# One-to-one correspondence between thermal structure factors and coupling
constants of general bilinear Hamiltonians

Bruno Murta  QuantaLab, International Iberian Nanotechnology Laboratory (INL),
4715-330 Braga, Portugal  Departamento de Física, Universidade do Minho,
4710-057 Braga, Portugal  J. Fernández-Rossier111On leave from Departamento de
Física Aplicada, Universidad de Alicante, Spain  joaquin.fernandez-
<EMAIL_ADDRESS>QuantaLab, International Iberian Nanotechnology Laboratory
(INL), 4715-330 Braga, Portugal

(August 28, 2024)

###### Abstract

A theorem that establishes a one-to-one relation between zero-temperature
static spin-spin correlators and coupling constants for a general class of
quantum spin Hamiltonians bilinear in the spin operators has been recently
established by J. Quintanilla, using an argument in the spirit of the
Hohenberg-Kohn theorem in density functional theory. Quintanilla’s theorem
gives a firm theoretical foundation to quantum spin Hamiltonian learning using
spin structure factors as input data. Here we extend the validity of the
theorem in two directions. First, following the same approach as Mermin, the
proof is extended to the case of finite-temperature spin structure factors,
thus ensuring that the application of this theorem to experimental data is
sound. Second, we note that this theorem applies to all types of Hamiltonians
expressed as sums of bilinear operators, so that it can also relate the
density-density correlators to the Coulomb matrix elements for interacting
electrons in the lowest Landau level.

Understanding the wonders and complexities of the microscopic world requires
tackling the notoriously hard quantum many-body problem. Given the ubiquity of
approximate methods in the state-of-the-art research on quantum many-body
phenomena, the existence of general theoremsHohenberg and Kohn (1964); Mermin
(1965); Mermin and Wagner (1966); Lieb (1989) is essential to set such
approximations on firm theoretical ground.

The present Letter follows a recent work by J. Quintanilla Quintanilla (2022),
which establishes a theorem valid for a general class of bilinear quantum spin
Hamiltonians,

$\hat{H}=\sum_{i,j,\alpha,\beta}J_{i,j}^{\alpha,\beta}\hat{S}_{i}^{\alpha}\hat{S}_{j}^{\beta},$
(1)

where $\hat{S}_{i}^{\alpha}$ is the $\alpha=x,y,z$ component of a spin
operator acting on site $i$ in an arbitrary lattice, and
$J_{i,j}^{\alpha,\beta}$ are the spin coupling constants. The general
Hamiltonian stated in eq. (1) encompasses most physically relevant types of
interactions, notably HeisenbergHeisenberg (1928), dipolar, IsingIsing (1925),
Dzyaloshinskii–MoriyaDzyaloshinsky (1958); Moriya (1960) and KitaevKitaev
(2006) interactions. As a result, a wide class of canonical quantum spin
models (e.g., IsingIsing (1925), HeisenbergHeisenberg (1928), XXZOrbach
(1958), Majumdar–GhoshMajumdar and Ghosh (1969a, b), Shastry-SutherlandShastry
and Sutherland (1981), HaldaneHaldane (1988), KitaevKitaev (2006)) are
particular cases of eq. (1). It should be noted, however, that this class of
Hamiltonians does not cover important quantum spin models such as the toric
codeKitaev (2003) or the bilinear-biquadratic Heisenberg model (including the
AKLT modelAffleck  _et al._ (1987)), which is known to describe some physical
systemsMishra  _et al._ (2021).

The theorem proven by J. QuintanillaQuintanilla (2022) for this class of
bilinear spin Hamiltonians asserts that there exists a one-to-one
correspondence between the exchange constants $J_{i,j}^{\alpha,\beta}$ and the
zero-temperature correlators

$\rho_{i,j}^{\alpha,\beta}(T=0)=\langle\Phi_{0}|\hat{S}_{i}^{\alpha}\hat{S}_{j}^{\beta}|\Phi_{0}\rangle$
(2)

for a physical system represented by the wave function $\ket{\Phi_{0}}$, which
corresponds to the non-degenerate222The proof of Quintanilla’s theorem assumes
the ground states of the two different bilinear spin Hamiltonians are both
non-degenerate, but a follow-up discussion relaxes this assumption. ground
state of a Hamiltonian of the form given in eq. (1). The proof of
Quintanilla’s theorem runs parallel to that of the Hohenberg-Kohn
theoremHohenberg and Kohn (1964) for density functional theory. Interestingly,
Mermin generalizedMermin (1965) the Hohenberg-Kohn theorem to finite
temperature, which motivates us to look for a finite-temperature
generalization of Quintanilla’s theorem as well.

