Phantom Bethe excitations and spin helix eigenstates in integrable periodic and open spin chains
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Phantom Bethe excitations and spin helix eigenstates in integrable periodic and open spin chains
Vladislav Popkov,1, 2 Xin Zhang,1 and Andreas Klümper1
1
Department of Physics, University of Wuppertal, Gaussstraße 20, 42119 Wuppertal, Germany
2
Faculty of Mathematics and Physics, University of Ljubljana, Jadranska 19, SI-1000 Ljubljana, Slovenia
We demonstrate the existence of a special chiral “phantom” mode with some analogy to a Goldstone mode
in the anisotropic quantum XXZ Heisenberg spin chain. The phantom excitations contribute zero energy to
the eigenstate, but a finite fixed quantum of momentum k0 . The mode exists not due to symmetry principles,
but results from non-trivial scattering properties of magnons with momentum k0 given by the anisotropy via
arXiv:2102.03295v2 [cond-mat.stat-mech] 26 Aug 2021
cos k0 = ∆. Different occupations of the phantom mode lead to energetical degeneracies between different
magnetization sectors in the periodic case. This mode originates from special string-type solutions of the Bethe
ansatz equations with unbounded rapidities, the phantom Bethe roots (PBR). We derive criteria under which
the spectrum contains eigenstates with PBR, both in open and periodically closed integrable systems, for spin
1/2 and higher spins, and discuss the respective chiral eigenstates. The simplest of such eigenstates, the spin
helix state which is a periodically modulated state of chiral nature, is built up from the phantom excitations
exclusively. Implications of our results for experiments are discussed.
Interacting quantum spin systems are a vibrant research mation” Uq (su(2)) of the symmetry algebra [13–15], involv-
field as fascinating kinds of order are realized with rather com- ing special possibly non-hermitian boundary terms.
plex order parameters or of topological nature. Even the spin- Remarkably, an analogue of a Goldstone mode scenario can
1/2 XXZ chain, despite its long history and being one of the happen in periodic spin systems with z-exchange anisotropy,
best studied paradigmatic models in quantum statistical me- namely a multiple occupation of a single mode can occur, but
chanics [1], remains a source of inspiration and fascinating now with a nonzero wave vector k0 fine-tuned to the system’s
new progress. This model is integrable and in principle allows anisotropy Jz /Jx = ∆ via cos k0 = ∆. The corresponding
for the calculation of objects that in generic systems are usu- excitation can be created at zero energetic cost. Like in the
ally not accessible in the thermodynamic limit. Among the isotropic case, the possibility of multiple occupations of the
relatively recent results the discovery of a set of quasi-local same zero-energy phantom mode leads to the high degeneracy.
conserved quantities [2] with strong implications on the the- Unlike in the isotropic case, the eigenstates form a multiplet
ory of finite-temperature quantum transport [3] and successes of degenerate chiral states carrying finite current.
in the calculation of finite temperature correlation functions Excitations with momentum mode ±k0 were discussed in
[4, 5] are exciting achievements. [16–21] for accounting for the energetical degeneracies of the
In this letter we are interested in phenomena of anisotropic spin-1/2 XXZ chain and related systems. For certain systems
quantum spin chains requiring the understanding of energeti- with commensurable values of k0 extended symmetry alge-
cal degeneracies in uncharted territory. A first example is the bras are realized and the completeness of the Bethe ansatz has
physics of so-called spin helix states (SHS) (4) which show been investigated [17–21].
