Multimagnon dynamics and thermalization in the S = 1 easy-axis ferromagnetic chain
←
→
Page content transcription
If your browser does not render page correctly, please read the page content below
Multimagnon dynamics and thermalization in the S = 1 easy-axis ferromagnetic chain
Prakash Sharma,1, 2 Kyungmin Lee,1, 2 and Hitesh J. Changlani1, 2, ∗
1
Department of Physics, Florida State University, Tallahassee, Florida 32306, USA
2
National High Magnetic Field Laboratory, Tallahassee, Florida 32310, USA
Quasiparticles are physically motivated mathematical constructs for simplifying the seemingly complicated
many-body description of solids. A complete understanding of their dynamics and the nature of the effective
interactions between them provides rich information on real material properties at the microscopic level. In this
work, we explore the dynamics and interactions of magnon quasiparticles in a ferromagnetic spin-1 Heisenberg
chain with easy-axis onsite anisotropy, a model relevant for the explanation of recent terahertz optics experi-
ments on NiNb2 O6 [P. Chauhan et al., Phys. Rev. Lett. 124, 037203 (2020)], and nonequilibrium dynamics in
ultra cold atomic settings [W.C. Chung et al., Phys. Rev. Lett. 126, 163203 (2021)]. We build a picture for the
properties of clouds of few magnons with the help of exact diagonalization and density matrix renormalization
arXiv:2107.09105v1 [cond-mat.str-el] 19 Jul 2021
group calculations supported by physically motivated Jastrow wavefunctions. We show how the binding energy
of magnons effectively reduces with their number and explain how this energy scale is of direct relevance for
dynamical magnetic susceptibility measurements. This understanding is used to make predictions for ultra cold-
atomic platforms which are ideally suited to study the thermalization of multimagnon states. We simulate the
nonequilibrium dynamics of these chains using the matrix product state based time-evolution block decimation
algorithm and explore the dependence of revivals and thermalization on magnon density and easy-axis onsite
anisotropy (which controls the strength of effective magnon interactions). We observe behaviors akin to those
reported for many-body quantum scars which we explain with an analytic approximation that is accurate in the
limit of small anisotropy.
I. INTRODUCTION finite frequency observables such as the dynamical magnetic
susceptibility. Adding to the richness of possible emergent be-
How does one characterize the low energy spectrum of a haviors from quasiparticle interactions is the effect of temper-
system of large number of electrons in systems? In many ature [14, 15], which must be accounted for to connect to real
cases, we are fortunate to afford a description of these seem- experiments. Thus, spin chains provide an ideal setting for
ingly complicated many-body systems in terms of quasiparti- exploring the dynamics and effective interactions of magnons,
cles. A comprehensive understanding of their dynamics and and their impact on measurements.
the nature of the effective interactions between them provides Spin-1 systems with predominantly Heisenberg interactions
us with rich information on real material properties at the mi- offer an interesting ground for exploring the physics of inter-
croscopic level. Quasiparticles can exist in various forms; acting magnons. For spin-1 and higher, terms such as the bi-
for example, effective electrons in a Landau-Fermi liquid [1], quadratic interaction and on-site anisotropy are allowed [16],
spinons in a quantum spin liquid [2] or magnons in a system both of which are forbidden for the spin-1/2 case. There is a
with conventional magnetic order [3, 4]. large class of materials and associated realistic models with
Magnetic spin chains provide particularly illuminating ex- high spin [13, 15, 17–21, 23–26]; with the ability to perform
amples of quasiparticle physics associated with many deep in- accurate measurements and theoretical simulations of these
sights for how strongly correlated electrons collectively act. systems, there is renewed interest in their physics. In addition
In the antiferromagnetic spin-1/2 chain, for example, neutron to the plethora of high spin compounds on the materials front,
scattering sees a continuum of excitations, providing a strik- the ability of cold atom systems to realize effective high spin
ing confirmation of spin fractionalization and the emergence models is an exciting opportunity to explore high spin physics
of spinon quasiparticles [5, 6], a consequence of correlated in new regimes [27, 28].
many-body effects. The spin-1 case is equally spectacular, Our work here is inspired by, but not limited to, recent tera-
giving rise to the Haldane spin gap associated with effective hertz (THz) optics experiments on NiNb2 O6 [15], and recent
spin-1/2 fractionalized degrees of freedom which are decon- realizations of spin-1 magnets with tunable anisotropy in ul-
fined [7–12]. The low dimensionality of spin chains makes tra cold atomic settings [27]. In the former experiment, the
magnetic order highly unstable to quantum fluctuations, and interaction between magnons was effectively tuned between
it is now known that even in higher dimensional systems, geo- attractive and repulsive by changing the direction and the
metrical effects such as frustration can achieve similar qualita- strength of external magnetic field (longitudinal versus trans-
tive outcomes [2, 13]. In the case of ferromagnets, more con- verse). This manifests itself as a significant shift in the peak
ventional magnon quasiparticles are expected and observed, feature which moved lower or higher in energy depending on
which are well described within the framework of spin wave the field direction and the temperature at which the measure-
theory. Interactions between magnons can lead to distinct sig- ment was made. Evidence of a critical point (analogous to that
natures in the excited state spectrum [4], and corresponding present in the S = 1/2 transverse field Ising model) was indi-
rectly inferred by comparisons of the experimental data to ex-
act diagonalization calculations. More recently, higher order
∗ hchanglani@fsu.edu magnon bound states arising out of magnon interactions have2
(a)
what imprint does anisotropy and magnon density leave on the
Veff bimagnon physics of thermalization of multimagnon states?
1 1 0 1 1 0 1 -1 With these objectives in mind, the paper is organized as
follows. In Sec. II, we visit the case of two magnons, and
- J S i Sj -D (Siz)2 magnon use a combination of the t-matrix method, the density ma-
trix renormalization group (DMRG) algorithm [31], and pre-
Single magnon dispersion
viously known exact results for two magnon bound states.
ωq In Sec. III, we transfer our lessons to the case of three and
higher magnon bound states, and study the nature of magnon
(b) (c) clouds using appropriately defined correlators. A simple Jas-
trow function captures all our numerical results surprisingly
D 1 0 -1 well, using which we provide both quantitative and qualitative
0 q
-Π Π characterization for magnon interactions. We then focus on
the formation of doublons in the magnon clouds, and how they
FIG. 1. (a) Schematic of a spin-1 chain showing the terms in the grow once they are completely saturated. In Sec. IV, we carry
Hamiltonian. A representative configuration in the Hilbert space is forth the acquired insights to address the findings of finite tem-
shown. The ground state corresponds to all sites in the |1i state (or
perature dynamical measurements in NiNb2 O6 . In Sec. V we
all |−1i). For this ground state, a |0i corresponds to a single magnon
and |−1i is a bimagnon. The magnons attract each other via effective study the effect of magnon-magnon interactions in the time
interactions mediated by the onsite anisotropy. (b) Single magnon domain, focusing on the protocol used in spin-1 cold atom se-
dispersion curve for the S = 1 FM spin chain obtained from linear tups [27]. We simulate the nonequilibrium dynamics of these
spin wave theory. Anisotropy gaps out the lowest energy excitation. chains using time-evolution block decimation (TEBD) [32].
