|  Non-collinear magnetism in the post-perovskite thiocyanate frameworks
CsM(NCS)3
---|---
|  Madeleine Geers,a,b Jie Yie Lee,a,b Sanliang Ling,c Oscar Fabelo,b Laura
Cañadillas-Delgado,b Matthew J. Cliffe,∗a †
|  AMX3 compounds are structurally diverse, a notable example being the post-
perovskite structure which adopts a two-dimensional framework with corner- and
edge-sharing octahedra. Few molecular post-perovskites are known and of these,
none have reported magnetic structures. Here we report the synthesis,
structure and magnetic properties of molecular post-perovskites: CsNi(NCS)3, a
thiocyanate framework, and two new isostructural analogues CsCo(NCS)3 and
CsMn(NCS)3. Magnetisation measurements show that all three compounds undergo
magnetic order. CsNi(NCS)3 (Curie temperature, $T_{\mathrm{C}}=8.5(1)\;$K) and
CsCo(NCS)3 ($T_{\mathrm{C}}=6.7(1)\;$K) order as weak ferromagnets. On the
other hand, CsMn(NCS)3 orders as an antiferromagnet (Néel temperature,
$T_{\mathrm{N}}=16.8(8)\;$K). Neutron diffraction data of CsNi(NCS)3 and
CsMn(NCS)3, show that both are non-collinear magnets. These results suggest
molecular frameworks are fruitful ground for realising the spin textures
required for the next generation of information technology.

##

††footnotetext: a School of Chemistry, University Park, Nottingham, NG7 2RD,
United Kingdom Email<EMAIL_ADDRESS>b
Institut Laue Langevin, 71 avenue des Martyrs CS 20156, 38042 Grenoble Cedex
9, France††footnotetext: c Advanced Materials Research Group, Faculty of
Engineering, University of Nottingham, University Park, Nottingham NG7 2RD,
United Kingdom††footnotetext: † Electronic Supplementary Information (ESI)
available: Additional experimental details for synthesis, single crystal X-ray
and neutron diffraction measurements and analysis, powder neutron diffraction
measurements and analysis, magnetic measurements, density functional theory
calculations; CIFs and mCIFs.. See DOI: 00.0000/00000000.

##  1 Introduction

A unifying goal in solid-state science is control over the physical properties
of materials, and the tunability of perovskites is perhaps the most striking
example. Traditionally these compounds are three-dimensional frameworks with a
general chemical formula AMX3 built from [MX6] corner-sharing octahedra.
Through substitution of the A, M and X ions, materials with remarkable
magnetic,1 electronic conductivity,2 photovoltaic3 and non-linear optical
properties,4 have been created.

Fig. 1: a) The structure of CaIrO3 with iridium polyhedra connected $via$
corner-sharing ($a$ axis) and edge-sharing ($c$ axis) [IrO6] octaheadra, Ca =
blue, Ir = brown, O = red. b) Crystal structure of CsMn(NCS)3 at 120 K
obtained from single crystal X-ray diffraction. c) A single [Mn(NCS)3]- layer.
d) Close-up view of of the structure highlighting the two chemically
independent metal ions (M1, M2). Cs = green, Mn = pink, N = blue, C = black, S
= yellow.

The perovskite structure is, however, only one of a wide-range of AMX3
structure-types, with perhaps the most closely related being the post-
perovskite structure. In contrast to perovskites, post-perovskites contain
[MX6] octahedra that both edge- and corner-share, resulting in two-dimensional
anionic [MX3] layers, rather than a three dimensional framework. These layers
are stacked with interstitial A cations positioned between them. Post-
perovskites are amongst the most abundant terrestrial minerals, as the MgSiO3
perovskite that makes up much of the Earth’s lower mantle undergoes a critical
high-pressure and temperature phase transition ($P_{C}\approx 125\;$GPa,
$T_{C}\approx 1250\;$K) to the post-perovskite structure-type near the mantle-
core boundary.5, 6 Due to the difficulty of reaching these extreme pressures,
post-perovskites which form at more accessible pressures and can be recovered
on quenching have proved useful analogues. These include the second- and
third-row transition metal oxides AMO3, A = Na, Ca, and M = Pt, Rh, Ir
($P_{\mathrm{syn}}\approx 5\;$GPa);7, 8, 9, 10 and first-row fluorides NaMF3,
M = Mg, Ni, Co, Fe, Zn.11, 12, 13, 14, 15 A handful of post-perovskite
compounds can even be obtained at ambient conditions: notably CaIrO3,16 the
post-actinide chalcogenides AMnSe3 (A = Th, U)17 and UFeS3,18 and TlPbI3.19 As
a result of this synthetic challenge, systematic tuning of the properties of
post-perovskites is much less well explored than for perovskites.

In particular, there are limits on our current understanding of the magnetic
properties of post-perovskites, in part because neutron diffraction studies
require large sample sizes. This is despite the fact that post-perovskites,
unlike perovskites, tend to have non-collinear magnetic structures.20, 16, 12,
15 As a result, both the exploration of fundamental properties of post-
perovskites and the potential utility of their non-trivial spin textures for
spintronic devices21, 22 or quantum memory storage23 remains limited.

In contrast to the relative scarcity of atomic post-perovskites, molecular
post-perovskites, where X is a molecular anion, are a growing class of
materials.24, 25, 26, 27, 28 Unlike their atomic analogues, the majority of
molecular post-perovskites are stable and synthesisable at ambient pressure.
Incorporating molecular components in these frameworks can also allow for
novel physical properties, arising from the additional degrees of freedom,
including non-linear optics,29 electric polarisation 30, 31 and complex spin
textures.32

Metal dicyanamides, AM(dca)3 dca = N(CN)2-, are the best established family of
molecular post-perovskites.33, 34, 35 The metal ions are separated by five-
atom-bridges and tend to be magnetically isolated, with no conclusive evidence
of long-range magnetic ordering.25, 36, 24 To explore collective magnetic
behaviour in molecular post-perovskite analogues we therefore focussed on
ligands capable of propagating stronger superexchange interactions.

Thiocyanate (NCS-) is a promising molecular ligand for creating magnetic
coordination frameworks with long-range magnetic ordering, even over extended
distances.37, 38, 39, 40, 41, 42 The thiocyanate also has asymmetric
reactivity, unlike both atomic ligands and most common molecular ligands,
_e.g._ formate or dca-.43 Applying the Hard-Soft Acid-Base principle,44 the
nitrogen terminus is less polarisable and so coordinates preferentially to
harder first-row transition metals, whereas sulfur has more diffuse orbitals,
and preferentially coordinates to softer main group second- and third-row
transition metals.38 Homoleptic framework structures are thus comparatively
rare, beyond the binary M(NCS)2.38, 37 Nevertheless, there are two reported
AM(NCS)3 homoleptic frameworks with the post-perovskite structure:
RbCd(NCS)345 and CsNi(NCS)3.26

In this work, we study CsM(NCS)3, M = Ni, Mn and Co. We describe the synthesis
of CsMn(NCS)3 and CsCo(NCS)3 and determine their structures, by single crystal
X-ray diffraction, to be post-perovskites isomorphic with CsNi(NCS)3. We used
bulk magnetometry to show that all three order magnetically. CsNi(NCS)3
($T_{\mathrm{C}}=8.5(1)\;$K) and CsCo(NCS)3 ($T_{\mathrm{C}}=6.7(1)\;$K) order
as weak ferromagnets with significant coercive fields, CsNi(NCS)3
$H_{\mathrm{C}}=0.331(2)\;$T and CsCo(NCS)3 $H_{\mathrm{C}}=0.052(2)\;$T,
whereas CsMn(NCS)3 orders as an antiferromagnet $T_{\mathrm{N}}=16.8(8)\;$K.
We then report neutron diffraction measurements, with single crystal and
powder samples. These found CsNi(NCS)3 to be a canted ferromagnet,
$\mathbf{k}=(0,0,0)$, and CsMn(NCS)3 to be a non-collinear antiferromagnet
which orders with $\mathbf{k}=(0,\frac{1}{2},\frac{1}{2})$ and four
sublattices which all order antiferomagnetically. Additionally, density
functional theory (DFT) calculations were undertaken, which confirm the
intralayer interactions are at least an order of magnitude stronger than the
interlayer interactions in these compounds.

Our results suggest that magnetic molecular frameworks, and thiocyanates in
particular, are a promising ground for exploring the magnetism of otherwise
challenging to realise structure types. The non-collinear structures we have
uncovered in these thiocyanate frameworks _via_ neutron diffraction suggest
that further studies of this family may uncover new routes to complex spin
textures.

##  2 Results

###  2.1 Synthesis and Structure

We synthesised CsNi(NCS)3, CsMn(NCS)3 and CsCo(NCS)3 by salt metathesis of the
metal sulfate and caesium sulfate with barium thiocyanate in aqueous solution.
CsNi(NCS)3 is stable at ambient conditions, however CsMn(NCS)3 and CsCo(NCS)3
are sensitive to humidity. Single crystals were obtained through slow
evaporation and the structures were determined using single crystal X-ray
diffraction to have the monoclinic space group $P2_{1}/n$, and to crystallise
in the post-perovskite structure. We found that, unlike the Ni and Mn
analogues, the crystallisation of CsCo(NCS)3 from solution typically occurs in
a two step process. Large, deep blue crystals of a second phase, believed to
be Cs2Co(NCS)4, are typically obtained, which recrystallise into small, deep
purple single crystals of CsCo(NCS)3 if left undisturbed for several weeks in
ambient conditions.

All three compounds are isomorphic and consist of anionic [M(NCS)3] layers in
the _ac_ plane in which the transition metal M2+ ions are connected through
$\mu_{1,3}$NCS ligands (Fig. 1$\mathrm{c}$). In between the layers, which are
stacked along the _b_ axis, lie the charge balancing caesium counterions. The
M2+ ions are octahedrally coordinated and there are two crystallographically
and chemically distinct metal sites. One transition metal ion (M1, Wyckoff
site 2 _c_) is coordinated by four nitrogen atoms and two sulfur atoms, whilst
the second transition metal ion (M2, Wyckoff site 2 _b_) is bonded to four
sulfur and two nitrogen atoms. The metal octahedra corner-share along the _a_
axis and alternate between M1 and M2. Along the _c_ axis, the metal octahedra
edge-share, and all the metal sites within an edge-sharing chain are the same.

Fig. 2: Magnetic susceptibility data with an applied field of $0.01$ T for a)
CsNi(NCS)3; b) CsMn(NCS)3; c) CsCo(NCS)3; and isothermal magnetisation
measurements at 2 K for d) CsNi(NCS)3 (field $-6$ to $+6$ T); e) CsMn(NCS)3
(field $0$ to $+7$ T); and f) CsCo(NCS)3 (field $-7$ to $+7$ T).

