Graph-theory treatment of one-dimensional strongly repulsive fermions
Jean Decamp, Jiangbin Gong, Huanqian Loh, Christian Miniatura
TL;DR
This paper uses spectral graph theory to analyze symmetry properties and predict energy gaps in one-dimensional strongly interacting fermionic systems, aiding experimental and quantum control efforts.
Contribution
It introduces a novel graph-theoretic approach to characterize symmetries and compute Tan contacts in strongly correlated fermion mixtures, surpassing brute force methods.
Findings
Complete symmetry characterization of 1D fermionic mixtures
Efficient prediction of energy gaps in complex spin systems
Guidance for experimental design and adiabatic control
Abstract
One-dimensional atomic mixtures of fermions can effectively realize spin chains and thus constitute a clean and controllable platform to study quantum magnetism. Such strongly correlated quantum systems are also of sustained interest to quantum simulation and quantum computation due to their computational complexity. In this article, we exploit spectral graph theory to completely characterize the symmetry properties of one-dimensional fermionic mixtures in the strong interaction limit. We also develop a powerful method to obtain the so-called Tan contacts associated with certain symmetry classes. In particular, compared to brute force diagonalization that is already virtually impossible for a moderate number of fermions, our analysis enables us to make unprecedented efficient predictions about the energy gap of complex spin mixtures. Our theoretical results are not only of direct…
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Graph-theory treatment of one-dimensional strongly repulsive fermions
Jean Decamp
MajuLab, CNRS-UCA-SU-NUS-NTU International Joint Research Unit, Singapore
Centre for Quantum Technologies, National University of Singapore, 117543 Singapore, Singapore
Department of Physics, National University of Singapore, 2 Science Drive 3, Singapore 117542, Singapore
Jiangbin Gong
Department of Physics, National University of Singapore, 2 Science Drive 3, Singapore 117542, Singapore
Huanqian Loh
Centre for Quantum Technologies, National University of Singapore, 117543 Singapore, Singapore
Department of Physics, National University of Singapore, 2 Science Drive 3, Singapore 117542, Singapore
Christian Miniatura
MajuLab, CNRS-UCA-SU-NUS-NTU International Joint Research Unit, Singapore
Centre for Quantum Technologies, National University of Singapore, 117543 Singapore, Singapore
Department of Physics, National University of Singapore, 2 Science Drive 3, Singapore 117542, Singapore
School of Physical and Mathematical Sciences, Nanyang Technological University, 637371 Singapore, Singapore
Yale-NUS College, 16 College Avenue West, Singapore 138527, Singapore
Université Côte d’Azur, CNRS, INPHYNI, Nice, France
Abstract
One-dimensional atomic mixtures of fermions can effectively realize spin chains and thus constitute a clean and controllable platform to study quantum magnetism. Such strongly correlated quantum systems are also of sustained interest to quantum simulation and quantum computation due to their computational complexity. In this article, we exploit spectral graph theory to completely characterize the symmetry properties of one-dimensional fermionic mixtures in the strong interaction limit. We also develop a powerful method to obtain the so-called Tan contacts associated with certain symmetry classes. In particular, compared to brute force diagonalization that is already virtually impossible for a moderate number of fermions, our analysis enables us to make unprecedented efficient predictions about the energy gap of complex spin mixtures. Our theoretical results are not only of direct experimental interest, but also provide important guidance for the design of adiabatic control protocols in strongly correlated fermion mixtures.
From quantum magnetism to the highly debated high-temperature superconductivity, many spectacular phenomena observed in condensed matter physics emerge from strong interactions among particles with a spin degree of freedom assa . When constrained to one dimension, the effect of correlations is even higher, leading to counter-intuitive behaviour such as spin-charge separation and the so-called fermionization giamarchi . Although the study of such strongly correlated systems is a notoriously complex task, recent experimental realizations of one-dimensional (1D) systems involving ultracold atomic fermions with spin degrees of freedom offer exciting opportunities, both in terms of our fundamental understanding of 1D quantum magnetism and in the prospect of quantum technological applications Gorshkov2010 ; Ye2014 ; Foelling2014 ; Pagano2014 ; Loft2016 ; Marchukov2016 . Indeed, in the limit of strong repulsion, it has been shown that these systems are equivalent to spin chains whose interaction parameters are experimentally tunable with the external potential Volosniev2014 ; Deuretzbacher2014 ; Murmann2015 . In particular, this non-trivial observation provides a mapping to a matrix diagonalization problem. However, the algebraic structure of this matrix has never been clearly identified. Therefore, although many recent theoretical articles have studied this model, the results were typically limited to 6 particles, beyond which the complexity of the system seemed to be an impassable barrier Cui2013 ; Massignan2015 ; Grining2015 ; Beverland2016 ; Decamp2016a ; Decamp2016b ; Pecak2017 ; Sowiski2019 .