sharp local polarization with respect to site dependent axes. In our letter we show why a macroscopic occupation of pre-
These states are routinely created, and widely used in coher- cisely ±k0 becomes possible, despite magnons of the wave
ent experimental protocols [6–8]. SHS can also be generated number k0 have non-trivial scattering. These states are real-
as non-equilibrium steady states via a dissipative quantum ized by non-standard string-type solutions of the Bethe ansatz
protocol [9–11] or via controlled local boundary dissipation. equations with infinite rapidities. The Bethe Ansatz equations
Remarkably, the needed boundary dissipation is of the type for singular roots are satisfied with a universal choice for their
which allows the system to retain, partly, its integrability [12]. arrangement (11), which makes them effectively “disappear”
The eigenvalue degeneracies of isotropic quantum spin from the set of Bethe Ansatz equations. For this reason we
chains are well understood on the basis of the su(2) symme- call the roots with infinite rapidities phantom Bethe roots and
try algebra. Simple eigenstates, that are fully polarized with the respective excitations phantom excitations.
respect to any axis form a multiplet of degeneracy N + 1 for We find phantom Bethe roots in other integrable systems
the spin-1/2 chain of length N . including open quantum systems and also for higher spins.
The high degeneracy of this ferromagnetic multiplet can Finally, we find that the role of the fully polarized eigen-
alternatively be explained by magnon excitations with soft states in the isotropic case is taken, in anisotropic systems, by
Goldstone mode at wavenumber k = 0. Contrary to the usual simple but rather nontrivial chiral states, the spin helix states.
situation when all magnons carry different momenta, this pre- The SHS have ballistic current and a harmonic modulation
cise k = 0 mode can be occupied up to N times. (with period 2π/k0 ) of transversal magnetization. Remark-
A z-anisotropy of the spin exchange interaction lifts the ably, the SHS are created with exclusively phantom excita-
high degeneracy, leaving just two degenerate eigenstates with tions, both in open and in periodic spin chains.
spins fully polarized in +z or −z direction. The su(2)-type Factorized eigenstates at commensurate values of
degeneracies can be restored by the so-called “quantum defor- anisotropy. We consider the XXZ spin-1/2 Hamiltonian for2
periodic and open boundary conditions. For the periodically condition is fulfilled, if the boundary interactions hl = ~hl ~σ1
closed chain we have and hr = ~hr ~σN satisfy
N
X hl |y0 i = κ σ z |y0 i + λ− |y0 i , (6)
HXXZ = hn,n+1 (∆) , (1) z
n=1
hr |yN −1 i = −κ σ |yN −1 i + λ+ |yN −1 i , (7)
y
hn,n+1 (∆) = J σnx σn+1
x
+ σny σn+1 + ∆ σnz σn+1
z
−I , where yN −1 = y0 + i(N − 1)γ, and λ± are some boundary-
dependent constants. The energy eigenvalue is E = λ− + λ+ .
with boundary conditions ~σN +1 ≡ ~σ1 . For convenience we
Note that in the open chain case a condition on the anisotropy
put J = 1 throughout this letter. For the open chain we have
∆ like in the periodic case is absent and ϕ in (4) can be real or
N
X −1 imaginary. Although having the same algebraic form, the SHS
HXXZ = hn,n+1 (∆) + ~hl ~σ1 + ~hr ~σN , (2) for the easy plane and easy axis cases have rather different
n=1 physical properties as visualized in Fig. 1.
The factorized SHS state is after the ferromagnetic state the
with boundary fields ~hl and ~hr on the first and on the last sites. simplest eigenstate of XXZ spin chains. Yet the SHS (4)
In both cases a shift −J∆ in the nearest-neighbour interaction is quite nontrivial, and describes a “frozen” spin precession
(1) is added for convenience. Both models (1), (2) are inte- around the z-axis with period 2π/ϕ, see Fig. 1. Due to the
grable and solvable via Bethe Ansatz methods [22–24]. We chiral nature, the SHS carries a remarkably high magnetiza-
parametrize the anisotropy ∆ of the exchange interaction as tion current, finite in the thermodynamic limit:
∆ = cos γ or ∆ = cosh η with η = iγ.
We want to construct factorized eigenstates of the Hamilto- − sin γ
hj z iSHS = h4i(σn+ σn+1 − h.c.)iSHS = ±2 ,
nians and introduce for each site the qubit state cosh2 (Re[y0 ])
where the sign ± corresponds to the choice ϕ = ±γ in (4).