Moving along the row from manganese to nickel, the average M-N and M-S bond
lengths, $d_{\mathrm{M-N}}$ and $d_{\mathrm{M-S}}$, decrease:
$d_{\mathrm{Mn-N}}(13\;\mathrm{K})=2.1607(14)\;$Å,
$d_{\mathrm{Co-N}}(120\;\mathrm{K})=2.072(3)\;$Å,
$d_{\mathrm{Ni-N}}(15\;\mathrm{K})=2.035(5)\;$Å and
$d_{\mathrm{Mn-S}}=2.674(3)\;$Å, $d_{\mathrm{Co-S}}=2.5652(10)\;$Å,
$d_{\mathrm{Ni-S}}=2.350(20)\;$Å. There are two intralayer M$\cdots$M
distances, between the edge-sharing octahedra ($d_{\mathrm{M1-M1}}$ =
$d_{\mathrm{M2-M2}}=c$) and the corner-sharing octahedra
($d_{\mathrm{M1-M2}}=a$). Again, moving from Mn2+ to Ni2+, the distance
decrease, with $d_{\mathrm{Mn1-Mn1}}=5.6540(4)\;$Å compared to
$d_{\mathrm{Co1-Co1}}=5.57860(11)\;$Å and $d_{\mathrm{Ni1-Ni1}}=5.5409(6)\;$Å;
and $d_{\mathrm{Mn1-Mn2}}=6.37315(5)\;$Å, $d_{\mathrm{Co1-Co2}}=6.30515(5)\;$Å
and $d_{\mathrm{Ni1-Ni2}}=6.2631(6)\;$Å. The interlayer spacing also
decreases, with the shortest M$\cdots$M distances, $d_{\mathrm{M,layer}}$,
decreasing from $d_{\mathrm{Mn,layer}}=7.20052(15)$ Å, through
$d_{\mathrm{Co,layer}}=7.18340(10)$ Å to $d_{\mathrm{Ni,layer}}=7.1050(8)$ Å.

Table 1: DFT predicted relative energies between the experimental post-
perovskite phase (CsM(NCS)3 = pPv) and the two known AM(NCS)3 perovskite
structure types: CsCd(NCS)3 = Cs[Cd]; (NH4)2NiCd(NCS)6 = NH4[NiCd]. The
perovskite structures were generated by swapping the A site cation for Cs+ and
the B site cation(s) by the appropriate transition metal.

M^2+ |  $E$(pPv)∗ |  $E$(Cs[Cd])∗ |  $E$(NH4[NiCd])∗
---|---|---|---
Ni | 0.0 | $+90.8$ | $+177.3$
Mn | 0.0 | $+84.8$ | $+137.0$
Co | 0.0 | $+286.7$ | $+114.5$

$\ast$ (meV per formula unit)

In addition to this post-perovskite structure, there are two different
perovskite structure-types with composition AM(NCS)3, CsCd(NCS)345 and
(NH4)2NiCd(NCS)6 (ESI Fig. 7).46 To understand the relative stability of the
post-perovskite structure compared to the perovskite-type structures, density
functional theory (DFT) calculations were performed. Hypothetical perovskites
were generated through atom-swaps, and the geometry optimised relaxation,
relative to the experimentally observed post-perovskite structure, was
calculated (Table 1). For simplicity, we focused on the spin-ferromagnetic
solution of the three different structure types for this comparison. We found
that the post-perovskite structure was more stable than the perovskite
structures by around 10 kJ mol-1 (0.1 eV per formula unit, approximately
$4\;kT$ at room temperature). This energy is consistent with the observed
exclusive formation of the post-perovskite structure, but suggests that more
unusual experimental conditions may allow access to the perovskite phases.

###  2.2 Magnetism

Fig. 3: Neutron diffraction data and the magnetic structure for CsNi(NCS)3. a)
Thermodiffractogram measured between 1.5 and 10 K on the D1b diffractometer
(ILL). The most intense magnetic Bragg reflection is indexed as the (100)
planes. The $F_{\mathrm{obs}}$ against $F_{\mathrm{calc}}$ plot obtained from
D19 data fit (nuclear and magnetic) collected at 1.8 K (b) and Rietveld fit
obtained from D1b data fit at 1.5 K (c) for the multi pattern refinement of
the $P2_{1}/c$ magnetic model describing the magnetic structure of CsNi(NCS)3.
d) The magnetic moment of the Ni2+ ions as a function of temperature obtained
by Rietveld refinements of the data collected at each temperature point
between 1.5 and 10 K. e) The magnetic structure of CsNi(NCS)3 viewed down the
$c$ axis. The two unique magnetic vectors are represented with purple arrows
(Ni1) and green arrows (Ni2). The Cs+ cations have been omitted for clarity
and the thiocyanate ligands are represented as a wire frame. f) The magnetic
structure viewed down the [111] direction. g) Angles describing the non-
collinearity of the moments. Fig. 4: Neutron diffraction data used to
determine the magnetic structure for CsMn(NCS)3. a) Thermodiffractogram
measured between 1.5 and 20 K on the D1b diffractometer (ILL). The most
intense magnetic Bragg reflection is indexed as the (100) planes. b) The
integrated intensity of the (100) magnetic reflection at 0.81
$\mathrm{\mathring{A}^{-1}}$ as a function of temperature. c) The magnetic
structure of CsMn(NCS)3. The magnetic vectors are depicted with red and orange
arrows for Mn1 (red = Mn1a and orange = Mn1b) and blue arrows for Mn2 (dark =
Mn2a and light = Mn2b). The Cs+ cations have been omitted for clarity and the
thiocyanate ligands are represented as a wire frame. d) $F_{\mathrm{obs}}$
against $F_{\mathrm{calc}}$ for the refinement of the $P_{S}\overline{1}$
magnetic model using the magnetic reflections observed at 2 K with the D19
diffractometer (ILL). e) The angles between the magnetic vectors.

Having synthesised an isostructural series of thiocyanate post-perovskites
containing paramagnetic ions, we next sought to understand their magnetic
behaviour. We measured the bulk magnetic susceptibility of each of these
compounds, in an applied field of $0.01$ T, to determine the magnetic ordering
temperature and average interaction strength. We used isothermal magnetisation
measurements at 2 K over the range $-7$ to $+7$ T ($-6$ to $+6$ T for M = Ni)
to assess the degree of magnetic hysteresis.

The zero-field cooled (_ZFC_) and field-cooled (_FC_) susceptibility of
CsNi(NCS)3 diverge below the ordering temperature $T_{\mathrm{C}}=8.5(1)\;$K
(Fig. 2$\mathrm{a}$). The Curie-Weiss fit of the inverse susceptibility above
$180$ K gives a value for the Curie-Weiss temperature, $\theta_{\mathrm{CW}}$,
of $-8.6(8)$ K. However, this value is particularly sensitive to the fitting
temperature range: $\theta_{\mathrm{CW}}=+1.5(4)$ K when $100<T<300\;$K,
whereas $\theta_{\mathrm{CW}}$ = $-12.7(8)$ K for a fit $200<T<300\;$ K. This
variation is likely due to the presence of significant single-ion anisotropy,
typical of Ni2+. The Curie constant, $C$, is $0.85(2)$ emu K mol-1, which is
lower than the spin only value, $C_{\mathrm{spin\;only}}=1$ emu K mol-1. The
isothermal magnetisation of CsNi(NCS)3 at 2 K shows hysteresis with a coercive
field of $H_{\mathrm{C}}=0.331(2)\;$T and a remnant magnetisation
$M_{\mathrm{rem}}=0.106(1)\;\mu_{B}$ per Ni, implying a canting angle of 6.1°
if there are only two distinct spin orientations. Beyond $1.19(1)$ T, the
hysteresis loop closes and there is a magnetic phase transition to a second
magnetic phase, reaching a magnetisation of $1.54(1)\;\mu_{B}$ per Ni at 6 T.

The magnetic susceptibility of CsMn(NCS)3 has a cusp at $16.8(8)$ K indicating
the onset of antiferromagnetic order (Fig. 2$\mathrm{b}$). An increase in
susceptibility at low temperature is likely due to a small fraction of a
hydrated impurity. Fitting the inverse susceptibility between $100<T<300$ K to
the Curie-Weiss law gives $\theta_{\mathrm{CW}}=-33.6(2)$ K, and $C=3.75(2)$
emu K mol-1, which is lower than the high-spin spin only expected value of
$C_{\mathrm{spin\;only}}=4.375$ emu K mol-1 for Mn2+. We did not observe any
evidence of hysteresis in the isothermal magnetisation data, consistent with
antiferromagnetic order. The magnetisation reaches $1.20(6)\;\mu_{B}$ per Mn
at the largest field measured ($7.00(1)$ T), well below the spin only
saturation magnetisation, $M_{\mathrm{sat.}}=5\;\mu_{B}$ per Mn, indicative of
the presence of significant antiferromagnetic interactions. The ratio of the
Curie-Weiss temperature to the ordering temperature is
$f=|\theta_{\mathrm{CW}}/T_{\mathrm{N}}|=2.1$, suggestive of slight
frustration or low-dimensionality.

Curie-Weiss fitting of the magnetic susceptibility of CsCo(NCS)3 between
$100<T<300\;$K suggests predominately antiferromagnetic interactions,
$\theta_{\mathrm{CW}}$ = $-19.7(2)$ K. The large Curie constant, $C=4.8(2)$
emu K mol-1, compared to the high-spin spin only value
$C_{\mathrm{spin\;only}}=1.875$ emu K mol-1, indicates that the unquenched
orbital moment remains significant in these compounds and the determined
$\theta_{\mathrm{CW}}$ therefore likely also includes the effects of the first
order spin-orbit coupling. d$\chi$/d _T_ shows two sharp minima, suggesting
that there are potentially two ordering temperatures for the compound at
$6.7(1)$ and $8.4(1)$ K. The isothermal magnetisation data measured at 2 K
show a hysteresis with $H_{\mathrm{C}}=0.052(2)\;$T and a remnant
magnetisation $M_{\mathrm{rem}}=0.400(1)\;\mu_{B}$ per Co, suggesting a
canting angle of $15.3$°. The hysteresis disappears at $1.86(1)$ T, when
$M=0.88(4)\;\mu_{B}$ per Co. Above this applied magnetic field, the
magnetisation steadily increases, reaching $2.00(7)\;\mu_{B}$ per Co at
$7.00(1)$ T, although the moment remains unsaturated, due to a combination of
single-ion anisotropy and antiferromagnetic interactions.