In this article, we claim that the spin chain model associated with 1D strongly repulsive fermionic mixtures has a natural interpretation in terms of spectral graph theory. This hitherto unobserved connection to a well-studied mathematical branch Mohar1991 ; Bacher1994 ; Friedman2000 ; Cesi2010 ; Poignard2018 provides a general and rigorous framework, which enables us to completely elucidate the symmetry structure of the spectrum for arbitrary external potentials and numbers of particles. This has strong implications: for example, it allows us to prove a generalized form of the Lieb-Mattis theorem, which implies in particular that the ground-state of the system is unmagnetized LiebMattisPR . Once again, this theorem has previously only been conjectured for 6-particles mixtures confined in a harmonic potential Decamp2016a ; Decamp2016b . More importantly, this framework and the symmetry structure we deduced allows one to split the problem, which is extremely complex, into lower dimensional irreducible representations. This enables us in particular to compute the energy gap with polynomial efficiency instead of exponential, thus providing a simple answer to a critical outstanding problem for adiabatic quantum computing Zhang2014 ; Albash2018 .
I Model and quantities of interest
We consider a 1D system of particles divided in fermionic components (e.g. spin orientations) with populations given by the partition of , i.e. such that and . We suppose that it is -symmetric, so that all the particles have the same mass , interact via the same -type potential with interaction strength , and are submitted to the same confining potential , which can be arbitrary. It can be described by the following Hamiltonian:
[TABLE]
where is the coordinate of particle and we suppose that the first particles belong to component , etc (Fig. 1a). Such a model has been realized experimentally using a harmonic potential to confine ultracold atoms, whose ground-state’s purely nuclear spin () guarantees the absence of spin flip collisions and symmetry with Pagano2014 . In the following, we will consider the strongly repulsive limit , which can be achieved by means of a confinement-induced resonance Olshanii1998 . In the formal regime, the system is said to be fermionized: it has the same spectrum (up to the degeneracy) as the spinless -particle fermionic system, with a totally antisymmetric Slater determinant of one-particle orbitals as a wavefunction. However, its permutation symmetry is a priori different. Following Volosniev2014 ; Deuretzbacher2014 and generalizing Girardeau’s Bose-Fermi mapping Girardeau1960 , we write the wavefunction of the system in the Bethe ansatz-like form
[TABLE]
where is the permutation group of , , is equal to 1 if and 0 otherwise, and the vector totally encodes the permutation symmetry of the state. For a given mixture the Pauli principle reduces the number of different coefficients to . The -dimensional vector space of classes of sectors that are equivalent up to a permutation of identical particles is called the snippet space Fang2011 . Noting that the system is no longer degenerate in the vicinity of the point, one can write a first-order expansion of the energy in :
[TABLE]
where is an eigenenergy of the non-interacting Hamiltonian associated with and the energy slope is a functional of . This quantity is equivalent to the so-called Tan’s contact, a pivotal and experimentally accessible quantity in -interacting systems that governs the large behaviour of the momentum distribution Olshanii2003 ; Tan2008a ; Hui11 ; Zwerger2011 ; Stewart2010 ; Chang2016 . Then, it can be shown that the values of and corresponding to the states belonging to the same degenerate manifold can be found by solving the following eigenvalue equation Volosniev2014 ; notespinchain :
[TABLE]
where is the vector of the independent coefficients and is a real symmetric matrix defined in the snippet space by
[TABLE]
The so-called nearest-neighbour exchange constants are given by
[TABLE]
if and are equal up to a transposition in positions and with the particles in and belonging to different components, and otherwise. Their values depend exclusively on and , which makes them experimentally tunable. Then, the ground state corresponds to the vector associated with the largest eigenvalue of , and the other states are given by the eigen-decomposition of .
II Connection with spectral graph theory
In what follows, we explain how symmetry and spectral properties of the spin chain model can be elucidated using spectral graph theory. The basic notions of finite group and representation theories are available from references Fulton2004 ; Liebeck ; Hamermesh_book .
