# Loop Shaping with
Scaled Relative Graphs

Thomas Chaffey  Fulvio Forni  Rodolphe Sepulchre  University of Cambridge,
Department of Engineering, Trumpington Street, Cambridge CB2 1PZ, {tlc37,
ff286<EMAIL_ADDRESS>
###### Abstract

The Scaled Relative Graph (SRG) is a generalization of the Nyquist diagram
that may be plotted for nonlinear operators, and allows nonlinear robustness
margins to be defined graphically. This abstract explores techniques for
shaping the SRG of an operator in order to maximize these robustness margins.

###### keywords:

Scaled Relative Graph, Nyquist, loop shaping, robustness

††thanks: The research leading to these results has received funding from the
European Research Council under the Advanced ERC Grant Agreement Switchlet n.
670645, and from the Cambridge Philosophical Society.

##  1 Introduction

Loop shaping is one of the earliest methods of controller design, originating
in the work of Nyquist, Bode, Nichols and Horowitz on feedback amplifiers
(Bode, 1960). The basic principle of loop shaping is to tune a system’s closed
loop performance by adjusting the open loop frequency response. Robustness is
captured by the distance of the Nyquist diagram from the point $-1$; closed
loop performance is captured by the sensitivity and related transfer
functions. Loop shaping is still widely used in industry today – the graphical
nature of the tool gives a clear view of the design tradeoffs between
performance and robustness. Even in the age of modern optimal and robust
control, loop shaping remains a core tool for the control engineer. The idea
of enlarging stability margins eventually led to the Zames’ formulation of $H$
control (Zames, 1981), and some of the most successful methods of robust
control combine $H$ control with classical loop shaping ideas (Vinnicombe,
2000; McFarlane and Glover, 1992).

The Scaled Relative Graph (SRG) is a graphical representation of a nonlinear
operator, recently introduced in the theory of optimization by Ryu et al.
(2021). The SRG allows simple, intuitive proofs of convergence for
optimization algorithms, and allows optimal convergence rates to be visualized
as distances on a plot. The authors have recently connected the SRG to
classical control theory, showing that it generalizes the Nyquist diagram of
an LTI transfer function (Chaffey et al., 2021). A range of incremental
stability results, including the Nyquist and circle criteria, small gain and
passivity theorems and secant condition, can be interpreted as guaranteeing
the separation of the SRGs of two systems in feedback, and this interpretation
has led to new conditions for incremental stability (Chaffey, 2022). The
distance between the two SRGs is an incremental disc margin, the reciprocal of
which bounds the incremental gain of the closed loop. The SRG makes the design
intuition afforded by the Nyquist diagram available for nonlinear systems.

This abstract describes ongoing research into the use of SRGs for loop shaping
nonlinear feedback systems. It has long been observed that introducing
nonlinearity can overcome fundamental limitations of LTI control – for
example, the describing function of the Clegg integrator has a phase lag of
only 38, rather than the usual $90$ of a linear integrator (Clegg, 1958). This
motivates a better understanding of how nonlinearities may be used to shape
the performance of a feedback system.

##  2 Review of Scaled Relative Graphs

We begin this extended abstract with a brief review of the theory of SRGs.

###  2.1 Signal Spaces

We describe systems using operators, possibly multi-valued, on a Hilbert
space. A Hilbert space $\mathcal{H}$ is a vector space equipped with an inner
product, $\bra{\cdot}\ket{\cdot}:\mathcal{H}\times\mathcal{H}\to\mathbb{C}$,
and the induced norm $\norm{x}\coloneqq\sqrt{\bra{x}\ket{x}}$.

We will pay particular attention to the Lebesgue space $L_{2}$. Given
$\mathbb{F}\in\\{\mathbb{R},\mathbb{C}\\}$, $L^{n}_{2}(\mathbb{F})$ is defined
as the set of signals $u:\mathbb{R}_{\geq 0}\to\mathbb{F}^{n}$ such that rCl
∥u∥ ≔(∫_0^ u(t) ¯u (t) dt)^12 ¡ , where $\bar{u}(t)$ denotes the conjugate
transpose of $u(t)$. The inner product of $u,y\in L_{2}^{n}(\mathbb{F})$ is
defined by rCl ⟨u—y⟩ ≔∫_0^ u(t) ¯y (t) dt. The Fourier transform of $u\in
L_{2}^{n}(\mathbb{F})$ is defined as rCl ^u(jω) ≔∫_0^e^-jωtu(t) dt. We omit
the dimension and field when they are immaterial or clear from context.

###  2.2 Relations

An _operator_ , or _system_ , on a space $\mathcal{H}$, is a possibly multi-
valued map $R:\mathcal{H}\to\mathcal{H}$. The identity operator, which maps
$u\in\mathcal{H}$ to itself, is denoted by $I$. The _graph_ , or _relation_ ,
of an operator, is the set $\\{u,y\;|\;u\in\mathrm{dom}\,{R},y\in
R(u)\\}\subseteq\mathcal{H}\times\mathcal{H}$. We use the notions of an
operator and its relation interchangeably, and denote them in the same way.

The usual operations on functions can be extended to relations. Let $R$ and
$S$ be relations on an arbitrary Hilbert space. Then: rCl S^-1 &= { (y, u) — y
S(u) }
S + R = { (x, y + z) — (x, y) S, (x, z) R }
SR = { (x, z) — y  s.t.  (x, y) R, (y, z) S }. Note that $S^{-1}$ always
exists, but is not an inverse in the usual sense. In particular, it is in
general not the case that $S^{-1}S=I$. The relational inverse plays a
fundamental role in the techniques described in this abstract. Rather than
directly shape the performance of a negative feedback interconnection, we will
shape the performance of its inverse relation – a parallel interconnection.

These operations will also be used on sets of operators, with the meaning that
the operations are applied elementwise to the sets (under the implicit
assumption that the operators have compatible domains and codomains).

###  2.3 Scaled Relative Graphs

We define SRGs in the same way as Ryu et al. (2021), with the minor
modification of allowing complex valued inner products.

Let $\mathcal{H}$ be a Hilbert space. The angle between $u,y\in\mathcal{H}$ is
defined as rCl (u, y) ≔acosRe⟨u—y⟩∥u∥∥y∥.

Let $R:\mathcal{H}\to\mathcal{H}$ be an operator. Given
$u_{1},u_{2}\in\mathcal{U}\subseteq\mathcal{H}$, $u_{1}\neq u_{2}$, define the
set of complex numbers $z_{R}(u_{1},u_{2})$ by rCl z_R(u_1, u_2) ≔&{∥y1\-
y2∥∥u1\- u2∥ e^j(u_1 - u_2, y_1 - y_2)
— y_1 R(u_1), y_2 R(u_2) }. If $u_{1}=u_{2}$ and there are corresponding
outputs $y_{1}\neq y_{2}$, then $z_{R}(u_{1},u_{2})$ is defined to be
$\\{\infty\\}$. If $R$ is single valued at $u_{1}$, $z_{R}(u_{1},u_{1})$ is
the empty set.

The _Scaled Relative Graph_ (SRG) of $R$ over
$\mathcal{U}\subseteq\mathcal{H}$ is then given by rCl
$\operatorname{$}{SRG}_{\,\mathcal{U}}({R})\coloneqq\tbigcup\slimits@_{u_{1},u_{2}\in\,\mathcal{U}}z_{R}(u_{1},u_{2}).If$U
=
H$,wewrite$$\operatorname{$}{SRG}_{\,}({R})\coloneqq\operatorname{SRG}_{\,\mathcal{H}}({R})$.
The SRG of a class of operators is defined to be the union of their individual
SRGs. Some examples of SRGs are shown in Figure 1, (a), (b) and (c).

###  2.4 Interconnections

The power of SRGs lies in the elegant interconnection theory of Ryu et al.
(2021). Given the SRGs of two systems, the SRG of their interconnection can be
bounded using simple graphical rules. Given two systems $R$ and $S$, and
subject to mild conditions, we have: rCl $\operatorname{$}{SRG}_{\,}({\alpha
R})&=\operatorname{SRG}_{\,}({R\alpha})=\alpha\operatorname{SRG}_{\,}({R})\\\
\operatorname{SRG}_{\,}({R+S})\subseteq\operatorname{SRG}_{\,}({R})+\operatorname{SRG}_{\,}({S})\\\
\operatorname{SRG}_{\,}({RS})\subseteq\operatorname{SRG}_{\,}({R})\operatorname{SRG}_{\,}({S})\\\
\operatorname{SRG}_{\,}({R^{-1}})=\operatorname{SRG}_{\,}({R})^{-1}.Fortheprecisemeaningsoftheseoperations,andtherequisiteconditionsonthesystems$R$and$S$,wereferthereaderto\cite[citep]{(\@@bibref{AuthorsPhrase1Year}{Ryu2021}{\@@citephrase{,
}}{})}.\par$

###  2.5 SRGs of systems

The SRGs of LTI transfer functions are closely related to the Nyquist diagram,
and the SRGs of static nonlinearities are closely related to the incremental
circle. These connections are explored in detail in (Chaffey et al., 2021;
Pates, 2021); below we recall the two main results. The h-convex hull is the
regular convex hull with straight lines replaced by arcs with centre on the
real axis – for a precise treatment, we refer the reader to (Huang et al.,
2020).