Our bulk magnetic properties measurements suggested that both CsNi(NCS)3 and
CsCo(NCS)3 are weak ferromagnets (canted antiferromagnets), or more generally
have uncompensated magnetic moments, with appreciable hystereses and field-
induced magnetic phase transitions. Comparatively, the absence of hysteresis
and a negative $\theta_{\mathrm{CW}}$ for CsMn(NCS)3 suggests it is an
antiferromagnet.

###  2.3 Neutron Diffraction

To explore the magnetic properties of these post-perovskites further, we
therefore carried out neutron diffraction experiments, both single crystal and
powder, to determine their ground state magnetic structures. Scale-up of our
initial synthetic routes produced high quality powder and single crystal
samples of CsNi(NCS)3 and CsMn(NCS)3 suitable for neutron diffraction, however
the two-stage synthesis of CsCo(NCS)3 meant we were unable to obtain either
large (mm3) single crystals or gram-scale pure phase microcrystalline powders,
precluding neutron studies for this compound.

We carried out single crystal neutron diffraction measurements of CsNi(NCS)3
($1.8\times 0.9\times 0.3$ mm3) using the D19 diffractometer at the Institut
Laue Langevin (ILL). A low temperature data collection at 2 K, below the
ordering temperature of 8.5 K, allowed us to determine the propagation vector
to be $\mathbf{k}=(0,0,0)$ by indexing of the magnetic Bragg reflections. We
combined these single crystal neutron diffraction (SCND) data with additional
powder neutron diffraction (PND) data collected on a polycrystalline sample
(1.1 g) using the powder neutron diffractometer D1b (ILL). The magnetic space
groups with maximal symmetry which permit magnetic moments to exist on the two
Ni^2+ ions were determined using the Bilbao Crystallographic Server47, 48, 49
to be $P2_{1}^{\prime}/c^{\prime}$ and $P2_{1}/c$ (BNS notation).48 Both of
these models have the same unit cell as the nuclear structure. We then refined
models in each space group against the combined PND and SCND data with a
multipattern refinement and found that only $P2_{1}/c$ was able to model the
additional intensity arising from the magnetic reflections. This refined
structure shows weak ferromagnetic order with two unique magnetic sites,
corresponding to the two crystallographic Ni2+ ions (Ni1 and Ni2) in the
nuclear structure. The two moments have a magnitude of $2.01(3)\;\mu_{B}$ and
were constrained to refine together with a negative correlation. The moments
are directed predominately along the _c_ axis with the canting only present
along the _b_ axis. The Ni1 moments (purple arrows Fig. 3) are canted at an
angle of $162$° along the $-b$ direction, whilst the Ni2 moments (green arrows
Fig. 3) are canted at an angle of $105$° in the $+b$ direction. The asymmetric
canting of the two sublattices results in a net magnetisation of
$0.116\;\mu_{B}$ per Ni along the $b$ axis ($+b$ direction). This is
equivalent to a single Ni2+ canting at an angle of $6.7$°, which is in
agreement with the magnetometry data. The magnetic moment of Ni1 and Ni2
present a relative tilt of $136$° between them.

We found that when subtracting the 2 K PND data from the 10 K data, there are
some nuclear peaks which do not directly overlap at the two temperatures. This
is particularly evident in the (020) and (200) reflections, which are the most
intense in the diffraction patterns (Fig. 3$\mathrm{c}$). On heating from 2 to
10 K, the _a_ axis increases by $0.028$ %, whereas the _b_ axis decreases in
length by $0.019$ %. In contrast, the _c_ axis remains constant within this
temperature range (ESI Fig. 6).

We were also able to measure a large single crystal of CsMn(NCS)3 ($6.3\times
2.5\times 1.1$ mm3) using D19 (ILL). We found in our diffraction data
collected at 2 K, below $T_{\mathrm{N}}$ = 16.8 K, a number of additional
Bragg reflections (2938 unique reflections) not present in our data collected
at 20 K, which could not be indexed to the nuclear structure. We were able to
index these additional magnetic Bragg reflections with
$\mathbf{k}=(0,\frac{1}{2},\frac{1}{2})$. Using the Bilbao Crystallographic
Server we determined the possible magnetic space groups for this propagation
vector to be $P_{S}\overline{1}$ and $P_{S}1$. We solved the magnetic
structures in both magnetic space groups, with the $P_{S}\overline{1}$ better
fitting the experimental data. The model comprises of four unique magnetic
sites: Mn1a and Mn1b, which arise from nuclear site Mn1 but are in alternate
layers (red and orange arrows Fig. 4$\mathrm{c}$), and Mn2a and Mn2b from
nuclear site Mn2 (dark and light blue arrows Fig. 4$\mathrm{c}$). The magnetic
unit cell is related to the nuclear cell as follows
$a_{\mathrm{mag}}=a_{\mathrm{nuc}}$, $b_{\mathrm{mag}}=2b_{\mathrm{nuc}}$ and
$c_{\mathrm{mag}}=2c_{\mathrm{nuc}}$. The magnitude of the moments for all Mn
sites is $4.63(9)\;\mu_{B}$ and were constrained to refine with a single
moment value. We found that allowing the moments to refine freely did not
significantly improve our fit (constrained refinement, $\chi^{2}=68.5$
compared to free refinement, $\chi^{2}=56.7$), and led to unphysically small
moment sizes and unstable moment angles. Constraining the moment angles of
Mn1b and Mn2b to be collinear (while allowing Mn1a and Mn2a to be non-
collinear) gave $\chi^{2}=120.3$, whilst constraining all four moments to be
collinear gave $\chi^{2}=807.6$. In the determined model, each of the four
magnetic sublattices, derived from a unique Mn2+ site, order
antiferromagnetically as expected from the bulk antiferromagnetic order
observed in the magnetisation data. Mn1a and Mn2a moments, are aligned
antiparallel within the anionic layer, while Mn1b and Mn2b are at an angle of
$130$° relative to each other (Fig. 4$\mathrm{e}$). Powder neutron diffraction
data were collected on the D1b diffractometer, which clearly shows the
additional magnetic Bragg reflections. However, the limited data quality
prevented quantitative refinement of these data.

###  2.4 Density Functional Theory Calculations

Our neutron analysis provided the ground states, but the energetics which
yield these states remained opaque. We therefore turned to DFT calculations.
In order to correct for the typical delocalisation errors encountered in DFT,
a Hubbard $U$ was incorporated.38, 50 There are four nearest-neighbour
interactions (Fig. 5): $J_{a}$, through the M1-NCS-M2 corner-sharing bridge;
$J_{c1}$, through the M1-NCS-M1 edge-sharing chain; $J_{c2}$, through the
M2-NCS-M2 edge-sharing chain; and $J_{b}$, between the layers. The magnetic
lattice therefore consists of rectangular lattices in which there are two
different kinds of chain along one direction, which are then coupled together
by $J_{b}$. Due to the offset of the layers, this means that if $J_{a}$,
$J_{c1}$ and $J_{c2}$ are antiferromagnetic, we would expect this lattice to
be frustrated. To calculate each of these interactions we therefore
constructed eight $2\times 1\times 1$ supercells with distinct ordering
patterns (ESI Table 1), and fitted their DFT+$U$ energies to the following
Hamiltonian:

$E=\sum_{ij}J_{ij}S_{i}\cdot S_{j}+E_{0}$ (1)

Fig. 5: The four nearest-neighbour magnetic interactions in CsM(NCS)3. M1 and
M2 are grey and black respectively and the thiocyanate ligands are represented
with sulfur (yellow) and nitrogen (blue) atoms only. Caesium cations and
carbon atoms have been removed for clarity.

where $J_{ij}$ denotes the superexchange interaction for the atom pair $ij$
(i.e. each interaction is counted once), $|S|=1$, and $E_{0}$ is the energy of
a hypothetical non-magnetic state. We found that for CsNi(NCS)3 and CsMn(NCS)3
we could obtain self-consistent results, but we were unable to achieve
acceptable self-consistency for CsCo(NCS)3. The residual unquenched orbital
moment for the ${}^{4}T_{1\mathrm{g}}$ ground state is likely responsible for
this and suggests that higher level calculations are likely to be required to
appropriately capture the magnetic behaviour of this compound.

Our calculations indicate that all four interactions are antiferromagnetic for
CsMn(NCS)3, with the interactions through the edge-sharing chains ($J_{c1}$
and $J_{c2}$) stronger than the corner-sharing bridge ($J_{a}$) between them
(Table 2). The magnitude of the $J_{b}$ interaction is much smaller than the
error and is thus not meaningful. We found that CsNi(NCS)3 has ferromagnetic
interactions within the edge-sharing chains, with the $J_{c2}$ chain notably
stronger, and the interactions between chains are antiferromagnetic. As with
CsMn(NCS)3, the $J_{b}$ interaction is small and zero within error. From these
interactions we were able to calculate Curie-Weiss temperatures: for Mn
$J_{\mathrm{CW,calc.}}=-22.3(5)\;$K and for Ni
$J_{\mathrm{CW,calc.}}=-0.05(65)\;$K which are broadly comparable with those
found experimentally. The ground states predicted by DFT are largely
consistent with those determined experimentally, allowing for the fact that
these non-relativistic calculations cannot predict spin-canting.

Table 2: DFT-derived superexchange energy, $J_{x}$, as defined in Eqn. 1 for
CsM(NCS)3, M = Ni, Mn.

M2+ | $J_{a}^{\ast}$ | $J_{b}^{\ast}$ | $J_{c1}^{\ast}$ | $J_{c2}^{\ast}$
---|---|---|---|---
Ni | 0.19(4) |  $-$0.02(10) |  $-$0.13(7) |  $-$0.25(7)
Mn | 0.78(3) | 0.03(6) | 2.40(6) | 1.79(6)

∗ (meV)

##  3 Discussion

The CsM(NCS)3 compounds all crystallise with the post-perovskite structure.
However, unlike most atomic analogues, these thiocyanate compounds adopt the
structure at ambient pressure. A common characteristic of atomic post-
perovskites is the presence of large octahedral tilt angles in the
corresponding perovskite phase, which leaves them more susceptible to undergo
the post-perovskite phase transition.11 Thiocyanate perovskites are already
very tilted, due to the shape of the frontier bonding orbitals, which could
explain the ease for formation of this structure type for CsM(NCS)3.38
Defining the plane as the $ac$ axes, the tilting occuring in these thiocyanate
compounds along the edge- and corner-sharing directions can be compared. The
corner-sharing octahedra, along the _a_ axis, have smaller deviations away
from the $ac$ plane, with angles of $\angle ac-\mathrm{M}-\mathrm{S}=42(1)$°
and $\angle ac-\mathrm{M}-\mathrm{N}=9(1)$° (ESI Fig. 9). Along the _c_ axis,
the edge-sharing octahedra adopt greater tilting angles as an inherent
consequence of having two thiocyanate ligands bridging each pair of metal
centres in this direction. The deviation away from the $ac$ plane is $\angle
ac-\mathrm{M}-\mathrm{S}=58(1)$° and $\angle ac-\mathrm{M}-\mathrm{N}=36(1)$°.
The incorporation of a molecular ligand permits access to this greater degree
of tilting without the need for external pressure. This is evident from our
DFT calcualtions which show that for these CsM(NCS)3 compounds the post-
perovskite structure-type is lower energy than other reported thiocyanate
perovskite-type structures (Table 1).