1
|yi = y . (3) Remarkably, the SHS (4) with adjustable wavelength can be
e
realized in cold atom experiments [6, 7].
The qubit state (3) with y = f + iF corresponds to a The very existence of an eigenstate (4) for the periodic spin
fully polarized spin 1/2 pointing into the direction ~n = chain, characterized by periodic modulations in the magneti-
(sin θ cos F, sin θ sin F, cos θ) with tan θ2 = ef . A site- zation profile seems to contradict the U (1) symmetry: XXZ
factorized state, the so-called spin-helix state (SHS) [9, 10] eigenvectors split in blocks with
P well defined values of the
global magnetization S z = σ
n n
z
and expectation values
|SHS(y0 , ϕ)i = |y0 i1 |y0 + iϕi2 . . . |y0 + i(N − 1)ϕiN , hσn+ i = hσn− i = 0 vanish, and so do hσnx i = hσny i = 0.
(4) This paradox is resolved by the energetical degeneracy of
eigenstates with different values of the total magnetization
subscripts indicating the site number, with uniformly increas- S z . We will show that a superposition of states from different
ing angles on some offset y0 becomes an eigenstate of the blocks yields the state (4) which is not an eigenstate of the
XXZ Hamiltonian if: (i) the increase ϕ of the angle is identi- operator S z .
cal to ±γ, the parameter of the anisotropy ∆ = cos γ, and (ii) Phantom Bethe roots at commensurate anisotropies in peri-
the boundary conditions can be accounted for. The parame- odic XXZ chains. The eigenstates and eigenvalues are given
ter ϕ is real (imaginary) for easy plane (easy axis) anisotropy in terms of rapidities µj (j = 1, 2, . . . n) whose total number
corresponding to a state with uniformly increasing azimuthal n may take any value out of 0, 1, . . . N . For any solution of
(polar) angle. the Bethe Ansatz equations (BAE)
The bulk interaction of the XXZ Hamiltonian applied to
any SHS state (4) results in 0 due to the “divergence” relation n
sinhN (µj − iγ/2) Y sinh(µj − µl − iγ)
= , (8)
h(∆)|yi ⊗ |y + iγi = sinhN (µj + iγ/2) l6=j sinh(µj − µl + iγ)
= |yi ⊗ (κσ z |y + iγi) − (κσ z |yi) ⊗ |y + iγi , (5) there is an eigenstate with energy and total momentum
where κ = i sin γ. For the periodic model (1), the SHS n
X n
X
will be an eigenstate if the periodic closure condition γN = E=− e(µj ), K= k(µj ) , (9)
2πm with integer m is satisfied. This can only happen for j=1 j=1
anisotropy |∆| ≤ 1.
For the open chain condition (ii) on the boundary can be with single particle energy and momentum defined by
satisfied not only in the case |∆| ≤ 1, but also for |∆| > 1.
4 sin2 γ sinh(µ + i γ2 )
For |∆| > 1 we may use expression (4) with the replacement e(µj ) = , eik(µ) = .
