Self-dual $S_3$-invariant quantum chains

We investigate a self-dual quantum chain with $S_3$ symmetry, the most general such three-state chain with nearest-neighbor interactions and time-reversal and parity symmetries. We find a rich phase diagram including gapped phases with order-disorder coexistence, integrable critical points with U(1) symmetry, and ferromagnetic and antiferromagnetic critical regions described by three-state Potts and free-boson conformal field theories respectively. We also find an unusual critical phase described by combining two conformal field theories with distinct"Fermi velocities".

We investigate a self-dual quantum chain with S3 symmetry, the most general such three-state chain with nearest-neighbor interactions and time-reversal and parity symmetries. We find a rich phase diagram including gapped phases with order-disorder coexistence, integrable critical points with U (1) symmetry, and ferromagnetic and antiferromagnetic critical regions described by threestate Potts and free-boson conformal field theories respectively. We also find an unusual critical phase described by combining two conformal field theories with distinct "Fermi velocities".
Introduction. Since even before the Time of Landau, a common strategy in statistical mechanics has been to stipulate the symmetries of the system and construct the simplest model obeying them. With a Z 2 symmetry, this strategy yields the much-studied Ising model. The resulting critical point separating ordered and disordered phases in the two-dimensional classical case and the corresponding one-dimensional quantum chain is self-dual 1 . "Parafermionic" quantum chains with n states per site and Z n symmetry 2 are natural generalizations that have been intensively studied recently, for reasons including the appearance of topological order and potential experimental realizations 3 .
However, very little systematic exploration of the simplest and most symmetric parafermionic chains has been done, a shame given their importance. We here aim to rectify the situation, by analyzing a one-parameter family of three-state quantum chains with S 3 symmetry and self-duality, the most general such chains with nearestneighbor interactions invariant under time-reversal and spatial parity. We find a rich variety of previously unknown critical and gapped phases, using detailed conformal field theory (CFT) and numerical techniques.
A sum of two Hamiltonians. The Hamiltonian we study is an arbitrary linear combination of two self-dual nearest-neighbor Hamiltonians where λ P = cos θ and λ 1 = sin θ. The well-known H P , given in (4), is the ferromagnetic three-state Potts chain at its critical point 4,5 . It is self-dual, with S 3 , translation, time-reversal and spatial parity symmetries. The lesser-known H 1 given in (5) has all of these symmetries and also a U (1) that when combined with selfduality yields Onsager-algebra symmetries and large degeneracies 6 . Both Hamiltonians are critical and integrable individually 7,8 , but their linear combination (1) is not integrable and not always critical.
Overview of Phase Diagram. Our results are summarized in the phase diagram in Figure 1. Four critical phases dominate the diagram, but very interesting gapped regions occur as well.
Three of the four large critical phases are described by well-known CFTs. The c =4/ 5   critical phases, as it is stable under all self-dual timereversal (or parity) invariant perturbations. The "Potts 1" region includes the integrable self-dual Potts Hamiltonian H P , but the existence of "Potts 2" region requires a more subtle analysis. The antiferromagnetic Potts region with a c =1 CFT also includes an integrable point, and its robustness contrasts with the instability of the corresponding critical point in the classical square-lattice model 9 . The fourth critical region, which we dubbed "c =3/2", is quite novel. We will explain how it is best thought of as a combination of two interacting CFTs with different "Fermi velocities". Another striking consequence of the self-duality is the existence of two unusual gapped phases with an emergent fractional supersymmetry. Both feature order-disorder coexistence, one between conventional Z 3 order and disorder, but the other between a novel "not-A" order and an SPT phase 10 .
