We develop a general operator algebraic method which focuses on projective representations of symmetry group for proving Lieb-Schultz-Mattis type theorems, i.e., no-go theorems that rule out the existence of a unique gapped ground state (or, more generally, a pure split state), for quantum spin chains with on-site symmetry. We first prove a theorem for translation invariant spin chains that unifies and extends two theorems proved by two of the authors in [OT1]. We then prove a Lieb-Schultz-Mattis type theorem for spin chains that are invariant under the reflection about the origin and not necessarily translation invariant.
For a large class of finite-range quantum spin models with half-integer spins, we prove that uniqueness of the ground state implies the existence of a low-lying excited state. For systems of linear size L, of arbitrary finite dimension, we obtain an upper bound on the excitation energy (i.e., the gap above the ground state) of the form (C\log L)/L. This result can be regarded as a multi-dimensional Lieb-Schultz-Mattis theorem and provides a rigorous proof of a recent result by Hastings.
Following recent developments in the classification of bosonic short-range entangled phases, we examine many-body quantum systems whose ground state fractionalization obeys the Lieb-Schultz-Mattis (LSM) theorem. We generalize the topological classification of such phases by LSM anomalies (arXiv:1907.08204) to take magnetic and non-symmorphic lattice effects into account, and provide direct computations of the LSM anomaly in specific examples. We show that the anomaly-free condition coincides with established filling constraints (arXiv:1705.09298, arXiv:1505.04193), and we also derived new ones…
We introduce a new boundary condition which renders the flux-insertion argument for the Lieb-Schultz-Mattis type theorems in two or higher dimensions free from the specific choice of system sizes. It also enables a formulation of the Lieb-Schultz-Mattis type theorems in arbitrary dimensions in terms of the anomaly in field theories of $1+1$ dimensions with a bulk correspondence as a BF-theory in 2+1 dimensions. Furthermore, we apply the anomaly-based formulation to the constraints on a half-filled spinless fermion on a square lattice with $π$ flux, utilizing time-reversal, the magnetic transla…
We propose a geometric {approach to Lieb-Schultz-Mattis theorem for} quantum many-body systems with discrete spin-rotation symmetries and lattice inversion or rotation symmetry, but without translation symmetry assumed. Under symmetry-twisting on a $(d-1)$-dimensional plane, we find that any $d$-dimensional inversion-symmetric spin system possesses a doubly degenerate spectrum when it hosts a half-integer spin at the inversion-symmetric point. We also show that any rotation-symmetric generalized spin model with a projective representation at the rotation center has a similar degeneracy under s…
We discuss quantum many-body systems with lattice translation and discrete onsite symmetries. We point out that, under a boundary condition twisted by a symmetry operation, there is an exact degeneracy of ground states if the unit cell forms a projective representation of the onsite discrete symmetry. Based on the quantum transfer matrix formalism, we show that, if the system is gapped, the ground-state degeneracy under the twisted boundary condition also implies a ground-state (quasi-)degeneracy under the periodic boundary conditions. This gives a compelling evidence for the recently proposed…
The $\mathrm{SU}(4)$ Heisenberg model on the honeycomb lattice is expected to host a quantum spin-orbital liquid at low temperature with an astonishing candidate material, $α$-ZrCl$_3$. We employed the canonical thermal pure quantum state method to investigate the finite-temperature phase of this model. Exploiting the full symmetry of $\mathrm{SU}(4)$, the calculation up to a 24-site cluster, which is equivalent to 48 sites in the spin-1/2 language, is possible. This state-of-the-art computation with large-scale parallelization enables us to capture the thermodynamic properties of the $\mathrm…
A generalization of the Lieb-Schultz-Mattis theorem to higher dimensional spin systems is shown. The physical motivation for the result is that such spin systems typically either have long-range order, in which case there are gapless modes, or have only short-range correlations, in which case there are topological excitations. The result uses a set of loop operators, analogous to those used in gauge theories, defined in terms of the spin operators of the theory. We also obtain various cluster bounds on expectation values for gapped systems. These bounds are used, under the assumption of a gap,…
Lieb, Schultz and Mattis (LSM) studied the S=1/2 XXZ spin chain. Theorems of LSM's paper can be applied to broader models. In the original LSM theorem it was assumed the nonfrustrating system. However, reconsidering the LSM theorem, we can extend the LSM theorem for frustrating systems. Next, several researchers have tried to extend the LSM theorem for excited states. In the cases $S^{z}_{T} = \pm 1,\pm 2 \cdots$, the lowest energy eigenvalues are continuous for wave number $q$. But we find that their proofs are insufficient, and we improve them. In addition, we can prove the LSM theory withou…