Our magnetometry and neutron diffraction measurements show that the compounds
magnetically order between 6 and 16 K, significantly lower than the closest
chemical analogues, the binary thiocyanates M(NCS)2 M = Mn, Fe, Co, Ni, Cu,38,
37 which order at $T_{\mathrm{N}}=29\;$K for Mn(NCS)2, $T_{\mathrm{N}}=20\;$K
for Co(NCS)242, 37 and $T_{\mathrm{N}}=54\;$K for Ni(NCS)2.37, 41 This
reduction in ordering temperature also occurs in atomic post-perovskites,
where post-perovskite NaNiF3 orders at $T_{\mathrm{N}}=22$ K, compared to
$T_{\mathrm{C}}=156$ K for the perovskite phase.12 The reported ordering
temperatures of the oxide post-perovskites are an order of magnitude larger
than those of the fluorides and thiocyanates,51, 8, 52 likely as the oxide
anions lie closer to the metal-insulator boundary.

One key difference between CsM(NCS)3 and M(NCS)2 is that the post-perovskites
only have three-atom connections between transition metals ($\mu_{13}$NCS
coordination mode), whereas M(NCS)2 have both one-atom and three-atom
connections ($\mu_{133}$NCS). The additional M-S-M superexchange pathway in
the binary thiocyanates likely strengthens the magnetic interactions in the
binary thiocyanates, although DFT calculations of Cu(NCS)2 suggest that
interactions through the M-S-C-N-M can be as strong or stronger than through
M-S-M bridges.38 Our DFT calculations further support this, showing
appreciable superexchange through the end-to-end bridging thiocyanates.

In contrast to these compounds, dca^- based post-perovskites containing
magnetic ions do not appear to order.25, 36, 24 The transition metals in these
compounds are separated by six bonds and $d$(Mn-NCNCN-Mn) = 8.9825(4) Å
([Ph4P]Mn(dca)3),24 compared to four bonds and $d$(Mn-NCS-Mn) = 6.37315(5) Å
(CsMn(NCS)3). This likely reduces the superexchange further. However,
Cr[Bi(SCN)6], with even longer superexchange pathways does order at
$T_{\mathrm{N}}=4.0\;$K,39 indicating that orbital overlap and orbital energy
matching are also playing a key role in this.

CsM(NCS)3, M = Ni, Mn, Co, all adopt non-collinear magnetic structures. Non-
collinearity also appears to be typical in the atomic perovskites. The only
previous experimentally reported magnetic structure of a post-perovskite is of
CaIrO3, which used magnetic resonant X-ray scattering to reveal a canted
stripe antiferromagnetic ground state.16 Octahedral tilting is predicted to be
a key parameter in determining the degree of non-collinearity,53 so it is
expected that the post-perovskite structures are sensitive to this factor as
well. Analysis of our isothermal magnetisation data for CsNi(NCS)3 (Fig.
2$\mathrm{d}$), assuming that there is only a single magnetic site (i.e. only
two distinct spin orientations), gives a canting angle of $6$°. However our
magnetic structure has two magnetic sites (i.e. four spin orientations), and
so there are in fact three ‘canting angles’, all of which are larger than $6$°
($9$°, $22$°, and $38$°). The symmetry constraints of the $P2_{1}/c$ magnetic
space group means that for each pair of canted moments (i.e. a single magnetic
site), the components of the magnetic moments along the _a_ and _c_ axes will
be of equal magnitude and so the uncompensated magnetisation lies only along
the _b_ axis. In CsNi(NCS)3, the uncompensated moments from each magnetic site
have opposite signs: $+0.80\;\mu_{B}$ per Ni2 and $-0.57\;\mu_{B}$ per Ni1,
with a net moment of $+0.114\;\mu_{B}$. Using bulk measurements for materials
with complex magnetic structures can therefore lead to underestimates of the
degree of non-collinearity.

The magnetic structure of CsMn(NCS)3, unlike the nickel and cobalt analogues,
orders as an antiferromagnet. The neutron data reveal a k =
$(0,\frac{1}{2},\frac{1}{2})$ propagation vector, which leads to four unique
sublattices. As a result of the anticentring translation in the
$P_{S}\overline{1}$ magnetic space group, each sublattice, and therefore the
overall structure, is antiferromagnetic. There are two distinct kinds of layer
within the magnetic structure, ‘a’ and ‘b’. Layer a, containing Mn1a and Mn2a,
is antiferromagnetically correlated; but Mn1b and Mn2b are non-collinear both
with respect to each other and also to Mn1a and Mn2a. The complexity of this
structure is perhaps surprising, considering the relative simplicity of the
nuclear structure and the lack of spin-orbit coupling expected for high spin
Mn2+. The layers stack so that each consecutive layer is offset by
$c_{\mathrm{nuc}}/2$, resulting in a triangular relationship between the
interlayer moments (Fig. 5). This layered stacking pattern may generate
frustration, which could explain the observed non-collinear structure.

##  4 Conclusion

In this paper, we have reported the synthesis, structure, magnetometry, single
crystal and powder neutron diffraction data for three isomorphic post-
perovskite thiocyanate frameworks, CsM(NCS)3 M = Ni, Mn, Co. Our magnetic
susceptibility measurements show that all the materials magnetically order,
CsNi(NCS)3 $T_{\mathrm{C}}=8.5(1)\;$K, CsMn(NCS)3 $T_{\mathrm{N}}=16.8(8)\;$K
and CsCo(NCS)3 $T_{\mathrm{C}}=6.7(1)\;$K. Our neutron diffraction experiments
on CsNi(NCS)3 and CsMn(NCS)3 revealed both compounds have complex non-
collinear ordering.

CsNi(NCS)3 orders as a weak ferromagnet with two magnetically distinct nickel
moments. CsMn(NCS)3, on the other hand, orders as an antiferromagnet with a
magnetic unit cell which is doubled along the nuclear _b_ and _c_ axes, and
has four unique sublattices.

Our neutron diffraction studies have shown that despite the relative
simplicity of the chemical structures, these thiocyanate post-perovskites are
a rich source of unusual magnetic orderings which cannot be recognised through
magnetometry data alone. Introducing molecular ligands into framework
structures may therefore provide a host of unexpected and complex spin
textures, motivating both future synthetic and neutron diffraction
investigations.

## Conflicts of interest

No conflicts of interest to declare.

## Acknowledgements

M.G. acknowledges the ILL Graduate School for provision of a studentship.
M.J.C. acknowledges the School of Chemistry, University of Nottingham for
support from Hobday bequest. We acknowledge the ILL for beamtime under
proposal numbers 5-41-1060,54 5-12-344,55 5-31-2726,56 5-31-2816.57 Raw data
sets from ILL experiments can be accessed via links provided in references.
Magnetic measurements were carried out in part using the Advanced Materials
Characterisation Suite, funded by EPSRC Strategic Equipment Grant
EP/M000524/1. S.L. acknowledges the use of the Sulis supercomputer through the
HPC Midlands+ Consortium, and the ARCHER2 supercomputer through membership of
the U.K.’s HPC Materials Chemistry Consortium, which are funded by EPSRC Grant
numbers EP/T022108/1 and EP/R029431/1, respectively.