Before proceeding to the extension of Quintanilla’s theorem to the case of
finite temperature, we first introduce some relevant concepts and notation.
The thermal spin correlators at temperature $T$ are defined as

$\rho_{i,j}^{\alpha,\beta}(\hat{W})\coloneqq\textrm{Tr}\left(\hat{W}\hat{S}_{i}^{\alpha}\hat{S}_{j}^{\beta}\right),$
(3)

where $\hat{W}=\sum_{n}\frac{e^{-\beta
E_{n}}}{Z}|\Phi_{n}\rangle\langle\Phi_{n}|$ is the density operator,
$Z=\sum_{n}e^{-\beta E_{n}}$ is the partition function, $\\{E_{n}\\}$ and
$\\{|\Phi_{n}\rangle\\}$ are the eigenenergies and eigenstates of a
Hamiltonian $\hat{H}$ of the form given in eq. (1), and $\beta^{-1}=k_{B}T$.
Expanding eq. (3) in the Hamiltonian eigenbasis gives

$\rho_{i,j}^{\alpha,\beta}(\hat{W})=\frac{1}{Z}\sum_{n}e^{-\beta
E_{n}}\langle\Phi_{n}|\hat{S}_{i}^{\alpha}\hat{S}_{j}^{\beta}|\Phi_{n}\rangle.$
(4)

Setting $T=0$, or $\beta\to\infty$, in eq. (4) results in eq. (2), as
expected. We note that spin correlators can be measured experimentally using
neutron diffractionShull and Smart (1949); Marshall and Lowde (1968).

In the following we show that the mapping between the coupling constants
$J_{i,j}^{\alpha,\beta}$ and the finite-temperature correlators (cf. eq. (4))
is bijective. We restrict ourselves to finite temperatures, since for
$T\to\infty$ all but local $\langle\hat{S}_{i}^{z}\hat{S}_{i}^{z}\rangle$
correlators vanish (cf. Appendix A).

The proof follows in a similar vein to the zero-temperature one by
QuintanillaQuintanilla (2022), but the Rayleigh-Ritz variational principle is
replaced by the Gibbs-Bogoliubov inequalityGibbs (2010); Mermin (1965) for the
Helmholtz free energy. Let a system described by a Hamiltonian $\hat{H}$ be in
contact with a thermal bath at temperature $T$. The Helmholtz free energy of
such system is

$F(\hat{W})=-k_{B}T\ln Z=\langle\hat{H}\rangle_{\hat{W}}-TS[\hat{W}],$ (5)

where
$\langle\hat{O}\rangle_{\hat{W}}=\textrm{Tr}\left(\hat{W}\hat{O}\right)=\frac{1}{Z}\sum_{n}e^{-\beta
E_{n}}\langle\Phi_{n}|\hat{O}|\Phi_{n}\rangle$ is the expectation value of
some operator $\hat{O}$ at finite temperature and
$S[\hat{W}]=-k_{B}\textrm{Tr}\left(\hat{W}\ln\hat{W}\right)$ is the von
Neumann entropy. The Gibbs-Bogoliubov inequality sets an upper bound on the
Helmholtz free energy,

$F(\hat{W})\leq\langle\hat{H}\rangle_{\hat{W}^{{}^{\prime}}}-TS[\hat{W}^{{}^{\prime}}],$
(6)

for any positive semidefinite operator $\hat{W}^{{}^{\prime}}$ of appropriate
dimensionality. The equality in eq. (6) only occurs either when
$\hat{W}=\hat{W}^{{}^{\prime}}$ or $T\to\infty$. A proof of the Gibbs-
Bogoliubov inequality can be found in MerminMermin (1965).

The proof of Quintanilla’s theorem at finite temperature proceeds by redutio
ad absurdum. We consider two different Hamiltonians of the form given in eq.
(1), $\hat{H}$ and $\hat{H}^{{}^{\prime}}$. Their corresponding coupling
constants, $J_{ij}^{\alpha,\beta}$ and $J_{ij}^{{}^{\prime}\alpha,\beta}$,
cannot therefore be all equal in pairs. Since the coupling constants determine
the energies and eigenstates, they determine $\hat{W}$ and
$\hat{W}^{{}^{\prime}}$ as well, the equilibrium density operators for
$\hat{H}$ and $\hat{H}^{{}^{\prime}}$, respectively. We then assume that both
$\hat{W}$ and $\hat{W}^{{}^{\prime}}$ are associated with the same finite-
temperature spin-spin correlators,

