully explored and60

has the potential to become a powerful framework to un-61

derstand a variety of phenomena, from superconductivity62

to topological properties, in multicomponent quantum63

materials.64

In this work we take a broader perspective and treat65

orbital-selective Mott physics as an example of the gen-66

eral concept of “flavour-selective Mott localization”12,67

which can be realized in a variety of multi-flavour sys-68

tems, where the flavour can be the spin, the orbital or any69

other quantum number. We realize a minimal instance of70

this phenomenon by means of an atomic quantum simu-71

lator based on the optical manipulation of nuclear-spin72

mixtures of ultracold two-electron 173Yb atoms. This73

platform allows the realization of multicomponent sys-74

tems with global SU(N) interaction symmetry13,14, as in75

recent works reporting the realization of SU(N) quantum76

wires15, SU(N) Mott insulators16,17 and, more recently,77

SU(N) quantum magnetism18. Here we introduce a novel78

approach to break the symmetry of SU(3) Fermi-Hubbard79

systems in a controlled way, which allows us to go beyond80

the investigations in the solid state and to observe di-81

rectly the two key signatures of selective-Mott physics12:82

an overall enhancement of Mott localization and the onset83

of flavour-selective correlations.84

The experiment is performed with three-component85

ultracold 173Yb Fermi gases with total atom number up86

to N = 4× 104 and temperature T ≃ 0.2TF . The atoms87

are trapped in a cubic 3D optical lattice (lattice constant88

d = λ/2 = 380 nm), which realizes the multi-flavour89

Hubbard Hamiltonian90

H =− t∑

〈i,j〉,α

(

c†αicαj + h.c.)

+ U∑

i,α,β 6=α

nαinβi + VT

+Ω

2

∑

i

(

c†1ic2i + h.c.)

(1)where α, β ∈ 1, 2, 3 indicate the fermionic flavours91

(corresponding to nuclear spin states m = +5/2,+1/292

and −5/2, respectively), t is the tunnelling energy be-93

tween nearest-neighboring sites 〈i, j〉, U is the onsite re-94

pulsion energy between two atoms of different flavours,95

and VT = κ∑

i,α R2i nαi describes the effects of a slowly-96

varying harmonic trapping potential (where Ri is the97

distance of site i from the trap center). In the absence98

of the 4th term of Eq. (1), H has an intrinsic global99

SU(3) symmetry, which is ensured by the invariance of100

atom-atom interactions on the spin state and by the re-101

alization of spin-independent optical potentials, i.e. U , t102

and κ do not depend on α. This symmetry is explicitly103

broken by the 4th term, which describes a coherent on-104

site coupling between flavours |1〉 and |2〉. This coupling105

is provided by a two-photon Raman process with Rabi106

frequency Ω/h20 (where h is the Planck constant). At107

the single-particle level, this coupling lifts the degeneracy108

between the flavours, creating two dressed combinations109

|±〉 = (|1〉 ± |2〉) /√2, energy-shifted from |3〉 by ±Ω/2,110

as sketched in Fig. 1c.111

We use an adiabatic preparation sequence to produce an112

equilibrium state of the atomic mixture in the optical lat-113

tice, with equal state populations N1 = N2 = N3 = N/3114

and in the presence of the coherent coupling Ω between115

states |1〉 and |2〉20. In order to characterize the degree116

3

U/t

/t

a) c)DMFT (homogeneous) 1D DMRG (trapped)

U/D

2 3 4

/D

10.0

0.2

0.4

0.8

0.6

Mott insulator

flavour-selectivephase

metal

DMFT (trapped)b)

FIG. 3. Theoretical analysis. a) DMFT phase diagram for a homogeneous system at T = 0 and 1/3 filling. The points referto different parameter runs. The boundary between the standard metal and the region of selective correlations marks a sharpcrossover where the quasiparticle weight of the coupled flavours rapidly drops to a value smaller than 0.05. b) Doublon fractionfor the trapped system at U = 2.6D and different Ω/D, obtained from a LDA analysis of the homogeneous DMFT results withthe experimental parameters. c) Doublon fraction obtained from DMRG for a 1D trapped system of N = 21 particles at T = 0.