###### Theorem 1

Let $g:L_{2}(\mathbb{C})\to L_{2}(\mathbb{C})$ be linear and time invariant,
with transfer function $G(s)$. Then
$\operatorname{SRG}_{\,}({g})\cap\mathbb{C}_{\imaginary\geq 0}$ is the
h-convex hull of $\text{Nyquist}\,({G})\cap\mathbb{C}_{\imaginary\geq 0}$.

###### Theorem 2

Suppose $S:L_{2}\to L_{2}$ is the operator given by a SISO static nonlinearity
$s:\mathbb{R}\to\mathbb{R}$, such that for all $u_{1},u_{2}\in\mathbb{R}$,
$y_{i}\in s(u_{i})$, rCl μ(u_1 - u_2)^2 (y_1 - y_2)(u_1 - u_2) λ(u_1 - u_2)^2.
Then the SRG of $S$ is contained within the disc centred at $(\mu+\lambda)/2$
with radius $(\mu-\lambda)/2$.

Theorems 1 and 2, and the SRG interconnection rules, allow us to construct
bounding SRGs for arbitrary interconnections of LTI and static nonlinear
components. A simple example is illustrated in Figure 1.

Figure 1: Bounding SRG for the composition of a first order lag and
saturation.

##  3 Incremental robustness and sensitivity

###  3.1 Stability and incremental gain

Given the negative feedback interconnection of Figure 2, incremental stability
is guaranteed by the separation of the SRGs of $P^{-1}$ and $-C$, and the
distance between them is an incremental robustness margin, the reciprocal of
which bounds the incremental gain of the feedback system. This is formalized
in (Chaffey et al., 2021, Thm. 2); we recall the result here.

Figure 2: Negative feedback control structure.

Let $\mathcal{H}$ be a class of operators. By $\bar{\mathcal{H}}$, we will
denote a class of operators such that $\mathcal{H}\subseteq\bar{\mathcal{H}}$
and $\operatorname{SRG}_{\,}({\bar{\mathcal{H}}})$ satisfies the chord
property: if $z_{1},z_{2}\in\operatorname{SRG}_{\,}({\bar{\mathcal{H}}})$,
then $\vartheta
z_{1}+(1-\vartheta)z_{2}\in\operatorname{SRG}_{\,}({\bar{\mathcal{H}}})$ for
all $\vartheta\in[0,1]$.

###### Theorem 3

Consider the feedback interconnection shown in Figure 2 between any pair of
operators $C\in\mathcal{C}$ and $P\in\mathcal{P}$, where $\mathcal{C}$ and
$\mathcal{P}$ are classes of operators on $L_{2}$ with finite incremental
gain. If, for all $\tau\in(0,1]$, rCl
$\operatorname{$}{SRG}_{\,}({\mathcal{C}})^{-1}\cap-\tau\operatorname{SRG}_{\,}({\bar{\mathcal{P}}})=\varnothing,thentheincremental$L_2$gainfrom$r$to$u$isboundedby$1/r_m$,where$r_m$istheshortestdistancebetween$$\operatorname{$}{SRG}_{\,}({\mathcal{C}^{-1}})$
and $-\operatorname{SRG}_{\,}({\bar{\mathcal{P}}})$.

Figure 3: Illustration of Theorem 3.

###  3.2 The sensitivity SRG

The operator $(I+PC)^{-1}$ maps $r$ to $e$ in the feedback system of Figure 2,
and the operator $(I-PC(-I))^{-1}$ maps $n$ to $y$. These two operators have
the same SRG, which we denote by $\mathcal{S}$ – the _sensitivity SRG_.

###### Definition 4

The _peak incremental sensitivity_ is the maximum incremental gain of the
operator $(I+PC)^{-1}$.

The peak incremental sensitivity is equal to the maximum modulus of
$\mathcal{S}$. The following theorem gives the peak incremental sensitivity an
interpretation as a robustness margin.

###### Theorem 5

Let $s_{m}$ be the shortest distance between
$\operatorname{SRG}_{\,}({PC})$ and the point $-1$. Then the peak incremental
sensitivity is equal to $1/s_{m}$.

##  4 Loop shaping

We demonstrate SRG loop shaping with two simple design examples for the
control structure shown in Figure 2.

###  4.1 Shaping for stability and robustness

As first design example, we show how to use SRGs to ensure incremental
stability of a closed loop system. Unlike traditional loop shaping, where the
return ratio $L=PC$ is modified, we graphically shape the inverse of the
feedback system, $(P+C^{-1})$, to improve the robustness of the closed loop.
The use of SRGs makes the design close to classical Nyquist analysis, despite
the nonlinearity of $P$.

Figure 4: Example control system. $\bar{P}=1/(s(s+1))$, $N$ is a nonlinear
operator and $C$ is the controller, to be designed. $r$ is the reference
input, $n$ represents sensor noise.

Consider the system in Figure 4. $C$ represents the controller, to be
designed. Suppose that the process consists of $\bar{P}$ with LTI dynamics
$1/(s(s+1))$, and a nonlinear operator $N$, whose SRG is known to be bounded
in the region illustrated in Figure 1 (c). We denote $C\bar{P}$ by $L$. The
controller $C$ is to be designed to stabilize the system and decrease the
incremental gain.

To ensure stability, we require the SRGs of $L^{-1}=(C\bar{P})^{-1}$ and $-N$
to be separated, for all scalings of $N$ between $0$ and $1$ (following
Theorem 3). With $C_{0}=1$, the closed loop is unstable, as shown in Figure 5
(a). Shifting $L^{-1}$ to the left, by designing $C$ to give
$L^{-1}=s(s+1)+1$, gives a stabilizing control. The controller reads
$C_{1}=s(s+1)/(1+s(s+1))$.

Figure 5: (a) SRGs of $-N$, $\bar{P}^{-1}$ and $(\bar{P}C_{1})^{-1}$. (b) SRGs
of $(\bar{P}C_{1})^{-1}$, $(\bar{P}C_{2})^{-1}$ and a scaled and rotated
nonlinearity, showing the improved robustness with $C_{2}$.

To improve robustness and reduce the incremental gain of the system, the
separation of $\operatorname{SRG}_{\,}({L^{-1}})$ and
$\operatorname{SRG}_{\,}({-N})$ must be increased (again, following Theorem
3). For example, setting $L^{-1}=s+1$ ($C_{2}=s$) gives good separation, and
an incremental gain bound from $r$ to $u$ of $8/7\approx 1.14$. As the
incremental gain of $N$ is bounded by $1$ (the maximum modulus of its SRG),
this value also bounds the incremental gain from $r$ to $y$. The increased
separation of the SRGs makes the system robust to uncertainties in the
nonlinearity $N$, as illustrated in Figure 5 (b).

###  4.2 Shaping for performance

We now focus on graphical methods for improving performance, and explore how
the sensitivity SRG can be shaped over particular sets of signals. We consider
a new system, again of the form of Figure 2, with $C=1/(ks+1)$, where $k$ is a
scalar to be designed, and $P$ is a unit saturation. The SRGs of $C$ and $P$
are shown in Figure 1 (a) and (b).

Tracking performance and noise rejection are both characterized by the
sensitivity SRG. Suppose that we would like this SRG to have a low modulus
(corresponding to incremental gain) for signals with a bandwidth of
$\omega_{0}=10$ rad$/$s and a maximum amplitude of $2$. The aim is to limit
the maximum amplification of $(I+PC)^{-1}$ over this range of signals.

A heuristic method is to maximize the distance between
$\operatorname{SRG}_{\,}({PC})$ and $-1$ over the frequency range
$[-\omega_{0},\omega_{0}]$ and amplitude range $[-2,2]$, following Theorem 5.
This corresponds to maximizing the minimum incremental gain of the inverse of
the sensitivity operator over this range of signals.

$\operatorname{SRG}_{\,}({PC})$ is bounded by the Minkowski product of
$\operatorname{SRG}_{\,}({P})$ and $\operatorname{SRG}_{\,}({C})$. Plotting
the SRG of the saturation $P$ over the amplitude range $[-2,2]$ gives the
half-disc shown in Figure 6 (b). The SRG of $C$ over
$[-\omega_{0},\omega_{0}]$ is described by rCl ( 11 + k2ω2, j -kω1 + k2ω2) for
$\omega\in[-\omega_{0},\omega_{0}]$. As a first design, we can set $k$ so that
the bandlimited SRG of $C$ is half the circle (Figure 6 (a)) – this is
achieved by setting $k=0.01$. This gives the bound on
$\operatorname{SRG}_{\,}({PC})$ shown in Figure 6 (c). The minimum distance to
the point $-1$ is $s_{m}=\sqrt{3}$.

Figure 6: Left: SRG of $1/(ks+1)$ over signals with bandwidth $[-1/k,1/k]$.
Right: SRG of saturation over signals with maximum amplitude $2$.

This method is, however, only a heuristic. The saturation introduces higher
harmonics, so the assumption that signals have a bounded spectrum is
invalidated when the loop is closed. However, given the lowpass properties of
the system, the approximation is reasonable. The higher order harmonics of the
output of the saturation have low magnitude, and the unit lag has a lowpass
behavior. The stability of the closed loop guarantees that these high
frequencies are indeed attenuated by the feedback system. This assumption is
similar to the lowpass assumption of describing function analysis Slotine and
Li (1991). The method here differs from describing function analysis, however,
in that arbitrary differences of bandlimited inputs are considered, not just
pure sinusoids.

