Dynamically generated quadrupole polarization using Floquet adiabatic evolution G. Camacho1C. Karrasch1and R. Rausch1

2025-08-18 0 0 981.32KB 15 页 10玖币

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Dynamically generated quadrupole polarization using Floquet adiabatic evolution

G. Camacho,1C. Karrasch,1and R. Rausch1

1Technische Universit¨at Braunschweig, Institut f¨ur Mathematische Physik,

Mendelssohnstrasse 3, 38106 Braunschweig, Germany

We investigate the nonequilibrium dynamics of the S= 1 quantum spin chain subjected to a

time-dependent external drive, where the driving frequency is adiabatically decreased as a function

of time (“Floquet adiabatic evolution”). We show that when driving the rhombic anisotropy

term (known as the “two-axis countertwisting” in the context of squeezed spin states) of a N´eel

antiferromagnet, we are able to induce an overall enhancement in the quadrupole polarization,

while at the same time suppressing the staggered magnetization order. The system evolves into a

new state with a net quadrupole moment and antiferroquadrupolar correlations. This state remains

stable at long times once the driving frequency is kept constant. On the other hand, we ﬁnd that we

cannot achieve a quadrupole polarization for the symmetry-protected Haldane phase, which remains

robust against such driving.

I. INTRODUCTION

In order to ﬁnd quantum states with desired properties,

we can look in various spaces: In the chemical space

we can investigate the multitude of natural compounds.

This space can be further extended by synthesis,

metamaterials or the replacement of chemical bonds

by magneto-optical traps in ultracold quantum gases.

Another possibility is to exploit the additional dimension

of time and engineer new states in nonequilibrium

conditions.

A particularly simple way to create a nonequilibrium

state is a quantum quench, where the system is suddenly

evolved with a new Hamiltonian. This can be used

to observe the melting of equilibrium order parameters,

such as string order [1,2], and can lead to quasi-steady

prethermalized states [3] before heating sets in, but does

not oﬀer optimal control and is not easily implementable

beyond ultracold-atom systems.

Alternatively, one can drive the system out of

equilibrium by a periodic external force, e.g., a

continuous laser beam with frequency Ω = 2π/T and

period T. In practical terms, this setup is well-controlled

in the high-frequency limit, where one can ﬁnd the

eﬀective Floquet Hamiltonian [4,5] by means of a

Magnus expansion of the original Schr¨odinger equation.

In leading order, this results in renormalized system

parameters, so that the problem can be analyzed using

equilibrium techniques (Floquet engineering). Heating

to an inﬁnite-temperature state should take place

eventually, but is shown to happen on exponentially

long time scales for large frequencies, leading to a stable

prethermalized state similar to the case of quenches [6,7].

Floquet engineering has been applied very

extensively [8–11] and a comprehensive listing of all

results is near-impossible. For non-interacting systems,

the electronic band structure is modiﬁed, which becomes

interesting if the topological character is changed [12–15].

For interacting systems, a lot of attention has been

devoted to the enhancement of superconducting

correlations [16–19] (often using intense pulses rather

than continuous beams) and the photo-inducement of

superconducting orders absent from equilibrium phases,

such as η-pairing [7,20,21]. Apart from that, there

have been eﬀorts to control the Kondo eﬀect [22],

exchange interactions [23], the Dzyaloshinskii-Moriya

interaction [24], the magnetization [25], or many-body

localization [26].

In equilibrium physics, the concept of adiabaticity is

fundamental. In practical terms, it can be used to

deﬁne and traverse phase diagrams or prepare complex

ground states by adiabatically changing the couplings of

a Hamiltonian, e.g. using quantum annealing. Extending

this concept to Floquet engineering, one can attempt

to adiabatically change the drive parameters to further

improve the degree of dynamic control of the system [27,

28].

In this work, we adopt the speciﬁc protocol of

initiating the system by driving a term with Ω = ∞,

followed by an adiabatic decrease of Ω [29,30]. This

adiabatically propagated state is called the “Floquet

ground state” [30], and Ω is freed up as an additional

control parameter in the procedure. This has been

ﬁrst studied for the integrable transverse-ﬁeld Ising

model, in which case the state was seen to undergo

topological phase transitions and Kibble-Zurek scaling

was observed [29,30].