$\rho_{i,j}^{\alpha,\beta}[\hat{W}]=\rho_{i,j}^{\alpha,\beta}[\hat{W}^{{}^{\prime}}],$
(7)

for all $i,j,\alpha,\beta$. We can use the Gibbs-Bogoliubov inequality to
write the following expression for the Helmoltz free energy of the unprimed
system:

$\displaystyle F(\hat{W})$
$\displaystyle=\sum_{i,j,\alpha,\beta}J_{ij}^{\alpha,\beta}\rho_{i,j}^{\alpha,\beta}[\hat{W}]-TS[\hat{W}]\leq\sum_{i,j,\alpha,\beta}J_{ij}^{\alpha,\beta}\rho_{i,j}^{\alpha,\beta}[\hat{W}^{{}^{\prime}}]-TS[\hat{W}^{{}^{\prime}}]=$
$\displaystyle=\sum_{i,j,\alpha,\beta}\left(J_{ij}^{\alpha,\beta}-J_{ij}^{{}^{\prime}\alpha,\beta}\right)\rho_{i,j}^{\alpha,\beta}[\hat{W}^{{}^{\prime}}]+F(\hat{W}^{{}^{\prime}}).$
(8)

We can now exchange the roles of $\hat{W}$ and $\hat{W}^{{}^{\prime}}$ to
obtain an identical expression for the primed system:

$F[\hat{W}^{{}^{\prime}}]\leq\sum_{i,j,\alpha,\beta}\left(J_{ij}^{{}^{\prime}\alpha,\beta}-J_{ij}^{\alpha,\beta}\right)\rho_{i,j}^{\alpha,\beta}[\hat{W}]+F(\hat{W}).$
(9)

Summing eqs. (8) and (9) yields:

$\displaystyle
F[\hat{W}]+F[\hat{W}^{{}^{\prime}}]\leq\sum_{i,j,\alpha,\beta}\left(J_{ij}^{{}^{\prime}\alpha,\beta}-J_{ij}^{\alpha,\beta}\right)\left(\rho_{i,j}^{\alpha,\beta}[\hat{W}]-\rho_{i,j}^{\alpha,\beta}[\hat{W}^{{}^{\prime}}]\right)+F[\hat{W}]+F[\hat{W}^{{}^{\prime}}].$
(10)

Using eq. (7) turns eq. (10) into: $F[\hat{W}]+F[\hat{W}^{{}^{\prime}}]\leq
F[\hat{W}]+F[\hat{W}^{{}^{\prime}}]$. The equality holds only in the two
trivial limits of infinite temperature or $\hat{H}=\hat{H}^{{}^{\prime}}$. For
finite temperature and $\hat{H}\neq\hat{H}^{\prime}$, we can replace the
symbol $\leq$ by a strict inequality, thus arriving at a contradiction:
$F[\hat{W}]+F[\hat{W}^{{}^{\prime}}]<F[\hat{W}]+F[\hat{W}^{{}^{\prime}}]$. It
follows, then, that the initial assumption stated in eq. (7) must be false, in
which case we can conclude that the finite-temperature correlators are single-
valued functions $\rho_{ij}^{\alpha,\beta}[\hat{W}]$ of the equilibrium
density operator $\hat{W}$, which is, in turn, uniquely determined by the
coupling constants $\\{J_{ij}^{\alpha,\beta}\\}$ of the model, so that
$\rho_{ij}^{\alpha,\beta}(J_{ij}^{\alpha,\beta})$ is injective.

Proving the injectivity of $\rho_{ij}^{\alpha,\beta}(J_{ij}^{\alpha,\beta})$
suffices to show it is bijective (i.e., a one-to-one mapping) since
$\rho_{ij}^{\alpha,\beta}(J_{ij}^{\alpha,\beta})$ is surjective by
construction, assuming, of course, that the physical system under study can be
described by a Hamiltonian of the form given in eq. (1). Indeed, given a model
defined by a set of coupling parameters $\\{J_{ij}^{\alpha,\beta}\\}$, we can
always determine, at least in principle, the respective equilibrium density
operator $\hat{W}$, which can then be used to compute the finite-temperature
spin-spin correlators $\\{\rho_{ij}^{\alpha,\beta}\\}$ per eq. (4). This is
entirely analogous to the trivial surjectivity of the mapping of the ground
state wave functions onto the set of number densities in the proof of the
Hohenberg-Kohn theorem Dreizler and Gross (1990): every number density must be
associated with a given wave function. Interestingly, the one-to-one relation
between the ground state wave function and the external potential in spin-
density-functional theory is not guaranteed to hold Capelle and Vignale
(2001).

