At odd generation number the sign problem of the Spin(10) mirror sector, written in flavour-singlet composite variables, lands on the complex vector-spinor 144, and Companion I of this series named one remaining check: whether a matter–mirror conjugation can pair the composite determinant when the interaction resolves the internal index. We settle this check, with a negative answer for the physical problem and a positive mathematical result along the way. Theorem. Any decoupled two-layer fermion problem on a bipartite lattice in which (A) the second layer's one-body matrix is the complex conjugate of the first, (B) the first satisfies the chiral condition η h† η = −h, and (C) the inter-layer pairing is real, symmetric and sublattice-preserving, has two anticommuting Majorana-time-reversal symmetries (T² = −1, the layer exchange; T² = +1, layer sign times sublattice sign) and is sign-free by the theorem of Li, Jiang and Yao. The theorem accounts for a full catalogue of admissible and forbidden terms, including five cases predicted in advance and confirmed. What it implies. Without pairing the weight of such a system is exactly |det(1+B₁)|²: the second layer is the conjugate copy of the first, and the sign-free model is the doubled ensemble of the one-layer problem. Condition (A) cannot be met by an interaction that acts on the mirror alone; a gapping interaction shared by the two layers gaps the matter layer as well. This is the positivity–selectivity tension of the lattice paper (Paper A), now in Majorana form, and it means that the construction does not provide a sampler for the physical, mirror-selective odd-generation problem. For that problem we prove that a gauge channel alone gives a real (not positive) weight, and we estimate the average sign on small lattices: mild for the gauge channel alone, severe once the density repulsion is included. All criteria marked "pre-registered" were fixed in writing before the corresponding runs. Verification and simulation scripts are included (companionII_skripte.zip). Derivations, code and drafting in collaboration with Claude (Anthropic).
At odd generation number the sign problem of the Spin(10) mirror sector, written in flavour-singlet composite variables, lands on the complex vector-spinor 144, and Companion I of this series named one remaining check: whether a matter–mirror conjugation can pair the composite determinant when the interaction resolves the internal index. We settle this check, with a negative answer for the physical problem and a positive mathematical result along the way. Theorem. Any decoupled two-layer fermion problem on a bipartite lattice in which (A) the second layer's one-body matrix is the complex conjugate of the first, (B) the first satisfies the chiral condition η h† η = −h, and (C) the inter-layer pairing is real, symmetric and sublattice-preserving, has two anticommuting Majorana-time-reversal symmetries (T² = −1, the layer exchange; T² = +1, layer sign times sublattice sign) and is sign-free by the theorem of Li, Jiang and Yao. The theorem accounts for a full catalogue of admissible and forbidden terms, including five cases predicted in advance and confirmed. What it implies. Without pairing the weight of such a system is exactly |det(1+B₁)|²: the second layer is the conjugate copy of the first, and the sign-free model is the doubled ensemble of the one-layer problem. Condition (A) cannot be met by an interaction that acts on the mirror alone; a gapping interaction shared by the two layers gaps the matter layer as well. This is the positivity–selectivity tension of the lattice paper (Paper A), now in Majorana form, and it means that the construction does not provide a sampler for the physical, mirror-selective odd-generation problem. For that problem we prove that a gauge channel alone gives a real (not positive) weight, and we estimate the average sign on small lattices: mild for the gauge channel alone, severe once the density repulsion is included. All criteria marked "pre-registered" were fixed in writing before the corresponding runs. Verification and simulation scripts are included (companionII_skripte.zip). Derivations, code and drafting in collaboration with Claude (Anthropic).