We consider many-body quantum systems on a finite lattice, where the Hilbert space is the tensor product of finite-dimensional Hilbert spaces associated with each site, and where the Hamiltonian of the system is a sum of local terms. We are interested in proving uniform bounds on various properties as the size of the lattice tends to infinity. An important case is when there is a spectral gap between the lowest state(s) and the rest of the spectrum which persists in this limit, corresponding to what physicists call a ``phase of matter". Here, the combination of elementary Fourier analysis with…
We review the Lieb-Schultz-Mattis theorem and its variants, which are no-go theorems that state that a quantum many-body system with certain conditions cannot have a locally-unique gapped ground state. We restrict ourselves to one-dimensional quantum spin systems and discuss both the generalized Lieb-Schultz-Mattis theorem for models with U(1) symmetry and the extended Lieb-Schultz-Mattis theorem for models with discrete symmetry. We also discuss the implication of the same arguments to systems on the infinite cylinder, both with the periodic boundary conditions and with the spiral boundary co…
We prove that a quantum spin chain with half-odd-integral spin cannot have a unique ground state with a gap, provided that the interaction is short ranged, translation invariant, and possesses time-reversal symmetry or ${\mathbb Z}_2 \times {\mathbb Z}_2$ symmetry (i.e., the symmetry with respect to the $π$ rotations of spins about the three orthogonal axes). The proof is based on the deep analogy between the matrix product state formulation and the representation of the Cuntz algebra in the von Neumann algebra $π({\mathcal A}_{R})''$ constructed from the ground state restricted to the right h…
The Lieb-Schultz-Mattis (LSM) theorem states that a spin system with translation and spin rotation symmetry and half-integer spin per unit cell does not admit a gapped symmetric ground state lacking fractionalized excitations. That is, the ground state must be gapless, spontaneously break a symmetry, or be a gapped spin liquid. Thus, such systems are natural spin-liquid candidates if no ordering is found. In this work, we give a much more general criterion that determines when an LSM-type theorem holds in a spin system. For example, we consider quantum magnets with arbitrary space group symmet…
Modulated symmetries from generalized Lieb-Schultz-Mattis anomalies
Yukawa Institute for Theoretical Physics, Kyoto University（京都大学汤川理论物理研究所）；Interdisciplinary Theoretical and Mathematical Sciences Program (iTHEMS) RIKEN（理化学研究所跨学科理论与数学科学项目）；Institute for Theoretical Physics, University of Cologne（科隆大学理论物理研究所）；Center for Quantum Mathematics at IMADA, Southern Denmark University（南丹麦大学IMADA量子数学中心）；Niels Bohr International Academy, Niels Bohr Institute, University of Copenhagen（哥本哈根大学尼尔斯·玻尔研究所尼尔斯·玻尔国际学院）
Matrix Product States for Modulated Topological Phases: Crystalline Equivalence Principle and Lieb-Schultz-Mattis Constraints
Hong Kong University of Science and Technology（香港科技大学）；The Chinese University of Hong Kong（香港中文大学）；RIKEN（理化学研究所）；Massachusetts Institute of Technology（麻省理工学院）；Stanford University（斯坦福大学）；University of Southern California（南加州大学）
We show that the exact ground state of the Lieb-Mattis Hamiltonian is an equal-weight superposition of all possible classical Néel states, and provide an exact formulation of this superposition in the $z$-spin basis for both $S=1/2$ and general $S$ using Schwinger bosons. In general, a superposition of possible rotations on a general initial state is symmetric if and only if the initial state has a nonzero overlap with a singlet state and is otherwise made up of states that vanish due to the symmetrization. Most notably, $|s, m=0 \rangle$ states will vanish if symmetrized, which explains how a…
Lieb-Schultz-Mattis (LSM) theorems provide powerful constraints on the emergibility problem, i.e. whether a quantum phase or phase transition can emerge in a many-body system. We derive the topological partition functions that characterize the LSM constraints in spin systems with $G_s\times G_{int}$ symmetry, where $G_s$ is an arbitrary space group in one or two spatial dimensions, and $G_{int}$ is any internal symmetry whose projective representations are classified by $\mathbb{Z}_2^k$ with $k$ an integer. We then apply these results to study the emergibility of a class of exotic quantum crit…
We prove ferromagnetic ordering of energy levels for XXX Heisenberg chains of any spin and XXZ spin chains with all spins equal to 1/2. Ferromagnetic ordering means that the minimum energies in the invariant subspaces of fixed total spin are monotone decreasing as a function of the total spin. This result provides a ferromagnetic analogue of the well-known theorem by Lieb and Mattis about ordering of energy levels in antiferromagnetic and ferrimagnetic systems on bipartite graphs.
