Ordering of magnetization-sector lowest energies and continuity of the lowest-energy branch at commensurate magnetization in one-dimensional U(1)-symmetric spin chains

The extended Lieb-Schultz-Mattis (LSM) theorems of Nomura, Morishige and Isoyama (J. Phys. A 48, 375001 (2015)) constrain the lowest-energy branch $E(S_T^z; q)$ — the lowest energy at fixed total magnetization $S_T^z$ and crystal momentum $q$ — for one-dimensional spin chains under only U(1) symmetry, translation invariance, and short-range interactions, thereby covering frustrated and nonsymmetric models. They prove continuity of $E(S_T^z; q)$ in $q$ for every incommensurate sector $S_T^z=(S-m/n)L+\Delta S_T^z$ with $\Delta S_T^z\neq 0$, and Oshikawa-Yamanaka-Affleck-type periodicity at commensurate sectors. Two linked questions are explicitly left open and remain open today: (i) at commensurate magnetizations $S_T^z=(S-m/n)L$, where the twist-operator orbit closes and the continuity proof fails, is $E(S_T^z; q)$ continuous in $q$ in the thermodynamic limit or genuinely discontinuous, as expected at magnetization plateaux; and (ii) what determines the relative ordering of lowest energies across different $S_T^z$ sectors — equivalently the structure of the zero-temperature magnetization curve (continuity, plateaux, cusps, macroscopic jumps) — beyond the non-frustrated regime where Bethe-ansatz and Perron-Frobenius methods apply. The problem asks for general, model-independent theorems, or explicit Hamiltonians with proofs, settling these sector-level questions.

Background

The Lieb-Schultz-Mattis (LSM) theorem (Lieb, Schultz and Mattis, 1961) states that the $S=1/2$ isotropic Heisenberg chain has, in the thermodynamic limit of infinite length, either a degenerate ground state or excited states arbitrarily close to the ground state; Affleck and Lieb (1986) extended it to chains of arbitrary half-integer spin. The modern setting is a one-dimensional (1D) ring of $L$ spins of magnitude $S$ whose Hamiltonian is invariant under global rotations of all spins about the $z$-axis (U(1) symmetry, so the total magnetization $S_T^z=\sum_j S_j^z$ is conserved) and under lattice translations, with short-range interactions. The conserved $S_T^z$ decomposes the Hilbert space into magnetization sectors; within a sector one can further diagonalize translations, whose eigenvalues are written $e^{iq}$, and imposing a momentum shift by hand ('twisted boundary conditions', introduced in this context by Kolb, 1985) gives access to $E(S_T^z; q)$, the lowest energy eigenvalue at total magnetization $S_T^z$ and crystal momentum $q$. LSM-type statements are statements about this function. Oshikawa, Yamanaka and Affleck (OYA, 1997) used the twist method to derive the commensurability criterion for magnetization plateaux — an interval of applied field over which the magnetization per site stays fixed; a plateau is equivalent to a finite gap for changing $S_T^z$ — namely that a plateau at magnetization per site $M$ can occur only when $pM$ is an integer, where $p$ is the period of the ground state. Oshikawa (2000) recast these arguments as a commensurability condition on the filling of lattice systems. Nomura, Morishige and Isoyama (2015) re-examined the twist-operator method (Kolb, 1985; Fáth and Sólyom, 1993, whose proof of dispersion continuity at $S_T^z=\pm 1,\pm 2,\ldots$ they found insufficient and corrected) and proved, assuming only U(1) symmetry, translation invariance, and short-range interactions: a convexity-type inequality linking $E(S_T^z;q)$ at three nearby momenta (Theorem 2), implying continuity of the lowest-energy branch in $q$ in the thermodynamic limit for every incommensurate sector $S_T^z=(S-m/n)L+\Delta S_T^z$ with $\Delta S_T^z\neq 0$ (Theorem 3); and periodicity $E(S_T^z;q)=E(S_T^z;q+2\pi/n)$ at commensurate sectors (Theorems 4, 5, 7, extensions of the OYA theorem). The proof needs neither uniqueness of the sector ground state nor space-inversion or spin-reversal symmetry, so it applies to frustrated chains and to models with Dzyaloshinskii-Moriya interactions — an antisymmetric exchange term of the form $D\cdot(S_j\times S_{j+1})$ that breaks spatial inversion. Why the commensurate case is different: the twist operator shifts a state's momentum by $2\pi S_T^z/L+2\pi S$; when $S_T^z=(S-m/n)L$ this shift is a rational multiple of $2\pi$, the orbit closes after $n$ steps, and the squeeze argument yields only periodicity, not continuity. Physically the commensurate sectors are exactly where magnetization plateaux, spontaneously dimerized states (neighboring spins pairing into singlets, which enlarges the spatial period), Néel-ordered states with enlarged period, and level crossings are expected (Section 8.2 and Fig. 5 of the source paper), so the missing continuity may genuinely fail — but no example is proven. The sector-ordering question has the same shape: the zero-temperature magnetization curve $M(h)$ as a function of applied field $h$ is obtained by a Legendre transform of the sector lowest energies $E_0(S_T^z)$, and its features — continuous segments, plateaux, cusp singularities, and macroscopic jumps (first-order transitions between magnetization sectors) — are known only model by model: exact Bethe-ansatz solutions (the Bethe ansatz is the standard exact-solution technique for integrable chains) for the unfrustrated XXZ chain — the anisotropic nearest-neighbor Heisenberg chain (Yang and Yang, 1966) — and Perron-Frobenius arguments in the Marshall-Lieb-Mattis (MLM) tradition (Lieb and Mattis, 1962), which use sign-definiteness of off-diagonal Hamiltonian matrix elements (i.e. absence of frustration) to fix the sector minimum at $q=0$ or $q=\pi$. In frustrated chains no general principle is known, and the LSM-type theorems themselves, as their authors state, 'say nothing' about the ordering of sector lowest energies.