We prove rigourously that the ground state of a class of interacting Majorana fermion models and its equivalent spinless fermion models is either unique or doubly degenerate if the lattice size N is even, and is always doubly degenerate if N is odd. Our proof is based on the reflection positivity of the system in the eigen-operator representation and holds in all dimensions with arbitrary lattice structures. Understanding of physical properties of strongly correlated quantum lattice models has long been a challenging problem. Because of the subtlety of interactions, it is generally difficult to solve these models. Rigorous results and exact solutions are clearly useful bench marks, but these are rare. In 1989, Lieb proved that the ground state of the Hubbard model is unique by utilizing a Perron-Frobenius-type argument and the spinreflection positivity of the system[1]. This extended the proof for the uniqueness of the ground state from an antiferromagnetic Heisenberg model[2] to an interacting fermion system and led to a more general application of reflection positivity in quantum flux phases[3], frustrated Heisenberg antiferromagnets[4], and other quantum lattice models[5]. The spin-reflection positivity is intimately connected with the sign rule of the groundstate wave function, which can be used to understand the minus sign problem caused by the fermion characteristics of itinerant electrons in quantum Monte Carlo simulations[5]. In this letter, we present a rigorous proof for two theorems on the degeneracy of ground states for a class of interacting quantum lattice models of two-species Majorana fermions. We will show that the ground state is doubly degenerate when the lattice size N is odd and at most doubly degenerate when N is even. Our proof utilizes a new kind of reflection positivity between the two-species Majorana fermions. These theorems can be applied to a number of interacting spinless fermion models which are equivalent to the Majorana fermion model. It sheds light on the understanding of topological degeneracy and symmetry-protected topological phases of Majorana fermions[6, 7], and the minus sign problem in an interacting spinless fermion model that was recently studied by a number of groups[8–10]. The model studied here is defined on a lattice with N sites,
Lattice regularization of chiral fermions has been a long-standing problem in physics. In this Letter, we present the density matrix renormalization group simulation of the 3-4-5-0 model of (1+1)D chiral fermions with an anomaly-free chiral U(1) symmetry, which contains two left-moving and two right-moving fermions carrying U(1) charges 3,4 and 5,0, respectively. Following the Wang-Wen chiral fermion model, we realize the chiral fermions and their mirror partners on the opposite boundaries of a thin strip of (2+1)D lattice model of multilayer Chern insulator, whose finite width implies the quantum system is effectively (1+1)D. By introducing two sets of carefully designed six-fermion local interactions to the mirror sector only, we demonstrate that the mirror fermions can be gapped out by the interaction beyond a critical strength without breaking the chiral U(1) symmetry, via the symmetric mass generation mechanism. We show that the interaction-driven gapping transition is in the Berezinskii-Kosterlitz-Thouless universality class. We determine the evolution of Luttinger parameters before the transition, which confirms that the transition happens exactly at the point when the interaction term becomes marginal. As the mirror sector is gapped after the transition, we check that the fermions in the light chiral fermion sector remain gapless, which provides the desired lattice regularization of chiral fermions.
Charge conjugation (C), mirror reflection (R), and time reversal (T) symmetries, along with internal symmetries, are essential for massless Majorana and Dirac fermions. These symmetries are sufficient to rule out potential fermion bilinear mass terms, thereby establishing a gapless free fermion fixed point phase, pivotal for symmetric mass generation (SMG) transition. In this work, we systematically study the anomaly of C-R-T-internal symmetry in all spacetime dimensions by analyzing the projective representation (i.e. the fractionalization) of the C-R-T-internal symmetry group in the quantum many-body Hilbert space on the lattice. By discovering the fermion-flavor-number-dependent C-R-T-internal symmetry's anomaly structure, we demonstrate an alternative way to derive the minimal flavor number for SMG, which shows consistency with known results from Kähler-Dirac fermion or cobordism classification. Our findings reveal that, in general spatial dimensions, either 8 copies of staggered Majorana fermions or 4 copies of staggered Dirac fermions admit SMG. By directly searching for 4-fermion interactions that form commuting stabilizers respecting all symmetry constraints, we can prove the explicit SMG gapping retained a unique ground state in the codespace. Furthermore, we establish the correspondence between the symmetry operators of staggered fermions and free fermions, which is instrumental in facilitating the analysis of symmetry fractionalization at the field theory level.