We present a possible definition of a mobility gap for a many-body quantum system, in analogy to definitions of dynamical localization for single particle systems. Using this definition, we construct "corrected" quasi-adiabatic continuation operators. Under an appropriate definition of a unique ground state, we show how to introduce virtual fluxes. Armed with these results, we can directly carry over previous results in the case of a spectral gap. We present a proof of decay of correlation functions and we present a proof of Hall conductance quantization under very mild density-of-states assum…
State Key Laboratory of Low-Dimensional Quantum Physics, Department of Physics, Tsinghua University（清华大学物理系低维量子物理国家重点实验室）；Qiuzhen College, Tsinghua University（清华大学邱成通数学科学中心）；Frontier Science Center for Quantum Information（量子信息前沿科学中心）；Hefei National Laboratory（合肥国家实验室）；Yau Mathematical Sciences Center, Tsinghua University（清华大学丘成桐数学科学中心）
Lieb-Mattis ordering theorem of electronic energy levels in the thermodynamic limit
University of Granada（格拉纳达大学）；Institute Carlos I of Theoretical and Computational Physics (iC1), University of Granada（格拉纳达大学理论计算物理卡洛斯一世研究所）；University CEU Fernando III（塞乌费尔南多三世大学）；Higher Polytechnic School, University CEU Fernando III（塞乌费尔南多三世大学高等理工学院）；University of Jaen（哈恩大学）；Department of Mathematics, University of Jaen（哈恩大学数学系）
We construct a model-independent framework describing stabilities of ferromagnetism in strongly correlated electron systems; Our description relies on the operator theoretic correlation inequalities. Within the new framework, we reinterpret the Marshall-Lieb-Mattis theorem and Lieb\rq{}s theorem; in addition, from the new perspective, we prove that Lieb\rq{}s theorem still holds true even if the electron-phonon and electron-photon interactions are taken into account. We also examine the Nagaoka-Thouless theorem and its stabilities. These examples verify the effectiveness of our new viewpoint.
We prove that in $(3+1)d$ quantum field theories with $\mathbb{Z}_N$ symmetry, certain anomalies forbid a symmetry-preserving vacuum state with a gapped spectrum. In particular, this applies to discrete chiral symmetries which are frequently present in gauge theories as we illustrate in examples. Our results also constrain the long-distance behavior of certain condensed matter systems such as Weyl-semimetals and may have applications to crystallographic phases with symmetry protected topological order. These results may be viewed as analogs of the Lieb-Schultz-Mattis theorem for continuum fiel…
We propose an index $\mathcal{I}_G$ which characterizes the degree of ingappability, namely the difficulty to induce a unique ground state with a nonvanishing excitation gap, in the presence of a symmetry $G$. $\mathcal{I}_G$ represents the dimension of the subspace of ambient uniquely-gapped in the entire $G$-invariant "theory space". The celebrated Lieb-Schultz-Mattis theorem corresponds, in our formulation, to the case $\mathcal{I}_G=0$ (completely ingappable) for the symmetry $G$ including the lattice translation symmetry. We illustrate the usefulness of the index by discussing the phase d…
These lecture notes focus on the application of ideas of locality, in particular Lieb-Robinson bounds, to quantum many-body systems. We consider applications including correlation decay, topological order, a higher dimensional Lieb-Schultz-Mattis theorem, and a nonrelativistic Goldstone theorem. The emphasis is on trying to show the ideas behind the calculations. As a result, the proofs are only sketched with an emphasis on the intuitive ideas behind them, and in some cases we use techniques that give very slightly weaker bounds for simplicity. This is a preliminary version of the lecture note…
The Lieb-Mattis theorem on the antiferromagnetic ordering of energy levels is generalized to SU(N) extended Hubbard model with Heisenberg exchange and pair-hopping terms. It is proved that the minimum energy levels among the states from equivalent representations are nondegenerate and ordered according to the dominance order of corresponding Young diagrams. In particular, the ground states form a unique antisymmetric multiplet. The relation with the similar ordering among the spatial wavefunctions with different symmetry classes of ordinary quantum mechanics is discussed also.
We review some recent results that express or rely on the locality properties of the dynamics of quantum spin systems. In particular, we present a slightly sharper version of the recently obtained Lieb-Robinson bound on the group velocity for such systems on a large class of metric graphs. Using this bound we provide expressions of the quasi-locality of the dynamics in various forms, present a proof of the Exponential Clustering Theorem, and discuss a multi-dimensional Lieb-Schultz-Mattis Theorem.
Anomalies are renormalization group invariants that constrain the dynamics of quantum field theories. We show that certain anomalies for discrete global symmetries imply that the underlying theory either spontaneously breaks its generalized global symmetry or is gapless. We identify an obstruction, formulated in terms of the anomaly inflow action, that must vanish if a symmetry preserving gapped phase, i.e. a unitary topological quantum field theory, exits with the given anomaly. Our result is similar to the $2d$ Lieb-Schultz-Mattis theorem but applies more broadly to continuum theories in gen…