ϕ = iη which results in a spin-helix state with fixed azimuthal cosh(2µj ) − cos γ sinh(µ − i γ2 )
angle and uniformly increasing polar angles. The eigenstate (10)3
The Bethe eigenvector Ψµ1 ,...µn = B(µ1 ) . . . B(µn )|0i is ob- Bethe roots µp for which all k(µp ) are exactly the same: ei-
tained by the application of magnon creation operators B(µj ) ther k(µp ) = +γ ≡ k0 or k(µp ) = −γ ≡ −k0 depending on
to the reference state |0i = | ↑↑ . . . ↑i of fully polarized spins the sign of the singular part in (11). Repeated action of B gen-
[23, 25]. erates “phantom” Bethe states (12) with “quantized” momenta
Definition. We shall call a Bethe root µp satisfying (8), a ±nγ and zero energy for all magnetization sectors n, yielding
phantom Bethe root, if it does not give a contribution to the the degeneracy of the eigenvalue E = 0 between different
respective energy eigenvalue (9) i.e. if Re[µp ] = ±∞. The sectors. Note that the E = 0 state is not a groundstate of (1),
next Lemma affirms that such phantom Bethe roots do exist: which is obtained by filling the Fermi sea with quasiparticles
Lemma 1: For anisotropy γ = 2πm/N with integer m there giving negative energy contributions to (9). The dimension of
exist the following “phantom” solutions of the BAE (8) for the degenerate subspace is deg = 2(N − 1) + 2 = 2N since
any given n = 1, 2, . . . N the states |+, ni, |−, ni for n = 1, 2, . . . N − 1 are linearly
p independent and for n = 0, N the states |+, ni, |−, ni coin-
µp = ±∞ + iπ , p = 1, 2 . . . n. (11) cide. The degeneracy between sectors with different magne-
n
tization leads to eigenstates with periodic modulations in the
These distributions remind of the string solutions to the Bethe density profile. Indeed, the SHS (4) with positive chirality
ansatz equations. Note however that (11) holds for any finite and ϕ = +γ = 2πm/N 6= π is a linear combination of phan-
system size N with a total number n of roots equidistantly tom Bethe states |+, ni, and SHS (4) with opposite chirality
distributed with separation π/n. Note that our lemma de- ϕ = −γ is a linear combination of |−, ni
scribes the precise arrangement of the infinite roots appearing
1/2 XN
in [1-6]. Upon introducing a finite magnetic flux resp. twisted N
boundary conditions, the roots become finite while the imagi- |SHS(y0 , ±2πm/N )i = ey0 n |±, ni, (13)
n n=0
nary parts stay close to the values of Lemma 1. This is relevant
for the dependence of the energy as function of the twist and see Supplemental [25] for the proof. Finally, note that the
has important consequences for the transport properties, see states (12) are chiral, which is evidenced by nonzero expec-
[25]. tation values of the magnetization current, see Supplemental
Proof. Assume µj = ±µ∞ + iπj/n, where µ∞ has a large material [25].
real part which we let to ∞ when evaluating the LHS of the
Bethe ansatz equations. As γ = 2πm/N the LHS of (8) be- 8n(N − n)
h±, n|j z |±, ni = ± sin γ , (14)
comes LHS → e∓iγN = 1. On the RHS the term µ∞ drops N (N − 1)
out leaving finite differences µj − µl = iπ(j − l)/n. Denoting
ω = eiπ/n , we have reaching its maximum of order |j z | → 2 sin γ for n = N/2.
Mixtures of regular and phantom excitations for the peri-
n
Y ω j−l e−iγ − ω −(j−l) eiγ odic XXZ model. Here we show that phantom Bethe roots
RHSj =
l=1
ω j−l eiγ − ω −(j−l) e−iγ can appear alongside with usual finite Bethe roots, for other
l6=j special values of the anisotropy.
n−1 l −iγ n−1 Let us assume that within a sector of n0 flipped spins, there
Y ωe − ω −l eiγ Y ω l e−iγ − ω −l eiγ
= = = 1. exists a BAE solution with n phantom Bethe roots µ1 , ..., µn
ω l eiγ − ω −l e−iγ −ω −l eiγ + ω l e−iγ and the remaining r = n0 − n Bethe roots are regular. We
l=1 l=1
denote the regular roots as x1 , . . . , xr where xj = µn+j . Let
Here we used that the set of ω j−l with l = 1, ..., n (and 6= j) us consider separately the BAE (8) subsets for phantom µp
is identical to the set of ω l with l = 1, ..., n − 1 as we have and for regular xj . Substituting (11) in (8) we obtain
ω n = −1.