The final step amounts to recognizing that bijectivity is a sufficient
condition for a function to be invertible. Hence, the coupling constants
$\\{J_{ij}^{\alpha,\beta}\\}$ are themselves single-valued functions of the
finite-temperature correlators $\\{\rho_{ij}^{\alpha,\beta}\\}$, which
concludes the generalization of Quintanilla’s theorem to the case of finite
temperature. Importantly, this result involves not only the ground state
manifold (regardless of its degeneracy) but excited states as well.

As in the case of Hohenberg-Kohn theorem, the present theorem does not give a
systematic method to obtain the functional that relates the coupling constants
$J_{ij}^{\alpha,\beta}$ to the correlators $\rho_{ij}^{\alpha,\beta}$. Hence,
Quintanilla’s theorem does not produce a practical short-term advantage to
tackle quantum spin Hamiltonians. Nevertheless, this theorem does provide a
solid theoretical basis for a novel approach to the important problem of
determining the parent Hamiltonian of experimental systems.

Artificial intelligence methods have been used to infer spin couplings out of
experimentally determined spin correlators in spin-ice compoundsSamarakoon
_et al._ (2021). The process includes the training of an artificial neural
network (ANN) based on classical spin model simulations. In a similar vein,
ANNs have been trained to infer spin couplings out of specific heat
measurements Yu  _et al._ (2021). We note that our finite-temperature theorem
provides not only a firm foundation but also a practical advantage to infer
spin couplings out of spin correlators, as it opens the possibility of
training a ANN with simulations of quantum spin models. As noted by Yu et
al.Yu  _et al._ (2021), this process is actually simplified at large
temperatures. The computational resources needed to carry out exact
diagonalizations of spin Hamiltonians scale exponentially with the system
size. At high temperatures, however, the spatial range of spin correlators is
expected to be shorter, which provides a natural cut-off for the size of the
simulation cellsYu  _et al._ (2021). We also note that the training could be
supported by digital quantum simulations of quantum spin modelsKandala  _et
al._ (2017) on noisy intermediate-scale quantum computersPreskill (2018) using
hybrid variational algorithmsBharti  _et al._ (2022). This approach may be
used to accurately determine the spin Hamiltonian of Kitaev materials, such as
RuCl3Trebst (2017); Winter  _et al._ (2017).

We finish by noting that, in all of the above, we never make use of the fact
that $\hat{S}_{i}^{\alpha}$ are spin operators, and therefore the theorem
applies to any Hamiltonian that can be expressed as a bilinear sum of
operators,

$\hat{H}=\sum_{a,b}J_{a,b}\hat{O}_{a}\hat{O}_{b}$ (11)

where $J_{a,b}$ describe couplings between operators $\hat{O}_{a}$ and
$\hat{O}_{b}$, with $a$ and $b$ general labels. For example, Hamiltonian (11)
includes the relevant case of interacting electrons that occupy a flat band or
a single Landau level, with $\hat{O}$ being the electronic density operator
and $J_{a,b}$ being the Coulomb interaction projected onto the lowest Landau
levelLaughlin (1983).

The theorem stated and proven above can be rephrased as follows. The couplings
$J_{a,b}$ are a single-valued functional of the thermal correlators:

$\rho_{a,b}=\frac{1}{Z}\sum_{n}e^{-\beta
E_{n}}\langle\Phi_{n}|\hat{O}_{a}\hat{O}_{b}|\Phi_{n}\rangle$ (12)

Thus, the theorem establishes a one-to-one correspondence between the finite-
temperature density-density correlations and the representation of the Coulomb
matrix elements in the lowest Landau level.

The Hohenberg-Kohn theoremHohenberg and Kohn (1964), and Mermin’s finite-
temperature extensionMermin (1965), became extremely useful when approximate
versions of the density functional, such as the Kohm-Sham local density
approximationKohn and Sham (1965), were developed. We hope that this paper
will inspire the quest for such approximate functionals within the context of
quantum spin Hamiltonian learning based on spin structure factors. We also
note the existence of a theorem relating the ground state energy to the
magnetization density in Heisenberg modelsLíbero and Capelle (2003); its
connection with our work remains to be explored.

In conclusion, we have demonstrated a theorem that establishes a one-to-one
relation between interaction couplings and finite-temperature correlators in a
general class of bilinear Hamiltonians. Our work generalizes a recent result
of QuintanillaQuintanilla (2022) in two ways. First, our theorem establishes
the validity of Quintanilla’s result for arbitrary temperatures. Second, we
note that the theorem is applicable beyond the realm of spin systems. Our
theorem puts the recent work that uses artificial intelligence to determine
spin couplingsSamarakoon  _et al._ (2021) on firm theoretical footing and may
provide a route to settle disputes about the nature of spin couplings in
quantum materials, such as Kitaev materials.