Every subregion of a quantum many-body state carries a canonical antiunitary — the Tomita–Takesaki modular conjugation J, its “modular mirror.” We prove two theorems about these mirrors on the lattice.(R1, the Lattice Bisognano–Wichmann Theorem.) For a ground state that is reflection-positive with respect to a geometric CPT reflection across a cut, the modular conjugation of the half-system is that reflection: J = Θ·R·K — on-site charge conjugation Θ, geometric reflection R, complex conjugation (time reversal) K. The proof is a two-line consequence of the uniqueness of the polar decomposition: reflection positivity makes the cut wavefunction matrix, in the CPT-paired basis, the positive factor of a polar decomposition whose unitary factor is the reflection itself. The modular pairing is extracted from the state —canonically, the transpose of the polar unitary of the cut matrix — and for the XXZ class it factorizes into (site reflection) × (⊗ᵢ iσʸᵢ) at machine precision: the charge conjugation is derived, not assumed. Reflection positivity itself is supplied for thisclass by Gibbs-state reflection positivity in the β → ∞ limit. A sharp converse criterion follows: a state’s modular mirror is geometric iff the polar unitary of its cut matrix factorizes geometrically — generic excited states fail the criterion by O(1).(R2, the Rotation-Character Theorem.) On a ring of N sites with local dimension d, for half-ring regions A and B = T_m A (the translate by m sites), the product of the two modular mirrors is an emergent rotation, J_A J_B = T_{−2m}, and its projected trace — the modular character of the companion papers — is exactly the group character of that rotation:χ(A, T_m A) = Tr[J_A J_B] = Tr[T_{−2m}] = d^{gcd(2m, N)}.The mirrors generate the full dihedral group D_N — the symmetry group of the discrete ring — realizing geometric modular action on the lattice, and the character’s quantized plateaus are representation theory: Lefschetz fixed-point counts of emergent rotations. Verified end-to-end at machine precision, including four parameter-free predictions (64, 729, 4096, 16) and the local-dimension law (d = 3 Haldane chain: 9.000 to twelve digits). Together, R1 and R2 give the criterion its meaning: a state carries emergent geometry precisely when its modular mirrors are spacetime reflections, and then the geometry’s symmetry group— and its characters — are computable.
The sign problem is a major obstacle in quantum Monte Carlo simulations for many-body fermion systems. We examine this problem with a new perspective based on the Majorana reflection positivity and Majorana Kramers positivity. Two sufficient conditions are proven for the absence of the fermion sign problem. Our proof provides a unified description for all the interacting lattice fermion models previously known to be free of the sign problem based on the auxiliary field quantum Monte Carlo method. It also allows us to identify a number of new sign-problem-free interacting fermion models including, but not limited to, lattice fermion models with repulsive interactions but without particle-hole symmetry, and interacting topological insulators with spin-flip terms.
A Parity Signature Distinguishing SO(5)1 from Spin(4)1 Boundaries
The level-one Wess-Zumino-Witten models SO(5)1 and Spin(4)1 share the same total quantum dimension and therefore the same topological entanglement entropy, yet they differ in the parity of their Majorana channel count: five (odd) versus four (even). We show that this even/odd difference is a sharp, falsifiable boundary signature even though the standard entropic diagnostic is blind to it. Under non-abelian bosonization SO(5)1 is five Majorana fermions, and the odd count leaves one mode unpaired and the spinor primary a single Ising-class sector of quantum dimension 2, where the even neighbor Spin(4)1 pairs cleanly and its spinor splits into two abelian sectors. On an explicit coupled-Majorana-chain realization, a class-D Z2 index and a two-parameter phase scan show that a protected unpaired-Majorana parity persists across an entire gapped phase for the odd (n=5) case and is absent for the even (n=4) control. On the interacting bilinear-biquadratic chains at the Reshetikhin point, a charge-conserving entanglement-spectrum computation resolves the two cases directly: the SO(5)1 tower carries the five-fold vector multiplet including the zero-weight state, the Spin(4)1 tower a four-fold shell without it, and the topological entanglement entropy γ=log2 is identical for both and cannot separate them where the parity and channel count can. A direct diagonalization of the genuine twisted sector resolves the spinor multiplet, which appears as the tensor square 4⊗4=1⊕5⊕10 rather than the bare chiral spinor, the expected consequence of the Nielsen-Ninomiya obstruction to a single chiral mode on a non-chiral lattice. We also propose a candidate mechanism in which the open non-abelian sector is completed non-locally through a three-dimensional Chern-Simons bulk; the demonstrated lattice results stand independently of this conjecture.