(c) Mapping between hyperfine states and S = 1 degrees of freedom We study revival and thermalization behaviors many of which
as used in recent ultra cold atomic setups. The pink and blue states resemble those seen for quantum many-body scars. We con-
correspond to hyperfine states of 87 Rb, and the chemical potential is clude by summarizing our findings and discussing avenues for
adjusted to two bosons per site. A microwave pulse can be used to possible future experiments.
prepare a superposition of hyperfine states, which in turn translates to
a superposition of |1i,|0i, |−1i (depicted with red, yellow and green
colors). More details are in the text.
II. TWO MAGNON BOUND STATES
been seen in FeI2 both in neutron [29] and terahertz optics In this section we review the elementary magnon quasipar-
experiments [30]. These findings suggest the importance of ticle excitations of the ferromagnetic (FM) spin chain, and
magnon-magnon interactions that arise from anisotropic terms use them to build a picture of two magnon bound states. We
in the Hamiltonian, which are consequences of spin-orbit cou- characterize properties such as their binding energy and their
pling. spatial extent. We will establish that the uniaxial anisotropy
In this paper we focus on a simple spin-1 Hamiltonian of D > 0 acts as an attractive interaction between magnons, and
direct experimental relevance, both from the point of view of show that this leads to magnon bound states.
real materials and cold atoms, For the spin-1 case a magnon is the lowest energy exci-
X X tation, which has Sz = ±1. (e.g., |1i → |0i) Since our
H = −J S~i · S~j − D (Siz )2 , (1) Hamiltonian in Eq. (1) has total Sz as a good quantum num-
hi,ji i ber, magnons can be used as building blocks to describe low
energy excitations of various Sz sectors. Using the Holstein-
where J > 0 is the ferromagnetic exchange interaction, D Primakoff transformation [33], we rewrite Eq. (1) in terms of
(> 0 in this paper) is the local onsite uniaxial anisotropy, Siµ bosonic magnon creation and annihilation operators a†i and ai :
for µ = x, y, z are spin-1 operators at site i. The ground state
is two fold degenerate, with the symmetry broken states be-
q
ing either |1, 1, . . . , 1i or |−1, −1, . . . , −1i. Working with Si+ = 2S − a†i ai ai , Siz = S − a†i ai . (2)
one of these ground states as our vacuum, the elementary
excitation of a ferromagnetic spin-1 chain is a magnon (e.g. Substituting the above transformations and expanding the
|1i → |0i) which has well defined energy and momentum. Hamiltonian up to quartic order in ai and a†i , we get
We generally focus on the case of D ≤ J, for NiNb2 O6 ,
J ≈ 14.8 K (0.308 THz) and D ≈ 5.2 K (0.108 THz) [15], H = H0 + H2 + H4 + . . . . (3)
i.e., D/J ≈ 0.35.
The questions we pose here are the following: What hap- where H0 = −LJS 2 − LDS 2 is the classical ground state
pens when there are multiple magnons in the system, as is energy and L is the total number of sites in spin chain.
expected at finite temperature (or with the introduction of a The single particle hopping term H1 is given by
magnetic field). Do the magnons attract and form composite
(a†i aj + H.c.) + ((2S − 1)D + 2JS)a†i ai ,
X X
bound states? What are the properties of these composite ob- H2 = −JS
jects, and is there a limit to how big they can be? And finally, hi,ji i3
12 0.8
0.04 0.8 D/J = 0.3
10 0.18 D/J = 1.0
E2/J
0.7
0.16 0.6 D/J = 2.0
EBJ/D 2
8 0.0 0.2 0.4 0.03
L (l, 0)
0.14
K
E2/J
6
EB/J
0.02 0.12 L= 0.4
0.0 0.2 0.4 32
4 D/J 64
96
0.01
128 0.2
2 160
192
224
0 0.00
256 0.0
0 /2 0.0 0.1 0.2 0.3 0.4 0.5 40 20 0 20 40
K D/J Lattice site (l)
(a) (b) (c)
FIG. 2. (a) Exact two magnon dispersion for D/J = 0.35, which is experimentally relevant. The shaded region indicates the two magnon
energy continuum and the two branches (red and blue) below the continuum represent the two stable bound states. The lower (blue) branch
represents the ‘Bethe type’ bound state while upper red branch represents ‘Ising type’ bound state. At K = 0, the two magnon bound state
energy lies below the continuum with energy difference 0.019 which is visible in the inset zoomed over a small region around K = 0 point.
(b) Binding energy EB /J as a function of anisotropy D/J calculated at for the lowest-energy two-magnon states. The inset shows EB J/D2
vs. D/J. The black dashed line in the inset is a linear fit to the data points excluding the upturn. The y intercept ∼ 0.122 agrees with the
t-matrix result that the leading order term in EB is quadratic in D with coefficient 1/8. The size of the bound state increases with decreasing
D, and the upturns of the curves in the inset show finite size effect. (c) Probability distribution of two magnons in the lowest two magnon
energy state with one magnon fixed at a reference site ‘0’. The circular dots at central site ‘0’ represent the probability of finding two magnons
on the same site (i.e., a bimagnon |−1i).
diagonalizing which gives the bare single magnon (spin wave) for the ferromagnetic chain with Sz conservation, since the
dispersion higher order terms in Holstein-Primakoff make no contribu-
tions to the case of one magnon.
~ωq = 2JS 1 − cos(qa) + (2S − 1)D (4) We now turn to the case of two magnons. This problem has
been solved exactly for arbitrary spin S ≥ 1 (in arbitrary di-
where a is the lattice constant of the spin chain. The mensions) at zero temperature by Tonegawa [34]. (Note that
anisotropy term vanishes for S = 1/2, since the (Siz )2 opera- the choice of J in Ref. [34] is different from ours by a fac-
tor cannot distinguish between up and down spins. However, tor of 2. The exact results have been adapted to match our
it is allowed for S = 1, and leads to a gap in the spin wave convention.) The key results are as follows. The energy of
dispersion (gives mass to the magnons), stabilizing ferromag- a two-magnon bound state in an S = 1 Heisenberg chain is
netic order. This result for one magnon excited states is exact given as a solution of a cubic equation
3 2
(1 + δ − ) + p2 (1 + δ − ) + p1 (1 + δ − ) + p0 = 0, (5)
where the two magnon energy is E2 = 4J, and
0 0
p0 = −[{(1 − 2α)ξ 2 cos2 (Ka/2) − 2αδ }2 + δ 2 ξ 2 cos2 (Ka/2)]/4α, (6a)
0 0 0
p1 = −(1 − 2α − δ )ξ 2 cos2 (Ka/2) + δ (2α + δ ), (6b)
0
2 2
p2 = {(1 − 4α)ξ cos (Ka/2) − 4α(2δ + α)}/4α. (6c)
K = q1 + q2 is the total momentum of the two magnons, and two branches when plotted as a function of K. For one of the
0
the parameters are given by ξ = 1, δ = δ = D/2J, and solutions, the two magnon-wavefunction has a large ampli-
α = 1/4. tude for magnons on nearest neighbor sites, and smaller am-
plitude for two magnons on the same site (i.e., a bimagnon
Even though Eq. (5) has three solutions, Tonegawa has ar-
corresponding to | − 1i). This solution is referred to as the
gued that only two of them are physically meaningful. Close
“Bethe type” bound state in analogy with the bound state so-
to the Brillouin zone boundary K = π (in units of 1/a the lat-
lution first found by Bethe on ferromagnetic S = 1/2 chains
tice constant), both solutions are real-valued, which appear as4
[35]. The other type of bound states are of the “Ising type,” within the t-matrix approximation is
where the amplitude for two magnons on the same site is
χ(0) (q, ω)
larger than the amplitude for nearest neighbor magnons. (This χt2 (q, ω) = , (8)
type of bound state is forbidden for S = 1/2 chains, and 1 − Dχ(0) (q, ω)
hence there is only one bound state branch for the two magnon where χ(0) is the Lindhard susceptibility
dispersion in this case).