The model. We study a quantum chain with three states for each of the L sites. The Hamiltonian is written in terms of operators σ j and τ j for j = 1, . . . , L, arXiv:1912.09464v1 [cond-mat.stat-mech] 19 Dec 2019 all acting on the 3 L dimensional Hilbert space. Each operator acts non-trivially only on a single site j, e.g. σ 2 = 1 ⊗ σ ⊗ 1 · · · 1, so that operators based on different sites commute. They obey the algebra with ω = exp 2πi/3 . In a σ j -diagonal basis, The Hamiltonian we study is Kramers-Wannier selfdual, with duality-broken cases analyzed in the companion paper 10 and in a future work. Although duality is not a symmetry (it is neither unitary nor invertible here), its action is simplified by ignoring subtleties with boundary conditions and by considering only its action on certain operators. The duality is then The self-dual quantum 3-state Potts Hamiltonian with periodic boundary conditions is the simplest one invariant under (3). It has an S 3 symmetry generated by charge conjugation, σ j ↔ σ † j , τ j ↔ τ † j , and by a Z 3 "shift": σ j → ωσ j , τ j → τ j . The latter is generated by Q 3 = j τ j , with the resulting charge r defined by Q 3 |ψ = ω r |ψ . It is also invariant under parity and time-reversal symmetries, namely P : σ j → σ L+1−j , τ j → τ L+1−j , and T : σ j → σ † j , τ j → τ j . Under the latter, complex numbers are conjugated as well.
The other nearest-neighbor self-dual Hamiltonian invariant under all symmetries of the Potts model is 10 Here the Z 3 is promoted to a U (1) symmetry. Moreover, it has a "dynamical" lattice supersymmetry as it obeys H 1 = Q 2 , with a fermionic Q that changes the number of sites 11 . Combining H P and H 1 yields the full model (1). Writing H(θ) in terms of Temperley-Lieb generators gives the same expression as in Ref. 12 but in a different representation with different physics. Other similar Hamiltonians 13,14 are distinct as well.
Potts phase 1. The continuum limit of the ferromagnetic self-dual three-state Potts point H(0) = H P is given by a CFT with c =4/5 15 . The toroidal partition function is known exactly 16 , and because the energy levels in a CFT are directly related to the dimensions of the scaling operators creating them 17,18 , one can find all the scaling dimensions exactly as well. Namely, the energy difference E a −E b ∝ (∆ a −∆ b )/L , where ∆ a and ∆ b are the dimensions of operator creating the states labelled by a and b respectively. The ratio of any two energy differences is universal, and so can be compared with lattice simulations as L → ∞.
In the region of such a critical point, the long-distance behavior is governed by an effective field theory found by perturbing the CFT by any relevant or marginal operators invariant under all the symmetries of the lattice model. No such self-dual operator obeying the symmetries of H(θ) exists in the Potts CFT, with the least irrelevant such operator having dimension 14/5. 15,19 Thus in the region of H P , perturbing by H 1 must be irrelevant, and the Potts CFT continues to describe H(θ) for |θ| small, as illustrated on the right of Figure 1.
Terminating Potts phase 1. Effective field theories also provide a nice way to understand the transitions out of this Potts phase. Namely, consider breaking the self-duality of H P by making the coefficients of the two types of terms in (4) unequal. When the coefficient of the first term is larger, σ j = 0, spontaneously breaking the S 3 symmetry. The self-dual H P then describes a transition between order and disorder. Including the irrelevant perturbation H 1 does not change the situation, so this critical order-disorder phase transition persists along the self-dual Potts line.
Standard arguments 20 indicate that in one direction the critical line terminates at a tricritical three-state Potts (TCP) point, a CFT with c = 6/7 21 . As opposed to the c =4/5 CFT, here there does exist a single relevant self-dual operator of dimension 10/7 invariant under all symmetries of H(θ) including the S 3 . Perturbing by this operator gives a field theory describing the corresponding flow from TCP to Potts 22,23 . The physics for θ positive is thus a close analog of the Ising chain perturbed by its least irrelevant self-dual parity-invariant operator 24,25 .
A similar flow occurs for θ negative. Here, however, the critical point terminating the phase is a free-boson CFT with c =1, the continuum limit of the U (1)-invariant integrable point H(−π/2) = −H 1 8,26,27 . Here a dimension 3/2 operator obeys all the symmetries of H(θ), but not the U (1) 10 . It is natural to identify this operator with H P , and so the U (1)-invariant critical point is unstable. Indeed perturbing by this operator results in a flow to the c =4/5 fixed point 28,29 , with no intervening phases.