##  5 Other types of systems

A significant advantage of the SRG is being able to place disparate system
types on an equal footing. Like continuous time LTI systems, finite
dimensional linear operators described by matrices lend themselves well to
shaping. Pates (2021) has shown that the SRG of a matrix is equal to the
numerical range of a closely related, transformed matrix. In the case of
normal matrices, the SRG is the h-convex hull of the spectrum (Huang et al.,
2020). These results pave the way for shaping a matrix’s SRG by matrix
multiplication and addition.

In cases where the analytic SRG is not available, the SRG can be sampled over
the signals of interest, and loop shaping methods can then be applied using
the sampled SRG. For example, Figure 7 shows a sampled SRG of the potassium
conductance of the Hodgkin-Huxley model of a neuron (Hodgkin and Huxley,
1952).

Figure 7: Sampling of the SRG of a potassium conductance.

## References

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# Continuous control of classical-quantum crossover by external high pressure
in the coupled chain compound CsCuCl3

Daisuke Yamamoto1,2<EMAIL_ADDRESS>Takahiro Sakurai3
<EMAIL_ADDRESS>Ryosuke Okuto4 Susumu Okubo5 Hitoshi Ohta5 Hidekazu
Tanaka6 Yoshiya Uwatoko7 1Department of Physics, Nihon University, Tokyo
156-8550, Japan  2Department of Physics and Mathematics, Aoyama Gakuin
University, Kanagawa 252-5258, Japan  3Research Facility Center for Science
and Technology, Kobe University, Kobe 657-8501, Japan  4Graduate School of
Science, Kobe University, Kobe 657-8501, Japan  5Molecular Photoscience
Research Center, Kobe University, Kobe 657-8501, Japan  6Department of
Physics, Tokyo Institute of Technology, Meguro-ku, Tokyo 152-8551, Japan
7Institute for Solid State Physics, The University of Tokyo, Chiba 277-8581,
Japan

## Abstract

In solid materials, the parameters relevant to quantum effects, such as the
spin quantum number, are basically determined and fixed at the chemical
synthesis, which makes it challenging to control the amount of quantum
correlations. We propose and demonstrate a method for active control of the
classical-quantum crossover in magnetic insulators by applying external
pressure. As a concrete example, we perform high-field, high-pressure
measurements on CsCuCl3, which has the structure of weakly-coupled spin
chains. The magnetization process experiences a continuous evolution from the
semi-classical realm to the highly-quantum regime with increasing pressure.
Based on the idea of “squashing” the spin chains onto a plane, we characterize
the change in the quantum correlations by the change in the value of the local
spin quantum number of an effective two-dimensional model. This opens a way to
access the tunable classical-quantum crossover of two-dimensional spin systems
by using alternative systems of coupled-chain compounds.

## Introduction

Since the inception of quantum mechanics, it was recognized that the apparent
dichotomy between quantum and classical physics was to be resolved, in the
sense that any consistent quantum theory should retrieve the predictions of
the classical theory in the limit of large quantum numbers bohr-20 . It just
so happens that unique quantum phenomena, such as quantum superposition and
quantum correlation, generally become unobservable when such regime is
approached. This fundamental aspect carries over to the second quantum
revolution, given that quantum information and quantum technologies are based
on the theory of quantum decoherence, which studies nothing but the
interactions of a quantum system with a system with a large number of degrees
of freedom (the environment) schlosshauer-19 . External control of the
classical-quantum crossover would be not only intriguing, but of primary
theoretical and experimental interest. A certain degree of success has been
obtained in this direction with photonic ra-13  or optomechanical systems
bai-17 . This work aims to demonstrate a way to achieve such control in much
less flexible systems, namely a class of solid-state materials.

High-pressure application is one of the few experimental tools that can
drastically change the microscopic physical parameters of materials. Effects
of high pressure on material characteristics have recently been studied with
considerable interest in the broad area of condensed-matter physics, having
led to intriguing phenomena including pressure-driven room-temperature
superconductivity snider-20 , topological phases bahramy-12 ; zhou-16 , and
the softening of Higgs mode in spin-dimer magnets ruegg-08 ; merchant-14 . In
particular, frustrated quantum many-body systems are promising examples
expected to feel significant pressure effects since the frustration due to
competing interactions gives rise to a large number of low-energy states with
small energy differences, which enhance the relative impact of external
pressure sera-17 ; sakurai-18 ; zvyagin-19 . Besides, even small quantum
fluctuations could also play an essential role in determining the physical
properties lacroix-11 ; chubukov-91 ; nikuni-93 . Therefore, operating with
external pressure on frustrated quantum materials could pave the way to
actively control the amount of quantum correlations across the classical and
quantum-mechanical regimes and explore exotic phenomena taking place in the
crossover.

Figure 1:  Ground state of easy-plane triangular-lattice antiferromangets. a
Schematic ground-state phase diagram in the space of the magnetic field $H$
scaled by the saturated field strength $H_{s}$, the reciprocal of spin quantum
number $S$, and easy-plane anisotropy $\perp H$. The phase boundaries on the
plane of no anisotropy (on the back face) are obtained by the $1/S$ expansion
method with the “cutting-at-1/3” procedure coletta-16  (solid lines) and
coupled-cluster method gotze-16  (red dots). Those on the planes of $S=1/2$
and $H/H_{\rm s}\rightarrow 1$ (on the right and top faces, respectively) are
sketches based on the predictions of Ref. yamamoto-14,  and Refs. starykh-15,
; marmorini-16, ; yamamoto-17-j, , respectively. The approximate locations of
some relevant materials inami-07 ; white-13 ; svistov-03 ; susuki-13 ;
koutroulakis-15  are indicated by the filled circles. b-e Illustrations of the
sublattice spin moments in each phase appearing in a.

One exciting yet challenging example of frustrated quantum systems is the
class of triangular-lattice antiferromagnets (TLAFs) collins-97 . The lattice
geometry based on triangle units prohibits the standard antiferromagnetic
order with an antiparallel alignment of neighboring spins. Owing to the
geometrical frustration combined with magnetic anisotropy, external magnetic
fields, fluctuations effects, etc., TLAF compounds exhibit a rich variety of
magnetic phases lacroix-11 ; chubukov-91 ; nikuni-93 ; kawamura-85 ;
collins-97 ; coletta-16 ; gotze-16 ; yamamoto-14 ; starykh-15 ; marmorini-16 ;
yamamoto-17-j . A schematic ground-state phase diagram of two-dimentional (2D)
TLAFs with exchange (or single-ion) anisotropy of easy-plane type under the
magnetic field $H$ applied perpendicular to the easy plane is shown in Fig.
1a, which summarizes the well-established kawamura-85 ; chubukov-91 ;
nikuni-93 ; collins-97 ; coletta-16 ; gotze-16  and the recently-predicted
yamamoto-14 ; starykh-15 ; marmorini-16 ; yamamoto-17-j  theoretical results.
The reciprocal of the spin quantum number, $1/S$, of magnetic ions in the
material usually serves as a good indicator of the quantum correlation
strength; specifically, $1/S=2$ is the most quantum while $1/S\rightarrow 0$
is classical.

Whereas the ground state of TLAFs at some fixed parameter planes is being
revealed, much less is known about what happens inside the three-variable
phase diagram of Fig. 1a. There also remain other open problems, especially on
essential differences between the classical (small $1/S$) and quantum (large
$1/S$) regime. For example, it should be interesting if one can examine the
continuous change in the nature of magnetic collective excitations from the
semi-classical regime of “magnons” carrying spin-1 to the highly-quantum
regime of “spinons” carrying spin-1/2 oh-13 ; kajimoto-15 ; ito-17 ; coldea-01
; balents-10 ; shen-16 ; paddison-17 ; shen-18 . Note that the latter is
expected to appear only with additional factors, such as a deformation of
triangular lattice coldea-01 ; balents-10  and longer-range couplings
paddison-17 , beyond the regular TLAF with nearest-neighbor interactions.
Whereas “$1/S$” has been often treated as a continuous variable in the widely-
used analysis method, called the $1/S$ expansion chubukov-91 ; nikuni-93 ;
coletta-16 , in real materials, however, the spin $S$ is basically fixed to a
certain integer or half-integer value at the chemical synthesize. This makes
it difficult to study the continuous change in the nature of materials from
the classical to quantum regime.

Here we propose the concept of actively controlling the amount of quantum
correlations, or more specifically, the value of “$1/S$,” in a continuous
manner by applying external pressure in the laboratory. The main idea is the
use of materials with a coupled-chain structure, such as $ABX_{3}$-type
hexagonal perovskites ($A=$Rb, Cs, $B=$V, Cr, Mn, Fe, Co, Ni, Cu, and $X=$F,
Cl, Br, I), achiwa-69 ; kakurai-84 ; maruyama-01 . Introducing a “squash”
mapping, we show that the magnetic properties of coupled spin chains can be
phenomenologically described by a single-layer TLAF model with effective spin
$\tilde{S}$. The crucial experimental step is a series of precise magnetic
measurements conducted under high pressure up to $P=1.21$ GPa on a CsCuCl3
single crystal nojiri-88 ; tazuke-81 ; hyodo-81 ; tanaka-92 ; ohta-93 ;
mekata-95 ; miyake-15 , which allows us to determine the exchange couplings to
great accuracy and, consequently, extract the parameters of the effective
model. We thus demonstrate that the value of effective spin $\tilde{S}$ can be
actually controlled by external pressure through the change in the material
parameters. This idea of controlling the classical-quantum crossover via the
squash mapping is expected to be applicable also to other platforms, including
cold atoms in optical lattices schafer-20 , trapped ions blatt-12 , and
Rydberg atoms in arrays of optical tweezers browaeys-20 , as well as directly
to the other materials of the $ABX_{3}$-type, such as CsNiF3 kakurai-84  and
RbCuCl3 maruyama-01 , and to the other coupled-chain compounds with different
lattice geometries.

