Spin peierls quantum phase transitions in coulomb crystals.

Abstract

The spin Peierls instability describes a structural transition of a crystal due to strong magnetic interactions. Here, we demonstrate that cold Coulomb crystals of trapped ions provide an experimental test bed in which to study this complex many-body problem and to access extreme regimes where the instability is triggered by quantum fluctuations alone. We present a consistent analysis based on different analytical and numerical methods, and we provide a detailed discussion of its experimental feasibility.

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Spin-Peierls Quantum Phase Transitions in Coulomb Crystals
A. Bermudez1and M. B. Plenio1
1Institut f¨ ur Theoretische Physik, Albert-Einstein Allee 11, Universit¨ at Ulm, 89069 Ulm, Germany
The spin-Peierls instability describes a structural transition of a crystal due to strong magnetic interactions.
Here we demonstrate that cold Coulomb crystals of trapped ions provide an experimental testbed in which
to study this complex many-body problem and to access extreme regimes where the instability is triggered
by quantum fluctuations alone. We present a consistent analysis based on different analytical and numerical
methods, and provide a detailed discussion of its feasibility on the basis of ion-trap experiments. Moreover, we
identify regimes where this quantum simulation may exceed the power of classical computers.
PACS numbers:
The beauty of low-dimensional quantum many-body sys-
tems (QMBS) relies on the complexity born of the combi-
nation of interactions, disorder, and quantum fluctuations.
However, these ingredients also conspire to render perturba-
tive techniques inefficient, posing thus a fundamental chal-
lenge that has inspired the development of a variety of ana-
lytical [1] and numerical [2] tools. Moreover, the synthesis
of low-dimensional materials has upgraded these challenges
from a theoretical endeavor into a discipline that underlies
some of the most exciting recent discoveries in condensed-
matter physics, such as the fractional quantum Hall effect.
The recent progress in the field of atomic, molecular, and op-
tical (AMO) physics presents a promising alternative to these
solid-state realizations of low-dimensional QMBS. This field,
which was originally devoted to the study of light-matter in-
teractionsatthescaleofasingleorfewatoms, isprogressively
focusing on the many-body regime in platforms such as neu-
tral atoms in optical lattices, cold Coulomb crystals of trapped
ions, or coupled cavity arrays (see [3], [4], and [5] for recent
reviews on each of these subjects). The possibility of experi-
mentally designing the microscopic Hamiltonians in order to
target a variety of complicated many-body models introduces
a novel approach to explore QMBS in a controlled fashion,
the so-called quantum simulations (QSs) [6].
InthisLetter, weexplorethecapabilitiesofAMOplatforms
for the QS of interaction-mediated instabilities in QMBS.
The standard playground for such instabilities is the one-
dimensional metal, where either the electron-electron interac-
tions destabilize the metal towards a superconducting state, or
the electron-phonon coupling leads to a charge-density-wave
condensate [7]. The latter instability is a consequence of the
so-called Peierls transition [8], where the electron-phonon in-
teractions induce a periodic distortion of the ionic lattice, and
open an energy gap in the conduction band of the metal.
ByvirtueoftheJordan-Wignertransformation[9], thisphe-
nomenon finds a magnetic counterpart: the spin-Peierls tran-
sition [10]. A spin-phonon-coupled antiferromagnet is unsta-
ble with respect to a dimerization of the lattice, which creates
an alternating pattern of weak and strong spin interactions,
and opens an energy gap in the spectrum of collective exci-
tations. From a theoretical perspective, the complete under-
standing of such a complex system, treating the dynamics of
the spins and phonons on the same footing, is still considered
to be an open problem [11]. From an experimental point of
a
b
z
x
y
x
y
z
x
y
z
kL
c
g > ˜ gc
g < ˜ gc
Figure 1.
stability accompanied by the crystal distortion. (a) In the param-
agnetic phase |P? = | ↑↑ ··· ↑? all the spins are parallel to a trans-
verse field g > ˜ gcthat points along the z-axis, and the Coulomb crys-
tals corresponds to an ion string. (b) The antiferromagnetic phase
g < ˜ gc corresponds to the two N´ eel-ordered groundstates |AF? ∈
{|+−···+−?,|−+···−+?}, where the spins are antiparallel in
thex-basis|±?=(|↑?±|↓?)/√2. Thisorder-disorderquantumphase
transition occurs with the linear-to-zigzag structural phase transition.