## Author Contributions

## Notes and references

  * Vishnoi _et al._ 2020 P. Vishnoi, J. L. Zuo, T. A. Strom, G. Wu, S. D. Wilson, R. Seshadri and A. K. Cheetham, _Angewandte Chemie International Edition_ , 2020, 59, 8974–8981.
  * Jaffe _et al._ 2020 A. Jaffe, S. A. Mack, Y. Lin, W. L. Mao, J. B. Neaton and H. I. Karunadasa, _Angewandte Chemie International Edition_ , 2020, 59, 4017–4022.
  * Kojima _et al._ 2009 A. Kojima, K. Teshima, Y. Shirai and T. Miyasaka, _Journal of the American Chemical Society_ , 2009, 131, 6050–6051.
  * Xu _et al._ 2020 J. Xu, X. Li, J. Xiong, C. Yuan, S. Semin, T. Rasing and X. Bu, _Advanced Materials_ , 2020, 32, 1806736.
  * Oganov and Ono 2004 A. R. Oganov and S. Ono, _Nature_ , 2004, 430, 445–448.
  * Murakami _et al._ 2004 M. Murakami, K. Hirose, K. Kawamura, N. Sata and Y. Ohishi, _Science_ , 2004, 304, 855–858.
  * Ohgushi _et al._ 2008 K. Ohgushi, Y. Matsushita, N. Miyajima, Y. Katsuya, M. Tanaka, F. Izumi, H. Gotou, Y. Ueda and T. Yagi, _Physics and Chemistry of Minerals_ , 2008, 35, 189–195.
  * Yamaura _et al._ 2009 K. Yamaura, Y. Shirako, H. Kojitani, M. Arai, D. P. Young, M. Akaogi, M. Nakashima, T. Katsumata, Y. Inaguma and E. Takayama-Muromachi, _Journal of the American Chemical Society_ , 2009, 131, 2722–2726.
  * Rodi and Babel 1965 F. Rodi and D. Babel, _Zeitschrift fur Anorganische und Allgemeine Chemie_ , 1965, 336, 17–23.
  * Bremholm _et al._ 2011 M. Bremholm, S. Dutton, P. Stephens and R. Cava, _Journal of Solid State Chemistry_ , 2011, 184, 601–607.
  * Martin _et al._ 2006 C. D. Martin, W. A. Crichton, H. Liu, V. Prakapenka, J. Chen and J. B. Parise, _American Mineralogist_ , 2006, 91, 1703–1706.
  * Shirako _et al._ 2012 Y. Shirako, Y. Shi, A. Aimi, D. Mori, H. Kojitani, K. Yamaura, Y. Inaguma and M. Akaogi, _Journal of Solid State Chemistry_ , 2012, 191, 167–174.
  * Dobson _et al._ 2011 D. P. Dobson, S. A. Hunt, A. Lindsay-Scott and I. G. Wood, _Physics of the Earth and Planetary Interiors_ , 2011, 189, 171–175.
  * Lindsay-Scott _et al._ 2014 A. Lindsay-Scott, D. Dobson, F. Nestola, M. Alvaro, N. Casati, C. Liebske, K. S. Knight, R. I. Smith and I. G. Wood, _Journal of Applied Crystallography_ , 2014, 47, 1939–1947.
  * Bernal _et al._ 2014 F. L. Bernal, K. V. Yusenko, J. Sottmann, C. Drathen, J. Guignard, O. M. Løvvik, W. A. Crichton and S. Margadonna, _Inorganic Chemistry_ , 2014, 53, 12205–12214.
  * Ohgushi _et al._ 2013 K. Ohgushi, J.-i. Yamaura, H. Ohsumi, K. Sugimoto, S. Takeshita, A. Tokuda, H. Takagi, M. Takata and T.-h. Arima, _Physical Review Letters_ , 2013, 110, 217212.
  * Ijjaali _et al._ 2004 I. Ijjaali, K. Mitchell, F. Q. Huang and J. A. Ibers, _Journal of Solid State Chemistry_ , 2004, 177, 257–261.
  * Noël and Padiou 1976 H. Noël and J. Padiou, _Acta Crystallographica_ , 1976, B32, 1593–1595.
  * Lin _et al._ 2021 W. Lin, J. He, K. M. McCall, C. C. Stoumpos, Z. Liu, I. Hadar, S. Das, H. Wang, B. Wang, D. Y. Chung, B. W. Wessels and M. G. Kanatzidis, _Advanced Functional Materials_ , 2021, 31, 2006635.
  * Bogdanov _et al._ 2012 N. A. Bogdanov, V. M. Katukuri, H. Stoll, J. van den Brink and L. Hozoi, _Physical Review B_ , 2012, 85, 235147.
  * Jia _et al._ 2020 F. Jia, S. Hu, S. Xu, H. Gao, G. Zhao, P. Barone, A. Stroppa and W. Ren, _The Journal of Physical Chemistry Letters_ , 2020, 11, 5177–5183.
  * Tokura and Kanazawa 2021 Y. Tokura and N. Kanazawa, _Chemical Reviews_ , 2021, 121, 2857–2897.
  * Li _et al._ 2021 S. Li, Y. Wang, Y. Wang, S. Sanvito and S. Hou, _The Journal of Physical Chemistry C_ , 2021, 125, 6945–6953.
  * Raebiger _et al._ 2001 J. W. Raebiger, J. L. Manson, R. D. Sommer, U. Geiser, A. L. Rheingold and J. S. Miller, _Inorganic Chemistry_ , 2001, 40, 2578–2581.
  * van der Werff _et al._ 2001 P. M. van der Werff, S. R. Batten, P. Jensen, B. Moubaraki and K. S. Murray, _Inorganic Chemistry_ , 2001, 40, 1718–1722.
  * Fleck 2004 M. Fleck, _Acta Crystallographica_ , 2004, C60, i63–i65.
  * Mączka _et al._ 2018 M. Mączka, A. Gągor, M. Ptak, D. Stefańska and A. Sieradzki, _Physical Chemistry Chemical Physics_ , 2018, 20, 29951–29958.
  * Wang _et al._ 2019 S.-S. Wang, R.-K. Huang, X.-X. Chen, W.-J. Xu, W.-X. Zhang and X.-M. Chen, _Crystal Growth & Design_, 2019, 19, 1111–1117.
  * Triki _et al._ 2020 H. Triki, B. Nagy, J. Overgaard, F. Jensen and S. Kamoun, _Structural Chemistry_ , 2020, 31, 103–114.
  * Boström _et al._ 2018 H. L. Boström, M. S. Senn and A. L. Goodwin, _Nature Communications_ , 2018, 9, 1–7.
  * Lee _et al._ 2021 J. Y. Lee, S. Ling, S. P. Argent, M. S. Senn, L. Cañadillas-Delgado and M. J. Cliffe, _Chemical Science_ , 2021, 12, 3516.
  * Lou _et al._ 2020 F. Lou, T. Gu, J. Ji, J. Feng, H. Xiang and A. Stroppa, _npj Computational Materials_ , 2020, 6, 114.
  * Biswas _et al._ 2006 M. Biswas, S. R. Batten, P. Jensen and S. Mitra, _Australian Journal of Chemistry_ , 2006, 59, 115.
  * Wang _et al._ 2015 H.-T. Wang, L. Zhou and X. Wang, _Acta Crystallographica_ , 2015, C71, 717–720.
  * Wang _et al._ 2019 S.-S. Wang, X.-X. Chen, B. Huang, R.-K. Huang, W.-X. Zhang and X.-M. Chen, _CCS Chemistry_ , 2019, 1, 448–454.
  * van der Werff _et al._ 2001 P. M. van der Werff, S. R. Batten, P. Jensen, B. Moubaraki, K. S. Murray and E. H.-K. Tan, _Polyhedron_ , 2001, 20, 1129–1138.
  * Bassey _et al._ 2020 E. N. Bassey, J. A. M. Paddison, E. N. Keyzer, J. Lee, P. Manuel, I. da Silva, S. E. Dutton, C. P. Grey and M. J. Cliffe, _Inorganic Chemistry_ , 2020, 59, 11627–11639.
  * Cliffe _et al._ 2018 M. J. Cliffe, J. Lee, J. A. M. Paddison, S. Schott, P. Mukherjee, M. W. Gaultois, P. Manuel, H. Sirringhaus, S. E. Dutton and C. P. Grey, _Physical Review B_ , 2018, 97, 144421.
  * Cliffe _et al._ 2022 M. J. Cliffe, O. Fabelo and L. Cañadillas-Delgado, _CrystEngComm_ , 2022, 24, 7250–7254.
  * Baran _et al._ 2019 S. Baran, A. Hoser, M. Rams, S. Ostrovsky, T. Neumann, C. Näther and Z. Tomkowicz, _Journal of Physics and Chemistry of Solids_ , 2019, 130, 290–297.
  * DeFotis _et al._ 1993 G. C. DeFotis, K. D. Dell, D. J. Krovich and W. W. Brubaker, _Journal of Applied Physics_ , 1993, 73, 5386–5388.
  * Shurdha _et al._ 2012 E. Shurdha, S. H. Lapidus, P. W. Stephens, C. E. Moore, A. L. Rheingold and J. S. Miller, _Inorganic Chemistry_ , 2012, 51, 9655–9665.
  * Boča _et al._ 2004 M. Boča, I. Svoboda, F. Renz and H. Fuess, _Acta Crystallographica_ , 2004, C60, m631–m633.
  * Pearson 1963 R. G. Pearson, _Journal of the American Chemical Society_ , 1963, 85, 3533–3539.
  * Thiele and Messer 1980 G. Thiele and D. Messer, _Zeitschrift fur Anorganische und Allgemeine Chemie_ , 1980, 464, 255–267.
  * Xie _et al._ 2016 K.-P. Xie, W.-J. Xu, C.-T. He, B. Huang, Z.-Y. Du, Y.-J. Su, W.-X. Zhang and X.-M. Chen, _CrystEngComm_ , 2016, 18, 4495–4498.
  * Aroyo _et al._ 2011 M. I. Aroyo, J. M. Perez-Mato, D. Orobengoa, E. Tasci, G. de la Flor and A. Kirov, _Bulg. Chem. Commun_ , 2011, 43, 183–197.
  * Gallego _et al._ 2012 S. V. Gallego, E. S. Tasci, G. de la Flor, J. M. Perez-Mato and M. I. Aroyo, _Journal of Applied Crystallography_ , 2012, 45, 1236–1247.
  * Perez-Mato _et al._ 2015 J. Perez-Mato, S. Gallego, E. Tasci, L. Elcoro, G. de la Flor and M. Aroyo, _Annual Review of Materials Research_ , 2015, 45, 217–248.
  * Wu _et al._ 2021 Y. Wu, W. Sun, S. Liu, B. Wang, C. Liu, H. Yin and Z. Cheng, _Nanoscale_ , 2021, 13, 16564–16570.
  * Ohgushi _et al._ 2006 K. Ohgushi, H. Gotou, T. Yagi, Y. Kiuchi, F. Sakai and Y. Ueda, _Physical Review B_ , 2006, 74, 241104.
  * Shirako _et al._ 2010 Y. Shirako, H. Satsukawa, H. Kojitani, T. Katsumata, M. Yoshida, Y. Inaguma, K. Hiraki, T. Takahashi, K. Yamaura, E. Takayama-Muromachi and M. Akaogi, _Journal of Physics: Conference Series_ , 2010, 215, 012038\.
  * Weingart _et al._ 2012 C. Weingart, N. Spaldin and E. Bousquet, _Physical Review B_ , 2012, 86, 094413.
  * 54 L. Cañadillas-Delgado, M. J. Cliffe, M. Geers and O. Fabelo, _ILL experiment 5-41-1060_ , http://doi.ill.fr/10.5291/ILL-DATA.5-41-1060.
  * 55 L. Cañadillas-Delgado, M. J. Cliffe and J. Y. Lee, _ILL experiment 5-12-344_ , http://doi.ill.fr/10.5291/ILL-DATA.5-12-344.
  * 56 L. Cañadillas-Delgado, M. J. Cliffe, O. Fabelo and M. Geers, _ILL experiment 5-31-2767_ , http://doi.ill.fr/10.5291/ILL-DATA.5-31-2767.
  * 57 L. Cañadillas-Delgado, M. J. Cliffe, C. V. Colin, O. Fabelo, M. Geers, V. Nassif and I. Puente Orench, _ILL experiment 5-31-2816_ , http://doi.ill.fr/10.5291/ILL-DATA.5-31-2816.


#  Many-body localization of 1D disordered impenetrable two-component fermions

M. S. Bahovadinov<EMAIL_ADDRESS>Russian Quantum Center, Skolkovo,
Moscow 143025, Russia  Physics Department, National Research University Higher
School of Economics, Moscow, 101000, Russia  D. V. Kurlov  Russian Quantum
Center, Skolkovo, Moscow 143025, Russia  B. L. Altshuler  Physics Department,
Columbia University, 538 West 120th Street, New York, New York 10027, USA
Russian Quantum Center, Skolkovo, Moscow 143025, Russia  G. V. Shlyapnikov
<EMAIL_ADDRESS>Russian Quantum Center,
Skolkovo, Moscow 143025, Russia  Moscow Institute of Physics and Technology,
Inst. Lane 9, Dolgoprudny, Moscow Region 141701, Russia  Université Paris-
Saclay, CNRS, LPTMS, 91405 Orsay, France  Van der Waals-Zeeman Institute,
Institute of Physics, University of Amsterdam,Science Park 904, 1098 XH
Amsterdam, The Netherlands

###### Abstract

We study effects of disorder on eigenstates of 1D two-component fermions with
infinitely strong Hubbard repulsion. We demonstrate that the spin-independent
(potential) disorder reduces the problem to the one-particle Anderson
localization taking place at arbitrarily weak disorder. In contrast, a random
magnetic field can cause reentrant many-body localization-delocalization
transitions. Surprisingly weak magnetic field destroys one-particle
localization caused by not too strong potential disorder, whereas at much
stronger fields the states are many-body localized. We present numerical
support of these conclusions.

