Figure 7.1: Single-particle spectrum of the Hofstadter Hamiltonian (7.6) forα= 1/2,γ = 0in the first magnetic Brillouin zonek∈[−π, π]×[−pi/q, π/q]withq= 2. At zero energy, the two magnetic bands are touching each other via Dirac cones, making the system a semi-metal for a Fermi energyF = 0. The quasi-momenta are shown in unitskG = (π, π/2).
fundamental Hofstadter problem. Remarkably, such a scenario is feasible using cold-atoms in artificial gauge fields [57, 54]. Thus, the semi-metallic Dirac dispersion for evenq becomes a generalization of graphene with a tunable number of Dirac cones. Energy gaps which were crossed by a single chiral edge mode in the QHE setup, are now traversed by a helical Kramer’s pair of edge states, corresponding to aZ2 topological insulator phase, as discussed in Sec.7.1. Note that one can use the same Gedankenexperiment to construct the Kane-Mele model [101] from two time-reversed copies of Haldane’s honeycomb model [76]. The Kane-Mele model with additional Hubbard interaction has recently been intensively studied [169, 89, 152, 197, 117, 201], in contrast to the Hofstadter problem, for which the effect of interactions has up to now not been considered.
In this section, we study the effect of interactions in the time-reversal invariantHofstadter-Hubbard model using real-space dynamical mean-field theory (RDMFT, see chapter 1). We explain our numerical results using analytical arguments obtained from perturbation theory and renormalization group approaches.
We consider interaction effects on both (semi-)metallic and gapped topological phases. Although Z2
topological insulators are known to be robust against disorder [138, 161, 148], rigorous and general results on the fate of topological insulators in the presence of Coulomb or Hubbard interactions are limited [198, 118, 71]. Some three-dimensional materials of the iridate family are possible candidates for systems where strong spin-orbit coupling and Coulomb interactions compete [145, 102]. In two spatial dimensions, however, topological insulator phases have so far only been found in HgTe/CdTe quantum wells [9, 110], where Coulomb interactions seem to be negligible.
7.2.1 Hofstadter-Hubbard Model
The non-interacting part of the Hamiltonian consists of two independent copies of the Hofstadter model (6.21), where the two distinct fermionic flavors (which we refer to as spins) feel an opposite effective magnetic field, i.e. a fieldB=Bσzez, whereezis the unit vector inz-direction andσzis the Pauli matrix.
These distinct copies are then coupled by a Rashba-type spin-orbit coupling in thex-direction, which
7.2. Time-Reversal Invariant Hofstadter-Hubbard Model with Ultracold Fermions 115
gives the possibility to flip the spin of a particle that propagates in thex-direction. This Hamiltonian can be realized in cold-atom experiments (see Sec. 5.5) and states
H0=−tX
x,y
c†x+1,yei2πγσxcx,y+c†x,y+1ei2πασzxcx,y+h.c.. (7.6)
Here,c†x,y = (c†↑,x,y, c†↓,x,y)at lattice site(x, y),σx, σz are Pauli matrices andγ ∈ [0,0.25], α = p/q are the spin-orbit coupling strength, the magnetic flux per unit cell in units of the Dirac flux quanta, respectively. In the following, we express all energies in units oft ≡ 1. The realization of the non-interacting Hamiltonian (7.6) in cold-atom experiments, was proposed by Goldmanet al.[57], where also the realizable topological phases of this model have been analyzed in terms of edge states forα= 1/6.
According to theZ2classification of TR-invariant systems, one finds non-topological (semi-) metallic and normal insulator phases and topological non-trivial QSH phases, depending on the parameterγand the Fermi energy of the system.
Figure 7.2: Magnetizationm=n↑−n↓in the Néel state is plotted versus interaction strengthU. Shown are lines corresponding to the valuesα = 1/2 (blue),1/4 (red),1/6 (magenta),1/8 (black), and1/10 (green). Inset: Fermi velocity2πvF (red symbols) for differentα= 1/qis shown versusq. AlsoUc(blue symbols) obtained within RDMFT versusqis shown. The magenta line is a fit ofvF to∝ 1/q and the cyan line ofUcto∝1/q2. Please note that the inset only holds forα= 1/qwith evenqand the fit is not to be understood as an interpolation for arbitraryq.