Given a finite group and a generating subset of , it is possible to associate a graph called a Cayley graph, where each vertex of is labelled by the elements of and such that two vertices are connected by an edge if and only if there is such that . If we assign a weight to each , the graph is weighted note1a . If we also consider a subgroup of , one can associate with , and a so-called Schreier graph , whose vertices are indexed by the left cosets and with the edges given by where , and note2 . These graphs are an essential tool in combinatorial and geometric group theory Brouwer2012 . Then, our connection can be established as follows: given a fermionic mixture defined by a partition of , the matrix defined in Eq. (5) is equal to the Laplacian matrix of the weighted Schreier graph . Here, is the Young subgroup associated with , with the set of permutations such that if does not belong to noteys ; and is the set of nearest-neighbour transpositions, where each is associated with a weight as defined in Eq. (6). An illustration in the case of a mixture of two spin up and two spin down is given in Fig. 1b, and all the non-trivial graphs in the case of particles are displayed in Fig. 2. Interestingly, this correspondence provides an interpretation of the Laplacian matrix as the generator of a random walk known as an interchange process Cesi2010 ; Lovász93randomwalks : The particles can be seen as a deck of cards, and, at rate 1, two adjacent cards are selected with a probability given by and exchanged.
III General structure of the spectrum
Considering two Schreier graphs and such that , one can show that the latter is a covering graph of the former, and therefore the Laplacian spectrum is contained in Bacher1994 ; Friedman2000 . In our situation, for example, the fact that implies that the graph in Fig. 2e is a covering of the one in Fig. 2c, and that . More generally, at fixed , for every partition of . In physical terms, this means that the high-spin limit where there are as many particles as components encompasses all the other cases of mixtures. The resulting question is to determine, for a given mixture , to which subset of the spectrum of corresponds. To answer this question, one has to note that the matrices can be seen as representations of the symmetric group . In the case where , each vertex of the Cayley graph corresponds uniquely to a permutation of the mixture. Then, acts on the vector space whose basis is indexed by the elements of , known as the group algebra. Thus, this matrix can be expressed in terms of the so-called regular representation , which is equivalent to the direct sum of all the irreducible representations (irreps) labelled by the partitions of James1984 . Denoting by the element of the group algebra corresponding to a transposition and by the set of all the irreps of , we get that is equal to
[TABLE]
where . Then, in the case of an arbitrary mixture , the key observation is that each snippet can be uniquely associated with a so-called tabloid of shape , and therefore that is acting on the vector space indexed by the tabloids, known as the permutation module note1sm . According to the so-called Young’s rule James1984 , is thus given by Eq. (7), where is now taken over the elements of such that . Here, is the dominance order: if for all . This central result, which can be understood as a consequence of the Pauli principle, completely elucidates the symmetry structure of the spectrum.
IV Symmetry ordering
Denote by the maximum eigenvalue of belonging to the symmetry class . From our previous discussion, we see that (since ) and thus that . In fact, using the fact that the graph is bipartite, one can always construct an eigenvector with eigenvalue that belongs to the symmetry class , so that note2sm . Therefore, we have shown that
[TABLE]
This fact has been numerically Decamp2016a ; Decamp2016b and experimentally Pagano2014 observed. It can be interpreted as a generalized Lieb-Mattis theorem LiebMattisPR , c.f. Eq. (3), which is salient in the context of quantum magnetism. Intuitively, the ground state is as spatially symmetric as possible and its total spin is minimized.
V Peculiar eigenvalues and spectral gap
From basic properties of the Laplacian matrix, we see that for every mixture . Moreover, the fact that is bipartite and -regular implies that , and more generally that if and only if Brouwer2012 . The simple eigenvalues [math] and are associated with the trivial and sign irreps, respectively.
In Bacher1994 , Bacher has studied in the unweighted case where for all , obtaining in particular an expression for the spectral gap, which is deeply related to the geometrical properties of Chung1996 ; Godsil2001 . His argument is based on a mapping between this graph and the cartesian product of path graphs (c.f. Fig. 2a).
We now proceed to generalize this non-trivial result to the weighted case. Let , where is given by the Laplacian matrix of the weighted path graph:
[TABLE]
Then, contains the following eigenvalue
[TABLE]
with . Such an eigenvalue is associated with the symmetry class , with a multiplicity of . Most importantly, the spectral gap of is given by
[TABLE]
where is the second largest eigenvalue of . Eq. (11) means for instance that the spectral gap of the Cayley graph in Fig. 2f is equal to the one of the path graph in Fig. 2a. This quantity can be related to the energy gap of the system through Eq. (3). In the context of adiabatic quantum computing, the minimum value of the energy gap along an adiabatic process is related to the minimum runtime of the algorithm by Albash2018 . It is therefore crucial to ensure that this gap remains sufficiently large along the adiabatic path, which is a priori an exponentially hard problem. Here, we have shown that it is sufficient to compute the lowest non-zero eigenvalue of the matrix defined in Eq. (9), instead of the whole matrix, of size notegap . Thus, by providing an efficient way to compute the energy gap for many configurations of a many-body setting, our method constitutes a huge practical advantage.