## Results

### Coupled-chain TLAF and its squash mapping

The hexagonal antiferromagnets of the $ABX_{3}$ type achiwa-69 ; kakurai-84 ;
maruyama-01  have spin chains along the $c$ axis, which form triangular
lattices on the $ab$ planes (see Fig. 2). We describe the magnetic properties
of the coupled-chain TLAFs under magnetic fields parallel to the $c$ axis by
the following Hamiltonian with spin-$S$ operators $\hat{\bm{S}}_{i,n}$ on site
$i$ of the $n$-th triangular layer:

$\displaystyle\hat{\mathcal{H}}$ $\displaystyle=$
$\displaystyle-2J_{0}\sum_{i,n}\left(\hat{\bm{S}}_{i,n}\cdot\hat{\bm{S}}_{i,n+1}-\Delta_{0}\hat{S}_{i,n}^{z}\hat{S}_{i,n+1}^{z}\right)$
(1) $\displaystyle+2J_{1}\sum_{\langle
i,j\rangle,n}\hat{\bm{S}}_{i,n}\cdot\hat{\bm{S}}_{j,n}-H\sum_{i{,n}}\hat{S}_{i,n}^{z},$

where the intrachain and interchain exchange couplings are assumed to be
ferromagnetic and antiferromagnetic, respectively ($J_{0},J_{1}>0$). Here, we
took into account the possible existence of easy-plane anisotropy
perpendicular to the $c$ axis ($\Delta_{0}>0$) in the intrachain coupling,
which is the case for CsCuCl3 nojiri-88 ; tazuke-81 ; hyodo-81 ; tanaka-92 ;
ohta-93 ; schotte-94 ; mekata-95 ; miyake-15 .

Figure 2:  Squash mapping. Illustration of the concept of mapping from
coupled-chain model with spin $S$, intrachain coupling $J_{0}$, and interchain
coupling $J_{1}$ to an effective single-layer model with spin $\tilde{S}$ and
coupling $J$.

The key of the squash mapping is the following intuitive idea. In weakly-
coupled spin chains ($J_{1}\ll J_{0}$), the time scale of the intrachain spin-
spin correlations along the $c$ axis is expected to be much shorter than that
of the interchain correlations in the $ab$ plane. The difference in the time
scales may be characterized by the ratio of the intrachain to interchain
coupling, $\alpha_{J}\equiv J_{0}/J_{1}$, which is $\sim 5.5$-$6.5$ for
CsCuCl3 tazuke-81 ; hyodo-81 ; tanaka-92 ; mekata-95 . From the standpoint of
the interchain interactions, therefore, the spins along each chain may appear
to move together to make up a single “large” spin $\hat{\bm{\mathcal{S}}}_{i}$
with an effective spin quantum number $\tilde{S}>S$, as illustrated in Fig. 2.
From this intuitive idea, one could introduce the following phenomenological
spin model:

$\displaystyle\tilde{\hat{\mathcal{H}}}$ $\displaystyle=$ $\displaystyle
2J\sum_{\langle
i,j\rangle}\hat{\bm{\mathcal{S}}}_{i}\cdot\hat{\bm{\mathcal{S}}}_{j}+A\sum_{i}\left(\hat{\mathcal{S}}_{i}^{z}\right)^{2}-H\sum_{i}\hat{\mathcal{S}}_{i}^{z}$
(2)

with spin $\tilde{S}>S$ on a “single layer” of triangular lattice. It is
natural to take into account the uniaxial two-ion exchange anisotropy along
the chains by introducing uniaxial single-ion anisotropy in the effective
model, given that the spins along each chain $i$ are squashed into
$\hat{\bm{\mathcal{S}}}_{i}$. The effective coupling constant $J$ and the
effective anisotropy $A$ should be related to the ones in the original model
as

$\displaystyle J=\frac{S}{\tilde{S}}J_{1}~{}~{}{\rm
and}~{}~{}A=\frac{4S}{2\tilde{S}-1}J_{0}\Delta_{0},$ (3)

such that the two models share the same value of the saturation magnetic
field:

$\displaystyle H_{\rm
s}=(18J_{1}+4J_{0}\Delta_{0})S=18J\tilde{S}+A(2\tilde{S}-1).$ (4)

The fitting method for the remaining parameter $\tilde{S}$ will be discussed
later for a specific case.

The above squash mapping constitutes effective dimensional reduction and spin
transmutation for coupled-chain models. The effective spin quantum number
$\tilde{S}$ will serve as a more suitable indicator of quantum correlation
strength in weakly-coupled spin chains, rather than the bare value of $S$.

### Pressure dependence of magnetic couplings in CsCuCl3

Figure 3:  Magnetization processes. Sketches of the magnetization possesses of
the coupled-chain compound CsCuCl3 (a) in the semiclassical regime under
ambient or low pressure and (b) in the highly-quantum regime under high
pressure. The phase transition points ($H_{\rm 1st}$, $H_{\rm c1}$, $H_{\rm
c2}$) and the saturation field ($H_{\rm s}$) are marked on the horizontal
axis.  Figure 4:  Magnetic susceptibility and magnetization curves under
pressure. a Longitudinal susceptibilities $\chi_{\parallel}$ at $H=1$ T and b
magnetization curves at $T=1.8$ K for a CsCuCl3 crystal under different
pressures, $P=0,0.14,0.34,0.49,0.82,1.05,1.21$ GPa (from top to bottom) when a
magnetic field is applied along the $c$ axis. The magnetization $M$ is scaled
by the saturation value $M_{\rm s}$.  Figure 5:  Pressure dependence of the
coupling parameters in CsCuCl3. a The estimated values of the intrachain
coupling constant $J_{0}$ and the anisotropy parameter $\Delta_{0}$. b The
estimated values of the intrerchain coupling constant $J_{1}$. The error bars
reflect six standard deviations for $J_{0}$ and are smaller than the symbol
size in the min-max values for $\Delta_{0}$ and $J_{1}$. The model functions
for each quantity, $J_{0}(P)$, $\Delta_{0}(P)$, and $J_{1}(P)$, are shown by
the solid curves. c and d The ratio of the intrachain to interchain coupling,
$\alpha_{J}(P)=J_{0}(P)/J_{1}(P)$, and the rescaled anisotropy parameter
$\Delta(P)=\alpha_{J}(P)\Delta_{0}(P)$. The reduction rate from the value at
$P=0$ is plotted for each with the dashed curves.

Hereafter, we take the $S=1/2$ coupled-chain TLAF compound CsCuCl3 as a
specific example to pursue the subject. In CsCuCl3, the intrachain coupling
possesses extra Dzyaloshinskii-Moriya (DM) interaction, which causes a long-
wavelength helical spin structure along the $c$ axis adachi-80 . However, one
can eliminate the DM interaction by performing a proper twist of the local
spin coordinates nikuni-93  (see Supplementary Note 1 for details). When
viewed in the twisted spin space, the intrachain helical spin structure
appears as uniform (ferromagnetic) spin alignment along the $c$ axis, allowing
us to use the model Hamiltonian in the form of Eq. (S1) and to apply the
squash-mapping picture shown in Fig. 2. This transformation is effectively
applicable for the magnetic field $H\parallel c$, since the form of the Zeeman
term is not affected by the twist along the $c$ axis.

It is well known chubukov-91  that the magnetization curve of TLAFs with
strong quantum correlations exhibits a plateau structure at one third of the
saturation magnetization $M_{\rm s}$ in a certain field range, $H_{{\rm
c}1}<H<H_{{\rm c}2}$. The previous high-field experiments for CsCuCl3 had
reported only the existence of a first-order phase transition with no plateau
for $H\parallel c$ at low temperatures nojiri-88 ; ohta-93 ; schotte-94 ;
miyake-15 , which has been interpreted as the transition from the “umbrella”
to “V-coplanar” state nikuni-93 ; schotte-94  (Fig. 3a). The transition point
$H_{\rm 1st}$ is shifted towards lower fields as the temperature increases;
specifically, $H_{\rm 1st}\approx 12.5$ T at 1.5 K and $H_{\rm 1st}\approx 6$
T at 10 K schotte-94 ; sera-17 . Recently, it has been reported that applying
high hydrostatic pressure $P\gtrsim 0.7$ GPa has induced the appearance of the
one-third magnetization plateau sera-17 , which has suggested the
stabilization of the collinear “UUD” state and possibly the “Y-coplanar” state
(Fig. 3b). The sublattice spin moments in each state are illustrated in Figs.
1c and 1d. The plateau formation indicates that the quantum correlations in
CsCuCl3 are drastically enhanced by external pressure. However, the specific
pressure dependence of the Hamiltonian parameters and the microscopic origin
of the plateau formation have not been revealed yet.