Introduction. Many-body localization (MBL) emerges from an interplay between
disorder and interparticle interactions, extending the phenomenon of Anderson
localization (AL) to many-body systems Basko  _et al._ (2006); Oganesyan and
Huse (2007). Numerous works revealed MBL-caused vanishing steady transport,
absence of thermalization, area-law scaling of entanglement entropy, etc.
Berkelbach and Reichman (2010); Barišić  _et al._ (2016); Bar Lev  _et al._
(2015); Prelovšek and Herbrych (2017); Pal and Huse (2010); Bar Lev and
Reichman (2014); Serbyn  _et al._ (2015); Luitz  _et al._ (2016); Luca and
Scardicchio (2013); De Luca  _et al._ (2014); Bauer and Nayak (2013); Serbyn
_et al._ (2013a); Buijsman  _et al._ (2019); Serbyn  _et al._ (2016, 2013b);
Rademaker and Ortuño (2016); Chandran  _et al._ (2015). For reviews, see e.g.
Luitz and Lev (2017); Abanin and Papić (2017); Alet and Laflorencie (2018).

MBL has been extensively studied in systems one-dimensional (1D) spinless
fermions and various spin chain models Luitz  _et al._ (2015); Kudo and
Deguchi (2018); Luitz (2016); Burin (2015). These models possess only a single
local degree of freedom (LDoF), which is not the case in more realistic
systems. For instance, the Fermi-Hubbard model possesses two LDoF, charge and
spin, both of which can be coupled to disorder. Recently, the disordered 1D
Hubbard model in the regime of a finite interaction strength was studied
numerically Mondaini and Rigol (2015); Zakrzewski and Delande (2018); Bonča
and Mierzejewski (2017); Lemut  _et al._ (2017); Kozarzewski  _et al._ (2018,
2019); Krause  _et al._ (2021). The results suggest that a sufficiently strong
random potential localizes the charge degree of freedom, whereas spin
excitations apperently exhibit a subdiffusive transport, i.e. remain
delocalized Protopopov and Abanin (2019); Prelovšek  _et al._ (2016);
Protopopov  _et al._ (2017); Środa  _et al._ (2019). A symmetric situation is
observed in a sufficiently strong random magnetic field coupled to spins: spin
excitations are localized whereas charge excitations are extended Leipner-
Johns and Wortis (2019).

This Letter is devoted to the 1D Hubbard model for two-component fermions in
the limit of infinitely strong on-site repulsion. Theoretical considerations
supported by the numerical analysis allowed us to demonstrate that arbitrarily
weak potential disorder localizes both charge and spin excitations. Indeed,
each site of the 1D lattice in this limit can be in 3 (rather than 4 as in the
case of finite on-site interaction) distinct states: empty and single occupied
with 2 possible spin orientations. The absence of the double occupation
reduces the possible dynamics to the motion of empty spaces (holons). The
holons behave themselves like non-interacting fermions Abarenkova  _et al._
(1997) even in the presence of a spin-independent disorder not (a) and thus
are subject to the conventional single-particle AL. A weak magnetic disorder
causes an effective interaction between the holons, which can result in the
delocalization, unless the disorder is sufficiently strong.

Model and its symmetries. The Hamiltonian of the conventional 1D Hubbard model
with an infinitely strong on-site repulsion on a ring with $L$ sites is,

$H_{0}=-t\sum_{i=1}^{L}\sum_{\sigma=\uparrow,\downarrow}{\cal
P}\left(c^{\dagger}_{i,\sigma}c_{i+1,\sigma}+\text{H.c.}\right){\cal P}.$ (1)

Here $c_{j,\sigma}$ is the annihilation operator of a fermion in the spin
state $\sigma$ in the site $j$, and $t$ is the nearest neighbor hopping
amplitude. The operator ${\cal
P}=\prod_{i}\left(1-n_{i,\uparrow}n_{i,\downarrow}\right)$ with
$n_{i,\sigma}=c^{{\dagger}}_{i,\sigma}c_{i,\sigma}$, projects out the states
with doubly-occupied sites. Below we impose periodic boundary conditions
($c_{L+1}=c_{1}$) and set $t=1$. The integrable Hamiltonian (1) was fully
analyzed by means of the Bethe Ansatz Essler  _et al._ (2005).

The disorder is represented by the additional term in the Hamiltonian:

$H_{D}=\sum_{i=1}^{L}\varepsilon_{i}\left(n_{i,\uparrow}+n_{i,\downarrow}\right)+\sum_{i=1}^{L}h_{i}\frac{n_{i,\uparrow}-n_{i,\downarrow}}{2},$
(2)

with $\varepsilon_{i}$ and $h_{i}$ being the random potential and magnetic
field, respectively. We assume that both are uniformly distributed:
$\varepsilon_{i}\in[-W,W]$ and $h_{i}\in[-B,B]$.

The disorder breaks translational and SU(2) symmetries, but a number of
symmetries are preserved. First, $H$ conserves the number of particles with a
given spin
$N_{\left\\{\uparrow,\downarrow\right\\}}=\sum_{i}n_{i,\left\\{\uparrow,\downarrow\right\\}}$.
Moreover, the infinite repulsion does not allow particles with opposite spins
to exchange their positions, i.e. the spin pattern is conserved up to cyclical
transmutations.

Localization measures. Quantum states of the model (1,2) with a fixed set of
conserved quantities form a ${\cal N_{H}}$-dimensional Hilbert space (${\cal
N_{H}}<3^{L}$). We will analyze many-body eigenstates in the basis of states
with a given occupation ($0$-empty, $\uparrow$,$\downarrow$-occupied by a
particle with a particular spin direction) on each lattice site (computational
basis):
$|s\rangle=|s_{1}\rangle\otimes|s_{2}\rangle\otimes...\otimes|s_{L}\rangle$
and
$|s_{i}\rangle\in\left\\{|0\rangle,|\uparrow\rangle,|\downarrow\rangle\right\\}$.

Several quantitative measures have been proposed to characterize localization
properties of the wavefunctions $\psi_{\alpha}(s)=\langle s|\alpha\rangle$.
Analysis of the scaling of the participation entropies $S_{q}$ with ${\cal
N_{H}}$ allows one to determine fractal dimensions $D_{q}$:

$S_{q}=\frac{1}{q-1}\ln\left(\sum_{s=1}^{\cal
N_{H}}|\psi_{\alpha}(s)|^{2q}\right)\xrightarrow{{\cal
N_{H}}\rightarrow\infty}D_{q}\ln\left({\cal N_{H}}\right).$ (3)

The eigenstates $|\alpha\rangle$ localized (LO) on a finite set of $|s\rangle$
have $S_{q}$ independent of ${\cal N_{H}}$ and thus $D_{q}=0$ for any $q>0$.
Extended ergodic (EE) states with $|\psi_{\alpha}(s)|^{2}\sim{\cal
N_{H}}^{-1}$ are characterized by $D_{q}=1$. The states with $0<D_{q}<1$ are
non-ergodic (NEE) albeit extended. The eigenstates which are supposed to be
localized when effective disorder is strong, can become extended as the
disorder is reduced. For the conventional single-particle AL in 3D space at a
critical strength of the disorder, the system undergoes a transition from LO
to EE behavior. The NEE states are believed to exist only at the critical
point. As for the MBL, there is a strong evidence of the existence of the NEE
phase intermediate between LO and EE, implying two consecutive transitions LO
$\rightarrow$ NEE and NEE $\rightarrow$ EE at two different strengths of the
disorder.

The fractal dimensions $D_{q}$ may be not very useful in the numerical
identification of the LO $\rightarrow$ NEE transition: it is difficult to
distinguish $D_{q}=0$ from $0<D_{q}\ll 1$ for a finite size system. More
sensitive to this transition is the Kullback-Leibler divergence $KL$ Kullback
and Leibler (1951); Khaymovich  _et al._ (2020); Pino  _et al._ (2019):

$KL=\sum_{s=1}^{\cal
N_{H}}|\psi_{\alpha}(s)|^{2}\ln\left(\frac{|\psi_{\alpha}(s)|^{2}}{|\psi_{\alpha+1}(s)|^{2}}\right),$
(4)

(the states $|\alpha\rangle$ are supposed to be ordered in energy). The NEE
states close to this transition can be viewed as the result of the
hybridization of LO states. The LO states are not correlated in space and the
ratio $|\frac{\psi_{\alpha}(s)}{\psi_{\alpha+1}(s)}|$ is exponentially large
if $|\psi_{\alpha}(s)|$ is not negligible, i.e. KL diverges with ${\cal
N_{H}}$. After the hybridization, NEE states $|\alpha\rangle$ and
$|\alpha+1\rangle$ involve mostly the same LO states. As a result
$|\psi_{\alpha}(s)|$ and $|\psi_{\alpha+1}(s)|$ are strongly correlated with
$|\frac{\psi_{\alpha}(s)}{\psi_{\alpha+1}(s)}|\sim O(1)$ and $KL<\infty$. An
abrupt change of $KL$ is therefore an indication of the LO to NEE transition.

Exact diagonalization. In our exact diagonalization of the Hamiltonian
$H=H_{0}+H_{D}$ in the computational basis, we choose $L$ to be multiple of 4,
$L\in\\{12,16,20\\}$, and confine ourselves to the sector with
$N_{\uparrow}=N_{\downarrow}=\frac{L}{4}$. The size of this sector is
${L\choose L/2}\times{L/2\choose L/4}$. The conservation of the spin pattern
further reduces the dimension of the Hilbert space to ${\cal N_{H}}={\cal
Z}{{L}\choose{L/2}}$, where ${\cal Z}$ is the number of distinct spin
sequences connected by the cyclical transmutations for a given spin pattern.
Below we only consider the Neél spin pattern (${\cal Z}=2$) with the
corresponding ${\cal N_{H}}\in\\{1848,25740,369512\\}$. We employ the shift-
invert exact diagonalization algorithm to obtain $m\in\\{20,20,100\\}$ states
from the central part of the energy spectrum. We average $KL$, $D_{1}$ and
$D_{2}$ over these $m$ states and then perform averaging over $N_{d}$
realizations of the disorder (from $N_{d}\sim 10^{4}$ for the smallest system
size up to $N_{d}\sim 10^{2}$ for the largest one). For later convenience, we
parametrize the strengths of the random potential and magnetic field as,

$B=\rho_{D}\cos(\theta),\qquad W=\rho_{D}\sin(\theta),$ (5)

where $\rho_{D}>0$ is the total disorder strength. We carry out the
calculations for various values of $\rho_{D}$ and $\theta$, ranging from the
random potential ($\theta=\pi/2$ and $B=0$) to the random magnetic field
($\theta=0$ and $W=0$).