(c)ArrangementofthelaserwavevectorkLlyingwithinthexy-plane.
Spin-Peierls transition: Scheme for the magnetic in-
view, the spin-Peierls phenomena observed so far [12] take
place at finite temperatures. Hence, the possibility of a spin-
Peierls transition only driven by quantum fluctuations remains
as an experimental challenge. We hereby present a theoreti-
cal proposal for a trapped-ion QS to tackle both problems.
In particular, by building on the recent experiments [13] on
the quantum Ising model (QIM) [14], we describe how to tai-
lor a spin-Peierls instability. We show that (i) the disordered
paramagnet in a linear ion chain changes into an ordered an-
tiferromagnet in a zigzag crystal [Fig. 1(a)-(b)], and (ii) the
spin-Peierls transition can be driven only by the quantum fluc-
tuations introduced by the transverse field of the QIM.
The system.– The advent of experimental techniques for the
confinement, cooling, and coherent manipulation of atomic
ions, underlies their suitability as a quantum-information ar-
chitecture [15]. This technology has also been exploited for
QS purposes [4], where the controlled increase of the num-
ber of trapped ions yields a genuine bottom-up approach to
the many-body regime. We consider a Coulomb gas formed
by an ensemble of N trapped ions of mass m, and charge e,
arXiv:1201.6671v1 [cond-mat.str-el] 31 Jan 2012

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which are described by the Hamiltonian
H0=ω0
2∑
i
σz
i+∑
i,α
?1
2mp2
iα+1
2mω2
αr2
iα
?
+e2
2∑
i?=j
1
|ri−rj|
(1)
where {ωα}α=x,y,zare the effective trapping frequencies of a
linear Paul trap. Here, ω0is the energy difference between
two electronic groundstates of the atomic structure |↑?i,|↓?i,
where ¯ h = 1, and σz
must be complemented by the laser-ion interaction responsi-
ble for coupling the electronic and the motional degrees of
freedom. We consider a pair of laser beams with frequencies
{ωl}l=1,2, wavevectors {kl}l=1,2, and phases {φl}l=1,2, which
are tuned close to the atomic transition. In the dipolar approx-
imation, the laser-ion Hamiltonian becomes
i= |↑i??↑i|−|↓i??↓i|. This Hamiltonian
HL=∑
l,i
(Ωlσ+
i+Ω∗
lσ−
i)cos(kl·ri−ωlt +φl),
(2)
where Ωlstands for the Rabi frequency of the transition, and
we have introduced the spin raising and lowering operators
σ+
some assumptions about the dynamics of this atomic plasma.
i= |↑i??↓i| = (σ−
i)†. To proceed further, we need to make
As evidenced in early experiments [16], a laser-cooled
plasma self-assembles in a Coulomb crystal, which undergoes
a series of structural phase transitions (SPTs) as the trapping
conditions are modified. In particular, when ωy? ωx,ωz, and
the ratio κx=(ωz/ωx)2is tuned across a critical value κc[17],
the geometry of the crystal changes from a linear string to a
zigzag ladder. Note that this SPT displays a rich phenomenol-
ogy that has recently revived the interest in the subject [18–
23]. Here, we focus on the linear regime close to the critical
point κx? κc, where the vibrations of the ions along each
of the confining axes are decoupled. We consider that the
laser wavevectors in (2) lie within the xy-plane [Fig. 1(c)],
kl= klxex+klyey, such that their frequencies are tuned close
to the resonance of the vibrational sidebands ωl≈ ω0±ωy.
Since these correspond to the strongly-confining y-axis, ωy?
ωx,ωz, the coupling of the laser beams to the x and z vibra-
tional modes becomes far off-resonant and can be neglected.