We confirmed this picture via DMRG 30,31 using ITensor 32 . By checking a variety of ratios (6), we located the tricritical Potts point at λ 1 ≈ 0.297λ P , i.e. θ =θ TCP ≈ 0.092π. Namely, we found for −π/2 < θ < θ TCP , these ratios approach the Potts CFT predictions as L → ∞, while for θ ≈ θ TCP , they approach the TCP values. We plot the ratio R 1 1 in Fig. 2, where Our computations of the scaling of the entanglement entropy 33 are consistent.
Potts phase 2. An elegant bosonic field theory describes H(θ) in the region near U (1)-invariant critical point with Hamiltonian −H 1 29 . As detailed in our companion paper 10 , this effective field theory is the same for either sign of λ P , implying that the same flow occurs on both sides of θ = −π/2. Thus somewhat surprisingly, a second critical phase is described by the same c =4/5 CFT as for the ferromagnetic Potts critical point, even though λ P < 0 means that the contribution of H P to H(θ < −π/2) is antiferromagnetic. Breaking the selfduality shows that this Potts critical line describes an unusual transition, between "not-A" order, where two of the three directions of spin are favored, and a symmetryprotected topological (SPT) phase 10 .
Even more surprisingly, the second Potts phase terminates at the far end in the same way as the first phase. Increasing the magnitude of λ P while keeping it negative, we encounter another TCP point at λ P ≈ 0.67λ 1 < 0 (θ ≈ −0.69π). One ratio (6) illustrating this behavior is shown in Figure 3.
Order-disorder coexistence. Changing θ away from θ TCP is a relevant perturbation by an operator of dimension 10/7. However, as opposed to the behavior at the c =1 point, the effective field theories are not the same for both signs of perturbation. In one direction, the flow is to the Potts CFT, while in the other direction, the flow is to a gapped region with order-disorder coexistence. In the latter, the physics is an S 3 analog of that in the tricritical Ising model 24,25,34 .
This coexistence is apparent directly in the lattice model. We show in Appendix A how four exact ground states occur at the frustration-free point λ 1 = λ P /3 where θ = θ ff ≈ 0.102π. Three of these four ground states are completely ordered, with σ j = ω A for A = 0, 1, 2. The fourth is the equal-amplitude sum over all states in the σ j -diagonal basis, and so is completely disordered. The latter ground state is dual to the other three, as hinted at by the fact that is a product state in the τ jdiagonal basis. The self-duality thus requires a coexistence between order and disorder (O-D) in this phase.
Our numerics confirm that O-D coexistence persists throughout a gapped phase. Moreover, as in the analogous phase surrounding the Majumdar-Ghosh point in a frustrated antiferromagnet 35 , it contains an incommensurate length scale past the frustration-free point 36 . Namely, for θ > θ ff , level crossings occur amongst excited states, and the correlators exhibit oscillations on top of the exponential decay. These oscillations are readily apparent in σ † i σ j plotted in Figure 4. The O-D phase in the analogous two-state case has an unbroken emergent supersymmetry 23 , with a lattice Hamiltonian that can be written as the sum of two supersymmetric ones 25 . In the S 3 -invariant case here, there is an emergent "fractional supersymmetry" 23 , a consequence of conformal spin ±4/3 operators 21 remaining symmetry generators off the TCP point. The effective field theory of the O-D phase thus can be written as the sum of two cubes of parafermionic operators 23,37 , On the lattice we find the intriguing structure that for a parafermionic operator Q, while in general Expressions for Q and Q 1,2,3 and their derivation are given in Appendix B. While the lattice expressions (7,8) hardly seem coincidental given the emergent symmetry, the precise relation seems hard to pin down. A second gapped O-D phase? Since the second Potts phase terminates in the TCP point at the bottom left of Figure 1, universality arguments imply also a gapped O-D phase where the three ground states of the not-A phase coexist with that of the SPT phase. A direct lattice derivation however is difficult, as no frustration-free point occurs here. Moreover, our DMRG numerics indicate a very long correlation length with substantial oscillations, although the ground states we find have a fairly small bond dimension. We take the latter as a strong sign of the existence of a gap in at least a small region, but very possibly a small gapless incommensurate region also occurs, labeled in Figure 1 as "IC?". At θ ≈ −0.73π, a transition occurs to the... Antiferromagnetic Potts phase. The third of the major critical regions surrounds the integrable antiferrogmagnetic three-state Potts (AFP) model H(π) = −H P . At this integrable point, the long-distance description is a c =1 free-boson CFT 9,38 as at θ =−π/2, although here the U (1) symmetry is emergent. The self-duality and S 3 symmetry require that both CFTs have the same bosonic compactification radius as well 10 .