To quantify the pressure effects, we first perform magnetic measurements on a
single crystal of CsCuCl3 under hydrostatic pressure conditions up to $P=1.21$
GPa for the temperature dependence (below 100 K) of the magnetic
susceptibility at magnetic field 1 T and the low-temperature (1.8 K)
magnetization curve up to 5 T. Using the measured data shown in Fig. 4 as well
as the previously-reported values of the first-order transition points $H_{\rm
1st}$ at the lowest temperature (1.5K) available in Ref. sera-17, , we
quantitatively estimate the pressure dependence of the magnetic coupling
parameters $J_{0}$, $\Delta_{0}$, and $J_{1}$ in the original model, Eq. (S1),
through the fittings with theoretical predictions for the ground state. For
the fittings, we employ the tenth-order high-temperature expansion lohmann-14
for the magnetic susceptibility and the $1/S$-expansion method nikuni-93  for
the magnetization curve and the first-order transition points. In the latter,
the energy is expressed in power series of $1/S$ and anisotropy $\Delta_{0}$
as

$\displaystyle E=S^{2}E_{0}+S^{2}E_{\Delta_{0}}+SE_{\rm LSW}+\cdots,$ (5)

where $S^{2}E_{0}$ is the classical energy for the isotropic system. Here, we
take into account up to the leading order corrections from the anisotropy,
$S^{2}E_{\Delta_{0}}$, and quantum effects within the linear spin-wave theory,
$SE_{\rm LSW}$ nikuni-93 ; chubukov-91 . The magnetization curve is obtained
by $M(H)=-dE(H)/dH$ coletta-16 . The theoretical values of the magnetic field
$H$ and the magnetization $M$ are converted into T (tesla) and $\mu_{B}/{\rm
Cu}^{2+}$, respectively, using the $g$ factor, which has been estimated to be
2.11 by the ESR measurements at room temperature, almost independently of
pressure within the experimental precision sakurai-17 . The saturation
magnetization per spin is thus given as $M_{\rm s}=g\mu_{\rm B}S=1.055\mu_{\rm
B}$. See Methods for more details.

Figures 5a and 5b show the values of $J_{0}$, $\Delta_{0}$, and $J_{1}$,
giving the best fits between experiment and theory. Applying the least squares
fittings to the values obtained at each pressure, we determine the following
model functions $J_{0}(P)$, $\Delta_{0}(P)$, and $J_{1}(P)$ for pressure $P$
in GPa:

$\displaystyle J_{0}(P)/k_{\rm B}$ $\displaystyle=$ $\displaystyle
28.45-10.49P~{}~{}{\rm[K]},$ (6) $\displaystyle\Delta_{0}(P)$ $\displaystyle=$
$\displaystyle 0.014+0.005P+0.005P^{2},$ (7) $\displaystyle J_{1}(P)/k_{\rm
B}$ $\displaystyle=$ $\displaystyle 4.86+2.03P~{}~{}{\rm[K]}.$ (8)

The values of the model functions at $P=0$, $J_{0}/k_{\rm B}=28.45$ K,
$\Delta_{0}=0.014$, and $J_{1}/k_{\rm B}=4.86$ K, are consistent with the
previous estimates at ambient pressure tazuke-81 ; hyodo-81 ; tanaka-92 ;
mekata-95 . In Figs. 5c and 5d, we plot the intrachain-to-interchain coupling
ratio $\alpha_{J}(P)=J_{0}(P)/J_{1}(P)$ and the rescaled anisotropy parameter
$\Delta(P)=\alpha_{J}(P)\Delta_{0}(P)$, nikuni-93 ; hosoi-18  which
characterize well the change of the material property. The parameter
$\alpha_{J}(P)$ is strongly reduced (by half at $P\sim 1$ GPa), which
indicates that a CsCuCl3 crystal with weakly-coupled quasi-1D spin chains
turns into a more 3D system by applying hydrostatic pressure. On the contrary,
the rescaled anisotropy $\Delta(P)$ experiences only a 20 percent reduction.

### Phase diagram and magnetization curve

Figure 6:  $H$-$P$ phase diagram. Theoretical ground-state phase diagram of
the model for CsCuCl3 in the plane of magnetic field $H$ and external pressure
$P$. We mark the points at which the magnetization anomalies have been
observed in the experiments of Ref. sera-17,  at temperature $T=1.5$ K by the
filled circles with error bars (blue: magnetization jump; orange: kink in
between the jump and plateau; green: end point of the plateau). The
corresponding values of the effective spin $\tilde{S}$ in the squashed model
are indicated on the upper axis.  Figure 7:  Magnetization curves. Theoretical
magnetization curves obtained by the $1/S$ expansion method nikuni-93  with
the “cutting-at-1/3” procedure coletta-16  using the model parameters
$J_{0}(P)$, $\Delta_{0}(P)$, and $J_{1}(P)$ at temperature $T=0$ for
$P=0,0.75,0.83,0.9$ GPa, together with the corresponding experimental data of
the present measurements at $T=1.8$ K, Miyake et al. at $T=1.4$ K (for
increasing fields) miyake-15 , and Sera et al. at $T=1.5$ K sera-17 . In a the
curves are vertically shifted by 0.1 from one another to avoid overlapping.
The inset shows the enlarged view of the transition region with no vertical
shift. In b-d, the magnetization scaled by the saturation value $M_{\rm s}$ is
plotted. Correspondingly, the curves of Sera et al. are scaled such that the
plateaux are located at $M=M_{\rm s}/3$.

Using the model parameters of Eqs. (6-8) and evaluating the energies of
different phases up to the leading order corrections from anisotropy and
quantum effects [Eq. (5)], we obtain the theoretical ground-state phase
diagram in the plane of magnetic field $H$ and pressure $P$ as shown in Fig.
6. The previous experimental observations by Sera et al. sera-17  on the
anomalies in the magnetization curves are plotted together. Note that in the
experimental data, the values of the pressure $P$ are reevaluated using the
calibration scheme that we use in the current work (see Methods). The plateau
endpoints for $P=0.83$ and 0.9 GPa are unclear within the experimental
precision in Ref. sera-17,  or out of the experimental field window $H<15$ T.

From the comparison between experiment and theory, the positions of the
observed anomalies are well identified as the transition points from Y to UUD
($H_{\rm c1}$), UUD to V ($H_{\rm c2}$), and umbrella to the other phases
($H_{\rm 1st}$), respectively. In particular, although a narrow field range
where the magnetization curve shows an almost linear increase between the
first-order jump and the 1/3-plateau has not been fully identified as the
Y-coplanar state only from the experiments of Ref. sera-17, , the agreement
with the theoretical prediction strongly supports its existence. On the upper
axis of Fig. 6, we mark the corresponding values of the effective spin
$\tilde{S}$ in the 2D squashed model (2) (which will be addressed in the
Discussion).

We also compare the theoretical and experimental magnetization curves at
$P=0$, $0.75$, $0.83$, and $0.9$ GPa in Figs. 7a-d. It can be seen that the
pressure-induced change in the magnetization processes are well reproduced by
the model calculations with Eqs. (6-8). While the agreement is excellent for
$P\lesssim 0.75$ GPa (and still good for $P=0.83$ GPa), it seems to get
slightly worse for larger values of pressure. Especially, looking at Fig. 7d,
we see that the plateau width is somewhat wider and the slope of the low-field
magnetization curve is smaller than the theoretical prediction for the
estimated pressure value. This might indicate that the pressure values of the
experiments were slightly underestimated due to pressure inhomogeneity in the
sample (see Supplementary Note 2 for a more detailed discussion).

### Mechanism for the plateau formation by applied pressure

Figure 8:  Two contributions to the plateau width. a Quantum fluctuation
measures $\eta$ and $\xi$ in the UUD state. b Classical and quantum
contributions, $W_{\rm p}^{\rm(cl)}$ and $W_{\rm p}^{\rm(qu)}$, to the plateau
width as functions of pressure $P$. We plot $W_{\rm p}^{\rm(cl)}$ with the
negative sign for convenience.