To include the effect of interactions, we add a Hubbard-type local interaction to the non-interacting Hamiltonian (7.6), such that the total Hamiltonian reads
H =H0+UX
x,y
n↑,x,yn↓,x,y, (7.7)
where the on-site interaction strengthU > 0 can be tuned experimentally using Feshbach resonances and by adjusting the lattice depth.
We first consider the Hubbard-Hofstadter problem for generalα = p/qat half-filling. For q odd the system is metallic with a nested Fermi surface, and antiferromagnetic Néel order occurs for infinitesimally small interactionU = 0+ as for the ordinary square lattice4. Forqeven the situation is very different because the system is a semi-metal (SM) at half-filling. The non-interacting band structure exhibitsqDirac cones (with a multiplicity of2due to the spin), which are separated by momentum2π/q in momentum space5. Theα= 1/2case is thus very similar to graphene (but note that the coordination number for the square lattice isz = 4rather thanz= 3for the case of graphene). For smallerαon the other hand, the
4This is in contrast to three dimensions, where a critical interaction strengthUc>0is required for antiferromagnetic ordering.
5The lattice constantaof the square lattice is set to unitya≡1.
system embodies a generalization of graphene with a tunable number of valleys.
We investigate the SM-insulator transition for variousα = 1/q (q even) within RDMFT. In Fig. 7.2, the magnetization is shown as a function of interaction strengthU. The insulating phase forU > Uc
is antiferromagnetically (AF) ordered with a magnetization pointing in thez-direction and an ordering wave-vectorQ = (π, π), which is the common Néel vector. We find that the critical value ofUc to enter the insulating and magnetically ordered phases decreases with increasingq. This is expected from the increasing scattering that can take place between the cones. AtUc we also observe a simultaneous opening of a single particle gap. Within our approach we thus find no sign of an intermediate non-magnetic gapped phase, which would indicate a possible spin-liquid phase [130, 8]. To understand the behavior ofUc(q), we make use of Herbut’s argument [85] (see Sec. 7.3). Herbut considered graphene and studied the SM-insulator transition within a large-N approach, and found thatUcdepends on2N, the number of Dirac cones (Nrefers to the spin degeneracy), and the Fermi velocityvF asUc∝vF/2N. As shown in detail in Sec. 7.3, we are able to match our results with Herbut’s analysis by replacing the Fermi velocities and2N =qN, whereqis again the number of Dirac cones in the single-particle spectrum. In fact, from the band structure atU = 0, we findvF ∝1/q. Consequently, settingN = 2for spin-1/2 particles,Uc should exhibit a 1/q2 behavior, which agrees very well with the RDMFT data, shown in the inset of Fig. 7.2. We further note thatα = 3/8, which exhibits a differentvF thanα = 1/8, is in agreement with our findings.
7.2.2 Tunable Magnetic Order
In this part, we consider the effect of finiteγin the Hamiltonian (7.6), i.e. the presence of Rashba-type spin-orbit coupling that breaks the axial spin symmetry, on the interaction induced magnetic ordering.
Finiteγdoes change the type of magnetic order in general. To demonstrate this, we consider fixedU = 5 atα = 1/6and calculate the magnetization pattern for γ = 0.125and γ = 0.25in Fig. 7.3 obtained within RDMFT. We obtain similar results for other values ofαandγ. Forγ= 0.125, the magnetization lies in theSy−Szplane and the spatially dependent magnetization reads
m(r) =stot(0,−cosπx
3 cosπy,sinπx
3 cosπy), (7.8)
wherestot is the modulus of the magnetization and is a function of the interaction strengthU and the spatial coordinater. Forγ= 0.25, the magnetic order is given by
m(r) =stot(0,0,cosπy). (7.9) Tuning the parameterγ, we thus pass from Néel order (γ= 0) to spiral order (γ= 0.125, shown in Fig. 7.3 and Eq. (7.8)) to collinear order (γ= 0.25, shown in Fig. 7.3 and Eq. (7.9)), thereby crossing two magnetic quantum phase transitions. Finally, we note that the modulus of the magnetizationstot is staggered for the intermediate value ofU shown in Fig. 7.3. The staggering decreases for larger values ofU, reducing its spatial dependence in the limitU −→ ∞.