Phantom Bethe vectors for periodic chains. The Bethe vec- eiγ(N −2r) = 1, (15)
tors corresponding to the phantom Bethe roots (PBR) solution
(11), under the conditions of Lemma 1, can be constructed as since each factor of the RHS containing a mixed pair µp , xj
described below (10). The two signs ± in (11) correspond to contributes a term exp(2iγ). The product over factors of the
different Bethe vectors which upon normalization read RHS involving two phantom roots results in +1 precisely as in
N −1
Lemma 1. The criterion (15) fixes the anisotropy parameter
1 X while the BAE subset for regular roots simplifies to
|±, ni = q e±iγ(l1 +...+ln ) σl−1 . . . σl−n |0i ,
N
n! l1 ,...,ln =0 r
n sinhN (xj −iγ/2) ±2iγn
Y sinh(xj −xl −iγ)
= e ,
n = 0, 1, . . . , N. (12) sinhN (xj +iγ/2) sinh(xj −xl +iγ)
l=1
l6=j
Each multiplication by a B(µj )-operator adds a quasiparticle
with momentum k(µj ) and zero energy. Within the standard for all j = 1, . . . , r, see also [19, 20]. This has the structure
picture [23, 24] quasi-particles obey a “Fermi rule”: all k(µj ) of the BAE of a twisted XXZ chain, because of the presence
are usually different. This property is violated for phantom of a constant phase factor. The signs ± match those in (11).4
Phantom excitations in the open XXZ chain. The energy SHS (4), with ϕ = γ, Re[y0 ] = β− and phase Im[y0 ] =
of Hamiltonian
PN (2) is given by (9) with an additional offset, π+iα− +iθ− (note that α− , θ− are imaginary and β− = −β+
E = j=1 e(µj ) + E0 , where are real to ensure hermiticity of H), is an eigenstate of H.
Indeed, one can check that (6), (7) are satisfied with λ± =
E0 = − sinh η (coth α− + coth α+ + tanh β− + tanh β+ ) , − sinh η(coth α± − tanh β± ), so that this SHS is an eigen-
(16) vector of (2) with eigenvalue λ− + λ+ , which coincides with
the phantom Bethe vector eigenvalue E0 (16). For the mag-
where the boundary fields hl,r are parametrized as netization profile of this SHS see Fig. 1, top panel. Unlike
for the periodic chain, here the eigenvalue E0 is generically
~h = sinh η
(cosh θ± , i sinh θ± , ∓ cosh α± sinh β± ), non-degenerate.
sinh α± cosh β± Simplest experimental setup. Using our results, long-
and +(−) corresponds to the right (left) field. The Bethe roots lived SHS can be obtained in experiments where effec-
µj satisfy BAE of a somewhat bulky form [25–28]. After tively one-dimensional spin 1/2 XXZ chains with tunable
some algebra [28] we find that if anisotropy are realized [6, 7]. A spin helix of the form (4)
with an adjustable wavelength is created within cold atoms
±(θ+ − θ− ) = (2M − N + 1)η + α− + β− + α+ + β+ setups by applying a magnetic field gradient in z direction
on an array of initially noninteracting qubits polarized along
mod 2πi, (17)
the x axis, see Methods of [6] for details. To make the SHS
is satisfied with some integer M = 0, 1, . . . , N − 1, each an eigenstate of the XXZ Hamiltonian, the wavelength Q
set of N Bethe roots contains n phantom Bethe roots of type of the spin-helix and the z-anisotropy ∆ must be related as
(11), where n takes one of two values n+ = N − M and ∆ = cos Qa where a is the lattice constant. Indeed under this
n− = M + 1 [28, 29]. The remaining N − n Bethe roots xj choice an SHS of type (4) |SHS± i := |SHS(iF0 , ±Qa)i
(= µn+j ) are regular and satisfy reduced BAE will remain invariant in the bulk and change initially only at
the boundaries, since
N −n±
G± (xj − η2 ) sinh2N (xj + η2 ) Y sinh(xj − xl + η) N −1
= × X
G± (−xj − η 2N
− η sinh(xj − xl − η) hn,n+1 (∆)|SHS± i = ∓i sin Qa (σ1z − σN
z
) |SHS± i,
2 ) sinh (xj 2) l=1
l6=j n=1
sinh(xj + xl + η) as follows from (5). The ends of the spin chain will thus play
× , j = 1, . . . , N − n± , (18)
sinh(xj + xl − η) the role of defects, and the state in the bulk will be altered
only by propagation of the information from the boundaries.