Figure 1:  $L$=4 ring with one holon (empty site) and spin pattern
$|\uparrow\downarrow\uparrow\rangle$. A clockwise detour ($0\circlearrowright
4$) of the holon leads to a cyclical transmutation
$|\uparrow\downarrow\uparrow\rangle\Rightarrow|\downarrow\uparrow\uparrow\rangle$.
The system returns to the original state only after 3 detours:
$|\uparrow\downarrow\uparrow\rangle\Rightarrow|\downarrow\uparrow\uparrow\rangle\Rightarrow|\uparrow\uparrow\downarrow\rangle\Rightarrow|\uparrow\downarrow\uparrow\rangle$.
Therefore, $\phi=\pm\frac{2\pi}{3}$. For a purely potential disorder the
states 0, 4 and 8 have the same potential energy, i.e. $\phi$ can be viewed as
a quasimomentum of the system.

Potential disorder. In the Bethe Ansatz solution of the disorder-free problem
governed by the Hamiltonian (1) each holon is characterized by momentum
$k_{j}$. The only difference of the holons from free spinless fermions on a
ring is the quasi-periodic rather than periodic boundary condition that
effects the quantization, $k_{j}L=2\pi n_{j}+\phi$, $n_{j}\in\mathbb{Z}$.
Qualitatively, the extra phase $\phi$ can be understood as follows. The
infinite Hubbard repulsion preserves the spin ordering but only up to cyclical
transmutations. Each detour of a holon around the ring causes such a
transmutation. To recover the original pattern the holon should make a number
of detours equal to the number of non-equivalent cyclical transmutations of
the pattern ${\cal Z}$ (see Fig. 1). For the spin pattern fixed up to the
cyclical permutations the many-body states of the clean system
$|\left\\{k_{j}\right\\},\phi\rangle$ are characterized by the momenta of the
holons $\left\\{k_{j}\right\\}$ and the total ”quasimomentum” $\phi$ of the
particles. The quantity $\phi$ should be a multiple of $2\pi/{\cal Z}$. For
the Neél pattern with an even number of particles we have ${\cal Z}=2$
allowing only $\phi=0$ and $\phi=\pi$.

All of the above applies to the disordered model $H_{0}+H_{D}$ as long as the
disorder is purely potential ($B=0,$ $\theta=\frac{\pi}{2}$ in Eq. (5)). The
potential disorder alters only one-particle states of the holons. The plane
waves are substituted by the eigenfunctions of a particle subject to the
random potential $\left\\{\varepsilon_{i}\right\\}$ with quasiperiodic
boundary conditions. These one-particle states are known to be localized for
arbitrarily small $W$. It is important to emphasize that as long as the random
magnetic field is absent the disorder does not lead to any interaction between
the holons and the occupation numbers of the one-particle states are
conserved. The many-body wave function is characterized by a macroscopically
large set $\left\\{\mu_{j}\right\\}_{\phi}$ instead of
$\left\\{k_{j}\right\\}$, where each $\left\\{\mu_{j}\right\\}_{\phi}$ labels
a localized one-particle wave function with a fixed $\phi$. This means that
from the many-body point of view the problem remains integrable even in the
presence of the potential disorder. The Bethe Ansatz based theory of the
Hubbard model with potential disorder will be published elsewhere not (a).

Figure 2:  The fractal dimensions $D_{1}$ (a), $D_{2}$ (b), and the Kullback-
Leibler divergence $KL$ (c) versus the total disorder strength $\rho_{D}$ for
the case of $\theta=0$ (random magnetic field only, no potential disorder) for
$L=12$ (black filled circles), $L=16$ (red open diamonds) and $L=20$ (blue
filled triangles). Insets (d)-(f) show the data near the critical point
$\rho_{D}\simeq 8.57$. In Region I the states are extended, whereas in Region
II they are localized. Error bars smaller than the symbol sizes are omitted.

Random magnetic field. In the presence of a spin-dependent disorder
($\theta<\pi/2$ in Eq. (5)) a cyclical transmutation of the spin pattern
caused by a holon detour changes the total energy (one can see, e.g., that the
potential energies of the states 0, 4 and 8 in Fig. 1 are all different). This
violates the conservation of the ’quasimomentum’ $\phi$, i.e. gives rise to
the off-diagonal matrix elements of the Hamiltonian,
$\langle\left\\{\mu^{\prime}_{j}\right\\},\phi^{\prime}|H|\left\\{\mu_{j}\right\\},\phi\rangle\neq
0$. We thus come to the conclusion that the random magnetic field plays a two-
fold role. First, like the potential disorder, it tends to transform the plane
waves into localized one-particle states. On the other hand, it hybridizes
different $|\mu_{j}\rangle$ \- states and can cause the many-body
delocalization in the $|\left\\{\mu_{j}\right\\},\phi\rangle$ -basis.

Below we present the results of numerical studies of the MBL transition in the
computational basis $\left\\{|s\rangle\right\\}$ rather than in the
$\left\\{|\left\\{\mu_{j}\right\\},\phi\rangle\right\\}$ basis, i.e. in the
clean limit ($H=H_{0}$, $\rho_{D}=0$) the many- body eigenstates
$|\left\\{k_{j}\right\\},\phi\rangle$ are apparently extended, whereas the
one-particle localization of the holon states $|\mu_{j}\rangle$ due to the
potential disorder implies MBL in the computational basis. We therefore have
to check numerically the following predictions: i) If the disorder is purely
magnetic, $\theta=0$ , the many-body states remain extended for a finite but
weak enough disorder because the problem is not a 1D one-particle one. With
increasing the randomness $\rho_{D}$ the system eventually becomes many-body
localized, i.e. it undergoes the MBL transition; ii) For the purely potential
disorder, $\theta=\pi/2$, the eigenstates are always localized in the
$\left\\{|s\rangle\right\\}$ \- basis. Sufficiently strong magnetic disorder
can destroy this localization if $\rho_{D}$ is small enough.

Fig. 2 presents the results of numerical diagonalization of the Hamiltonian
(1), (2) in the absence of the potential disorder ($W=0$, $\theta=0$,
$\rho_{D}=B$) for different values of the system size $L$. One can see that
for a weak disorder the fractal dimensions $D_{1,2}$ are close to the ergodic
value $1$. Moreover, they increase with $L$, i.e. it is plausible that the
quantum states remain not only extended but also ergodic. The deviations from
$D_{1,2}=1$ can be viewed as a finite size effect. If the disorder is strong,
$\rho_{D}\gtrsim 8.6$, the fractal dimensions $D_{1,2}$ decrease as $L$
increases and it is likely that $D_{1,2}\rightarrow 0$ for
$L\rightarrow\infty$, which corresponds to the LO phase. At the transition
between extended and LO regimes the curves $D_{1,2}(\rho_{D})$ for different
$L$ should cross each other at a critical disorder $\rho_{D}=\rho^{C}_{D}$ and
we find $\rho^{C}_{D}\approx 8.57$.

For intermediate magnetic disorder, $D_{1,2}$ increase with $L$ but are much
smaller than 1 and decrease as the disorder gets stronger. One may assume that
in the limit $L\rightarrow\infty$ the fractal dimensions remain between $1$
and $0$. This would signal the existence of the NEE phase. However, it is
impossible to exclude that $D_{1,2}\xrightarrow{L\rightarrow\infty}0$. The
divergence $KL$ is sensitive to the transition to the LO state but not to the
EE $\rightarrow$ NEE transition. We found that $KL$ is practically
$L$-independent for $\rho_{D}<\rho^{C}_{D}$ as it should be if the states are
extended. The $KL(L)$ \- dependence at $\rho_{D}>\rho^{C}_{D}$ is another
evidence of the LO phase at large disorder. Note that with the accuracy
limited by the finite size of the system the analysis of the data for both the
fractal dimensions $D_{1,2}$ and $KL$ yields the same value of the critical
disorder.

Figure 3:  The fractal dimensions $D_{1}$ (a), $D_{2}$ (b), and the Kullback-
Leibler divergence $KL$ (c) versus the angle $\theta$ [see Eq. (5)] at the
total disorder strength $\rho_{D}=2$ for $L=12$ (black filled circles), $L=16$
(red open squares), and $L=20$ (blue open triangles). Right panel (d)-(f)
shows the data near $\theta=\pi/2$. Error bars smaller than the marker size
are not shown. Figure 4:  A sketch of the phase diagram in the $B-W$ space,
based on the finite-size calculations of the fractal dimensions $D_{1}$. The
colorbar indicates the value of $D_{1}$ for $L=16$.

Two-component disorder. Fig. 3 presents the numerical calculation of
localization properties in the presence of both random magnetic field and
random potential. We show the dependence of $D_{1,2}$ and $KL$ on $\theta$ at
a fixed total disorder $\rho_{D}=\sqrt{W^{2}+B^{2}}=2$. First note that at
$\theta=\pi/2$, i.e. in the potential limit, $KL$ increases with the system
size $L$ and both $D_{1}$ and $D_{2}$ decrease as $L$ grows. This behavior is
an indication of the LO phase, which we expect. This localization is well
pronounced being compared to the extended phase, exhibited in purely magnetic
disorder. This can be seen from Fig. 2 and is also clear from the dependence
of $D_{1,2}$ and $KL$ on $\theta$ close to $\theta=0$ in Fig. 3. It is
difficult to find any deviation of $KL$ from the ergodic value $KL=2$, whereas
both $D_{1}$ and $D_{2}$ are close to $1$ and increase with $L$.

The most interesting and somewhat unexpected is the fact that this extended
behavior persists at $\rho_{D}=2$ for almost all values of $\theta$.
Deviations are visible only when $\pi/2-\theta$ becomes smaller than
$0.045\pi$ (which corresponds to $B<0.3$ ). The behavior of $D_{1,2}$ and KL
indicates the transition to the LO phase at $\theta^{C}\approx 0.497\pi$ ,
i.e. $B^{C}\simeq 0.01$. This tiny field is sufficient to destroy the LO phase
created by the potential disorder $W\approx 2$. This can be seen from Fig. 4
presented in terms of the potential disorder strength $W$ and magnetic field
strength $B$. In particular, the LO phase can be recovered by increasing the
amplitude of the random magnetic field up to $B\simeq 8$. In the interval
$B^{C}<B\lesssim 8$ the eigenstates of the disordered Hamiltonian
$H_{0}+H_{D}$ are extended. It is possible that close to the edges of this
interval the states loose ergodicity and become NEE. Unfortunately, the
available sizes $L$ of the system for our computation are too small to justify
this guess. However, the reentrance to the localized regime with increasing
$B$ is hard to dispute.