Fortunately, in the case of one-dimensional chains,

this “adiabatic Floquet” protocol lends itself to an

eﬃcient simulation even for non-integrable systems

using matrix-product states (MPS). The initial state is

guaranteed to have low entanglement for the class of

gapped chains in accordance with the area law. This is a

key property that is exploited by the MPS formalism [31].

Furthermore, as long as the change of frequency is

slow enough, the entanglement is expected to grow only

slowly and long propagation times may be reached.

This stands in contrast to quench dynamics, where the

entanglement entropy increases linearly with time [32],

while the MPS bond dimension (i.e., the number of

variational parameters to represent the state) has to

increase exponentially.

In this paper, we show that the adiabatic Floquet

arXiv:2210.16088v2 [cond-mat.str-el] 14 Apr 2023

protocol can be used to convert a conventional

N´eel antiferromagnetic state into an unconventional

antiferroquadrupolar state. More speciﬁcally, we apply

the protocol to the non-integrable S= 1 spin

chain, a system which is mainly interesting for its

symmetry-protected “topological” Haldane phase and

the potential for spin-nematic (quadrupolar) order.

The latter is a state with nonvanishing anisotropic

second-order expectation values of the type DSα

jSβ

jE6= 0,

while having a vanishing ﬁrst-order expectation Sα

j=

0 (where Sα=x,y,z

jis a spin operator). Thus, it is

an interesting quantum state that carries no magnetic

moment, but still breaks the spin-rotational symmetry

via a more complicated order parameter. A spin nematic

can be regarded as something between a ferromagnet and

a spin liquid: While it lacks magnetic order like the

latter, it still breaks the rotational symmetry like the

former and has a preferred axis. The name derives from

the physics of nematic liquid crystals, which in a similar

sense constitute a phase between a liquid and a solid [33].

While quadrupolar order is in principle possible in the

S= 1 chain, in the following section we discuss that

it is not easily achievable in equilibrium, motivating an

extension to driven systems.

Starting from the ground state of an initial

Hamiltonian with Ω = ∞, we slowly drive the system

from the high-frequency to the mid-frequency region,

representing the wavefunction as a MPS. We ﬁnd that

if the initial state is in the symmetry-protected Haldane

phase, it still remains remarkably robust against the drive

and no new phase transition is found. On the other hand,

if the initial state is in a trivial ordered phase, then we

are able to induce an overall quadrupolar moment and

enhanced correlations, eventually reaching a stable phase

with long-range antiferroquadrupolar order, where the

staggered magnetization is suppressed.

II. MODEL

A. Initial Hamiltonians

We consider a one-dimensional chain of localized

spins with S= 1 at zero temperature. The real

system under consideration might in fact be a two-

or three-dimensional array of such chains, where the

interchain coupling is captured on the mean-ﬁeld level

by a staggered magnetic ﬁeld h[35–38]. Experimentally,

such systems are realized in various Ni- and V-based

compounds [39–57] (see also Fig. 1).

The Hamiltonian is an extended variant of the

Heisenberg spin chain:

HHeis =JX

Sj·~

Sj+1 +DX

jSz

j2−hX

(−1)jSz

(1)

where Jis the exchange interaction parameter. ~

Sj=

NC-Ni(CN)2-CN

z²(Ni)

x²-y²(Ni)

z²(N)

FIG. 1. The compound Ni(C2H8N2)2Ni(CN)4(NENC)

under the action of a periodic driving in the x2−y2orbital

of the Ni(II) atoms. The ﬁgure is inspired by Fig. 1 in

Ref. [34], but we have added a possible experimental driving

setup. The structure of the chain repeats in the horizontal

direction. Molecular orbitals from the Ni and N atoms

have been represented in the x, y, z geometry by the lobes

(see legend). Each Ni(II) is attached to the next one by a

NC-Ni(CN)2-CN conﬁguration. The magnetic properties of

such Ni compounds have been studied experimentally. A

theoretical description of these compounds is proposed by

eﬀective Hamiltonians in the form of Eq. (1) representing

the S= 1 chain of the Ni(II). The experimental realization

sketched in this ﬁgure corresponds to the driving protocol

given by Eq. (12) of this work, where the drive in the xy-plane

(represented as a sinusoidal wave) continuously changes the

probability distribution of electronic x2−y2orbital in the

Ni(II) atoms.