dd k −1
Z
Below a certain threshold momentum Kth , the cubic χ(0) (q, ω) = . (9)
equation yields a pair of complex-valued solutions (with (2π)d ω + i0+ − (−k + k+q )
nonzero imaginary part), and only one real-valued solution. We define the binding energy of two magnons as
The complex-valued solution occurs when the corresponding
EB ≡ (E2 − E0 ) − 2(E1 − E0 ) = E2 − 2E1 + E0 (10)
branch of bound state is no longer stable. Thus, Kth is the mo-
mentum at which the bound state dispersion joins the two par- where E0 , E1 , and E2 are respectively the energies of the
ticle continuum. To demonstrate this, we numerically solve lowest lying states in the 0-magnon, 1-magnon, and 2-magnon
Eq. (5) for D/J = 0.35 and plot the two-magnon dispersion sectors. At q = 0, where the two-magnon bound state energy
in Fig. 2(a). Indeed, we find two branches of two-magnon is minimized, we find that EB , identified by the location of
bound states, with one branch merging with the two-particle poles in χt2 , is
continuum. At K = 0 we see that the two-magnon bound
D2
state is separated from the continuum (see inset). EB = . (11)
8J
The order of appearance of the two magnon branches de-
The result for the ferromagnet is in sharp contrast to bound
pends on the strength of D/J: the origin of this effect is the
states in high spin antiferromagnets, where the binding energy
touching of the Bethe and Ising branches at K = π. For p
D/J . 0.5, the Bethe (Ising) type constitutes the lower (up- was found to be EB = JD/2S 2 [37]. We also note that
per) branch, and vice versa for D/J & 1.1. (This is to be ex- when the magnon interaction is repulsive, χt2 does not allow a
pected, after all, for small D/J it is energetically unfavorable pole, and thus no bound state exists.
for two magnons to want to give up a significant part of their We assess the validity of the above approaches with the help
kinetic energy in order to be perfectly localized on the same of (almost) exact DMRG calculations for two magnons (in the
site, while at large D/J this bimagnon/doublon formation Sz = L−2 sector) for various system sizes. Figure 2(b) shows
does become favorable.) For intermediate D/J the behavior our results from DMRG as functions of the anisotropy D/J.
is more subtle, for the lowest branch the small |K| wavefunc- Since the two magnons are only weakly bound for small D/J,
tions are of the Bethe type. For larger |K|, the lower energy we observe significant finite size effects in the binding energy
branch is of the Ising type, and the higher energy branch is of in this regime (the upturn). However, by carefully extrapolat-
the Bethe type. (This subtlety has been discussed in a note in ing EB /D2 to the D → 0 limit, we do find that it approaches
proof by Ref. [34] in response to the study of Ref. [36].) The a value close to 1/8, which is consistent with the t-matrix
two branches cross at the zone boundary K = π/a: The en- results. Furthermore, Taylor expansion of the exact solution
ergy of the Bethe type bound state is given by 3J + 2D while from Ref. [34] is given by
the energy of the Ising type bound state is fixed at 4J and does EB (D/J)2 5(D/J)3 57(D/J)4
not depend on D. = + + + · · · . (12)
J 8 64 2048
The formation of bound states for D > 0 can also be cap- This confirms the importance of the on-site interaction pro-
tured by treating the quartic order term H4 in the expansion vided by D for magnons with near the Brillouin zone center,
in Eq. (3) that describes the interaction between the magnons, an assumption made in the t-matrix approach.
given by Our results are further strengthened by evaluating the scalar
components ψ(l, l0 ) (in the notation of [34]) of the two
JX † † magnon wavefunction |ψ2 i,
a†i a†i ai ai −
X
H4 = −D ai aj aj ai
2 1X
i hiji |ψ2 i = ψ(l, l0 )Sl− Sl−0 |ψ0 i (13)
2 0
J X † † l ≤l
+ ai ai ai aj + a†j a†j aj ai + H.c. , (7) 0
8 where l, l are site indices and |ψ0 i is the ferromagnetic
hiji −
ground state (with zero magnons), and Sl(l 0 ) are the spin low-
ering operators. We focus on the ground state which we find
using the t-matrix approach. The details of our computations to be in the K = 0 sector. Fixing a reference site 0 our
are discussed at length in Appendix A. Here we highlight the plot in Fig. 2 (c) confirms that for small D/J, the bound
key ingredients of our calculation. state is indeed of the Bethe type (|ψ(0, 0)| < |ψ(1, 0)|), and
The Heisenberg term J contributes to interaction between for large D/J it is of the Ising type (|ψ(0, 0)| > |ψ(1, 0)|)
magnons on neighboring sites, while the anisotropy term D with |ψ(0, 0)| ≈ |ψ(1, 0)| for D/J ≈ 1.15. We will
serves as attractive interaction between the magnons on the see later that when the magnon number increases, this bi-
same site. Keeping only the on-site interaction D, since the magnon/doublon formation becomes increasingly important
J term vanishes for q → 0, the two-magnon susceptibility even for small D/J.5
D =0 D = 0.35 D = 0.7
J J J
18
0.0035
3rd magnon position (j)
9 0.0030
0.0025
0.0020
|Cijo(3)|2
0
0.0015
9 0.0010
0.0005
18 0.0000
18 9 0 9 18 18 9 0 9 18 18 9 0 9 18
2nd magnon position (i) 2nd magnon position (i) 2nd magnon position (i)
(a)
cut along j = i 1 cut along j = i
cuts
10 4
20 j=i 1
8
j= i
10
15 linear fit
(3) 2
|
12
10
|Cijo
0 j i
X 10
16
10 D = 0.1 D = 0.5
fit param
j 0 i
20
D = 0.2
D = 0.3
D = 0.6
D = 0.7
X 5 slope = 1.99
10
D = 0.4 D = 0.8
0
0 10 20 30 40 50 0 5 10 15 20 25 30 0.0 2.5 5.0 7.5 10.0
site i site i J/D
(b) (c)
(3)
FIG. 3. (a) The three-magnon correlator Cijo , obtained by fixing one magnon at a reference site o, computed for the ground state of three
magnons, for representative values of anisotropy parameter D, in a periodic chain of length L = 40 using ED. The colors indicate the
magnitude of the correlator. (b) The spatial dependence of the correlator for two one-dimensional cross sections, respectively corresponding
to the two magnons at sites i and j with j = i − 1 (left panel) and j = −i (right panel), while the third magnon is located at ‘o0 . Both cross
sections show exponential decay with increasing spatial dependence between the magnons. Black solid lines are the fits with Jastrow factor
Q |xa −xb |/ξ
a6
cut along j = i 1, k = 1 cut along j = i, k = i + 1
20 cuts
10 3
j = i 1, k = 1
10 7 15 j = i, k = i + 1
linear fit
(4) |2
10 11
|Cijko
10
10 15 D = 0.1 D = 0.5 fit param
D = 0.2 D = 0.6 5 slope = 2.01
10 19 D = 0.3 D = 0.7
D = 0.4
0
0 6 12 18 24 30 36 0 4 8 12 16 20 0.0 2.5 5.0 7.5 10.0
site i site i J/D
(a) (b)
(4)
FIG. 4. The four-magnon correlator Cijko , obtained by fixing one magnon at a reference site o, computed for the ground state of four magnons
for representative values of anisotropy parameter D, in a periodic chain of length L = 76. (a) The spatial dependence of the correlator for
two one-dimensional cross sections, respectively corresponding to the two magnons at sites i and j with j = i − 1 and k = 1 (left panel), and
j = −i and k = −i + 1 (right panel). Both cross Qsections show exponential decay with increasing spatial dependence between the magnons.
