Since the discovery of solitons carrying half of the electron charge [1,2] it has been widely recognized [3][4][5] that some states of matter can be characterized by fractional quantum numbers. Maybe the most celebrated example is the fractional quantum Hall state where quasiparticles carry fractional charges [6,7]. Other prominent examples coming from topological phases with short-range topological order, such as symmetry-protected topological (SPT) systems in one dimension, include spin- 1 2 edge states in the spin-1 Haldane chain [8,9] as well as spin- 1 4 zero-energy modes (ZEMs) localized at the edges of one-dimensional spin-triplet superconductors [10][11][12]. In higher dimensions, surface states in topological insulators as well as disordered magnetic systems such as spin ice [13], certain spin liquids [14], and the corners of certain ionic insulators and collinear antiferromagnets [15,16] also exhibit signatures of fractionalization.
Gapped one-dimensional (1D) systems with symmetries can be broadly classified into trivial phases, SPT phases, and symmetry-broken phases [17]. In this respect, to the best of our knowledge, known 1D systems that exhibit fractionalization are SPT phases. In this work, we show that fractionalization also occurs in a system which exhibits a symmetry-broken phase. To this end we shall consider the paradigmatic XXZ spin- 1 2 chain which exhibits spontaneous symmetry breaking of discrete spin-flip symmetry. We apply magnetic fields at the edges and solve the system exactly with the Bethe ansatz and numerically using the density matrix renormalization group (DMRG) for both even and odd numbers of site chains, and show that in the low-energy sector it hosts quantum spin- 1 4 states localized at the edges. In the * Contact author: pparmesh@umd.edu low-energy sector, in addition to the fractionalization of the spin that occurs in the bulk of the chain, where the fundamental spin-1 magnon excitations fractionalize to spin-1 2 spinon excitations, there exists further fractionalization where spins 1 4 are localized at the edges. We shall further argue that these fractional quarter spins are sharp quantum observables. We believe that this result might have some impact on understanding the dynamics [18][19][20][21][22][23][24], heat, and spin transport [25][26][27].
We consider the XXZ Hamiltonian with boundary magnetic fields (h L , h R ) at the left and the right edges of an open chain,
where σ x,y,z j are the Pauli matrices and > 1 is the anisotropy parameter. In the limit where the boundary fields are zero, on top of being U (1) symmetric, (1) is space-parity P and time-reversal T invariant. It is also invariant under the Z 2 = {1, τ } spin-flip symmetry, i.e., [H, τ ] = 0 where τ = N 1 σ x j . For generic nonzero boundary fields h L,R , = 0, both P and Z 2 symmetries are explicitly broken. However, on the two lines h L = ±h R the Hamiltonian (1) displays P and P • Z 2 symmetries, respectively.
The Hamiltonian in Eq. ( 1) is integrable by the method of the Bethe ansatz for arbitrary boundary fields h L,R and [28], which is used in the present paper to determine the low-energy eigenstates analytically. The system with periodic boundary conditions was first solved by Bethe [29] in the isotropic limit, → 1. The solution was later extended to include anisotropy along the z direction [30][31][32][33][34][35]. In the gapped regime ( > 1) it exhibits a continuous U (1) symmetry and also a discrete Z 2 spin-flip symmetry. The discrete Z 2 symmetry is spontaneously broken [36] and in the thermodynamic limit the system exhibits two degenerate symmetry-broken ground states [37]. The Bethe ansatz method to include the boundaries was developed in Refs. [38][39][40] and the ground state and boundary excitations in various bulk phases exhibited by the XXZ spin chain were found in Refs. [41][42][43][44]. An independent method to diagonalize the Hamiltonian using vertex operators was developed in Refs. [45,46], and was later extended to include the boundary fields in Ref. [47] where the boundary S matrix and the integral formula for correlation functions have been found. Recently, new band structures in the spectrum at large anisotropies have been found [48]. The system was shown to exhibit a strong zero-energy mode by Fendley [49] and it was recently constructed in Ref. [50] using the algebraic Bethe ansatz.