(Sx

j, Sy

j, Sz

j) represents the spin-1 operator at the j-th

site, with Sα=x,y,z

jrepresenting the diﬀerent spin

projections. As the local basis |σi, we take the eigenbasis

of Sz

jand denote the eigenvectors as |σi=+,0,−

with the eigenvalues of +1, 0 and −1, respectively.

Finally, his the staggered magnetic ﬁeld, while Dis the

anisotropy in the z-direction (easy-axis for D < 0 and

easy-plane for D > 0). We set ~= 1 and J= 1, thereby

measuring all energies in units of Jand times in units of

~/J.

For D=h= 0, there exists a gapped

phase (the “Haldane phase”), which (for periodic

boundary conditions) has a unique symmetry-protected

ground state with exponentially decreasing spin-spin

correlations. The robustness of the Haldane phase has

been a focal point of previous studies in equilibrium [58–

64], where it was found that it can be characterized by a

non-local string order parameter [65–67]

Oα=x,y,z

string = lim

|j−k|→∞hSα

jeiπPk−1

l=j+1 Sα

lSα

ki,(2)

and by a global twofold degeneracy in the entanglement

spectrum. It is protected by a combination of inversion

symmetry, time-reversal (in the sense Sx,y,z → −Sx,y,z)

and combined rotations of πabout a pair of axes [63].

Since a ﬁnite hbreaks all these symmetries at once, even

a small value destroys the Haldane phase [36]. However,

it remains robust against the anisotropy term, which

does not break any of the above symmetries [62–64].

In this case, the Haldane phase is a thermodynamic

phase, which is stable in an extended region of the phase

diagram, eventually losing in competition to strong-D

phases (see below) once Dexceeds a critical value. An

interesting question is thus how this robustness extends

into non-equilibrium.

The other limiting cases of Eq. (1) are as follows: For

D→+∞, the ground state is given by a product state of

local 0projections. In the D→ −∞ limit, the ground

state is two-fold degenerate, given by the N´eel state . . . +

−+−. . . and the N´eel state shifted by one lattice

site: . . . −+−+. . .. For h→ ±∞, the ground

state is given by a unique N´eel state.

The S= 1 chain is also arguably the simplest system

that allows for quadrupolar exchange. The quadrupole

operator is deﬁned as the traceless tensor

Qαβ

j=Sα

jSβ

j+Sβ

jSα

j−2

3S(S+ 1)δαβ .(3)

It has ﬁve linearly independent components that can be

grouped into a vector:

Qj=





Qx2−y2

Q3z2−r2

Qxy

Qyz

Qxz







=





(Sx

j)2−(Sy

j)2

√33(Sz

j)2−S(S+ 1)

jSy

j+Sy

jSx

jSz

j+Sz

jSy

jSz

j+Sz

jSx







,(4)

so that Pαβ Qαβ

jQαβ

j= 2 ~

Qj·~

Qj. Quadrupolar

exchange thus requires a product of four spin operators

and is usually discussed within the bilinear-biquadratic

model [68,69], given by:

Hblbq =JX

Sj·~

Sj+1 +JqX

j~

Sj·~

Sj+12.(5)