In the case of a box potential of size , we can use the exact expression of Volosniev2017 and obtain:
[TABLE]
Quite surprisingly, we note that in this case the spectral gap behaves as in the large limit, and is thus an increasing function of the number of particles. As displayed in Fig. 3, this scaling behaviour is in net contrast with the cases of a harmonic potential, where is a decreasing function of , or a quartic potential, where is almost constant. We observe that the spectral gap is larger for more confining potentials, i.e. potentials that grow faster when approaches infinity.
VI Discussion
Our results suggest that strongly confined 1D systems may become promising candidates for quantum adiabatic computing, allowing: (i) Complex encoding due to the large number of spins; (ii) An adiabatic tuning of the exchange constants of the effective spin chain through the external potential; (iii) A well-controlled energy gap that can be computed efficiently with our method. This paves the way to further theoretical and experimental investigations that may eventually lead to technological applications note3sm .
There are also many theoretical perspectives to this study. For instance, one may combine this framework with random matrix theory in order to study the subtle interplay between disorder and interactions, by considering the graphs we have introduced in this work with a random choice of weights accounting for the presence of a random external potential Abanin2019 . Furthermore, noting that the case when the external potential is homogeneous is integrable Guan2013 , our connection suggests that these graphs play an important role in quantum integrable systems, and, conversely, that one could apply results derived from this theory to the study of Cayley graphs. More generally, this hitherto unobserved link between different communities, ranging from cold atom and condensed matter physics to pure mathematics (for the graph-theoretical and probabilistic interpretations) and even quantum technologies, should stimulate fruitful collaborations.
Acknowledgements.
We thank D. Zhang and T. Nicholson for useful discussions. This work has been partially funded by the Singapore NRF Grant numbers NRF-NRFI2017-04 (WBS No. R-144-000-378-281) and NRF-NRFF2018-02.
Appendix A Basic notions of graph theory
Here we recall a few basic definitions of graph theory Bondy2008 . A graph is a pair where is a set of vertices and is a set of edges that are elements of . We say that is undirected when if and only if . The degree of a vertex is the number of vertices that are connected by an edge, or adjacent, to . A weighted graph is a graph where each edge is associated with a positive real number — the unweighted case corresponding to for all . In this case is the sum of the weights of the edges between and its adjacent vertices. A graph is said to be regular when all his vertices have the same degree. It is said to be bipartite when can be divided into two disjoint sets and such that each edge connects an element of with an element of . If all the vertices of (resp. ) have the same degree (resp. ), the graph is said to be -biregular. A graph is a covering graph of if there is a surjective map such that, for each , the restriction of to a neighbourhood of is a bijection onto a neighbourhood of .
Given a graph , one can canonically associate a set of matrices, being the number of vertices, whose spectra can be related to the structural properties of the graph Brouwer2012 . The adjacency matrix of is such that entry is equal to (or if the graph is weighted) if vertex “” is adjacent to vertex “” and [math] otherwise. The degree matrix is simply the diagonal matrix whose diagonal entries are given by the degree of the corresponding vertex. The Laplacian matrix is then defined by
[TABLE]
This matrix can be seen as a discrete version of the (negative) continuous Laplacian operator, hence the name. It can also be interpreted as the generator of a random walk on Lovász93randomwalks .
Let us now enunciate some basic properties of the Laplacian matrix. It is easy to see that, for an undirected graph, , is symmetric and therefore it can be diagonalized in an orthonormal basis and has a real spectrum . Moreover, the fact that all the diagonal entries of are positive and that each diagonal entry is equal to the sum of the absolute values of the non-diagonal entries in that row imply that . Furthermore, since every row sum and column sum is zero, the vector always satisfies , and thus . Noting that if has connected components note1 , is a block-diagonal matrix with blocks, we see that the multiplicity of [math] is equal to . In particular, for a connected graph, can be written:
[TABLE]
Many bounds have been obtained on these eigenvalues, and in particular on and on the spectral gap , which can be related to geometrical properties of (see e.g. Chung1996 or Godsil2001 ). For example, one has Mohar1991
[TABLE]
with equality if and only if is biregular (in the case of a connected graph).