of localization of the particles in the lattice, we measure117

the number of atoms Nd in doubly-occupied sites (called118

“doublons” in the following) with photoassociation spec-119

troscopy, as in previous experiments that demonstrated120

the onset of Mott localization for ultracold fermions16,19,20.121

In Fig. 2a we report typical measurements of Nd as a122

function of the total atom number N . In a harmonically123

trapped system, the rate of change of Nd vs N gives infor-124

mation on the core compressibility19,21. A vanishing value125

of Nd over an extended range of N signals the presence126

of an incompressible state with one atom per site in the127

center of the trap (since adding particles does not lead128

to a proportional increase of doublons), while the critical129

N above which Nd then departs from zero can be con-130

nected to the magnitude of the energy gap protecting the131

localized phase. Fig. 2a shows two datasets for U = 2.6D132

(where D = 6t is the tunnelling energy times the lattice133

coordination number), and two different Rabi couplings134

Ω = 0 and Ω = 1.6D, respectively. The range of N in the135

figure spans a range of local fillings up to ≈ 0.8 atoms/site136

per state (for the largest N) in the center of the trap in137

the noninteracting case. The comparison between the two138

datasets clearly shows that the Rabi coupling Ω results139

in an enhanced suppression of doublons, enlarging the140

region of N where the incompressible state forms.141

In the following we take the doublon fraction fd =142

〈Nd/N〉, averaged over the N interval marked by the gray143

region in Fig. 2a, as an indicator of the degree of Mott144

localization of the system. The measured values of fd145

are shown as a function of U and Ω in the diagram of146

Fig. 2b, clearly revealing the cooperative effect of Rabi147

coupling and repulsive interactions driving the system148

towards a Mott-localized state. The same data are plotted149

with error bars in Figs. 2c,d along two different line cuts150

of the diagram in Fig. 2b. Fig. 2c shows the effect151

of an increasing U in the transition towards an SU(3)152

Mott insulator for Ω = 0, while Fig. 2d shows a similar153

localization effect induced by Ω at a fixed interaction154

strength U = 2.6D.155

We now show the comparison of the experimental re-156

sults with different theoretical analysis of the model in157

Eq. (1). Fig. 3a shows a zero-temperature phase diagram158

obtained from Dynamical Mean-Field Theory (DMFT)22159

for the homogeneous system (VT = 0) with a uniform160

1/3 filling (one atom per site) and equal number of parti-161

cles for each flavour. The phase diagram clearly has the162

same shape of the experimental one, showing that the163

Hubbard U and the coupling Ω cooperate in driving the164

system from a metallic phase to a more localized state.165

Between the standard metal and the Mott phase, we find166

a region where the degree of correlations (as measured by167

the quasiparticle weight) is strongly selective, the coupled168

flavours being much more localized than the uncoupled169

one. It is evident that both selective and global Mott170

localization occur at a smaller U if Ω is included.171

In order to connect this effect to the experimentally172

measured signal, in Fig. 3b we show the doublon fraction173

fd obtained from the homogeneous DMFT results after174

a local-density approximation (LDA) analysis, to take175

into account the effect of the harmonic trapping in VT .176

The reduction of fd with increasing Ω is in agreement177

with the experimental observations reported in Fig. 2.178

The lack of a quantitative matching with Fig. 2d can be179

attributed to imperfections in the initial state preparation180

and to the finite temperature of the experiment, resulting181

in an average entropy per particle S/N ≈ 2.5kB , which is182

known21 to produce an effect on the double occupancies183

in a trapped system already at Ω = 0.184

In Fig. 3c we finally show the result of a zero-185

temperature DMRG calculation of fd for a harmonically186

trapped 1D system20. Although a quantitative agreement187

with the experimental data should not be seeked (because188

of the different dimensionality and the finite temperature189

of the experimental realization), the overall behavior, i.e.190

4

FIG. 4. Evidence of state-selective correlations. Ex-perimental values of γ(12) as a function of Ω/D for a fixedU = 2.6D. The circles are averages over an interval N =(5 . . . 35) × 103, error bars are obtained from a bootstrapanalysis20, while the color shade is a guide to the eye. Thecrosses are obtained from a LDA analysis of the DMFT re-sultes. The dashed line shows the expected value for anSU(3)-symmetric system.