We can qualitatively understand this type of magnetic order by rigorously deriving a quantum spin Hamiltonian for even stronger interactions when charge fluctuations freeze out completely at half-filling.
For the derivation, we consider a Hubbard model with a finite number ofN lattice sites at half-filling, which reads
H=HU+Ht=U
N
X
i=1
n↑,in↓,i−tX
hi,ji
c†ieiϕˆijcj, (7.10) where the phaseϕˆijis a lattice site dependent2×2matrix, according to Eq. (7.6). In the strong coupling limit, i.e. U >> t, the hopping HamiltonianHtis considered as the perturbation and the ground state of the unperturbed system is the2N times degenerate6. To obtain a low-energy effective Hamiltonian, we apply a unitary transformation to the Hamiltonian (7.10), which is known as the Schrieffer-Wolff transformation. Consider therefore the hermitian operatorS, such that
H˜ =e−iSHeiS (7.11)
6The ground state at half-filling consists ofNsingly occupied sites, where two spin degrees of freedom are possible at every lattice site.
7.2. Time-Reversal Invariant Hofstadter-Hubbard Model with Ultracold Fermions 117
Figure 7.3: Real space magnetization profilem(r)inSy−Szplane forα= 1/6,U = 5andγ = 0.125 (a) andγ = 0.25, respectively. The spiral order in (a) and collinear order in (b) can be explained by the effective spin model (7.22). The color scheme indicates the modulus of the magnetization, which is staggered for intermediateU.
is a unitary transformation of the original Hamiltonian. Expanding this transformation up to second order in the operatorSleads to
H˜ =H+i[H, S]−1
2[[H, S], S] +O(S3). (7.12) Choosing the operatorSsuch that
i[HU, S] =−Ht (7.13)
results for the Hamiltonian in
H˜ =HU+ i
2[Ht, S] +O(S3). (7.14)
By making use of (7.13), the matrix elements of the operatorScan be expressed in the eigenbasis ofHU, which we callB, leading to
hm|S|ni=ihm|Ht|ni Em−En
, (7.15)
where|mi,|ni ∈ Band Em, En are the eigenenergies of HU for the corresponding eigenstates. The diagonal elements ofS can be chosen equal to zero, sinceHthas vanishing diagonal elements in the eigenbasis ofHU. The matrix representation ofScan be exploited, leading to
HtS =X
m,n
Ht|mihm|S|nihn|=iX
m,n
Ht|mihm|Ht|ni Em−En
hn|. (7.16)
After inserting (7.16) and its hermitian conjugate, the Hamiltonian (7.14) transforms to H˜ =HU −1
2 X
m,n
Ht|mihm|Ht|ni Em−En
hn|+|nihn|Ht|mi Em−En
hm|Ht
. (7.17)
Projecting the Hamiltonian (7.17) to the manifold of degenerate ground states, where the low energy physics takes place, leads to the states|nibelonging to the ground state manifold, withEn = 0. On the other hand, the application ofHtalways takes states out of the ground state manifold, such that|mimust always be an excited eigenstate with eigenenergyEm =U. We can therefore drop both sums in (7.17) and write
H˜ =HU− 1
UP Ht2P, (7.18)
wherePis the projector onto the ground state manifold. Since the first operator on the right of Eq. (7.18) is equal to zero in the ground state manifold, only contributions from the second operator have to be taken into account. The only non-zero contributions to the Hamiltonian can be expressed as
H˜ =−t2 U
X
hi,ji
c†ieiϕˆijcjc†jeiϕˆjici. (7.19)
This operator can now be transformed into a sum of spin-1/2operators, resulting in a Heisenberg-type model and recovering the antiferromagnetic Heisenberg Hamiltonian foreiϕˆij =1. For the Hamiltonian (7.6), the matrix elements for coupling in thex-direction andy-direction have to be treated separately.