Let us remark that this argument has one possible exception,
theremight beavibrational softmodewherethe phononscon-
dense at the SPT. As identified in [18], this corresponds pre-
cisely to the zigzag mode along the x-axis. This affects the
laser-ion coupling (2) regardless of the soft-mode frequency
(see Supplementary Material)
The above considerations allow us to extract the relevant
part of the Hamiltonian (1) after introducing ri= lz(˜ z0
qixex+qiyey+qizez), where ˜ z0
in units of lz= (e2/mω2
ments along the corresponding axes. Following [22], the dis-
placements along the x-axis have been adapted to the afore-
mentioned zigzag mode qix= (−1)iδqix, such that δqixis a
smooth function that allows a gradient expansion. The Hamil-
iez+
iare the equilibrium positions
z)1/3, and qiαare the small displace-
tonian then becomes H0=1
?ml2
?ml2
2∑iω0σz
i+Hx+Hy, where
?
Ky
ij
2(∂jqiy)2,
Hx=∑
i
z
2
(∂tδqix)2+rx
i
2δq2
ix+ux
i
4δq4
ix
+∑
i?=j
Kx
2(∂jδqix)2,
ij
Hy=∑
i
z
2
(∂tqiy)2+ry
i
2q2
iy
?
+∑
i?=j
(3)
and we have introduced the gradient ∂jfi= fi− fj. In these
expressions, the coupling energies for the vibrations are
rx
i= mω2
xl2
z
?1−1
xl2
z
2κxζi(3)?, ux
∑
l?=i
i= mω2
?
xl2
z
?3
4κxζi(5)
?
,
Kx
ij= mω2
?
(−1)l+i+1κx
2|˜ z0
i− ˜ z0
l|
δj,i+1,
(4)
along the x-axis, expressed in terms of the inhomogeneous
function ζi(n) = ∑l?=i[(−1)i−(−1)l]n−1|˜ z0
Z, and the Kronecker delta δlm. In the limit of tight confine-
ment along the y-axis, κy= (ωz/ωy)2? 1, we find
i− ˜ z0
l|−nwith n ∈
ry
i= mω2
yl2
z, Ky
ij=
mω2
2|˜ z0
yl2
i− ˜ z0
zκy
j|3,
(5)
Accordingly, Hy corresponds to a set of dipolarly-coupled
harmonic oscillators, whereas Hxdescribes a set of nearest-
neighbor-coupled anharmonic oscillators.
The coupled harmonic oscillators (3) are diagonalized
yielding a set of collective modes with frequencies ωn, whose
excitations, created and annihilated by a†
to as the hard phonons. This yields the quadratic Hamilto-
nian Hy= ∑nωna†
red and blue vibrational sidebands of the atomic transition,
ω1≈ ω0−ωnand ω2≈ ω0+ωn, we can express the laser-ion
interaction (2) as follows (see Supplementary Material)
?
where we have introduced the sideband coupling strengths
Fr
Lamb-Dicke parameters ηln= kly/√2mωn? 1, the normal-
mode vibrational amplitudes Min, and θr= k1xlz,θb= k2xlz.
In contrast, the anharmonic oscillators (3) correspond to an
inhomogeneous version of the φ4model on a lattice, namely,
an interacting scalar field theory that cannot be exactly diag-
onalized. This model yields a SPT that can be understood as
follows. The regime rx
frequency ratios fulfilling κx< κc,i= 2/ζi(3), yields the lin-
ear ion configuration, which respects the Z2symmetry of the
model. Conversely, when rx
ions self-organize in the the zigzag ladder corresponding to
the broken-symmetry phase, whereby ?δqix? ?= 0 signals the
condensation of the soft phonons in the zigzag mode. We note
that the φ4model fulfills rx
geneous SPT setting in at the center of the trap [20]. As out-
lined previously, when the soft phonons condense ?δqix? ?= 0,
n,an, shall be referred
nan. By setting the laser frequencies to the
HL=∑
in
Fr
ineiθrqixσ+
ian+Fb
ineiθbqixσ+
ia†
n+H.c.
?