An important distinction between the two, however, is that the AFP Hamiltonian is stable under symmetrypreserving self-dual perturbations. The stability arises because the lattice analog of the relevant dimension-3/2 CFT operator has momentum π relative to the antiferromagnetic ground state, according to our numerics. Since H(θ) is invariant under translation invariance, such an operator cannot appear in the effective field theory around θ = π. All other relevant operators are disallowed as before, resulting in an AFP phase. This stability does not occur in the corresponding square-lattice classical model 9,38 , presumably because its interactions are antiferromagnetic in both space and "time" directions, while in our Hamiltonian setup, interactions in the "time" direction are effectively ferromagnetic.
where e.g. ∆ a,B is a scaling dimension in the free-boson theory. We give in Appendix C a list of all relevant and marginal operators with their symmetry properties. From the list it is apparent that this c =3/2 CFT has no relevant self-dual symmetry-preserving operators. Numerics indicate that indeed H(θ) remains critical as θ is varied from π/2. Namely, exact diagonalization indicates that energy differences of low-lying states remain proportional to 1/L, as in a CFT. Moreover, our DMRG calculation of entanglement-entropy scaling 33 in this region remains consistent with that in a c =3/2 CFT. The criticality therefore extends to a full phase.
However, we find that the critical theory changes as θ is varied. Although energy differences still can be fitted to (6) as L → ∞, the scaling dimensions no longer obey (9) for θ = π/2. Instead, where, crucially, the ratio of "Fermi velocities" v TCI /v P does not depend on the level a (to reasonably good numerical accuracy). 42 Indeed, using exact diagonalization we show in Figure 5 various velocities found from energies E j a,k for the j th excited state in the Z 3 = a, momentum k sector. If (5) were exact, a decomposition into two CFTs H(θ) → v P (θ)H P + v TCI (θ)H TCI would presumably hold throughout this critical region, as a consequence of an exactly marginal perturbation. However, two marginal perturbations of conformal spin 2 are possible and so (10) may only be approximate (as the data at the extremes of Figure 5 suggest); the two critical theories may be coupled via another marginal operator. For this reason, we include quotes in the "c =3/2" denoting this critical region in Figure 1.
The reason such interactions are possible is the presence at θ = π/2 of three marginal self-dual operators at having dimension 2 and conformal spin 2 invariant under all symmetries (along with their partners of conformal spin −2). One such operator is simply the energymomentum tensor of the full CFT, and the decomposition into two CFTs then accounts for a second. The presence of the third such operator is a fascinating consequence of the two distinct decompositions: we find it must be present for there to be two sets of two energymomentum tensors 43 .
Incommensurate transitions The only other bits of our phase diagram are the transitions out of the "c =3/2" phase. In the direction of the anti-ferromagnetic Potts phase, the Fermi velocity v TCI in Figure 5 is vanishing, indicating a phase transition. Our numerics indicate a small gapless incommensurate region centered around θ ≈ 0.9π. As the "c =3/2" phase is approached from the gapped O-D phase, the incommensurabilities get larger and the gap gets smaller, and again numerics indicate the possibility of a small gapless incommensurate region between the two at θ ≈ 0.19π.