The coupling in they-direction is described by the matriceseiϕˆij = ei2πxiσz, such that for a specific summand of (7.19)
c†ieiϕˆijcjc†jeiϕˆjici = c†iei2παxiσz (1−n↑,j) c↑,jc†↓,j c↓,jc†↑,j (1−n↓,j)
!
e−i2παxiσzci
= c†iei2παxiσz
−Szj −S−j
−Sj+ Szj
e−i2παxiσzci
= c†i
−Szj −Sj−ei4παxi
−S+je−i4παxi Szj
ci
= −2SzjSzi−cos(4παxi)
S+jS−i +Sj−S+i
−isin(4παxi)
S+jS−i −S−jS+i , (7.20) where we have defined the spin operators
Szi = n↑,i−n↓,i
2 , S+i =Sxi +iSiy=c†↑,ic↓,i. (7.21) For the coupling in thex-direction, we find a similar expression, which in combination with (7.20) leads to the Heisenberg-type effective spin Hamiltonian
H˜ = JX
x,y
Sxx,ySxx+1,y+ cos(4πγ)
Syx,ySyx+1,y+Szx,ySzx+1,y + sin(4πγ)
Szx,ySyx+1,y−Syx,ySzx+1,y
+ JX
x,y
Szx,ySzx,y+1+ cos(4παx)
Sxx,ySxx,y+1+Syx,ySyx,y+1 + sin(4παx)
Syx,ySx,y+1x −Sxx,ySyx,y+1 , (7.22) where we have introduced the coupling constantJ ≡4t2/U.
The first part of Eq. (7.22) describes a spin exchange inx-direction. Forγ= n2 withn∈Z, we obtain a simple antiferromagnetic Heisenberg term. Other values ofγ, however, break theSU(2)symmetry and cause anisotropy of XXZ-type withSxas the anisotropy direction in spin space. Forγ 6= n4, there is an additional Dzyaloshinskii-Moriya (DM) interaction term in the YZ-plane [43, 139], which is responsible for the spiral spin order in Fig. 7.3(a). Spin exchange in they-direction is periodic with an extended unit cell in thex-direction depending on the fluxα=p/q: for oddqthe unit cell containsqlattice sites, but for evenq it only containsq/2 lattice sites, reflecting second order perturbation theory. For instance, one finds for theπ-flux lattice (α= 1/2) an ordinary Heisenberg exchange term. For other values ofα, the XY-term exhibits a modulation of its amplitude depending onα, while theZ-term always favors AF Ising order. This rich magnetic order predicted by the spin Hamiltonian is in agreement with our RDMFT findings.
7.2.3 Topological Phases in the Hofstadter-Hubbard Model
In this section, we study the effects of interactions on systems that have a bulk gap, but possible gapless edge excitations, i.e. the effect of interactions on theZ2 classification. ForU = 0, we distinguish the normal (NI) and topological (TI) insulating phases by calculating theZ2 invariantν using Hatsugai’s method [52].
ForU >0, we identify the phases by computing the spectral function in a cylindrical geometry using RDMFT and counting the number of gapless helical edge states crossing the bulk gap. The TI phase exhibits an odd number of helical Kramer’s pairs per edge while the NI phase has an even number,
7.2. Time-Reversal Invariant Hofstadter-Hubbard Model with Ultracold Fermions 119
Figure 7.4: EF-λ-phase diagram forα = 1/6 atγ = U = 0. We find insulating phases for fillings nF = l/6, with l ∈ {0,1,2,4,5,6} and (semi-)metallic phases otherwise. The insulating phases are either normal (NI) or quantum spin Hall (QSH) insulators. The staggering potential induces a phase transition from NI to QSH forl= 2,4and also from semi-metal to NI at half-fillingl= 3.
including zero. Edge states are also crucial for detecting topological phases in cold-atom experiments, and we numerically study how robust they are with respect to interactions. In the following, we focus on fixedα= 1/6, which qualitatively captures all phenomena that occur in this system for generalα=p/q.
We also consider an additional term in the Hamiltonian, that is available in cold-atom setups [57, 54]: a staggering of the optical lattice potential along thex-direction
Hλ=λX
j
(−1)xc†jcj, (7.23)
which is added to the Hofstadter-Hubbard Hamiltonian (7.7).
In the axial symmetric case ofγ= 0, there exist TI phases only away from half-filling, since the system is a (semi-)metal fornF = 1(and not too largeλ, U). This is shown in Fig. 7.4, and is expected as the spinless Hofstadter problem atα = 1/6 exhibits a QHE with Chern numberC =±2ifF lies within the two energy gaps closest to zero and a QHE withC=±1forF in the other gaps7(see Fig. 7.5). The Chern number corresponds to the number of chiral edge modes in an open geometry. In the time-reversal invariant system at hand, we thus find an according number of helical Kramer’s pairs within the gaps.