Finally, one can get the spectrum of graphs obtained by operations on other graphs with a known spectrum Mohar1991 . For example, given two graphs and , we can define the Cartesian product with vertices in and such that there is an edge between and if either or . Then, writing and , one has
[TABLE]
This formula is crucial in Bacher’s proof for the unweighted version of Eq. (11) of the main text Bacher1994 .
Appendix B Representation theory of
In what follows we describe the construction of the set of irreducible representation (irreps) of the symmetric group (see e.g. James1984 ).
The set is in bijection with the conjugacy classes of , which are characterized by a given structure for their decomposition in disjoint cyclic permutations. Thus, there is a one-to-one correspondence between and the set of partitions of . A convenient way of representing a partition note2a of is through a Young diagram, a left-justified set of boxes with rows, where each row contains boxes. For example, an irrep of is characterized by the partition , or equivalently by the following Young diagram:
[TABLE]
is called the shape of , and we denote the corresponding (class of) irrep(s) by . The conjugate of a Young diagram of shape is the diagram with columns of lengths . E.g. the conjugate of the previous example is
[TABLE]
The set of Young diagrams is partially ordered by the so-called dominance order , such that (where the last terms of one of the partitions may be equal to zero) if and only if
[TABLE]
For example, one has
[TABLE]
Intuitively, it means that one can go from the left diagram to the right one by moving a certain number of boxes from upper rows to lower rows. Note that it is not a total order when . For instance, it is not possible to compare
[TABLE]
Let us be a little more precise on how an irrep can be constructed from a Young diagram. A Young tableau is a Young diagram whose boxes are labelled by integers. Two tableaux of same shape are said to be row-equivalent if they are equal up to permutations of the rows. For example,
[TABLE]
are row-equivalent. This defines equivalence classes on the set of -tableaux that are called tabloids. For the previous example, it is represented as
[TABLE]
Note that each permutation of a fermionic mixture, or snippet, can be regarded as a tabloid with rows of lengths and with entries of row corresponding to the positions of the particle of type . For example, if we consider an -particle mixture with particles of type , particles of type and of type , the tabloid given in Eq. (23) corresponds to the following snippet:
[TABLE]
We now define the permutation module as the -vector space whose basis is indexed by the set of -tabloids. Note that its dimension is , as defined in the main text. Note also that is in bijection with , the group algebra of , whose basis is indexed by the elements of and such that . Associated with is the (left) regular representation of : writing a vector in as with , and given , the linear map is defined by
[TABLE]
which can be linearly extended on by writing . Similarly, one can identify any permutation module with a representation of by considering the natural action of on the vector space .
Then, for a given -tableau , one can associate the following element of :
[TABLE]
where is the subgroup of permutations preserving all columns of , is the sign of the permutation , and is the tabloid corresponding to . is called a polytabloid. Then, the Specht module is defined as the subspace of generated by the elements when runs through all the -tableaux. It can be shown that the basis of is given by the elements when runs though all the standard Young tableaux of shape , that is the tableaux whose entries are increasing from left to right along the rows and up to down along the columns. In particular, its dimension is given by the number of standard Young tableaux, which can be easily obtained from the so-called hook length formula. Furthermore, the so-called Young’s rule states that the permutation module can be decomposed in the following way:
[TABLE]
where are positive integers.
Here again, can be seen as a representations of . Then, it can be shown that the set of all the Specht modules is, in fact, . Note that when taking , Eq. (27) implies in particular that the regular representation is a sum of all the irreps. In this case, the numbers are given by the dimensions of the irreps. Intuitively, keeping in mind Eq. (26), an irrep can be seen as symmetrizing the rows and anti-symmetrizing the columns of the -tableaux.
Appendix C Generalized Lieb-Mattis theorem — A detailed proof of
Denoting by the maximum eigenvalue of corresponding to the symmetry class , and by the maximum eigenvalue of , we have shown in the main text that , and then we claimed that in fact
[TABLE]
In order to prove this assertion, let us first observe that the graphs are bipartite. This is immediately clear for the Cayley graph , where the vertices associated with are separated between the ones corresponding to and the ones corresponding to . In the general case of a Schreier graph, the vertices are the left cosets , or equivalently the set of tabloids. With each , we can associate the representative such that the indices corresponding to particles belonging to a same component are displayed in increasing order. For example, the representative of the tabloid displayed in Eq. (24) corresponds to
[TABLE]
with particles of type , of type , and of type . Defining the sign of by , we see that the vertices of are also separated between the positive and the negative ones.