When > 1 the ground state |g displays antiferromagnetic (AFM) order with nonzero staggered magnetization σ = lim N→∞ N -foot_0 N j=1 (-1) j g|σ z j |g and is gapped. Indeed, for all values of the edge fields, there is a gap (m) in the spectrum to single-particle spin- 1 2 spinon excitations
However, at low fields, i.e., |h L,R | < -1, the lowest excited state is a midgap state |e which lies below the continuum. We obtain the bound state energies 1 which are given by
where α = (L, R). This midgap state is reminiscent of the existence of spin-1 2 boundary bound states, localized at the left and the right edges. The spin quantum numbers and energies of the ground state as well as the midgap state depend on both the parity of the number of sites, (-1) N , as well as on the boundary fields h L,R . When N is odd, the two states |g and |e have opposite total spins S z = ±1/2. Taking as a reference state the |- 1 2 state with energy E 0 , the |+ 1 2 state is obtained by adding a localized bound state at each edge. This state has energy E 0 + m L + m R . Depending on the edge magnetic fields, and hence on the sign of m L + m R , the ground state and the midgap state
Notice that on the line h L + h R = 0, the two states |± 1 2 are degenerate. In the N → ∞ limit, there is spontaneous symmetry breaking (SSB) of the P • Z 2 symmetry. In the particular case of zero edge fields, both P and Z 2 symmetries are spontaneously broken. For N even both (|g , |e ) states have total spins S z = 0 and the bound state construction is presented in the Supplemental Material (SM) [51]. We display in Fig. 1 the phase diagram for low fields for an odd number of sites N. states. On the separatrix h L + h R = 0 there is spontaneous symmetry breaking and the edge spin operator becomes a zero-energy mode.
Due to the open boundaries and the presence of the edge fields (h L , h R ), the spin profiles S z j = σ z j /2 in both the ground state and the midgap state differ from the bulk antiferromagnetic order close to the boundaries. The Bethe ansatz results suggest that for large enough N we may write (see SM [51] for more details)
where
is the exact staggered magnetization of the XXZ chain in the thermodynamic limit and S z ( j) is the relative deviation with respect to the anitferromagnetic (AFM) bulk profile. Due to the gap in the bulk these deviations are expected to be localized close to both the left and the right edges
where S z L,R ( j) are localized close to j = 1 and j = N, respectively [i.e., S z L,R (N/2) ∼ e -N/foot_1 ]. This is indeed what we find, where we clearly observe an exponential localization of the relative spin accumulation for various values of at constant boundary fields h L = h R = 0.2 (see SM [51]).
The spin accumulations, or depletions, do not come as a surprise and are expected due to the open boundaries and the presence of the edge fields. What is nontrivial is that they correspond to a genuine spin fractionalization in both the ground state and the midgap state. As we shall now demonstrate, in the thermodynamic limit and for all > 1, h L,R , there exist fractionalized quarter spin operators associated with each edge, Ŝz L and Ŝz R , which have well-defined fractional eigenvalues
In the basis (|g , |e ) the above fractional spin operators commute with each other, and anticommute with the spin-flip operator, i.e., [ Ŝz
Since the edge spin operators have fractional spin ±1/4 one may verify that the Ŝz have eigenvalues 0 or ±1/2 depending on whether N is even or odd. For the fractional spin operators (7) to describe sharp quantum observables in the subspace spanned by (|g , |e ), not only do they have to average to ±1/4 in both states, but also their variance must vanish in the thermodynamic limit, i.e.,
and
where the average • • • is taken in each of the two states (|g and |e ). Following the authors of Refs. [4,5] we define the fractional spin operators as their convolution with a decaying function f (x), where we take f (x) = e -αx to write
which takes the limit α → 0 after the limit N → ∞. We stress that the order of limits in ( 12) is important since by taking the limit α → 0 first, both S z L,R would identify with the total magnetization S z .
Due to the AFM long-range order it is convenient to distinguish between the contributions of the staggered part of the spin profile and that of the exponentially localized contributions,
where the relative accumulation operators are given by
) j e -α(N+1-j) . ( 14)
We have used the identity lim α→0 ∞ j=1 (-1) j e -α j = -1 2 . In practice, one may set α = 0 in (14) provided the summation over j extends to the middle of the chain j max = N/2. Convergence is then expected to be of order e -N/2 . Before going further, it is worthy to point out that although the relative accumulations defined in Eq. ( 14) have the same variance as the fractional spin operators (12) they do not qualify as spin operators in the sense that they do not anticommute with the spin-flip operator τ , i.e., { ŜL,R , τ } = 0. 2 These relative accumulations would have fractional eigenvalues which depend on the anisotropy parameter . Taking into account the AFM long-range order in the bulk is essential for the spin accumulations to have fractional eigenvalues ±1/4 independently of the model parameters, as we shall see.
Evaluating ( 14) would require the knowledge of the wave function of the ground state and the midgap states which is a formidable task within the Bethe ansatz approach. Hence, we resort to the complementary DMRG approach which is implemented through the TENPY software [52], that allows access to the ground state and midgap state with arbitrary precision owing to the gapped nature of both of these states. We take a maximum bond dimension of 400, with a minimal singular value decomposition cutoff of 10 -10 , and converge the energy up to a maximal energy error on the order of ∼10 -10 . We have computed the edge spin accumulations S L = Ŝz L and S R = Ŝz R in both the ground state and the midgap state for a wide range of boundary fields |h L,R | < -1 and parameters > 1.