The width of the magnetization plateau associated with the UUD phase can be
expressed as

$\displaystyle W_{\rm p}=H_{\rm c2}-H_{\rm c1}=W^{({\rm cl})}_{\rm p}+W^{({\rm
qu})}_{\rm p}$ (9)

with

$\displaystyle W^{({\rm cl})}_{\rm p}$ $\displaystyle=$
$\displaystyle-16J_{1}S\Delta,$ (10) $\displaystyle W^{({\rm qu})}_{\rm p}$
$\displaystyle=$ $\displaystyle
12J_{1}\left(\eta(\alpha_{J})-\xi(\alpha_{J})\right),$ (11)

following the method used in a seminal work by Chubukov and Golosov
chubukov-91 . Here,
$\eta\equiv-\langle\hat{a}_{i_{\uparrow}}\hat{a}_{j_{\downarrow}}\rangle$
(resp.
$\xi\equiv\langle\hat{a}^{\dagger}_{i_{\uparrow}}\hat{a}_{j_{\uparrow}}\rangle$)
indicates the anomalous (resp. normal) quantum correlations between the
magnons $\hat{a}_{i}$ on the neighboring “up” and “down” sites (resp. on the
two neighboring “up” sites) in a unit triangle (see Fig. 8a). Since the
relation $\eta-\xi>0$ always holds, the quantum term $W^{({\rm qu})}_{\rm
p}>0$ contributes to the emergence of the magnetization plateau whereas the
classical term $W^{({\rm cl})}_{\rm p}<0$ works in the opposite way,
reflecting the easy-plane anisotropy in the classical interactions between
spins. The separation between the two lines, $W^{({\rm qu})}_{\rm
p}-(-W^{({\rm cl})}_{\rm p})$, in Fig. 8b, indicates the estimation of the
potential plateau width. The pressure dependence of $W^{({\rm qu})}_{\rm p}$
and $W^{({\rm cl})}_{\rm p}$ show that the emergence of the plateau in CsCuCl3
by applying pressure is predominantly attributed to the enhancement of quantum
correlations rather than the reduction of anisotropy, reflecting the behaviors
of $\alpha_{J}$ and $\Delta$ shown in Figs. 5c and 5d.

The above result shows an essential difference from the previous study
hosoi-18  in the understanding of the mechanism underlying the pressure-
induced plateau formation. In the analysis of Ref. hosoi-18, , the intrachain
coupling $J_{0}$ was assumed to be constant with the applied pressure, and the
plateau formation was explained as resulting from the reduction of the
effective anisotropy $\Delta$. Our present analysis based on the parameter
fittings with the experimental data has revealed that the change in $\Delta$
is not enough to explain the emergence of the plateau, but the enhancement of
the quantum effects associated with the strong reduction of $J_{0}$ plays a
key role as mentioned above. This finding leads us to the concept of the
pressure-induced classical-quantum crossover, which we will discuss in the
Discussion section.

## Discussion

Figure 9:  Quantum correlation measure for coupled spin chains. a Increase
rate of the effective spin $\tilde{S}$ in the squashed model from the original
value $S$ as a function of the ratio of the intrachain to interchain coupling,
$\alpha_{J}=J_{0}/J_{1}$. b Energy differences between the umbrella and
Y/UUD/V states, $\delta E_{\rm LSW}$ for the original model with
$\alpha_{J}=1,5,10,15$ (blue-solid lines) and $\delta\tilde{E}_{\rm LSW}$ for
the squashed model with $\tilde{S}/S=1.24,2.02,2.69,3.23$ (orange-dashed
lines), within the leading $1/S$ (linear spin-wave) corrections.

We have studied the pressure effects on the magnetization process of the 3D
material CsCuCl3 with weakly coupled spin chain structure. Let us connect the
results to the physics of the 2D TLAF model, Eq. (2), via the squash mapping
illustrated in Fig. 2. The energy of the squashed 2D model is also expanded in
power series of $1/\tilde{S}$ and anisotropy $A$ as

$\displaystyle\tilde{E}=\tilde{S}^{2}\tilde{E}_{0}+\tilde{S}^{2}\tilde{E}_{A}+\tilde{S}\tilde{E}_{\rm
LSW}+\cdots,$ (12)

in a similar fashion to Eq. (5). Substituting the correspondence relations
(3), one can easily see that the classical part of the energy (scaled by the
spin length) is identical for the original and effective models apart from a
constant shift, that is,
$S(E_{0}+E_{\Delta_{0}})=\tilde{S}(\tilde{E}_{0}+\tilde{E}_{A})+{\rm const.}$
for any $\tilde{S}$. Therefore, the effective spin $\tilde{S}$ should be
determined in such a way that it reflects the strength of quantum correlation
effects. The stabilization of Y/UUD/V orders against the classical umbrella
order is the most significant role of quantum correlations in TLAFs
chubukov-91 . Therefore, it should be reasonable to find the value of
$\tilde{S}$ such that the energy difference between the umbrella and Y/UUD/V
states, $\delta E_{\rm LSW}=E^{\rm umbrella}_{\rm LSW}-E^{\rm Y/UUD/V}_{\rm
LSW}$, is well reproduced by the corresponding quantity $\delta\tilde{E}_{\rm
LSW}$ of the effective model (2). This is done by minimizing the quantity

$\displaystyle\int_{0}^{H_{\rm s}}\left|\delta E_{\rm
LSW}(H)-\delta\tilde{E}_{\rm LSW}(H)\right|^{2}dH.$ (13)

A similar procedure has been used to mimic quantum fluctuation effects in 2D
TLAF models by a classical-spin biquadratic coupling griset-11 .

Before showing the result, let us comment on the difference of the squash-
mapping procedure from the Weiss-field treatment in which the interactions of
the spin on a given layer ($ab$ plane) with its neighbors on adjacent layers
are replaced by effective magnetic fields. Whereas such a treatment may give a
reasonable description for quasi-2D materials with small interlayer coupling
yamamoto-15 , it fails to capture the quantum correlations in the intrachain
couplings of the coupled-chain materials. The squash mapping takes into
account the quantum correlations through the value of $\tilde{S}$, and more
importantly, the 2D squashed model (2) is written in the same form as the
model for a realistic 2D TLAF material, while the Weiss-field model includes
extra terms of effective local magnetic fields with the strength and direction
determined in a self-consistent fashion.

The fitting of $\delta E_{\rm LSW}$ and $\delta\tilde{E}_{\rm LSW}$ in the
same scale of $J_{1}$ with respect to $\tilde{S}/S$ only depends on the
intrachain/interchain coupling ratio $\alpha_{J}$ [under the correspondences
(3)]. As expected, the value of $\tilde{S}/S$ is larger (more classical) for
larger $\alpha_{J}=J_{0}/J_{1}$ as shown in Fig. 9a. Figure 9b are typical
examples of the comparison between $\delta E_{\rm LSW}$ and
$\delta\tilde{E}_{\rm LSW}$ with the optimized $\tilde{S}/S$ at several values
of $\alpha_{J}$, showing a good agreement between the original (3D) and
effective (2D) models. Of course, the spin operator
$\hat{\bm{\mathcal{S}}}_{i}$ in Eq. (2) is properly defined only when
$\tilde{S}$ is an integer or half-integer value in a strict sense beyond the
$1/\tilde{S}$ expansion. Nevertheless, the value of $\tilde{S}$ can still be
taken as an indicator for the strength of quantum fluctuations existing in the
coupled-chain compound under consideration. For example, a material with the
intrachain coupling $J_{0}$ being five times larger than the interchain
coupling $J_{1}$ is expected to exhibit the same extent of quantum effects as
the corresponding 2D material with the spin being about two times larger than
the original one.

Figure 10:  Phase diagram in an extended parameter space. Theoretical ground-
state phase diagram of the model for CsCuCl3 in the plane of magnetic field
$H$ scaled by the saturation value $H_{\rm s}$ and the reciprocal of the
effective spin $\tilde{S}$. The solid and dashed curves in the left panel are
the phase boundaries obtained for the original model of CsCuCl3 with the ratio
of the intralayer to interlayer interaction, $\alpha_{J}$ and the effective 2D
model with the corresponding values of the effective spin $\tilde{S}$,
respectively. The range of the pressure application in the current experiments
is indicated by the vertical red dashed-dotted lines. The right illustration
schematically shows the corresponding parameter plane in the three-variable
phase diagram for 2D easy-plane triangular-lattice antiferromangets, shown in
Fig. 1.

.

Using the result of Fig. 9a with the original spin value $S=1/2$, we can
translate the pressure dependence of the intrachain/interchain coupling ratio
$\alpha_{J}$ for CsCuCl3, shown in Fig. 5c, into continuous change of the
effective spin $\tilde{S}$ in terms of the 2D TLAF model. The obtained values
of $\tilde{S}$ are indicated on the upper axis of the phase diagram in Fig. 6.
Now let us discuss the extension of the model calculations beyond the
parameter range of the current experiments, with the caveat that the
extrapolation is in general less reliable. Figure 10 shows the predicted phase
diagram in an extended parameter space, where the horizontal axis is converted
from $P$ to $1/\tilde{S}$. The corresponding values of $\alpha_{J}$ are
indicated on the upper axis. When $\alpha_{J}=0$, the model is trivially
reduced to the spin-1/2 Heisenberg model for a purely 2D TLAF with isotropic
exchange coupling. Therefore, the pressure-induced stabilization of the
magnetization plateau can be interpreted by means of the effective 2D TLAF
model as a consequence that the pressure pushes the value of $1/\tilde{S}$
from the semi-classical ($1/\tilde{S}<1$) regime towards the highly-quantum
($1/\tilde{S}=2$) regime. Although the change in $1/\tilde{S}$ was not
significantly large in the current experiment with a piston cylinder cell, it
was fortunate that the magnetic parameters of CsCuCl3 at ambient pressure were
located in the vicinity of the crossover regime between the semi-classical and
highly-quantum magnetization processes, which are shown in Figs 3a and 3b,
respectively.

Note that, as shown in Fig. 10, whereas the effective 2D model reproduces well
the phase boundaries $H_{{\rm c}1}$ and $H_{\rm 1st}$ in low fields, the value
of $H_{{\rm c}2}$ is somewhat overestimated. This is caused by the fact that
the fitting of the zero-point energies, $\delta E_{\rm LSW}$ and
$\delta\tilde{E}_{\rm LSW}$, is relatively less satisfactory in the high-field
region, as seen in Fig. 9b, which could be improved by considering the $H$
dependence of the effective spin $\tilde{S}$ but with extra complexity.