Black solid lines are the fits with Jastrow factor a 0, this picture is dramati-
(3)
X
|ψn i = C(x1 , x2 , . . .)Sx−1 Sx−2 · · · Sx−n |ψ0 i (14) cally altered, Cijo is now localized around i = j = o, and de-
x1 ≤x2 ≤...≤xn cays with distance. The magnons clearly cluster together more
strongly along the i ≈ j line and particularly near i ≈ j ≈ o,
where we have generalized the notation ψ(l, l0 ) from Eq. (13) the larger D is, indicating bound state formations.
to the case of n ≥ 2, with site indices x1 , x2 , . . . , xn and These qualitative assertions are made more precise by sys-
absorbed the factor of 2 from Eq. (13). For the case of S = 1, tematically studying one dimensional cross sections along
no three indices can be the same since there are maximum j = i + 1 and j = −i, as shown in Figs. 3(d–f). The cor-
of two magnons on a given site. C(x1 , . . .) represents the relator is found to be exponentially decaying with |i| (as may
amplitude of a particular configuration of magnons, and are be anticipated); the precise exponent depends on the cross sec-
complex-valued in general. Here we focus on the ground state tion under consideration, as we will explain shortly. For the
of each multimagnon sector on a periodic chain, which we regime of interest (D < J), we find that the amplitude for two
find to have a net momentum of zero, and C(x1 , . . .) to be magnons on being at the same site (i.e. when two of the in-
positive. dices ‘i’, ‘j’, or ‘o’ are equal) is smaller than the two magnons
We numerically determine the exact ground state of n being at adjacent sites (i = j ± 1), suggesting the magnons
magnons, |ψn i. We then measure the correlators form a three-magnon analogue of Bethe type bound state in-
(3) stead of a single-ion type bound state.
Cijo = hψ0 |Si+ Sj+ So+ |ψ3 i, (15a)
Based on the above observations, the formation of higher-
(4)
Cijko = hψ0 |Si+ Sj+ Sk+ So+ |ψ4 i, (15b) order bound states (as ground states of the n-magnon prob-
lem) may be anticipated, but is not a priori obvious. This is
where i, j, k and o are site indices, with o a fixed reference because there is an inherent competition between the possi-
(3) (4)
site. These correlators Cijo and Cijko directly correspond to bility that the system forms a heavy “droplet” or “cloud” of
the first-quantized wavefunctions of three- and four-magnon multiple magnons which stick together, giving up their indi-
states. vidual kinetic energy for their collective good, and the pos-
As in the case of two magnons, we find that uniaxial sibility that the magnons split apart into smaller clouds, each
anisotropy D plays an essential role in stabilizing multi- of which has its own kinetic energy. From the four-magnon
magnon bound states in the spin chain. Figure 3(a-c) shows correlators shown in Fig. 4, we find many qualitative similar-
three-magnon correlator calculated in the lowest-energy three- ities with the cases of two and three magnons, supporting the
magnon state for three representative values of D. For D = 0, former scenario.7
1.0 100 1.00
n=2 5 D = 0.0 D = 0.6 bimagnon increases,
n=3 10 1 4 0.75 D = 0.2 D = 0.8 spacing decreases
1
0.5 n=4 3 D = 0.4 D = 1.0
E5
En = En En
n=5 10 2 2 0.50 E4
0.0 10 1
n=6 0.2 0.3 0.4 0.5
(Siz)2
N ij
Siz
1/n 0.25
n2
n=7 E3
10 3 10 3
n=2 ν3
n=8 0.00 E2
0.5 1 n=9 10 4 n=3
10 5
n = 10 n=4 ν2
1 4 7 10 13
n=5
0.25
1.0 sites 10 5 E1
1 4 7 10 13 16 0 10 20 30 0 5 10 15 20 25 ν1 = D
sites (i) r = |i j| magnon number (n) E0
(a) (b) (c) (d)
FIG. 5. (a) hSiz i and 1 − h(Siz )2 i calculated with DMRG for D/J = 0.35 in the lowest-energy states of various magnon number sectors, on
a chain of length L = 100 with the open boundary condition. The initial states are chosen such that the magnons are on the left end of the
chain. (b) Normalized pair correlation function ρij ≡ hni nj i for D/J = 0.35 in a periodic chain of length L = 80 for 2 and 3 magnons,
L = 68 for 4 magnons, and L = 62 for 5 magnons. Solid lines are fits to the exponential form e|i−j|/ζ and dotted lines are the extrapolations
from the long distance fits. Inset is the pair correlation length ζ vs 1/n obtained from the fits. (c) Successive energy gap between consecutive
magnon sectors as a function of magnon number for various anisotropy parameters D with J = 1 fixed in a periodic chain of length L = 180.
For small n, ∆En decreases with increasing n. With increasing n, oscillatory behavior is observed for D/J & Dc /J ≈ 0.5. (d) Schematic
showing the reduction in transition frequency due to multimagnon bound state formation in different magnon sectors.
We now unify our findings to qualitatively and quantita- also extract the size of the magnon cloud without resorting to
tively understand what happens for larger number of magnons. the Jastrow form by
Motivated by the observation of the correlator’s exponential X
decay with separation between two magnons, we conjecture hXn i ≡ (xn − x1 )|C(x1 , . . . , xn )|2 . (17)
that for sufficiently small number of magnons (which depends x1 ≤···≤xn
on the system size as well as D/J) we can think of the prob-
lem in the dilute limit. (We ignore the formation of the single- This yields hX2 i = ξ for the two magnon Jastrow function
ion bound state.) In this limit, the n-magnon wavefunction as expected. We find that for n = 2, 3, 4, the magnon extent
can be written as a product of pairwise Jastrow factors hXn i does not vary significantly with n, and is still (roughly)
Y ∼ ξ across a range of D/J.
C(x1 , x2 , . . . , xn ) ∝ e−|xa −xb |/ξ . (16) For much higher n, the computation of higher magnon cor-
a8
1 even 100 where En and En−1 are the ground state energies of the n-
D=0.05
D=0.1 and n−1-magnon sectors, respectively. Our calculation on the
(Siz)2
10 4 D=0.2
0 D=0.3 spin chain with various anisotropy D/J, shown in Fig. 5(c),
Siz
D=0.4
10 8 D=0.5 clearly shows that the energy difference between successive
D=1.0
even
1
11 odd 100 magnon sectors for small n decreases with magnon num-
bers, as expected from the effective attraction between the
(Siz)2
0 D= 10 4
0.05 0.4 magnons. Starting at ∆E1 = D, ∆En decreases with in-
Siz
0.1 0.5
0.2 1.0 10 8
creasing n up to a particular D-dependent number nc . Said
0.3 odd
1
1 differently, the energy cost to put an additional magnon into
85 90 95 100 100 120 140
site i site i the cloud decreases with increasing n.