Y
G± (u) = sinh(u ∓ ασ ) cosh(u ∓ βσ ) ,
σ=± Thus the state can be destroyed only after times of order t =
N a/vchar , where vchar is the sound velocity, N is the number
while the total eigenvalue has contributions from the regular of spins and a is the lattice constant. For example, in [6, 7],
Bethe roots only. We like to note that (16) holds literally for the process of the expansion of the defect in the bulk can be
case n = n+ . For n = n− the +E0 contribution in (16) is to monitored. On the other hand, if the SHS period does not
be replaced by −E0 , see [28]. We findPMthat the BAE (18) for match the anisotropy ∆ 6= cos Qa, then the initial SHS will be
N
n = N − M describes dim G+ M = m=0 m Bethe states, destroyed after times of order t = a/vchar . On one hand, the
while the remaining 2N − dim G+ M eigenstates are contained effect is robust (w.r.t. the phase of the helix and chain length
in the other, complementary BAE set for n = M + 1 [28, 29]. N ), and on the other hand, it is sensitive w.r.t. the matching
Unlike in the periodic setup, where some Bethe eigenstates condition for the anisotropy ∆. This sensitivity can be used as
contain PBR modes, and other eigenstates are fully regular, a benchmark for calibrating the anisotropy, or the wave-length
in open systems, satisfying criterion (17), all 2N eigenstates of the produced SHS, or both.
include phantom Bethe roots. Remarkably, the condition (17) Phantom Bethe states: Easy axis ∆ = cosh η > 1. The
appears in [30–33] as a condition for the application of the SHS of the form (4) with y0 = iπ − θ− + α− + β− satisfies
Algebraic Bethe Ansatz. The BAE set (18) coincides with (6), (7) with κ → − sinh η and λ± = − sinh η(coth α± +
that found by an alternative method [30, 31, 33]. tanh β± ). Consequently, state (4) is an eigenstate of H with
Now we focus on the simplest Bethe states, corresponding eigenvalue λ+ + λ− = E0 . Thus, state (4) is a phantom Bethe
to all Bethe roots being phantom, n+ = N , the respective vector. It describes spins on the lattice with fixed azimuthal
energy given by E0 (16). We demonstrate that such “phan- angle and changing polar angle along the chain, see Fig. 1,
tom” Bethe states are spin-helix states (4) with appropriately lower panel. Unlike the “azimuthal” spin helix state (4), the
chosen parameters. The phantom Bethe states for mixtures of “polar” SHS carries no spin current, hj z iSHSpolar = 0.
phantom and regular Bethe roots can be also obtained explic- Discussion. We have described a novel scenario of excita-
itly and show chiral features [28, 29]. tions in integrable systems, namely phantom excitations with
Phantom Bethe states: open XXZ chain. Easy plane phantom Bethe roots corresponding to unbounded rapidities.
regime |∆| < 1. It is straightforward to verify that the The existence criterion for these states is formulated and de-5
⟨σn x,y,z ⟩SHS grable models, e.g. [41, 42].
1.0 Financial support from the Deutsche Forschungsgemein-
schaft through DFG project KL 645/20-1, is gratefully ac-
0.5 knowledged. X. Z. thanks the Alexander von Humboldt Foun-
dation for financial support. V. P. acknowledges support
by European Research Council (ERC) through the advanced
10 20 30 40 50
n Grant No. 694544—OMNES. V. P. thanks S. Essink for indi-
cating the work in [6]. We thank W. Ketterle for drawing our
-0.5 attention to his newest experiment [7], where the time evolu-
tion of a transversal spin helix state (4) has been studied.
-1.0
⟨σn x,y,z ⟩SHS polar
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