To conclude, through high-pressure magnetic measurements and theoretical
investigations on a CsCuCl3 crystal, we have developed a scientific concept
for the control of quantum-mechanical correlations in weakly-coupled spin
chain materials by applying external pressure. The parameter fitting for the
model Hamiltonian of CsCuCl3 has shown that the ratio of the intrachain to
interchain spin coupling, $\alpha_{J}$, is strongly reduced by hydrostatic
pressure application. From an intuitive idea of mapping the spins along each
chain into a single large spin $\tilde{S}>S$, we introduce an effective spin
model that is “squashed” onto a 2D plane and establish the correspondence
between the parameters of the original and effective model Hamiltonians. Since
the spin quantum number can take only an integer or half-integer value in
nature, one can in principle access the phase diagram only with discrete
values of $S$ in experiments. Our observations open up an interesting
possibility of performing quantum simulation studies that can interpolate the
properties of 2D spin models at discrete spin values by performing high-
pressure experiments on coupled-chain compounds. Moreover, the spin value $S$
has been actually treated as a continuous variable in theoretical studies
using analytical methods such as the $1/S$ expansion and the Schwinger-boson
mean-field theory (with parameter $\kappa=2S$ lacroix-11 ; arovas-88 ; wang-10
; kargarian-12 ; samajdar-19 ). The interpretation based on the squash mapping
opens a way for high-pressure experiments on coupled-chain compounds to
directly realize a huge variety of the theoretical phase diagrams that has
been predicted so far (and will be obtained in the future) for 2D models with
continuous $S$.

Considering the variety of coupled-spin-chain compounds, including the other
materials in the $ABX_{3}$-type hexagonal perovskite family achiwa-69 ;
kakurai-84 ; maruyama-01  and those with different lattice geometries, this
concept also provides us with a unique opportunity to study the continuous
classical-to-quantum crossover of the ground state and the elementary
excitations in a wide variety of 2D frustrated quantum antiferromagnets. For
example, the spatially-anisotropic TLAF model has been extensively studied in
the literature balents-10 ; trumper-99 ; manuel-99 ; wang-06 ; yunoki-06 ;
heidarian-09 ; merino-14 ; ghorbani-16  as a model showing a rich phase
diagram including quantum spin liquids. High-pressure experiments on a
coupled-chain compound, e.g., RbCuCl3, in which spin-1/2 chains form a
spatially anisotropic triangular lattice maruyama-01 , could enable us to
simulate the theoretical phase diagram with active and continuous control of
effective spin $\tilde{S}$. Such an experiment may allow access to the spin
liquid quantum critical point via the melting of magnetic long-range order by
tuning pressure (or the value of $\tilde{S}$). Future research in such a
direction would be promising to shed new light into the connection between the
semi-classical “magnon” and highly-quantum “spinon” descriptions of magnetic
quasiparticles oh-13 ; kajimoto-15 ; ito-17 ; coldea-01 ; balents-10 ; shen-16
; paddison-17 ; shen-18 . Finally, note that although not a few compounds in
the family of $ABX_{3}$-type hexagonal perovskites have antiferromagnetic
intrachain coupling achiwa-69 , there is still every chance that the pressure
application changes it to ferromagnetic one, allowing for the squashed model
description we proposed here.

## Methods

### Sample setting and magnetization measurements under pressure

Single crystal samples of CsCuCl3 were prepared by following the procedure
described in Ref. tanaka-85, . A clamp-type piston-cylinder pressure cell made
of CuBe alloy with an outer diameter of 8.7$\phi$, an inner diameter of
2.7$\phi$ and a cylinder length of 72 mm was used sakurai-12 . A sample is
enclosed in a Teflon capsule with a pressure medium Daphne 7373 (Idemitsu
Kosan Co., Ltd.). A plate-like CsCuCl3 sample with the long axis along the
c-axis was prepared. The dimension was 2 mm$\times$ 6 mm and the thickness was
about 1 mm ($\sim$18 mg). The pressure was calibrated by the change of the
superconducting transition temperature of tin smith-67 . A tin foil with a
thickness of 0.2 mm was formed into a tube shape ($\sim$30 mg), and the sample
was placed in this tube.

Magnetization was measured by a commercially available magnetometer equipped
with a superconducting quantum interference device (MPMS-XL, Quantum Design,
Inc.). The measurement was performed using the option “background subtraction”
of MultiVu software attached to MPMS. First, to obtain the background data,
temperature variation and magnetic field variation sequences were run at
ambient pressure for the pressure cell including tin without sample. Then, the
magnetization of CsCuCl3 at each pressure was obtained by subtracting the
background from the total magnetization including CsCuCl3 sample in the same
sequences. The background data at ambient pressure was used for all
measurements. The magnetic field is applied parallel to the c-axis. The
temperature variation measurements were done at 1 T below 100 K, and the field
variation measurements were done at 1.8 K up to 5 T.

In temperature variation measurement, the temperature range was limited below
100 K to avoid change in pressure. The clamp-type pressure cell has a
relatively large pressure drop when the temperature is decreased, especially
between the room temperature and 100 K (at most 0.2 GPa), whereas it hardly
has change in pressure below 100 K murata-97 .

In this study, the pressure was calibrated using the relationship between the
pressure and the superconducting transition temperature of tin given in Ref.
smith-67, . In the magnetization measurement under pressure by Sera et al.
sera-17 , the pressure was also calibrated by the superconducting transition
temperature of tin, but by a different formula given in Ref. boughton-70, .
The pressure values stated when we referred to the data of Sera et al. sera-17
, including Figs. 6 and 7, were the ones recalibrated by the former
calibration formula; specifically, $P=0.25,0.50,0.68,0.75,0.81$ GPa in Sera et
al. were reevaluated as $P=0.27,0.55,0.75,0.83,0.90$ GPa, respectively, and
$P=0.1$ MPa was regarded as $P=0$.

### Parameter fitting of magnetic susceptibility data

Figure 4a shows the temperature dependence of the magnetic susceptibility
parallel to the $c$ axis, $\chi_{\parallel}$, measured at $H=1$ T under
different pressures, $P=0,0.14,0.34,0.49,0.82,1.05,1.21$ GPa. The core
diamagnetic ($\chi_{\rm dia}=-1.09\times 10^{-4}$ emu/mole) and Van-Vleck
paramagnetic ($\chi_{\rm v}=0.48\times 10^{-4}$ emu/mole) contributions
tazuke-81  are already subtracted. As can be seen, the overall value of
$\chi_{\parallel}$ decreases considerably as the pressure increases, which
indicates that the dominant coupling parameter for the magnetic energy scale,
namely the intrachain coupling strength $J_{0}$, significantly decreases. The
peak of each curve is located at the Néel temperature $T_{\rm N}$.

To quantify the pressure dependence of $J_{0}$, we perform a fitting of the
experimentally measured $\chi_{\parallel}(T)$ in the temperature range
$50$-$100$ K to the expression

$\displaystyle\frac{T}{C}\chi_{\parallel}(T)={\rm
Pad\acute{e}}(4,5)\left[1+\sum_{n=1}^{9}a_{n}\left(\frac{J_{0}}{k_{\rm
B}T}\right)^{n}\right],$ (14)

where, $C=N_{0}g^{2}\mu_{\rm B}^{2}/4k_{\rm B}$ is the Curie constant with
$N_{0}$ being the Avogadro number and ${\rm
Pad\acute{e}}(4,5)\left[\cdots\right]$ means the Padé approximant of order
[4/5]. The coefficients $a_{n}$, which are (lengthy) functions of
$\alpha_{J}$, are obtained by the tenth-order high-temperature expansion
method lohmann-14  (see Supplementary Fig. 1). Here, we ignored the small
contributions from $\Delta_{0}$. In Fig. 5a, the values of $J_{0}$ obtained by
the fittings were shown. Note that the values of $J_{1}$ fitted to the
magnetic susceptibility data strongly vary depending on the temperature range
used for the fittings. Therefore, we adopt only the model function of
$J_{0}(P)$, which is the most dominant parameter for the susceptibility
measurements, from the above fittings.

### Parameter fitting of magnetization curves

Figure 4b shows the scaled magnetization curves $M/M_{\rm s}$, which are
measured under static magnetic field up to 5 T at temperature $T=1.8$ K for
different pressures, $P=0,0.14,0.34,0.49,0.82,1.05,1.21$ GPa. It can be seen
that the curves are almost linearly proportional to $H$ in this field range.
The magnetization curves have also been measured by Miyake et al. miyake-15
(at $P=0$) and Sera et al. sera-17  (up to $P=0.90$ GPa in our calibration).
There is a little variability in the slope of $M(H)$ among the experiments.