(a) (b)
As we have discussed previously, the extent of the magnon
cloud does not increase appreciably with increasing n. It is
2.0 thus expected that the Ising type bound states become energet-
1.5 even
odd ically more favorable than the Bethe type. This is unlike the
1.5 case of two and three magnons where the Bethe type bound
1.0 states are lower in energy than the Ising type bound states for
EDW/J
DW
1.0 D < J/2.
e )/J
EDW
0.25 When a large number of magnons form a bound state, ki-
0.5
0.5 netic energy (XY terms) is highly suppressed, and thus its
(EDW
even
o
0.00
odd 0 D/J 1 energetics can be understood from considering the interac-
0.0 0.0 tions between the magnons (Ising terms). The z compo-
0 2 4 6 8 10 0.0 0.2 0.4 0.6 0.8 1.0
J/D D/J nentPof Heisenberg exchange P and uniaxial anisotropy (i.e.,
(c) (d) −J hi,ji Siz Sjz and −D i (Siz )2 ), both lower the energy
for single-ion Ising like (| · · · , −1, −1, −1, · · · i) bound state.
FIG. 6. (a), (b) Spatial profiles of the domain walls at various val- However, both terms contribute nothing to the sites where the
ues of D for L = 180 chain (J = 1). The labels ‘even’ and ‘odd’ Bethe-like (| · · · , 0, 0, 0, · · · i) bound state is located. The
refer respectively to the total Sz = 0 and −1 sectors. (c) Thick- even-odd oscillations observed in Fig. 5(c) beyond certain
ness ξDW of the domain wall vs. D, from fits to 1 − h(Siz )2 i ∼ numbers of magnons for large values of D/J are due to
e−|x−xDW |/ξDW . (d) Energy of a domain wall EDW vs. D. The in- the fact that an unpaired magnon (|0i) in the odd magnon
set shows the energy difference between the ‘odd’ domain wall and sectors raises energy because system favors bimagnon state
‘even’ domain wall. (|−1i). Furthermore, the oscillatory behavior is observed for
D & 0.5J, which coincides with the range of D where,
by measuring the magnon pair correlation, which we define as in the two-magnon bound state dispersion, the lower energy
branch begins to have Ising type boundstates with significant
ρij = hni nj i, (18) bimagnon contribution.
where ni is the local number operator for magnon at site i.
Substituting ni = 1 − Siz , the pair correlation takes the form
(in terms of local spin operators) ρij = 1 − hSiz i − hSjz i + C. Dense magnon limit: Domain walls
hSiz Sjz i. In Fig. 5(b), we plot the pair correlation function for
up to five magnons. We find that the average separation be- A complementary picture to the magnons is provided by
tween magnons scales as 1/n. This suggests that magnons domain walls, which captures the behavior of systems with
get closer to each other (on average) on introducing more large number of magnons since the two ends of a mul-
magnons in the system. This is broadly consistent with the timagnon droplet can each be viewed as a domain wall
cloud being constant in size (for a small number of magnons) (DW). In the Ising limit for S = 1, there are two types of
with its size being D dependent. Said differently, the magnon DWs: |· · · , 1, 1, −1, −1, · · · i with its center on a bond, and
cloud “absorbs” additional magnons, i.e. it gets heavier and |· · · , 1, 1, 0, −1, −1, · · · i with its center on a site. Easy-axis
its size does not expand significantly. This cannot continue anisotropy term prefers the former. With the inclusion of XY
indefinitely for arbitrarily large number of magnons, since bi- interaction, they acquire thicknesses larger than one lattice
magnon formation eventually saturates the cloud. constant. The two types, however, reside in different Sz sec-
tors, and tunneling between one type and the other is allowed
only when there are multiple DWs in the system. The oscil-
B. Energetics of magnon clouds latory behavior in ∆En , shown in Fig. 5(c), can then be as-
cribed to the fact that the even magnon sector allows for two
We now discuss the energetics of multimagnon states. The bond-centered DWs, which have lower energy, while the odd
energy to introduce an additional magnon into the cloud of magnon sector forces one of the DWs to be site-centered.
n − 1 magnons is Figures 6(a) and 6(b) show spatial profiles of the two types
of DWs at various values of D on a L = 180 lattice with open
∆En = En − En−1 , (19) boundary condition, calculated using DMRG. A DW here is9
defined as the lowest energy state in the total Sz = 0 (for linearly polarized light is inferred from the transmission co-
‘even’) or −1 (for ‘odd’) sector, starting from the correspond- efficient; using this information, the dynamical susceptibility
ing configuration in the Ising limit as the initial state. The is determined. The locations of peaks in the dynamical sus-
DWs have exponential profiles, and their thicknesses decrease ceptibility reveal information about the energy levels of the
with increasing D [see Fig. 6(c)]. At small D with thick DWs, system, allowing indirect inference of which transitions are
the two types of DWs track each other closely; the two start most active at a given temperature. In a longitudinal applied
diverging significantly at D ∼ 0.3, where ξDW ∼ 1 and on- field, the most prominent peak in the low-frequency suscep-
site correlations between magnons become important. Fur- tibility moves to lower frequency on increasing temperature.
thermore, the amplitude of the oscillation in ∆En for large n This observation was not reported in CoNb2 O6 with effective
matches the energy difference between the two types of DWs S = 1/2 magnetic ions [45] (albeit with different exchange
[compare Fig. 6(d) and its inset with Fig. 5(c)]. interactions [46, 47]), strongly hinting that the S = 1 nature
of the magnetic ions is at the heart of the effect. Additionally,
the direction of the temperature-dependent shift was reported
IV. RECAP OF FINITE TEMPERATURE DYNAMICAL to depend on the direction of the applied static magnetic field.
EXPERIMENTS AND CONNECTION TO OUR RESULTS For the case of longitudinal field with field strength B,
magnon number is a good quantum number and all our analy-
Till this point, our focus has been on multimagnon states ses in the previous sections apply straightforwardly, with −D
which are ground states of their respective magnon number being replaced by −D −gµB B. [The additional Zeeman term
sectors. However, an explanation of the finite temperature dy- −gµB B does not alter the wavefunctions in a given magnon
namical susceptibility of the spin-1 chain requires us to de- sector, all it contributes in a given sector is an overall energy
velop the connection to excited states (and corresponding ma- shift.] For the case of transverse applied magnetic fields, ap-
trix elements) which enter the response functions. In this sec- plied along the x axis, (which we have not considered in this
tion, we briefly recap crucial aspects of the terahertz optics ex- paper), magnon number is not a well-defined at small field
periment [15] on the spin-1 chain compound NiNb2 O6 (which strengths. For large field strengths (i.e., past the critical point)
we refer here on as the “JHU experiment”) and summarize the it is reasonable to choose the quantization axis to be along the
key findings. We build on results presented in earlier sections direction of the applied field, and it is this axis which serves
with the objective of explaining the findings of the JHU ex- as the natural quantization axis. In this setup, magnon num-
periment. ber is approximately conserved and their mutual interactions
In the JHU experiment, the direction of placement of the are repulsive. Due to this repulsion, the peak frequency in
chain of magnetic atoms was referred to as the z axis, the the low-frequency dynamical susceptibility increases with in-
direction of incidence of light the x axis, and the light is lin- creasing temperature.