Problem Statement

Consider a one-dimensional ring of $L$ spins of magnitude $S$ with periodic boundary conditions, whose Hamiltonian is invariant under the continuous U(1) spin-rotation symmetry about the $z$-axis (so the total magnetization $S_T^z=\sum_j S_j^z$ is conserved) and under lattice translations, and has short-range interactions; frustration and the absence of space-inversion, spin-reversal, or time-reversal symmetry are all allowed, matching the generality of the extended Lieb-Schultz-Mattis theorems of Nomura, Morishige and Isoyama (2015). Let $E(S_T^z; q)$ denote the lowest energy in the sector of total magnetization $S_T^z$ and crystal momentum $q$. Those theorems establish continuity of $E(S_T^z; q)$ in $q$ for all incommensurate sectors $S_T^z=(S-m/n)L+\Delta S_T^z$ with coprime integers $m,n$ independent of $L$ and integer $\Delta S_T^z\neq 0$ with $|\Delta S_T^z|\ll L$, and periodicity $q\to q+2\pi/n$ at commensurate sectors $S_T^z=(S-m/n)L$ with $m,n$ coprime integers, but leave the sector-level energetics in exactly the commensurate regime undetermined. Determine: (1) Commensurate-sector continuity — at $S_T^z=(S-m/n)L$ (the case $\Delta S_T^z=0$ excluded from the continuity theorem), is $E(S_T^z; q)$ continuous as a function of $q$ in the thermodynamic limit $L\to\infty$, or can it be genuinely discontinuous? A complete answer either proves continuity under explicitly stated assumptions at the same level of generality as the existing theorems, or constructs an explicit Hamiltonian in the model class for which the discontinuity of the lowest-energy branch is proven. (2) Sector ordering — writing $E_0(S_T^z)=\min_q E(S_T^z;q)$ for the lowest energy in sector $S_T^z$, determine the relative ordering of the sector lowest energies $E_0(S_T^z)$, equivalently the structure of the zero-temperature magnetization curve $M(h)$ obtained by Legendre transform (continuity, magnetization plateaux, cusp singularities, macroscopic jumps, and the momentum location — $q=0$ versus $q=\pi$ versus other symmetry-allowed momenta — of each sector minimum): either general model-independent constraints, i.e. theorems giving conditions on the interactions under which each feature must or cannot occur, or explicit constructions delimiting which behaviors are possible within the model class. A complete answer to (2) must settle each of the listed features, either by a general constraint or by explicit constructions delimiting that feature's possible behaviors; a single construction may settle more than one feature. The common objective is to complete the magnetization-resolved low-energy portrait of U(1)-symmetric one-dimensional spin chains in the commensurate regime — the regime the symmetry-based twist machinery cannot reach because the twist orbit closes. The ideal outcome is a model-independent statement at the generality of the existing LSM-type theorems, but either route offered above — a general theorem or an explicit construction with proof — counts as a complete answer.