The model functions of $\Delta_{0}$ and $J_{1}$ [Eqs. (7) and (8)] were
determined such that the low-temperature magnetization curves obtained by the
different experiments could be all reasonably reproduced (see Fig. 7 and
Supplementary Fig. 2). The theoretical calculations were based on the
evaluation of the energy up to the leading orders of the anisotropy and $1/S$.
Each term in Eqs. (5) and (12) is obtained by following the procedure of Ref.
nikuni-93,  for each phase (umbrella, Y, or V). It should be noted that the
magnetic field $H$ and single-ion-type anisotropy $A$ have to be treated as
order of $S$ and $S/(2S-1)$ tsuru-86 , respectively, to obtain the correct
expression for the saturation field. The slope of the magnetization curve in
low fields can be calculated from the thermodynamic relation $M=-dE/dH$
coletta-16  with $E$ for the umbrella phase.

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## Acknowledgements

We would like to thank Prof. Masafumi Sera for giving us the details of their
pressure calibration method. We also thank Dr. Giacomo Marmorini for a careful
reading of the manuscript. This work was carried out by the joint research
program of Molecular Photoscience Research Center, Kobe University, and
supported by KAKENHI from Japan Society for the Promotion of Science: Grant
Numbers 18K03525 (D.Y.), 19K03746 (T.S.), 19K21852 (H.O.), 17H01142 (H.T.),
and 19H00648 (Y.U.), and “Early Eagle” grant program from Aoyama Gakuin
University Research Institute (D.Y.).

## Supplementary Information

### Supplementary Note 1: Transformation into a twisted spin frame

In the laboratory frame, the coupled-chain triangular-lattice antiferromagnet
(TLAF) CsCuCl3 in a magnetic field $H$ parallel to the $c$ axis is well
described by the following Hamiltonian Stanaka-92 :

$\displaystyle\hat{\mathcal{H}}$ $\displaystyle=$
$\displaystyle-2J_{0}^{\perp}\sum_{i,n}\left(\hat{s}_{i,n}^{x}\hat{s}_{i,n+1}^{x}+\hat{s}_{i,n}^{y}\hat{s}_{i,n+1}^{y}\right)-2J_{0}^{\parallel}\sum_{i,n}\hat{s}_{i,n}^{z}\hat{s}_{i,n+1}^{z}-\bm{D}\cdot\sum_{i,n}\left(\hat{\bm{s}}_{i,n}\times\hat{\bm{s}}_{i,n+1}\right)$
(S1) $\displaystyle+2J_{1}\sum_{\langle
i,j\rangle,n}\hat{\bm{s}}_{i,n}\cdot\hat{\bm{s}}_{j,n}-H\sum_{i,n}\hat{s}_{i,n}^{z},$

where $\hat{\bm{s}}_{i,n}$ denotes the local $S=1/2$ spin on site $i$ of the
$n$-th triangular layer. The ferromagnetic interaction between the transverse
(longitudinal) components of the nearest-neighbour spins along the $c$ axis is
denoted by $J_{0}^{\perp}$ ($J_{0}^{\parallel}$) and the isotropic
antiferromagnetic interaction between the spins in the $ab$ plane is $J_{1}$.
The Dzyaloshinskii-Moriya (DM) interaction $\bm{D}=(0,0,d)$, which favors
helical spin structures along the $c$ axis, can be eliminated by the unitary
transformation Snikuni-93

$\displaystyle\left(\begin{array}[]{c}\hat{s}^{x}_{i,n}\\\
\hat{s}^{y}_{i,n}\\\
\hat{s}^{z}_{i,n}\end{array}\right)=\left(\begin{array}[]{ccc}\cos nq&-\sin
nq&0\\\ \sin nq&\cos nq&0\\\
0&0&1\end{array}\right)\left(\begin{array}[]{c}\hat{S}^{x}_{i,n}\\\
\hat{S}^{y}_{i,n}\\\ \hat{S}^{z}_{i,n}\end{array}\right)$ (S11)

with the twist angle $q$ along the $c$ axis. By setting
$q=\arctan(d/2J_{0}^{\perp})$, we can rewrite the Hamiltonian in the form with
no DM term as

$\displaystyle\hat{\mathcal{H}}$ $\displaystyle=$
$\displaystyle-2J_{0}\sum_{i,n}\left(\hat{\bm{S}}_{i,n}\cdot\hat{\bm{S}}_{i,n+1}-\Delta_{0}\hat{S}_{i,n}^{z}\hat{S}_{i,n+1}^{z}\right)+2J_{1}\sum_{\langle
i,j\rangle,n}\hat{\bm{S}}_{i,n}\cdot\hat{\bm{S}}_{j,n}-H\sum_{i,n}\hat{S}_{i,n}^{z},$

which is Eq. (1) of the main text. The parameters are related to those of the
original Hamiltonian by $J_{0}=J_{0}^{\perp}\sqrt{1+(d/2J_{0}^{\perp})^{2}}$
and $J_{0}(1-\Delta_{0})=J_{0}^{\parallel}$. In the main text, we determine
the pressure dependencies of the three parameters $J_{0}$, $\Delta_{0}$, and
$J_{1}$ of the model Hamiltonian in the twisted spin space. Note that if the
magnetic field has a finite transverse component (say,
$\propto\sum_{i,n}\hat{s}_{i,n}^{x}$), this unitary transformation cannot
simplify the model since one has to deal with a non-uniform magnetic field
with different directions for different layers
($\propto\sum_{i,n}(\hat{S}^{x}_{i,n}\cos nq-\hat{S}^{y}_{i,n}\sin nq)$) as a
trade-off for eliminating the DM term.

### Supplementary Note 2: Differences in the pressure environment from the
previous experiments by Sera et al.

In the comparison of the theoretical magnetization curves with the
experimental data shown in Fig. 7, the agreement seems to get slightly worse
for larger values of pressure, especially at $P=0.9$ GPa (Fig. 7d). A possible
reason for this is a slight underestimation of pressure due to the pressure
inhomogeneity in the sample used in Ref. Ssera-17 .

The large differences in the experimental conditions from the previous
experiment by Sera et al. are the amount of sample and the setting way of tin
whose superconducting transition temperature is used for pressure calibration
at low temperature Ssmith-67 . As for the amount of sample, we used a sample
of 17.7 mg, while Sera et al. used a sample of 62.5 mg to gain the S/N ratio
in their homemade magnetization measurement equipment Ssera-17 . Since the
dimensions of the pressure cells are similar, this causes a difference in
sample length of about 3 times. In general, the pressure inhomogeneity in a
sample becomes larger as the sample gets longer since it occurs along the
cylindrical axis in the piston-cylinder type pressure cell when pressure-
transmitting fluid freezes. Therefore, it can be said that the pressure
inhomogeneity in the sample was expected to be larger in the previous
experiment by Sera et al. Ssera-17 , compared to that in the measurement of
the present work. Regarding the setting of tin, we made a tube with almost the
same height as the sample, and put the sample into the tin tube (see Methods
in the main text) in order to correctly detect the pressure experienced by the
sample. On the other hand, Sera et al. placed tin just below the sample [M.
Sera, private communication]. It means that they measured the pressure only at
the bottom of the sample with larger pressure distribution.

Supplementary Fig. 2 shows the comparison of the pressure dependence of the
reciprocal of slope of low-field magnetization curves. It can be seen that the
value obtained from the experiment by Sera et al. Ssera-17  becomes clearly
larger than that from our experiment for high pressures ($\gtrsim 0.75$ GPa).
Indeed, this fact suggests that the pressure value measured in the experiment
by Sera et al. may be slightly underestimated in the high pressure region,
owing to the pressure inhomogeneity in their long sample and the above-
mentioned measurement method of the pressure. Moreover, the pressure
inhomogeneity could also make unclear the plateau structure and the inflection
points in the experimental magnetization curve for high pressures.

## References

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  * (2) Nikuni T. & Shiba, H. Quantum Fluctuations and Magnetic Structures of CsCuCl3 in High Magnetic Field. J. Phys. Soc. Jpn. 62, 3268 (1993).
  * (3) Sera, A., Kousaka, Y., Akimitsu, J., Sera, M. & Inoue, K. Pressure-induced quantum phase transitions in the $S=1/2$ triangular lattice antiferromagnet CsCuCl3. Phys. Rev. B 96, 014419 (2017).
  * (4) Smith, T. F. & Chu, C. W. Will Pressure Destroy Superconductivity? Phys. Rev. 159, 353 (1967).
  * (5) Lohmann, A., Schmidt, H.-J. & Richter J. Tenth-order high-temperature expansion for the susceptibility and the specific heat of spin-$s$ Heisenberg models with arbitrary exchange patterns: Application to pyrochlore and kagome magnets. Phys. Rev. B 89, 014415 (2014).
  * (6) Miyake, A., Shibuya, J., Akaki, M., Tanaka, H. & Tokunaga, M. Magnetic field induced polar phase in the chiral magnet CsCuCl3. Phys. Rev. B 92, 100406(R) (2015).

Supplementary Fig 1:  Fitting for magnetic susceptibility under pressure.
Enlarged view of Fig. 4a in the temperature range $50<T<100$ K together with
the best-fitting theoretical curves (solid lines) according to the tenth-order
high-temperature expansion lohmann-14  combined with the Padé approximation.
Supplementary Fig 2:  Fitting for magnetization curves under pressure. The
reciprocal of the slopes of the low-field magnetization curves shown in Fig.
4b is plotted as a function of pressure $P$, together with the ones extracted
from the experiments of Miyake et al. miyake-15  and Sera et al. sera-17 . The
theoretical curve obtained with the model parameters of Eqs. (6-8) is plotted
by the solid line.