early polarized with its oscillating magnetic field along the The JHU experimental findings call for a closer look at the
y axis. (As a first approximation, we will ignore any possi- mechanism by which this temperature dependent energy shift
ble canting of the easy-axis of the spins with respect to the z occurs for the S = 1 chain. Within linear response theory
axis, i.e., the easy-axis anisotropy is perfectly along the one (Kubo formalism), the dynamical susceptibility at finite tem-
dimensional chain of spins.) The absorption cross section of perature is given by,
π(1 − e−βω ) X −βEp 2
χyy (ω, T ) = e hq|Sy |pi δ(Ep − Eq + ω) (20)
Z p,q
where ω is the frequency being probed, T and β = 1/kB T ence decreases from D to zero on increasing magnon num-
are
P the temperature and inverse temperature respectively, Z = ber, followed by even-odd oscillations. (For D/J = 0.35
−βEp
p e is the partition function, |pi are the eigenstates of we find these oscillations to be fairly weak, however they are
the
P y yHamiltonian with the energy eigenvalues Ep , and Sy = significantly strengthened at larger D/J, especially around
S
i i (S i being local spin operator acting at site i). There- D/J = 0.5 which is when the Ising type/bimagnon bound
fore, the transition matrix element hq|Sy |pi couples only those states become important. This effect is potentially observable
states (|pi and |qi) which differ in total Sz quantum number in systems where a large anisotropy can be realized.) Since
by one unit of angular momentum. Thus, the transition fre- higher magnon sectors are entropically favored on increasing
quency ω = Eq − Ep is the energy difference that involves temperature, our calculations suggest that the peak frequency
states in two consecutive magnon sectors. (We will attach an should be reduced by an amount of D when the temperature
additional label to the state label to indicate the magnon num- is increased from low temperature T J, D to the high tem-
ber sector it belongs to.) perature limit T J, D. This observation is consistent with
the findings of the JHU experiment for the case of longitudi-
In the previous section we showed that this energy differ-10
nal fields (their Fig. 3 for B = 65 kG)–the peak in the dy- 8.0
| j (2)|Sy|i (1) |2
namical susceptibility moves from 0.28 THz to 0.20 THz on 0.125
increasing from the temperature from 5 K to 50 K. Given the 8.5 0.100
i, max(THz)
j (THz)
simplistic modeling of the spin chain, this observed shift of 9.0 0.075
0.08 THz is very reasonable compared to the theoretical esti-
E(2)
0.050
mate of D = 0.108 THz. 9.5 Data
0.025
The above argument relies on simplifying assumptions meani i, max
10.0 0.000
about the energetics of the transitions that contribute to the dy- 10.0 9.5 9.0 10.0 9.5 9.0
namical susceptibility, only ground state to ground state tran- E(1)
i (THz) E(1)
i (THz)
sitions between n and n + 1 magnon sectors were considered. (a)
However, according to the Kubo formula Eq. (20), all possi-
ble contributions arising from transitions must be accounted | j (3)|Sy|i (2) |2
8.50 0.125
for. χyy (ω, T ) measures the appropriately weighted sum of
all possible transitions, which are consistent with the selec- 8.75 0.100
i, max(THz)
j (THz)
tion rules (no change in linear momentum, and change of spin 9.00 0.075
9.25
E(3)
angular momentum by one quantum). From χyy (ω, T ) alone 0.050
one can not reconstruct the transitions that contribute signifi- 9.50 0.025 Data
meani i, max
cantly to it. 9.75 0.000
9.75 9.50 9.25 9.00 8.75 9.75 9.50 9.25 9.00 8.75
This lack of information is remedied with the numerical E(2) E(2)
i (THz) i (THz)
(here, exact diagonalization) calculations. Figure IV shows
transition matrix elements between 1 → 2 magnon and 2 → 3 (b)
magnon energy levels for the material specific parameters.
2
The many-body eigenstates in the n- (n + 1-) magnon sector, FIG. 7. Matrix elements hj (n+1) |Sy |i(n) i for transitions between
organized by increasing energy, are on the horizontal (verti-
(a) (left panel) for n = 1 and (b) (left panel) n = 2 magnon levels
cal) axis. The size of each black dot is proportionate to the calculated in a periodic chain of length L = 24 using ED and for
2
matrix element hj (n+1) |S y |i(n) i representing the transition parameters relevant to NiNb2 O6 , i.e., J = 0.308 THz and D =
(n)
0.108 THz. The size of the black dots represent the square of the
between a pair of energy levels |i(n) i (with energy Ei in the absolute value of the matrix elements. Right panels of both (a) and
(n+1) (b) represent the frequency ωi,max at which the transitions in the
n-magnon sector) and |j (n+1) i (with energy Ej in the
n + 1-magnon sector) (Here, the matrix elements for 1 → 2 left panels are most active for a given |i(n) i i.e. corresponding to
magnon and 2 → 3 magnon sectors are plotted.) the |j (n+1) i with the largest absolute value of the matrix element
2
The results suggest that not all transitions are equally im- hj (n+1) |Sy |i(n) i .
(n)
portant: For each Ei , most of the weight is concentrated
(n+1)
on a single Ej , and other contributions are small. For
each |i(n) i we identify the frequency ω = Ej
(n+1)
− Ei for
(n) Hamiltonian using a Mott insulator of doubly occupied sites
which the matrix element is largest and refer to it as ωi,max . and demonstrated the dynamical properties associated with
For the cases considered (n = 1 and n = 2) we find that the presence of single-ion anisotropy. In this setup, two hy-
ωi,max is largely independent of |i(n) i in a given n−magnon perfine states are mapped to spin-1 degrees of freedom via
sector. Importantly, this average/typical value of ωi,max (indi- Siz = 21 (a†i ai − b†i bi ), Si+ = a†i bi , Si− = b†i ai , where ai and
cated by the red line in each panel) decreases with increasing bi are boson annihilation operators at site i for state a and state
magnon number. Although this shift across 1 → 2 and 2 → 3 b, respectively, with the constraint that a†i ai + b†i bi = 2. We
magnon sectors is small, the general trend is consistent with henceforth refer to this setup and the associated experiment as
our earlier findings. Said differently, the effective energy cost the “MIT experiment.”
to add an additional magnon (with zero additional momentum, The MIT experiment studied spin dynamics by first prepar-
as dictated by the matrix element selection rules) decreases ing the state of all atoms as an equal superposition of the two
with increasing magnon number. This holds not only for the hyperfine states using a combination of microwave pulses,
ground state each n-magnon sector but also for excited states.
N
O |ai − i|bi |ai − i|bi
|ψi = √ ⊗ √ (21a)
i=1
2 i,atom1 2 i,atom2
V. NONEQUILIBRIUM DYNAMICS OF MAGNONS IN N
ULTRA-COLD ATOMIC SETTINGS O 1 √
= |1i − i 2|0i − | − 1i (21b)
i=1
2
We now consider the implications of our findings on re-
cent ultra-cold atomic experiments performed with the same This initial state was allowed to time-evolve and the operator
PL
spin Hamiltonian as in Eq. (1) (the sign convention of D in A = 2−3 L1 i=1 h(Siz )2 i was measured as a function of time.