Conclusion. We have found the phase diagram of the one-dimensional self-dual 3-state Potts model perturbed by the only self-dual nearest-neighbor interaction obeying all of its symmetries. As well as critical ferromagnetic and antiferromagnetic Potts phases, we find an unusual "c =3/2" critical phase, and at least one gapped orderdisorder coexistence phase. The rich behavior extends off the self-dual line; for example, four gapped phases meet at the U (1) invariant c =1 point at θ = π/2 10 . Elsewhere, breaking self-duality can preserve gapless behavior, as we will describe in another companion paper. 306, 470 (1988). 42 To obtain (10), we consider only levels not degenerate at θ = π/2; degenerate levels have a more complicated mixing. 43 We have found the explicit linear relations between the two sets, making it possible to also relate certain combinations of primary fields in the Potts and TCI CFTs to free fermions and bosons. These relations make possible the derivation all sorts of marvelous identities for CFT correlators, a topic we will return to in a future publication, along with an extension to more general theories.  We here derive the four exact ground states present at the frustration-free point λ P = 1, λ 1 = 1/3. They are where σ |A = ω A |A for A = 0, 1, 2, while |0 ≡ (|0 + |1 + |2 )/ √ 3 so that τ |0 = |0 . The first three ground states are completely ordered in the σ-diagonal basis, while the last is the equal-amplitude sum over all states.
The derivation relies on writing the corresponding Hamiltonian H(θ ff ) as a sum over projectors. First we write it as where P B (j) and P C (j) are two-site projectors: As |00 is annihilated by both two-site projectors, |00...0 must be a ground state of the Hamiltonian as a whole. Similarly, applying B j and C j , we see that |AA is also annihilated by both projectors for A = 0, 1, 2. Considering three sites in a row, j, j + 1, j + 2, it can be shown that the only states annihilated by P 2l (j), P 2r (j) and P 2l (j + 1) are |000 , |AAA , analogously to the twostate per site case in Ref. 25. Continuing this along the chain we obtain the four ground states given.
It is worth noting that using the Temperley-Lieb formulation 12,44 , we have also found analogous frustration-free points for the general perturbed n-state Potts model for n ≥ 3 with a nearest-neighbor S n preserving perturbation. These have n+1 degenerate ground states, n of which are competely ordered and one completely disordered. Supersymmetric Hamiltonians can be written in the form H = Q 2 for some fermionic Q 45 . In our earlier work 25 we showed that in the critical Ising chain with a self-dual perturbation, the Hamiltonian can be written as H 2 = Q + 2 + Q − 2 . The fermionic Q ± are sums over products of odd numbers of Majorana fermions γ a on neighboring sites: The couplings λ I and λ 3 give the strengths of the Ising term and the perturbation respectively in H 2 . Since Q + and Q − do not commute, the supersymmetry is not exact on the lattice, but it emerges in the continuum in the O-D phase.
To generalize this construction to our 3-state model, it is natural to consider parafermions ψ a instead of Majorana fermions. They are defined by (B2) They obey the algebra ψ † j = ψ 2 j , ψ 3 j = 1, ψ j ψ k = ωψ k ψ j , for j < k . We need a candidate Q to be a sum over terms with parafermion number 1 modulo 3 for it to cube to something local. The simplest Q giving something non-trivial when cubed is Such a Q is not Hermitian. To obtain a Hamiltonian invariant under charge conjugation, parity and timereversal requires β 3 ∈ R and α = 2ω 2 β. Then some tedious algebra gives where we ignore a constant term. This expression without its last line is exactly the frustration-free Hamiltonian (A2) in the order-disorder coexistence phase when β = −1, as noted above in (7).
The last line of (B4) is self-dual but not nearestneighbor like H(θ) is. To remove it and to be able to match H(θ) away from the frustation-free point, we instead consider analogously to the sum in the Ising case. We then define , β a,j = βe iθa e 2πaji 3 , µ a = θ a + 2θ a .
In terms of the Q 3 formulation there are clearly three special values ν = 0, 1/2, ∞, where we treat ν = −∞ and ν = ∞ as the same point. ν = 0 corresponds to the ferro/antiferromagnetic Potts model, depending on the sign of α. This is analogous to the Ising case, where the Ising point was recovered by taking one of the terms in Q to ∞ and the other to 0. ν = ±∞ corresponds to λ P = λ 1 and is the point where the τ j +σ † j σ j+1 +h.c. term vanishes. ν = 1/2 is more of a mystery and corresponds to λ p = 5λ 1 . This just appears to be a random point in the ordered Potts phase.