Given a mixture , we consider an eigenvector of with eigenvalue . Then, we can define a vector such that, for every component corresponding to some coset :
[TABLE]
Seeing as an element of the permutation module , we deduce from the definition of that is an element of the Specht module , i.e. that the corresponding state belongs to the symmetry class of type . Moreover, since is a real symmetric matrix, we can write as a sum of orthogonal eigenvectors with eigenvalues . Therefore, we have
[TABLE]
and
[TABLE]
Since is the maximum eigenvalue of , we deduce that belongs to the eigenspace with eigenvalue and we obtain Eq. (28).
In fact, we can prove that, in generic cases, the only vectors with maximum eigenvalue belong to the symmetry class . This can be seen as a consequence of a general theorem Poignard2018 , which states that, for a given weighted graph, the spectrum of its Laplacian matrix is simple for generic choices of weights noteleb . In our situation, this means that a small random perturbation of the external potential will lead to a spectrum that contains no accidental degeneracies between different symmetry classes — although it usually does contain degeneracies associated with , which denote the number of time the irrep appears in the decomposition of the permutation module according to Young’s rule (cf Eq. (27)).
Using the decomposition of the spectrum according to the irreps of as described in the main text, Eq. (28) implies that
[TABLE]
Writing the energy of the corresponding ground state with symmetry in the strongly repulsive limit, we thus have:
[TABLE]
This is a generalized version of the Lieb-Mattis theorem, which has important consequences in the theory of magnetism LiebMattisPR . Intuitively, the ground-state wavefunction “wants” to be as symmetric as possible. For spin- particles ( case), this means that the total spin of the system is minimized, and thus that the ground-state is unmagnetized. Eq. (33) has been checked for several few-body systems Decamp2016a ; Decamp2016b . Our approach provides a rigorous and general proof.
Note that Eq. (28) provides a way to compare ground-state energies , when and are not comparable according to the dominance order. It is indeed sufficient to compare the spectral radii and of the real symmetric matrices and (resp.), which can be obtained effectively, e.g. using the Lanczos algorithm Lanczos:1950zz .
Appendix D Proposed experimental implementation
The model of strongly-correlated one-dimensional spinor gases can be realized in the laboratory by working with group-II atoms like strontium and ytterbium. These atoms in their singlet ground state only have nuclear spin components, therefore they exhibit symmetry Gorshkov2010 ; Ye2014 ; Foelling2014 ; Pagano2014 . To prepare a specific number of atoms , one can use optical tweezers to prepare one atom per tweezer in the collisional blockade regime Grangier2001 , and use acousto-optical modulators Lukin2016 , spatial light modulators Browaeys2016 , or microlens arrays Birkl2019 to scale up to a desired number of tweezers. Alternatively, one can use a quantum gas microscope to determine the number of atoms loaded into isolated chains Greiner2009 ; Bloch2010 . To obtain a specific superposition of spins, one can coherently address different spin states using clock transitions with long-lived coherence times. Having prepared the atom number and quantum states, these atoms can then be loaded into an overall 1D potential while the tweezer or lattice potentials separating the individual atoms are turned off. A harmonic 1D potential can be created by interfering red-detuned lattice beams in two dimensions. A confinement-induced resonance, where the transverse confinement length is tuned to match the scattering length, can be used to reach the strongly repulsive regime Jochim2012 . To vary , one can change the external 1D potential. For example, instead of a harmonic potential, a box potential can be created with the help of steep repulsive walls formed from blue-detuned light shaped by spatial light modulators Hadzibabic2013 or other diffractive optics Zwierlein2017 .
The contact can be determined from a range of experimental methods, including Bragg spectroscopy Vale2010 ; Vale2011 , RF spectroscopy Stewart2010 ; Cornell2012 ; Jin2012 ; Zwierlein2019 , RF Ramsey interferometry Fletcher2017 , and photoassociation experiments Castin2009 ; Hulet2005 . Of these, measuring the molecular fraction from photoassociation is likely to be one of the most sensitive methods Ni2018 in the limit of one atom per spin component, where . Beyond the ground state, the contact for the first excited state will become accessible as the temperature of the ultracold atoms increases. In particular, our graph theory analysis predicts the difference between the maximum contact and the second largest contact where to be the same as for the case of particles in spin down and one particle in spin up. Such symmetry allows us to gain insight on spectral gaps in the latter case, especially for highly excited many-body states that are otherwise experimentally challenging to access.
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