For a filling ofnF = 1/6,5/6the system is thus a TI. We observe this topological phase to be stable even for large interactions up toU = 10. A NI-TI phase transition can be induced in the other gap for nF = 2/6,4/6 by applying a sufficiently large staggered lattice potentialλ ≤ 1 (see Fig. 7.4). Fixing nF = 2/6, we now turn on interactions, and observe that this phase is quite stable as shown in Fig. 7.6.
Eventually, large enough interactions reverse the effect of the staggering potential and drive the system into the NI phase. Note that a static Hartree-like approximation (red dashed line) yields comparable re-sults for smallU but overestimates the effect of staggering for larger values ofU.
A topological phase at half-filling occurs only if we break the axial symmetry in the system by con-sideringγ > 0. We present both the non-interactingλ-γphase diagram8, shown in Fig. 7.7(a), and the interactingλ−Uphase diagram for different values ofγ, which is shown in Fig. 7.7(b). Both semi-metal and QSH phases are robust up to interaction strengths of orderU ≈3−5, at which point larger inter-actions drive the system into a magnetically ordered state. A qualitative understanding of the interacting phase diagram follows from the observation that interactions mainly reverse the effect of staggering.
Prominently, we observe an interaction-driven NI to QSH transition forγ = 0.25andλ ≥1.5, and a metal-QSH transition for0.22≤γ <0.25andλ≥1. Using RDMFT for a cylinder geometry, we are able to directly observe the behavior of the edge states in the interacting system. Gapless edge states are key ingredients to different detection schemes of topological phases in cold-atom systems [173, 172, 200, 3].
Since topological (QSH) phases are uniquely characterized by their helical edge states [196], a probe of
7The plus, minus sign for the Chern numberChere refers to spin-up, spin-down fermions, which feel opposite magnetic fields and therefore the bulk gaps belong a Chern number with opposite sign.
8The non-interactingλ-γphase diagram has also been obtained in [57], however, they have made a small mistake in their calculations, such that their result differs from the one that is presented here.
Figure 7.5: Spectrum of the spin-up fermions for the caseα= 1/6, showing QH phases for the first and second bulk gap with Chern numbersC=−1,−2, respectively. The corresponding topological phase for a system described by the time-reversal invariant Hofstadter-Hubbard model has the topological invariant ν=C mod 2 = 1,0, i.e. a QSH phase for the first bulk gap and a NI phase in the second bulk gap.
Figure 7.6: Left: U-λ-phase diagram at nF = 2/6 and γ = 0 obtained within RDMFT, the red line indicates the phase boundary obtained by a Hartree-Fock-type static mean-field theory, which coincides with RDMFT for small interactions. Interactions reverse the staggering induced phase transition from a NI to a TI. This can be understood within a simple mean-field picture, as the interactions prefer a uniform density distribution, while the staggering prefers a staggered density distribution. Right: Mean-field picture of compensation of the staggering potential by interactions.
these states is the most direct measurement [57, 58, 21]. In Fig. 7.8, we give an example of the spectral functionA(ky, ω)for the interaction driven NI-QSH transition atγ = 0.25, λ = 1.5. ForU = 0.5, we find no gapless edge modes that connect the two bulk bands, which correspond to the NI. On the other hand, atU = 2, we clearly find a single pair of helical edge modes traversing the bulk gap, which corre-sponds to the QSH phase.
In this section, we investigated the Hofstadter-Hubbard model using RDMFT complemented by analytical arguments. We quantitatively determined the interacting phase diagram including two additional terms relevant in cold-atom experiments, a lattice staggering and Rashba-type spin-orbit coupling. Interactions drive various phase transitions. Similar to graphene, we find that a semi-metal at half-filling turns into a magnetic insulator at a critical finite interaction strength. Rashba-type spin-orbit interactions lead to tunable magnetic order with collinear and spiral phases. We explicitly demonstrate the stability of the topological phases with respect to interactions, and verify the existence of robust helical edge states in the strongly correlated TI phase, which is crucial for experimental detection schemes.