For $\rho_{D}\gg 1$ the extended phase is not expected, since both disorder
components damp effective interparticle interactions not (a). From Fig. 2 one
expects this limit to be achived at $\rho_{D}\simeq 8.57$. In the other limit,
$\rho_{D}\ll 1$, the states are localized at $\theta=\frac{\pi}{2}$.
Sufficiently weak random magnetic field induces effective interholon
interaction, which completely delocalizes the states. This implies
$\theta_{c}\rightarrow 0.5\pi$ with $\rho_{D}\rightarrow 0$. Our final results
for the phase diagram are presented in Fig. 4.

Conclusions. Compared to the potential static disorder, the random magnetic
field has a dramatically different effect on the quantum many-body states of
the 1D Hubbard model with infinitely strong on-site repulsion. The potential
disorder does not violate the integrability of the problem - the holons remain
free spinless fermions albeit subject to the random potential. Accordingly,
the one-particle wave functions are localized at arbitrarily weak disorder. To
complete the mapping onto the standard free fermion problem one has to take
into account the quasiperiodic boundary conditions (QBC) for these functions.
The QBC reflect the cyclical permutation in the spin sequence caused by the
motion of the holons. Each stationary state of the system is characterized by
a particular occupation of single-holon states as well as by the phase $\phi$
of the QBC, which can take several values determined by the spin pattern.

The infinite on-site repulsion preserves the spin pattern up to cyclical
permutations irrespective of the type of disorder. In the potential case
distinct cyclical transmutations do not change the potential energy giving
rise the “quasimomentum”-phase $\phi$ of the QBC. Violation of this symmetry
by the random magnetic field hybridizes the states with different $\phi$ and
different holon quantum numbers making the system non-integrable and possibly
extended in the MBL sense.

A very strong disorder of any type practically excludes the cyclical
transmutations and many-body states are localized in the computational basis.
For a purely potential disorder it means that the eigenenergies are degenerate
in $\phi$. If the potential disorder is weak but finite, a gradual reduction
of the random magnetic field from a large value eventually causes the many-
body delocalization. However, a further decrease should recover the localized
phase.

These predictions are supported by the Bethe Ansatz based analytics not (a)
and clearly demonstrate the reentrant MBL behavior caused by the random
magnetic field. Unfortunately, the size of the system available for the exact
diagonalization turns out to be too small for the analysis of the ergodicity
of the extended states. The Hamiltonian (1),(2) can be implemented in systems
of ultracold atoms or existing multi-qubit arrays. Being qualitatively
transparent with well-developed analytical approaches, this model has
advantages over e.g. spin chains as a laboratory for investigating MBL and
searching for Quantum Supremacy (Arute  _et al._ , 2019).

###### Acknowledgements.

We thank V.E. Kravtsov, V. Gritsev, O. Gamayun, and V. Smelyanskiy for useful
discussions. This research was supported in part through computational
resources of HPC facilities at NRU HSE and by the Russian Science Foundation
Grant No. 20-42-05002. We also acknowledge support of this work by Rosatom.

## References

  * Basko  _et al._ (2006) D. Basko, I. Aleiner, and B. Altshuler, Annals of Physics 321, 1126 (2006).
  * Oganesyan and Huse (2007) V. Oganesyan and D. A. Huse, Phys. Rev. B 75, 155111 (2007).
  * Berkelbach and Reichman (2010) T. C. Berkelbach and D. R. Reichman, Phys. Rev. B 81, 224429 (2010).
  * Barišić  _et al._ (2016) O. S. Barišić, J. Kokalj, I. Balog, and P. Prelovšek, Phys. Rev. B 94, 045126 (2016).
  * Bar Lev  _et al._ (2015) Y. Bar Lev, G. Cohen, and D. R. Reichman, Phys. Rev. Lett. 114, 100601 (2015).
  * Prelovšek and Herbrych (2017) P. Prelovšek and J. Herbrych, Phys. Rev. B 96, 035130 (2017).
  * Pal and Huse (2010) A. Pal and D. A. Huse, Phys. Rev. B 82, 174411 (2010).
  * Bar Lev and Reichman (2014) Y. Bar Lev and D. R. Reichman, Phys. Rev. B 89, 220201 (2014).
  * Serbyn  _et al._ (2015) M. Serbyn, Z. Papić, and D. A. Abanin, Phys. Rev. X 5, 041047 (2015).
  * Luitz  _et al._ (2016) D. J. Luitz, N. Laflorencie, and F. Alet, Phys. Rev. B 93, 060201 (2016).
  * Luca and Scardicchio (2013) A. D. Luca and A. Scardicchio, EPL (Europhysics Letters) 101, 37003 (2013).
  * De Luca  _et al._ (2014) A. De Luca, B. L. Altshuler, V. E. Kravtsov, and A. Scardicchio, Phys. Rev. Lett. 113, 046806 (2014).
  * Bauer and Nayak (2013) B. Bauer and C. Nayak, Journal of Statistical Mechanics: Theory and Experiment 2013, P09005 (2013).
  * Serbyn  _et al._ (2013a) M. Serbyn, Z. Papić, and D. A. Abanin, Phys. Rev. Lett. 110, 260601 (2013a).
  * Buijsman  _et al._ (2019) W. Buijsman, V. Gritsev, and V. Cheianov, Phys. Rev. B 100, 205110 (2019).
  * Serbyn  _et al._ (2016) M. Serbyn, A. A. Michailidis, D. A. Abanin, and Z. Papić, Phys. Rev. Lett. 117, 160601 (2016).
  * Serbyn  _et al._ (2013b) M. Serbyn, Z. Papić, and D. A. Abanin, Phys. Rev. Lett. 111, 127201 (2013b).
  * Rademaker and Ortuño (2016) L. Rademaker and M. Ortuño, Phys. Rev. Lett. 116, 010404 (2016).
  * Chandran  _et al._ (2015) A. Chandran, I. H. Kim, G. Vidal, and D. A. Abanin, Phys. Rev. B 91, 085425 (2015).
  * Luitz and Lev (2017) D. J. Luitz and Y. B. Lev, Annalen der Physik 529, 1600350 (2017).
  * Abanin and Papić (2017) D. A. Abanin and Z. Papić, Annalen der Physik 529, 1700169 (2017).
  * Alet and Laflorencie (2018) F. Alet and N. Laflorencie, Comptes Rendus Physique 19, 498 (2018).
  * Luitz  _et al._ (2015) D. J. Luitz, N. Laflorencie, and F. Alet, Phys. Rev. B 91, 081103 (2015).
  * Kudo and Deguchi (2018) K. Kudo and T. Deguchi, Phys. Rev. B 97, 220201 (2018).
  * Luitz (2016) D. J. Luitz, Phys. Rev. B 93, 134201 (2016).
  * Burin (2015) A. L. Burin, Phys. Rev. B 92, 104428 (2015).
  * Mondaini and Rigol (2015) R. Mondaini and M. Rigol, Phys. Rev. A 92, 041601 (2015).
  * Zakrzewski and Delande (2018) J. Zakrzewski and D. Delande, Phys. Rev. B 98, 014203 (2018).
  * Bonča and Mierzejewski (2017) J. Bonča and M. Mierzejewski, Phys. Rev. B 95, 214201 (2017).
  * Lemut  _et al._ (2017) G. Lemut, M. Mierzejewski, and J. Bonča, Phys. Rev. Lett. 119, 246601 (2017).
  * Kozarzewski  _et al._ (2018) M. Kozarzewski, P. Prelovšek, and M. Mierzejewski, Phys. Rev. Lett. 120, 246602 (2018).
  * Kozarzewski  _et al._ (2019) M. Kozarzewski, M. Mierzejewski, and P. Prelovšek, Phys. Rev. B 99, 241113 (2019).
  * Krause  _et al._ (2021) U. Krause, T. Pellegrin, P. W. Brouwer, D. A. Abanin, and M. Filippone, Phys. Rev. Lett. 126, 030603 (2021).
  * Protopopov and Abanin (2019) I. V. Protopopov and D. A. Abanin, Phys. Rev. B 99, 115111 (2019).
  * Prelovšek  _et al._ (2016) P. Prelovšek, O. S. Barišić, and M. Žnidarič, Phys. Rev. B 94, 241104 (2016).
  * Protopopov  _et al._ (2017) I. V. Protopopov, W. W. Ho, and D. A. Abanin, Phys. Rev. B 96, 041122 (2017).
  * Środa  _et al._ (2019) M. Środa, P. Prelovšek, and M. Mierzejewski, Phys. Rev. B 99, 121110 (2019).
  * Leipner-Johns and Wortis (2019) B. Leipner-Johns and R. Wortis, Phys. Rev. B 100, 125132 (2019).
  * Abarenkova  _et al._ (1997) N. I. Abarenkova, A. G. Izergin, and A. G. Pronko, Zapiski Nauchnykh Seminarov POMI 249, 7 (1997).
  * not (a) D. V. Kurlov, M. S. Bahovadinov, B. L. Altshuler, G. V. Shlyapnikov (unpublished).
  * Essler  _et al._ (2005) F. H. Essler, H. Frahm, F. Göhmann, A. Klümper, and V. E. Korepin,  _The one-dimensional Hubbard model_ (Cambridge University Press, 2005).
  * Kullback and Leibler (1951) S. Kullback and R. A. Leibler, The Annals of Mathematical Statistics 22, 79 (1951).
  * Khaymovich  _et al._ (2020) I. M. Khaymovich, V. E. Kravtsov, B. L. Altshuler, and L. B. Ioffe, Phys. Rev. Research 2, 043346 (2020).
  * Pino  _et al._ (2019) M. Pino, J. Tabanera, and P. Serna, Journal of Physics A: Mathematical and Theoretical 52, 475101 (2019).
  * not (a) See Supplemental Material for details.
  * Arute  _et al._ (2019) F. Arute et al., Nature 574, 505 (2019).

Supplemental material

##  .1 Exact diagonalization (ED) results of $D_{1}$ for various disorder
amplitudes $\rho_{D}$

To obtain a qualitative sketch of the phase diagram, we repeated our ED
calculations of $D_{1}$ for the system sizes $L=\left\\{12,16\right\\}$ and
the values of $\rho_{D}=\left\\{2,3,4,5,6,7,8,9,10,11\right\\}$. Based on the
following data, we sketched the phase diagram (in the thermodynamic limit),
which is represented as Fig. 4 in the main text.

Figure S1:  The fractal dimensions $D_{1}$ versus the angle $\theta$ [see Eq.
(5) of the main text] for various values of $\rho_{D}$. The considered system
sizes are $L=12$ (black filled circles) and $L=16$ (red crosses). Error bars
smaller than the symbol size are omitted.