a reduction of doublons for increasing U and Ω, is cor-191

rectly captured and agrees also with DMFT. This also192

indicates that the phenomena we are exploring are generic193

and qualitatively independent on the dimensionality.194

In addition to the enhancement of Mott localization,195

the DMFT phase diagram of Fig. 3a shows that Ω is196

expected to result in a flavour-dependent localization of197

the many-body system. The flavour-selective behavior198

can be experimentally detected by resolving the spin199

character of the doublons, i.e. counting how many atoms200

form doublons in each of the three pairs |12〉, |23〉, |31〉201

by means of state-selective photoassociation at a high202

magnetic field20. In Fig. 4 we plot the quantity γ(12) =203

Nd(12)/Nd, where Nd(12) is the number of atoms forming204

doublons in the |12〉 channel, as a function of Ω and fixed205

U = 2.6D. The measured value at Ω = 0 agrees with the206

expectation for an SU(3)-symmetric system, for which207

γ(12) = 1/3. As Ω is increased and the SU(3) symmetry is208

broken, γ(12) diminishes, eventually approaching zero for209

Ω ≈ D. The doublons acquire a strongly flavour-selective210

behavior.211

This suppression of |12〉 doublons is triggered by the212

polarization effect in the rotated |±〉 basis, which can213

be understood, at a qualitative level, already from the214

simplified, noninteracting case sketched in Fig. 1c: while215

|23〉 and |31〉 doublons can be formed by two fermions in216

the lowest single-particle states |+〉 and |3〉, |12〉 doublons217

can be formed only if the two fermions occupy states |+〉218

and |−〉, therefore with an additional energy cost Ω/2.219

Interactions then increase this effect12, leading to strong220

flavour-selective results even for small values of Ω.221

The crosses in Fig. 4 are the result of a DMFT cal-222

culation in which the flavour populations are kept equal223

and the harmonic trapping VT has been taken into ac-224

count in a LDA approach. The results of this numerical225

calculation are in good agreement with the experimental226

findings, with a larger degree of selectivity for the theoret-227

ical calculation. We argue that a better agreement could228

be seeked by including finite-temperature effects in the229

calculation: indeed, we expect the state-selective behavior230

to be reduced by the thermal occupation of higher-energy231

states, leading to an effective reduction of the polarization232

in the |+〉, |−〉 basis.233

We note that the experimental observation of a finite234

number of |12〉 doublons at small, but finite Ω is an235

indication of the validity of the protocol used for the236

preparation of the atomic state, which is different for Ω =237

0 and Ω > 020. We also note that, despite the polarization238

in the dressed |+〉, |−〉 basis discussed before, we have239

verified that the populations of the bare states |1〉, |2〉, |3〉240

remain always equal under all the experimental conditions241

we have considered. This is indeed an important aspect of242

our experiment. If the populations of the bare states were243

not fixed (and in particular N3 was left free to adapt), the244

dressed states would be populated according to the scheme245

in Fig. 1c, leading to N1 = N2 > N3, which favours246

flavour-selective physics already in the non-interacting247

system. The experimental state preparation procedure248

counteracts the trivial differentiation between flavours by249

forcing an even occupation. Therefore, we conclude that250

the flavour selectivity we observed is essentially due to251

quantum correlations induced by interactions.252

In the theoretical DMFT calculations, the equal pop-253

ulation constraint is enforced including an external field254

h20 that favours the occupation of |3〉 in such a way to255

match the experimental condition N1 +N2 +N3 = N/3.256

Comparing with Ref. 12, where the populations were left257

free, we observe that the quantum correlations leading to258

the selective regime survive to the inclusion of the field259

h. In general terms, single-particle effects trigger flavour260

selectivity, but the inclusion of interactions strongly en-261

hances the differentiation, turning a minor modulation262

of kinetic energy into a quantitative phenomenon which263

can also lead to a selective Mott transition24. This is a264

very general framework, underlying many investigations265

of multicomponent models.266

Exploiting an idealized quantum simulator we have ob-267

tained a clear-cut evidence for correlation-induced flavour-268

selective physics, where the SU(N) symmetry-breaking269

coupling Ω is only the trigger of a flavour-selective phe-270

nomenon which is fundamentally driven by correlation271

effects. Our first realization of multicomponent Hubbard272

physics with coherent internal couplings opens new paths273

for the quantum simulation of new classes of materials274

ranging from high-temperature superconductors to inter-275

acting topological insulators as described by the Bernevig-276

Hughes-Zhang model23. As the coupling is realized with277

5

a nonzero momentum transfer (i.e. non collinear Raman278

beams), a full range of possibilities will emerge, including279

the study of magnetic crystals25, fractional quantum Hall280

states26, and the effect of interactions of topological phase281

transitions27 and on the associated edge states28.282

Acknowledgments. We acknowledge insightful dis-283

cussions with M. Dalmonte, D. Clement, F. Scazza and284

financial support from projects TOPSIM ERC Consolida-285

tor Grant, TOPSPACE MIUR FARE project, QTFLAG286

QuantERA ERA-NET Cofund in Quantum Technologies,287

MIUR PRIN project 2017E44HRF, MIUR PRIN project288

2015C5SEJJ, MIUR PRIN project 20172H2SC4 004,289

INFN FISh project. L.D.R was supported by the U.S.290

Department of Energy, Office of Science, Basic Energy291

Sciences, Division of Materials Sciences and Engineering292

under Grant No. DE-SC0019469.293

Authors contribution. L.Fa., J.C. and M.C. con-294

ceived the experiments. D.T., L.F.L., L.Fr., D.B.O.295

and G.C. carried out the measurements. L.Fa., L.F.L.,296

L.Fr. and D.T. analyzed the experimental results. K.B.,297

L.D.R. and M.C. performed DMFT calculations. R.E.B.298

performed DMRG calculations. L.F.L. and L.Fa. per-299

formed exact diagonalization calculations. All authors300

contributed extensively to the discussion of the results301

and to the writing of the manuscript.302

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