Ref. [27] is the opposite of what we have considered here and When written out in terms of spin degrees of freedom, this
elsewhere [15]). The authors of Ref. [27] implemented this wavefunction is a superposition of multiple magnon sectors11
1.0
= 10 1.0
= 10 1.0
= 20 1.0
= 20
D/J = 0.1 D/J = 0.1
0.8 D/J = 0.2 0.8 D/J = 0.2
0.5 D/J = 0.3 0.5 D/J = 0.3
| (t)| (0) |2
| (t)| (0) |2
Sy(t) / Sy(0)
Sy(t) / Sy(0)
D/J = 0.4 D/J = 0.4
0.6 D/J = 0.6 0.6 D/J = 0.6
0.0 0.0
0.4 0.4
0.5 0.2 0.5 0.2
1.0 0.0 1.0 0.0
0 2 4 6 0 2 4 6 0 2 4 6 0 2 4 6
tD/2 tD/2 tD/2 tD/2
(a) (b)
1.0
= 30 1.0
= 30 1.0
= 90 1.0
= 90
D/J = 0.1 D/J = 0.1
0.8 D/J = 0.2 0.8 D/J = 0.2
0.5 D/J = 0.3 0.5 D/J = 0.3
| (t)| (0) |2
| (t)| (0) |2
Sy(t) / Sy(0)
Sy(t) / Sy(0)
D/J = 0.4 D/J = 0.4
0.6 D/J = 0.6 0.6 D/J = 0.6
0.0 0.0
0.4 0.4
0.5 0.2 0.5 0.2
1.0 0.0 1.0 0.0
0 2 4 6 0 2 4 6 0 2 4 6 0 2 4 6
tD/2 tD/2 tD/2 tD/2
(c) (d)
FIG. 8. Time profiles of hSy (t)i/hSy (0)i and the Loschmidt echo (revival fidelity) for various representative θ, corresponding to different
starting states |ψ(θ, t = 0)i, calculated using TEBD on spin chains with open boundary conditions. A maximum bond dimension of m = 100
was used for θ ≤ 20◦ (L = 200), and m = 50 (L = 100) was used for the rest of the calculations. (Times beyond which the bond
dimension reaches the maximum value m with truncation cutoff of = 10−8 are not shown to rule out any truncation errors from growth of
entanglement). In each panel representative small D/J and large D/J values at fixed θ are shown. Thermalization is slow (or virtually absent
on the time scale of the plot) for small θ and small D/J, but becomes rapid when either parameter is made large.
with the most dominant contribution coming from the Hilbert the n-magnon sector, and |ni is the n-magnon wavefunction
space that corresponds to Sz = 0. (The material equivalent of whose amplitude is given by cn = hn|ψi.
the above experiment will require measurements of oscillation What should one expect to observe in the above setup given
and thermalization time scales ≈ 2π~/D of the order of 10 the framework developed in the previous sections? If the
picoseconds.) magnons were truly noninteracting, the energy spacings be-
Motivated by the MIT experiment, we propose a modifi- tween the n- and n + 1-magnon sectors would be exactly
cation with the objective of demonstrating the importance of D. This has a direct measurable consequence in the time
magnon-magnon interactions and magnon density on spin dy- domain. The measurement of the Sy operator as a function
namics and thermalization. We prepare an initial state which of time hSy (t)i ≡ hψ(t)|Sy |ψ(t)i would yield the charac-
corresponds to spins rotated about the x-axis by angle θ with teristic frequency En − En+1 = D i.e. the oscillation time
respect to the z-axis i.e. with direction vector (0, sin θ, cos θ). scale T = 2π/D. However, this energy spacing is renormal-
(In our notation, the angle realized in the MIT experiment is ized with magnon number (density) and the value of D itself.
θ = −90◦ .). The starting ket is given by At low magnon density (small θ) the magnons are essentially
N
cos θ + 1 i sin θ cos θ − 1 noninteracting. At higher magnon density, this energy differ-
|−1i . ence decreases due to magnon-magnon interactions, and thus
O
|ψ(θ, t = 0)i = |1i+ √ |0i+
i=1
2 2 2 a larger time scale of oscillation is expected. While oscil-
(22) lations dominate the short time behavior, signatures of ther-
Rotation by an arbitrary angle θ (which can be controlled by malization are to be expected at long times. This manifests
applying the microwave pulse for a shorter duration) has the itself in multiple metrics, for example, at large t, hSy (t)i → 0
2
effect of introducing a tunable finite density of magnons. (We and the Loschmidt echo (revival fidelity) hψ(0)|ψ(t)i → 0.
will drop the θ label in |ψ(θ, t)i from here on). This initial (We note that the results for θ < 0 are directly related to the
product state is a linear combination of states with definite case of θ > 0. For example, hSy (t)i differs by an overall mi-
magnon number n, nus sign for θ → −θ, and thus we discuss only the case of
X X θ > 0.)
|ψ(t = 0)i = Pn |ψi = cn |ni (23)
To go beyond the qualitative arguments presented above,
n n
we perform matrix product state-based TEBD calculations,
where Pn is the operator which projects the wavefunction to preparing |ψ(0)i as in Eq. (22). A maximum bond dimen-12
= 30.0 = 30.0
1.2 L = 50 1.0 1.0
1.0 L = 100
L = 200
0.5 0.9
0.8 L = 400
Sy(t) / Sy(0)
0.8
|cn|2 L
Deff/D
0.6 0.0 0.7
0.4 TEBD D/J = 0.1
0.5 0.6
0.2 = 10
0.5 = 20
0.0 1.0 = 30
0.0 0.1 0.2 0.3 0.4 0.4 0.0 0.1 0.2 0.3 0.4
0.0 2.5 5.0 7.5 10.0 12.5 15.0
Magnon density n/L tD/2 Magnon density n/L
(a) (b) (c)
√
FIG. 9. Representative results from the approximation in Eq. (24). (a) the amplitude |cn |2 L associated with every magnon sector for
θ = 30◦ for representative L. (b) hSy (t)i/hSy (0)i for θ = 30◦ . (c) Deff /D as a function of magnon density; determination of D̄eff /D using
the average magnon density n̄/L for the case of θ = 10◦ , 20◦ , 30◦ is represented by the dashed lines. Comparison with the D̄eff /D from the
periods of the first oscillations in TEBD results for D/J = 0.1 is also shown.
sion of m = 100 and a time step of t = 0.02 (in units of considered as essentially unchanged from those for the D = 0
J = 1) were employed. We rescale the time axis to be in model. Thus, we have
units of tD/2π, in these units the maxima of the Loschmidt
echo and hSy (t)i must occur at every integer for perfectly non exp(−iHt)Pn |ψi ≈ exp(−iẼn t)Pn |ψi (24)
interacting magnons. where Ẽn ≡ hψ|Pn HPn |ψi/hψ|Pn Pn |ψi is the energy of the
Figure 8 shows our results for various representative values state Pn |ψi. The computation of arbitrary operator expecta-
of θ and confirms many of our qualitative expectations. At tion values is straightforward: For example, hSy (t)i is
large θ, i.e., high average density of magnons, (see θ = 30◦ X
and θ = 90◦ ), only a few (or no) coherent oscillations are hSy (t)i = hψ|Pn eiHt Sy e−iHt Pm |ψi (25a)
observed, and thermalization is rapid. At small θ (i.e., low n,m
average magnon density), on the other hand, the oscillation X
≈ c∗n cm ei(Ẽn −Ẽm )t hn|Sy |mi (25b)
time is very nearly 2π/D, a more refined renormalized esti-
n,m
mate can be obtained with an approximation we will discuss
shortly. Our investigations suggest that there is a possibil- (Similar computations can be carried out for other operators
ity of a prethermal phase [48–50] for small θ-dependent D/J using the algebra of coherent states, see for example [54, 60,
(for example, D/J . 0.33 for θ = 10◦ , D/J . 0.17 for 61].)