,
(6)
in=i
2Ω1η1nMineiφ1,Fb
in=i
2Ω2η2nMineiφ2, the collective
i>0, ux
i>0 corresponding to trapping-
i< 0, ux
i> 0 for κx> κc,i, the
i?= rx
j, which leads to an inhomo-

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there is a non-trivial effect in the laser-ion Hamiltonian (6)
that must be considered. We show below that, in this case,
the hard phonons mediate a spin-spin interaction, whereas the
soft condensed phonons are responsible for a dimerization of
the coupling strengths. This turns out to be the key ingredi-
ent for a zero-temperature spin-Peierls transition. Let us also
emphasize that this model, which is a cornerstone in the mi-
croscopic description of SPTs [24], has not been combined
with a spin-Peierls distortion to the best of our knowledge.
Dimerized quantum spin model.– The spin-phonon model
in (3) and (6) yields an extremely complex QMBS. We an-
alyze the onset of a spin-Peierls quantum phase transition
by performing a series of simplifications. First, we neglect
the time-dependence of the zigzag distortion (3). This adi-
abatic approximation, which is standard in the treatment of
spin-Peierls phenomena [10], is justified if the zigzag mode is
much slower than the effective spin dynamics, which is valid
close to the critical point. Hence, we treat the SPT classi-
cally by setting qix= (−1)i?δqix? self-consistently. Second,
we consider a homogeneous zigzag distortion, which amounts
to neglecting the nearest-neighbor couplings in Eq. (3). Third,
when the coupling of the spins to the hard phonons (6) is
weak, they can be integrated out yielding an effective quan-
tum spin model. In the linear string, this leads to a dipolar
version of the celebrated QIM [25], whereas the frustrated J1-
J2QIM arises in the zigzag configuration [26]. In this work,
we show that in the vicinity of the critical point κx≈ κc,i, the
quantum spin model corresponds to a dipolar QIM with addi-
tional spin-spin couplings whose sign alternates periodically
when the soft phonons condense.
In analogy to the Mølmer-Sørensen gates [27], we consider
that the red- and blue-sideband terms (6) have opposite de-
tunings δnr= −δnb=: δn, where δnr= ω1−(ω0−ωn), and
δnb= ω2−(ω0+ωn). Besides, their Rabi frequencies fulfill
Ω1k2
In this limit, the sidebands (6) create a virtual hard phonon
which is then reabsorbed by a distant ion, leading thus to
the aforementioned effective spin-spin interaction. The above
constraints are responsible for the destructive interference of
the processes where a phonon is created and then reabsorbed
by the same ion, a crucial property that underlies the availabil-
ity of an effective spin Hamiltonian that is decoupled from the
hard phonons. Finally, by considering that the pair of laser
beams are counter-propagating k1= −k2=: k, which implies
that the parameters are θr= −θb=: θ, it is possible to obtain
the following spin Hamiltonian (see Supplementary Material)
?
where the coupling strengths are the following
?+cos(θ(qix−qjx))+cos(θ(qix+qjx)+φ−)?,
Jxy
Jyx
1y=Ω2k2
2y, andattainvaluessuchthat|Fr
in|=|Fb
in|?δn.
Heff=∑
i?=j
Jxx
ijσx
iσx
j+Jyy
ijσy
iσy
j+Jxy
ijσx
iσy
j+Jyx
ijσy
iσx
j
?
, (7)
Jxx
Jyy
ij= Jij
ij= Jij
ij= Jij
ij= Jij
?+cos(θ(qix−qjx))−cos(θ(qix+qjx)+φ−)?,
?−sin(θ(qix−qjx))−sin(θ(qix+qjx)+φ−)?,
?+sin(θ(qix−qjx))−sin(θ(qix+qjx)+φ−)?,
(8)
and we have introduced the relative phase between the lasers
φ−= φ1−φ2. Here, the spin-spin coupling strengths are
Jeff
2|˜ z0
where we made a expansion for κy? 1, and introduced the
bare detuning δy= ω1−(ω0−ωy) = −ω2+(ω0+ωy), the
bare Lamb-Dicke parameter ηy= ky/?2mωy, and the com-
interaction leads to a generalization of the famous XY quan-
tum spin model [28]. In order to obtain the promised dimer-
izedspinmodel, onehastophase-lockthelaserbeamsφ−=0,
and consider the critical region where the zigzag distortion is
small enough θ?δqi? ? 1. Then, we can Taylor expand and
obtain an antiferromagnetic Ising interaction characterized by
Jxx
Jij=
i− ˜ z0
j|3, Jeff=Ω2
Lη2
16δ2
yκy
y
?