Appendix C: The c =3/2 CFT The CFT representing the continuum limit of H 1 has central charge c =3/2 and corresponds to the orbifold of the product of a free boson and free fermion CFT: Z s-a ( √ 3) in the conventions of Ref. 41. This theory has a variety of remarkable properties, including an exact lattice supersymmetry 11 , and an Onsager-algebra symmetry 6 (our Hamiltonian H 1 here is −H 0 there). An even more remarkable property is the two ways of splitting into two CFTs 41 .
Here we confine ourselves to presenting a list of all lowdimensional operators and their symmetry properties, to show that the only perturbing operator allowed in the effective field theory is marginal. This list follows from using the partition functions presented in Ref. 41 along with a careful analysis of the discrete symmetries.
All marginal and relevant operators are listed in Table I. Each field is given in terms of its dimension ∆, along with its spin, s, Z 3 charge, r, momentum on the lattice, k, left and right dimensions for both TCI+Potts and boson+fermion electric and magnetic charges [m, n], and behavior under both dualities D and D . The dimension ∆ is the sum of the left and right dimensions (the holomorphic and antiholomorphic parts) , while the spin is the difference. The spin gives the difference in momentum relative to either k = 0 or k = π when multiplied by 2π/L. The electric and magnetic charges [m, n] arise from the U (1) and dual U (1) symmetries of the free-boson CFT respectively. Horizontal lines across the whole table separate states with different dimension, spin, Z 3 charge or lattice momentum, while those split by lines stretching only from column five onwards differ only under duality. Those states not separated by any lines obey all of the same symmetries (ignoring the U (1)). Such states can and do mix when finding the eigenstates of the TCI and Potts Hamiltonians.
The Z 3 charge ω r is specified solely by the Potts field in the TCI + P language, and by m in the B+F language. For Potts the two (1/15, 1/15) fields have r = ±1, as do the two (2/3, 2/3) fields, while all others have r = 0. For the boson, the Z 3 charge is given by m modulo 3. The momentum k is given by m + 2n in the B + F picture and is specified by the TCI field for TCI + P. We explain this fact in a future work by breaking the duality. The momentum derives from the Z 2 in the TCI model: 0, 1/10, 3/5, 3/2 have k = 0 while 3/80, 7/16 have k = π, as can be seen by consistency with the boson and fermion representation.
Thus in addition to the two stress-energy tensors, there is a third marginal operator obeying every symmetry of the Hamiltonian. This mischievous operator is, presumably, the reason why the model may not split exactly to a combination of the Potts and TCI models with different Fermi velocities as we perturb from the integrable point. As we perturb along the self-dual line, we might expect the dimensions of the operators to change as the field theory changes due to the marginal operators. As the only marginal operators are spin 2, we see that they can never have dimension 2 and so can never become relevant, meaning that the phase is preserved. The least irrelevant fields obeying all symmetries of the identity and with spin s < 2 are a pair of s = 0 fields with ∆ = 3. In terms of TCI + P these are (3/2 + 0, 3/2 + 0) and (1/10 + 7/5, 1/10 + 7/5). In terms of B + F, these are some linear combinations of (3/2 + 0, 3 H(0, 1). ∆ and s give the dimension and spin of the field, while r gives the Z3 sector. k gives the momentum of the corresponding state in the lattice model. TCI + P gives the weights in the TCI and Potts basis in the form (hTCI + hP,hTCI +hP). A primed number denotes the Virasoro raising operator L−1 (L−1) acting on the primary, while a double-primed number has L−2 (L−2) acting on it. B + F gives the weights in the boson and fermion basis in the form (hB + hF,hB +hF). [m, n] gives the electric and magnetic charge of the boson (the linear combinations are chosen to behave nicely under duality). D and D give the two dualities, which are described in the text. For fields with Z3 charge r = 1, there is a corresponding field with r = −1 not given. For fields with k = 0, π, there is a corresponding field with momentum −k not given.