θ = 15◦ , and D/J . 0.11 for θ = 20◦ .). Such long ther- The only nonzero contributions to hSy (t)i are from m =
malization time scales occur in systems that are close to in-
P
n ± 1. Note that hn| hi,ji S ~i · S
~j |ni for an L-site periodic
tegrability or have scar-like states in the spectrum (see, for chain equals L, i.e., it is independent of n, which follows from
example, Refs. [51–59]). the fact that the normalized |ni ∝ Pn |ψi are the different Sz
A qualitative explanation of this effect is as follows. If projections of a superspin of length L. Hence the energy dif-
the prepared initial state has finite overlaps with eigenstates ference Ẽn − Ẽn±1 arises purely from the on-site anisotropy
which form a tower of states (states uniformly spaced in en- term and does not depend on J. Hence it is convenient to
ergy), it will result in perfectly coherent oscillations in several define
time-dependent observables. This arises due to the preces- (n)
Deff ≡ Ẽn − Ẽn−1 (26a)
sion of a superspin of length L (for spin-1), whose 2L + 1 X
Sz projected states (appropriately normalized) are Pn |ψi for = −D hn|(Siz )2 |ni − hn−1|(Siz )2 |n−1i .
n = 0, 1, . . . , 2L. For D = 0, all Pn |ψi are exactly de- i
generate, a consequence of SU(2) symmetry. For D/J small (26b)
but nonzero, these states (which strictly speaking, do not re-
main exact eigenstates) have a spacing which is approximately An exact computation yields,
(but not exactly) D. Most importantly, these energy spacings D X E
P[n/2] m L
are nonuniform, which, in turn, leads to thermalization in the z 2 m=0 22m L−n+m,n−2m,n
n (Si ) n = L − n + 2 P[n/2] L
1
large time limit, the smaller the nonuniformity the longer the i m=0 22m L−n+m,n−2m,n
thermalization scale. (27)
We demonstrate these arguments with an approximation L
L!
which is quantitatively accurate for short time, especially for where i,j,k = i!j!k! for i + j + k = L is the trinomial
function. Note that n i (Siz )2 n and hence the plot for
P
small D/J and small θ. In this limit, the eigenstates can be13
(n)
Deff /D versus magnon density n/L is independent of θ. θ 1.0
D/J = 0.1 1.0
D/J = 0.1
has the effect of selecting the average magnon density and = 10
0.8 = 20
(n) 0.5 = 30
| (t)| (0) |2
hence the value of Deff which controls the time period of the
Sy(t) / Sy(0)
= 90
oscillations. 0.6
0.0
Figure 9 shows our representative results for periodic 0.4
chains within the framework√ of the approximation. For θ = 0.5 0.2
30◦ , the amplitude |cn |2 L associated with every magnon 0.0
1.0
sector for representative L is shown as a function of magnon 0 2 4 6 0 2 4 6
density. (The profile tD/2 tD/2
√ is expected to approach Gaussian, and
hence the factor L is introduced when comparing different (a)
system sizes.) Since the system sizes are finite, nonzero con-
tributions arise from a range of magnon densities, however, 1.0
D/J = 0.3 1.0
D/J = 0.3
in the thermodynamic limit the only nonzero contribution will = 10
0.8 = 20
be from n̄/L, the average magnon density. θ thus controls the 0.5 = 30
| (t)| (0) |2
Sy(t) / Sy(0)
= 90
average magnon density in the wavefunction |ψi. 0.6
0.0
(n)
n̄/L in turn determines D̄eff , the Deff with the largest con- 0.4
tribution. The right panel of Fig. 9 shows this connection with 0.5 0.2
the help of dashed lines for the case of θ = 10◦ , 20◦ , 30◦ . The 0.0
1.0
value of D̄eff /D decreases on increasing θ (and hence magnon 0 2 4 6 0 2 4 6
density). Since this D̄eff sets the time period of oscillations tD/2 tD/2
T ≈ 2π/D̄eff , we are able to infer its value from the TEBD (b)
calculations. We compare the TEBD result for the time of the
first oscillation for D/J = 0.1 with the approximate result, 1.0
D/J = 0.6 1.0
D/J = 0.6
and confirm that the agreement is good and the discrepancy is = 10
0.8 = 20
less than a percent. However, the approximation does not cap- 0.5 = 30
| (t)| (0) |2
Sy(t) / Sy(0)
= 90
ture higher order effects in D and θ and long time behavior, 0.6
0.0
for example, in the (exact) TEBD calculations oscillations are 0.4
not seen for large θ & 30◦ (see Fig. 8). 0.5 0.2
The central panel shows hSy (t)i/hSy (0)i calculated within
1.0 0.0
our approximation. For short time, the finite size effects are 0 2 4 6 0 2 4 6
essentially negligible. At longer times hSy (t)i/hSy (0)i ap- tD/2 tD/2
pears to decay, but this is a finite size effect. All nonzero (c)
contributions in hSy (t)i emerge from a single Deff in the ther-
modynamic limit, for any finite size system there is always a FIG. 10. Time profiles of hSy (t)i/hSy (0)i and the Loschmidt echo
spread of contributing√ energy scales as discussed previously for various D/J. For each value of D/J representative θ (which
in reference to |cn |2 L. Hence there is no thermalization in controls the average magnon density) are shown. A maximum bond
this approximation for the infinite chain limit. dimension of m = 100 was used for θ ≤ 20◦ (L = 200), and
Finally, in Fig. 10, we plot the time profiles of hSy (t)i and m = 50 (L = 100) was used for the rest of the calculations. At
the Loschmidt echo, calculated with TEBD, for various rep- short times, there is an increase in the effective oscillation time scale
resentative values of D and compare their θ dependence. In on increasing θ due to a reduction in Deff as discussed in the text.
all cases where an oscillation can be clearly identified, we ob-
serve that the time period is larger for larger θ (i.e., larger
magnon density). This is consistent with the increased role understood using a simple pair Jastrow function which cap-
of magnon-magnon attraction, which effectively reduces D̄eff tures our numerical DMRG results.
the spacing between energy levels. Among our primary findings were the observation that,
for a small number of magnons, the cloud does not signif-
icantly alter its size (spatial extent) on the introduction of
VI. CONCLUSION additional magnons. The energy cost for having additional
magnons decreases from D towards zero. Once the magnon
In summary, we have studied the properties of multiple cloud is saturated entirely, the formation of Ising type bound
interacting magnons in a spin-1 chain with ferromagnetic states becomes important, and manifests itself as an even-odd
Heisenberg interactions and easy-axis anisotropy using both magnon number effect. The lessons learnt from studying the
analytic and numerical methods. The model is of direct rele- ground state of the few magnon problem were used to extend
vance to both real materials [15] and cold-atom setups [27]. our understanding of the origin of the temperature dependent
The easy-axis anisotropy (D > 0, in this paper) serves as a frequency shift observed in previous experiments. The dy-
source of attractive magnon-magnon interactions which leads namical Kubo formula, involving matrix elements and energy
to the formation of magnon clouds, whose characteristics scales was analyzed, and the energy scales effective at high
were explored. Many of the properties of these clouds can be temperature were identified.You can also read