All together our results are consistent with an accumulation of a spin S L,R = ±1/4 at the two edges of the system in both the ground state and the midgap state. Furthermore, we verify explicitly that these quarter spins reconstruct the total spin S z , as given by Eq. ( 8), of the ground state and the midgap state for both N even and N odd. We show here our results for an odd number of sites fixing = 3 and an anisotropic edge field configuration h L = 0 with varying h R in Fig. 2. To check that the quarter spins observed so far do not depend on the value of > 1, we also show the spin accumulations fixing h L = h R = 0.2 (in this case S L = S R owing to the P symmetry) and varying . More results are given in the SM [51].
We also calculated the spin variance to directly verify that the quarter spins found so far are sharp quantum observables. In the thermodynamic limit, the variance, as defined in Eq. (10), is then obtained as
Taking the N → ∞ is challenging and we circumvent this issue by assuming an ansatz relating δS 2 L (N, α) and
We have verified this ansatz by taking the difference of δS 2 L (N, α) for different N's. This is shown in Fig. 3. The fitted parameter B ≈ 2 is nearly independent of the boundary fields, while A takes a nonuniversal value. With the above ansatz we can calculate δS 2 L without explicitly taking the thermodynamic limit yielding δS 2 L = 0 and hence find that the variance in Eq. ( 10) vanishes.
In summary we find that in the low-energy subspace spanned by the ground state |g and the midgap state |e one can assign to the left and the right edges a fractional spin state with eigenvalues S L,R = ± 1 4 . On the basis of our results we find it safe to expect that this is to be the case irrespective of the anisotropy parameter > 1 and the values of the edge fields [h L,R | < -1. Due to the zero variance of the fractional spin operators (12), the quarter spins S L,R are not simple quantum averages of half-integer spins at different sites but rather sharp quantum observables. The orientations of these quarter spins depend on the boundary fields and on the parity of the number of sites N in such a way that ( 8) is satisfied in all ground states. Since the fractional spins at each edge are good quantum numbers we may then label the ground state and the midgap state as |g(e) = |S L , S R . For odd N spin chains these states are given by |± 1/4, ±1/4 whereas for even chains they are given by |± 1/4, ∓1/4 . One can easily verify that the total spin is S z = ±1/2 and S z = 0 for the odd and even cases. We want to point out that the existence of a gap above the ground state and the midgap state seems to be crucial for the quarter spins to be sharp quantum observables, and hence fractional spins ±1/4 only exist in the low-energy states which are separated from the continuum. Indeed, in the limit → 1 where the mass gap goes to zero, we end up with the XXX Heisenberg chain. In this case it was found in Ref. [53] that although the fractional spins ±1/4 exist in the ground state, their variance is not zero and hence the fractional spins are not genuine quantum observables.
The first natural question that arises is whether or not the quarter spins found so far survive in the higher excited states of the spectrum of the XXZ chain. However, excited states above the midgap state contain propagating spinons. In such case, even if a quarter spin can be defined on average, we do not expect its variance to be zero as found for the XXX spin chain with edge fields [53]. Another related question is whether these quarter spins survive edge fields higher than the critical value h c = -1. In this regime there are no midgap states [41][42][43][44] but we believe that sharp quarter spins exist in the ground state due to the existence of the spectral gap.
We shall end by commenting about the relation between the quarter spins found in this work with spontaneous symmetry breaking of the Z 2 symmetry in the case of zero edge fields, i.e., h L = h R = 0. In the limit of zero edge fields the two states |g(e) become degenerate in the thermodynamic limit as the bound state energies is not independent as Ŝz L = ± Ŝz R in both states, there exists only one ZEM, say Ŝz L . At this point it is worth mentioning that the Hamiltonian (1) displays the remarkable property, discovered by Fendley [49], of having a strong zero-energy mode F in the thermodynamic limit satisfying the following properties,
The existence of the latter operator ensures that in the N → ∞ limit the Hilbert space associated with the XXZ spin chain fractionalizes into two degenerated towers with eigenvalues τ = ±1 which are mapped onto each other by the action of F . We may therefore conclude that when projected in the low-energy subspace spanned by the ground state and the midgap state the Fendley operator identifies with the fractional spin operator
Of course, since we do expect a quarter fractional spin to be sharp in all the excited states, Ŝz L is not a strong ZEM in contrast with the Fendley operator F , but rather a soft ZEM. We finally notice that, following the same lines of arguments as given above, the fractional spin Ŝz L is also a soft ZEM on the two lines h L = h R for N even and h L = -h R for N odd where the symmetries P • Z 2 and P are spontaneously broken. It would be interesting to know if a strong zero mode similar to (17) exists on these two symmetric lines. We hope that the quarter spins found in this work could be probed in experiments using ultracold atoms in optical lattices [54] and Josephson junction arrays [55].
Note that a different expression was obtained in[42,43]. We also use the same notation as in[43].
For instance it would not couple to an external magnetic field penetrating smoothly near the left edge whereas ŜL would.