Acknowledgements. J.F.R. acknowledges financial support from FCT (Grant No.
PTDC/FIS-MAC/2045/2021), FEDER / Junta de Andalucía — Consejería de
Transformación Económica, Industria, Conocimiento y Universidades, (Grant No.
P18-FR-4834), and Generalitat Valenciana funding Prometeo20XXX. MICIIN-Spain
(Grant No. PID2019-109539GB-C41). B.M. acknowledges support from the FCT PhD
scholarship No. SFRH/BD/08444/2020.

##  Appendix A Infinite-Temperature Spin-Spin Correlators

The spin-spin correlators $\rho_{ij}^{\alpha\beta}(\hat{W})$ at a nonzero
temperature $T$ are given by eq. (4). Setting $T\to\infty$, or $\beta\to 0$,
gives $e^{-\beta E_{n}}=1$ for all eigenenergies $\\{E_{n}\\}$, in which case

$\rho_{ij}^{\alpha\beta}(T\to\infty)=\frac{1}{Z}\sum_{n}\langle\Phi_{n}|\hat{S}_{i}^{\alpha}\hat{S}_{j}^{\beta}|\Phi_{n}\rangle\equiv\frac{1}{Z}\textrm{Tr}\left(\hat{S}_{i}^{\alpha}\hat{S}_{j}^{\beta}\right).$

Being a scalar, $\rho_{ij}^{\alpha\beta}(T\to\infty)$ is invariant under a
change of basis. Since the density operator $\hat{W}$ now only contributes a
constant prefactor $\frac{1}{Z}$, we can replace the Hamiltonian eigenbasis
$\\{|\Phi_{n}\rangle\\}$ with the product basis
$\\{\bigotimes_{i}|S_{i}\rangle\\}$, where we define the quantization axis
such that $\hat{S}_{i}^{z}|S_{i}\rangle=S_{i}|S_{i}\rangle$ at every site $i$,
with $S_{i}\in\\{-S,-S+1,...,-1,0,1,...,S-1,S\\}$ for a local spin-$S$.
Computing the trace in this product basis gives

$\rho_{ij}^{\alpha\beta}(T\to\infty)=\frac{\delta_{\alpha z}\delta_{\beta
z}}{(2S+1)^{2}}\sum_{S_{i},S_{j}=-S}^{S}\langle
S_{i}|\hat{S}_{i}^{\alpha}|S_{i}\rangle\langle
S_{j}|\hat{S}_{j}^{\beta}|S_{j}\rangle,$

where the Kronecker deltas follow from the fact that the only spin component
with nonzero entries along the diagonal is $\hat{S}^{z}$, so for any choice
other than $(\alpha,\beta)=(z,z)$ the correlator vanishes. The prefactor
results from the fact that the trivial sums over the remaining $N-2$ spins-$S$
give a factor $(2S+1)^{N-2}$, which cancels with the partition function
$Z=(2S+1)^{N}$.

Considering, for the moment, the nonlocal case $i\neq j$, we realize that, for
every configuration where $\langle S_{i}|\hat{S}_{i}^{z}|S_{i}\rangle\langle
S_{j}|\hat{S}_{j}^{z}|S_{j}\rangle$ takes a value $c$, there is another one
taking the symmetric value $-c$. In other words, any term $\langle
S|\hat{S}_{i}^{z}|S\rangle\langle
S^{\prime}|\hat{S}_{j}^{z}|S^{\prime}\rangle$ is cancelled out by another term
$\langle-S|\hat{S}_{i}^{z}|-S\rangle\langle
S^{\prime}|\hat{S}_{j}^{z}|S^{\prime}\rangle$. Hence, all nonlocal spin-spin
correlators vanish at infinite temperature:

$\rho_{ij}^{\alpha\beta}(T\to\infty)=\frac{\delta_{ij}\delta_{\alpha
z}\delta_{\beta z}}{2S+1}\sum_{S_{i}=-S}^{S}\langle
S_{i}|(\hat{S}_{i}^{z})^{2}|S_{i}\rangle.$

The remaining sum can be computed explicitly:
$\sum_{S_{i}=-S}^{S}S_{i}^{2}=\frac{S(S+1)(2S+1)}{3}$. Replacing in the
expression above yields

$\rho_{ij}^{\alpha\beta}(T\to\infty)=\delta_{ij}\delta_{\alpha z}\delta_{\beta
z}\frac{S(S+1)}{3}.$

Of course, this result is valid for any Hamiltonian, since neither the
eigenspectrum nor the eigenstates appear in any step of this calculation at
infinite temperature.

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