1+δy
ωy
?
ωy,
(9)
mon Rabi frequency ΩL:=Ω1=Ω2. Hence, the spin-phonon
ij= 2Jij, and Jyy
ij= 0, but also
Jxy
ij= 2Jij(−1)j+1θ?δqjx? = Jyx
which give rise to the quantum dimerization (i.e. a magnetic
interactionthatdoesnotcommutewiththeIsingcoupling, and
alternates between a ferromagnetic-antiferromagnetic sign
Jxy
field h that can be obtained from a microwave that is far off-
resonant with respect to the atomic transition. Altogether, the
spin Hamiltonian becomes Heff= HIsing+Hdimer, where
ji,
(10)
ij,Jyx
ji∝ (−1)i+1). Besides, we also consider a transverse
HIsing=∑
i?=j
Jeff
i− ˜ z0
Jeff(−1)j+1ξj
|˜ z0
|˜ z0
j|3σx
iσx
j−h∑
i
σz
i,
Hdimer=∑
i?=j
i− ˜ z0
j|3
σx
iσy
j+Jeff(−1)i+1ξi
|˜ z0
i− ˜ z0
j|3
σy
iσx
j,
(11)
and we have introduced ξi= θ?δqi? ? 1. Let us remark that
the couplings of the dimerization Hamiltonian depend on the
condensation of the soft phonons ?δqix?, which is in turn de-
scribed by the φ4theory. Below, we show how this model
leads to the desired Spin-Peierls instability.
Spin-Peierls quantum phase transition.– To demonstrate
that the introduced scheme yields a QS of the spin-Peierls in-
stability, we simplify the model by neglecting its long-range
and inhomogeneities. The spin model can be solved by means
of a Jordan-Wigner transformation after setting ξ = ξi ∀i,
which is later used as input to the φ4model self-consistently.
In the limit ξ ? 1, the groundstate energy fulfills
Eg(ξ) ≈ Eg(0)−2JN
π
where J > 0 is the nearest-neighbor antiferromagnetic cou-
pling, g = h/J, and we have introduced a monotonically-
decreasing positive-definite function ϕ(g). We have com-
pared this expression to numerical DMRG calculations (see
Supplementary Material). The above lowering of the ground-
state energy pinpoints the instability towards the lattice distor-
tion. Besides, the spectrum of magnetic excitations displays
the following energy gap ∆ ∝??g−?1+4ξ2??. With respect
sition of the standard QIM at gc= 1 [14], the dimerization
ϕ(g)ξ2< Eg(0),
(12)
totheparamagnetic-to-antiferromagneticquantumphasetran-

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4
breaks the self-duality of the model and allows the critical
point to flow towards higher values of the transverse field
gc→ ˜ gc(ξ) =
?
1+4ξ2.
(13)
Therefore, the disordered paramagnet close to criticality gc<
g ≤ ˜ gc will be unstable towards the ordered antiferromag-
net if the lowering of the energy (12) compensates the struc-
tural change. For self-consistency, we incorporate this en-
ergy change in the φ4model.
Eg(ξ)−Eg(0) ∝ ∑iδq2
parameters of the scalar field theory (4). In analogy to the
magnetic quantum phase transition, the critical value of the
SPT is also displaced, but this time towards a smaller value
By noticing that it fulfills
ix, it becomes clear how to modify the
κc,i→ ˜ κc,i=
?ζi(3)
2
+
2J
mω2
zl2
z
θ2ϕ(g)
π
?−1
.
(14)
Therefore, the linear ion string close to criticality ˜ κc,i≤ κx<
κc,iis unstable towards the symmetry-broken zigzag phase.