References

1. Kiyohide Nomura, Junpei Morishige, Takaichi Isoyama, 'Extension of the Lieb-Schultz-Mattis theorem', J. Phys. A: Math. Theor. 48, 375001 (2015). DOI: 10.1088/1751-8113-48-37-375001, arXiv:1503.05662. https://doi.org/10.1088/1751-8113-48-37/375001
2. E. Lieb, T. Schultz, D. Mattis, 'Two soluble models of an antiferromagnetic chain', Annals of Physics 16, 407-466 (1961). DOI: 10.1016/0003-4916(61)90115-4. https://doi.org/10.1016/0003-4916(61)90115-4
3. I. Affleck, E. H. Lieb, 'A proof of part of Haldane's conjecture on spin chains', Letters in Mathematical Physics 12, 57-69 (1986). DOI: 10.1007/BF00400304. https://doi.org/10.1007/BF00400304
4. M. Kolb, 'Symmetry and boundary condition of planar spin systems', Phys. Rev. B 31, 7494-7496 (1985). DOI: 10.1103/PhysRevB.31.7494. https://doi.org/10.1103/PhysRevB.31.7494
5. G. Fáth, J. Sólyom, 'Isotropic spin-1 chain with twisted boundary condition', Phys. Rev. B 47, 872-881 (1993). DOI: 10.1103/PhysRevB.47.872. https://doi.org/10.1103/PhysRevB.47.872
6. M. Oshikawa, M. Yamanaka, I. Affleck, 'Magnetization Plateaus in Spin Chains: "Haldane Gap" for Half-Integer Spins', Phys. Rev. Lett. 78, 1984-1987 (1997). DOI: 10.1103/PhysRevLett.78.1984. https://doi.org/10.1103/PhysRevLett.78.1984
7. M. Oshikawa, 'Commensurability, Excitation Gap, and Topology in Quantum Many-Particle Systems on a Periodic Lattice', Phys. Rev. Lett. 84, 1535-1538 (2000). DOI: 10.1103/PhysRevLett.84.1535. https://doi.org/10.1103/PhysRevLett.84.1535
8. Takaichi Isoyama, Kiyohide Nomura, 'Discrete symmetries and the Lieb-Schultz-Mattis theorem', Progress of Theoretical and Experimental Physics 2017, 073I01 (2017). DOI: 10.1093/ptep/ptx139, arXiv:1605.03385. https://doi.org/10.1093/ptep/ptx139
9. Hal Tasaki, 'Lieb-Schultz-Mattis Theorem with a Local Twist for General One-Dimensional Quantum Systems', J. Stat. Phys. 170, 653-671 (2018). DOI: 10.1007/s10955-017-1946-0, arXiv:1708.05186. https://doi.org/10.1007/s10955-017-1946-0
10. Yoshiko Ogata, Yuji Tachikawa, Hal Tasaki, 'General Lieb-Schultz-Mattis Type Theorems for Quantum Spin Chains', Communications in Mathematical Physics 385, 79-99 (2021). DOI: 10.1007/s00220-021-04116-9, arXiv:2004.06458. https://doi.org/10.1007/s00220-021-04116-9
11. C. N. Yang, C. P. Yang, 'One-Dimensional Chain of Anisotropic Spin-Spin Interactions. II. Properties of the Ground-State Energy Per Lattice Site for an Infinite System', Phys. Rev. 150, 327-339 (1966). DOI: 10.1103/PhysRev.150.327. https://doi.org/10.1103/PhysRev.150.327
12. Elliott Lieb, Daniel Mattis, 'Ordering Energy Levels of Interacting Spin Systems', J. Math. Phys. 3, 749-751 (1962). DOI: 10.1063/1.1724276. https://doi.org/10.1063/1.1724276
13. Santanu Pal, Siddhartha Lal, 'Magnetization plateaus of the quantum pyrochlore Heisenberg antiferromagnet', Phys. Rev. B 100, 104421 (2019). DOI: 10.1103/PhysRevB.100.104421, arXiv:1902.06475. https://doi.org/10.1103/PhysRevB.100.104421