Because of the identity

Qi·~

Qj= 2(~

Si·~

Sj)2+~

Si·~

Sj−2

3[S(S+ 1)]2,(6)

the Hamiltonian Eq. (5) boils down to a competition

of ordinary exchange interaction and quadrupolar

exchange. In 1D, quantum ﬂuctuations generally prevent

spontaneous ordering unless the order parameter is

conserved. In this case, a quadrupolar state may rather

be deﬁned via quadrupolar correlations that dominate

over spin-spin correlations. Such correlations are found

with a 3-site period for J, Jq>0 and Jq/J > 1 [70]. For

J < 0, a ferroquadrupolar order was initially predicted

close to the ferromagnetic phase [71], but highly accurate

MPS calculations demonstrate that it either does not

exist (with a dimerized phase found instead), or only

exists in a very narrow parameter regime [70,72]. In

2D, quadrupolar phases are better deﬁned and a ﬁnite

quadrupole moment generally arises for Jq/J>1 [73,

74].

We note that the choice

HAKLT =X

Sj·~

Sj+1 +1

j~

Sj·~

Sj+12(7)

yields the famous Aﬄeck-Kennedy-Lieb-Tasaki (AKLT)

ground state [60,61], which belongs to the Haldane

phase, but is exactly representable by an MPS with very

low entanglement. From a technical point of view, it

is thus a more convenient representative member of the

Haldane phase than Eq. (1) with h=D= 0.

A possibility to generate ﬁnite quadrupolar moments

in one dimension is the explicit breaking of the

spin-SU(2) symmetry via the so-called “rhombic

single-ion anisotropy” [75–77]:

δHE=EX

jhSx

j2−Sy

j2i

=EX

Qx2−y2

j=E

jhS+

j2+S−

j2i,(8)

where we have introduced the standard spin-ﬂip

operators S±

j=Sx

j±iSy

j. A more general coupling to

the square of the spin operator is also possible [77,78].

In the context of squeezed spin states, the term in

Eq. (8) is known as the “two-axis countertwisting”

(TACT) [79] and there are several proposals of how to

implement it [80]. In equilibrium, a strong value of E

will induce a ﬁnite quadrupole moment in the direction

of DQx2−y2E, similar to how an external ﬁeld induces a

ﬁnite magnetization [75]. However, this requires ﬁnding

a material with large Eand small Dat the same time.

Recently, attention has shifted to a diﬀerent regime,

where bond-nematic rather than local order might be

found. For an S= 1/2 system in a strong magnetic

ﬁeld close to saturation, deviations in magnetization

are given by magnons. If the eﬀective interaction

between these magnons is attractive, they may form pairs

and condense, with non-zero quadrupolar correlations

S+

iS+

jplaying the same role as anomalous expectation

values of fermion-pair creation operators Dc†

ic†

jEin a

superconductor [81]. The best-documented experimental

example where this may occur LiCuVO4[82,83]. The

corresponding experiments are quite challenging and

need to be performed in magnetic ﬁelds of 45-50T.

Summarizing, a spin-nematic state is an exotic

nonmagnetic state with a higher-level order parameter.

The minimal spin value to observe it locally is S= 1.

A stabilization of this state requires strong biquadratic

exchange or a strong anisotropy. However, both are

expected to be weak in real materials or require ﬁnding

a ﬁne-tuned point [84–86], so that the part of the phase

diagram where spin-nematic phases are predicted could

not be explored in practice. Attention has therefore

shifted to the diﬀerent physical regime of magnon pairing

in high magnetic ﬁelds for S= 1/2 materials, which

has its own challenges. Here, we pursue an alternative

idea, namely the purposeful enhancement of quadrupolar

interactions using nonequilibrium driving, starting from

an ordinary S= 1 system (Haldane chain or N´eel

antiferromagnet).

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DynamicallygeneratedquadrupolepolarizationusingFloquetadiabaticevolutionG.Camacho,1C.Karrasch,1andR.Rausch11TechnischeUniversitatBraunschweig,InstitutfurMathematischePhysik,Mendelssohnstrasse3,38106Braunschweig,GermanyWeinvestigatethenonequilibriumdynamicsoftheS=1quantumspinchainsubjectedtoatime-d...

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Dynamically generated quadrupole polarization using Floquet adiabatic evolution G. Camacho1C. Karrasch1and R. Rausch1

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