We have thus proved our claim (i) that the paramagnetic
phase in the linear ion string will be unstable towards the anti-
ferromagnetic zigzag ladder [Fig. 1]. Moreover, by fixing the
ratio of the trapping frequencies in the linear regime, κx<κc,i,
we can drive both the structural and the magnetic phase transi-
tions by only modifying the transverse magnetic field g across
˜ gc. The necessary condition is that the trapping frequencies
are tuned according to the following expression
mω2
zl2
z=
2Jθ2ϕ(˜ gc)
π?κ−1
x −1
2ζi(3)?.
(15)
Hence, the zigzag antiferromagnet g < ˜ gcis driven onto a lin-
ear paramagnet by increasing the quantum fluctuations g > ˜ gc
(and viceversa). This supports our claim (ii) that the spin-
Peierls transition can be driven by quantum fluctuations alone.
Experimental considerations.– We focus on25Mg+and se-
lecttwohyperfinelevelsforthespinstates|↑i?=|F =2,mF=
2?,|↓i? = |3,3?, such that the resonance frequency in (1) is
ω0/2π = 1.8 GHz. We consider a N = 30 ion register with
trapping frequencies ωz/2π ≈ 300 kHz, ωx/2π ≈ 7 MHz,
and ωy/2π = 10 MHz. The phase-locked laser beams lead-
ing to (2) are blue-detuned δy/2π ≈1 MHz, such that the
two-photon Rabi frequencies are ΩL/2π ≈ 1 MHz, and the
Lamb-Dicke parameter ηy≈ 0.2. With these values, the re-
quired constraints detailed in the Supplementary Material are
fulfilled, and we obtain a nearest-neighbor spin coupling with
the typical strength J = 2Jeff/|˜ z0
experiments [13]. By considering these parameters, the condi-
tion (15) imposes the following constraint over the anisotropy
(κc,i−κx)/κc,i∼ 10−4η2
close to the structural phase transition. In practice, the soft ra-
dialtrappingfrequencymustbecontrolledwithanaccuracyof
∆ωx∼ 10−6ωx≈1-10 Hz, which coincides with the the preci-
sion required to observe quantum effects in the SPT [22]. Pro-
vided that this precision is achieved in the experiments, one
could optically pump thelinear ion register to|ψ(0)?=⊗|↑i?,
and then study its adiabatic evolution towards the AF phase as
the transverse field g(t) is decreased. The corresponding AF
order can be measured by fluorescence techniques, whereas
the structural phase transition could be directly observed in
a CCD camera, or inferred from spectroscopy of the vibra-
tional modes. A simpler experiment would require to set the
the anisotropy parameter within the instability regime given
by the displacements in Eqs. (14) and (13), which leads to
(κc,i−κx)/κc,i∼ η2
magnet would be directly unstable towards to zigzag AF with-
out the need of adiabatically tuning the transverse field g(t),
and the demanding accuracy over the trap frequencies.
i−˜ z0
i+1|3≈ 1 kHz observed in
y, which requires to be sufficiently
y≈ 10−2and h/J ≈ 10−2.The linear para-
Conclusions and Outlook.– A sensible QS must address
questions that are difficult to assess by other analytical or nu-
merical methods. In this work, we have proposed a trapped-
ion QS that fulfills this requirement, since there are regimes
where the many-body model in Eqs. (3) and (6) is no longer
tractable. In particular, it would be fascinating to explore the
consequences of quantum fluctuations and non-adiabatic ef-
fects of the zigzag distortion. Besides, this QS may address
the effects of the inhomogeneities, the long-range dipolar tail
of the spin-spin interactions, and even the dynamics across
such a magnetic structural quantum phase transition. We em-
phasize that the incorporation of all these effects compromises
the efficiency of existing numerical methods for QMBS, mak-
ing our QS of the utmost interest.
Acknowledgements.– This work was partially supported by
the EU STREP projects HIP, PICC, and by the Alexander von
Humboldt Foundation. We acknowledge useful discussions
with J. Almeida and S. Montangero.
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