arXiv subjects

Pramod Padmanabhan

Publications and source records attributed to Pramod Padmanabhan.

At least 19 recordsLinked to original sources

arXiv2026

Multiparameter Quantum Affine Spaces and the Scalene Yang--Baxter Equation

We construct representation-independent families of solutions of the non-braided scalene Yang--Baxter equation from quadratic noncommutative algebras. Beginning with two anticommuting generators, we obtain a continuous deformation in terms of quantum-plane algebras and extend the construction to an arbitrary number of generators. In the latter case the scalene Yang--Baxter relation fixes the multiparameter exchange matrix to an exact multiplicative form, thereby selecting a distinguished subclass of multiparameter quantum affine spaces. We construct finite-dimensional realizations using singular matrices and finite Heisenberg--Weyl operators, as well as infinite-dimensional realizations in terms of bilateral weighted shifts and multiplicative-shift operators. The explicit realizations are generically \emph{purely scalene}: although the ordered triple satisfies the scalene Yang--Baxter equation, its individual constituent operators do not satisfy the ordinary non-braided Yang--Baxter equation. These results provide a representation-independent algebraic framework for constructing Yang--Baxter-irreducible scalene triples and a starting point for investigating their possible applications to quantum integrability.

nlin.SI↗

arXiv2026

Scalene Yang--Baxter triples as a source of hidden symmetries beyond the ordinary Yang--Baxter equation

We study a nearest-neighbor non-Hermitian spin chain obtained from one member of an exact non-braided scalene Yang--Baxter triple. Its local Hamiltonian density violates both the difference-form Reshetikhin condition and its general non-difference counterpart, obstructing its realization by a differentiable homogeneous regular solution of the ordinary Yang--Baxter equation. The transfer matrices constructed from the regular member do not commute among themselves at distinct spectral parameters. Nevertheless, the scalene Yang--Baxter relation implies cross-commutativity with another transfer matrix constructed from the third member of the scalene triple. We evaluate the latter for arbitrary chain length and show that, on even periodic chains, it is a finite generating function of a non-obvious staggered nilpotent symmetry. The resulting conserved hierarchy belongs entirely to the algebra generated by this single symmetry and hence does not constitute an extensive family of algebraically independent charges. Nevertheless, this example demonstrates that scalene Yang--Baxter triples can act as an algebraic symmetry-discovery mechanism beyond the ordinary self-commuting transfer-matrix framework.

cond-mat.stat-mech↗

arXiv2026

Symmetries of the Generalized Yang--Baxter Equations

The generalized Yang-Baxter equations are multi-site versions of the standard Yang-Baxter equation. When spectral parameters are included, such equations are expected to lead to integrable Hamiltonians with local interactions involving multiple degrees of freedom. In this work we characterize both the continuous and discrete symmetries of these equations required to establish an equivalence class of solutions. We find that the set of such symmetries depends on the number of sites on which the equation is supported. In several cases there are more symmetries than the standard Yang-Baxter equation, thus placing heavy constraints on the number of inequivalent solutions and the associated integrable models. As an application we show how to restrict the generic $16\times16$ ansatz to the 30-dimensional subspace invariant under a chosen discrete symmetry subgroup. An explicit $16\times 16$ solution is also written down.

nlin.SI↗

arXiv2026

Hidden Ising models from the generalized Yang-Baxter equation

We introduce a one dimensional spin $\frac{1}{2}$ Hamiltonian with multi-site interactions, but still local. The algebra of its Hamiltonian densities resembles that of the transverse field Ising model. Using this fact we show that its spectrum is free-fermionic but with a huge degeneracy for each level. The source of the degeneracy is a set of local conserved quantities that act like a classical background field for the quantum system. The thermodynamics of this system is contrasted with the standard Ising model. At the gapless points in the energy spectrum, we show that this system can be derived from the quantum inverse scattering method adapted to a multi-site generalization of the Yang-Baxter equation as introduced by E. Rowell and Z. Wang. The $R$-matrix is constructed using generators of extraspecial 2-groups. This helps us extract all the conserved charges and lay the framework for a general mechanism to generate such multi-site interaction spin systems that are transverse field Ising models under the hood. A remark on how to obtain P. Fendley's free-fermion in disguise models in this formalism is also included.

cond-mat.stat-mech↗

arXiv2025

Noninvertible Kramers-Wannier duality symmetries for the discrete-time quantum Ising chain

Integrable trotterization} provides a method to evolve a continuous time integrable many-body system in discrete time, such that it retains its conserved quantities. Here we explicitly show that the first order trotterization of the critical {\it transverse field Ising model} is integrable. The discrete time conserved quantities are obtained from an inhomogeneous transfer matrix constructed using the {\it quantum inverse scattering method}. The inhomogeneity parameter determines the discrete time step. We then focus on the non-invertible {\it Kramers-Wannier} duality-symmetry for the trotterized evolution. We find that the discretization of both space and time leads to a doubling of these duality operators. They account for discrete translations in both space and time. As an interesting application, we find that these operators also provide maps between trotterizations of different orders. This helps us extend our results beyond the trotterization scheme and investigate the Kramers-Wannier duality-symmetry for finite time Floquet evolution of the critical transverse field Ising chain. {Finally, we investigate how these non-invertible operators shape the phase diagram of the discrete-time evolution. This question is particularly interesting in the Floquet setting, which is known to host a richer phase structure than its undriven counterpart. We systematically construct the necessary operators which relate different phases away from criticality for both trotterized and Floquet evolutions.

quant-ph↗

arXiv2025

Clifford Solver for the Tetrahedron Equation and its Variants

The different forms of the tetrahedron equation appear when all possible ways to label the scattering process of infinitely long straight lines are considered in three dimensional spacetime. This is expected to lead to three dimensional integrability, analogous to the Yang-Baxter equation. Among the three possibilities, we consider two of them and their variants. We show that Clifford algebras solve both the constant and the spectral parameter dependent versions of all of them. We also present a scheme for canonically solving higher simplex equations using tetrahedron solutions.

hep-th↗

arXiv2025

The Yang-Baxter integrability of the critical Ising chain

We show that the one dimensional, critical transverse field Ising model is Yang-Baxter integrable. This is done by constructing commuting transfer matrices built out of a $R$-matrix satisfying the Yang-Baxter equation with additive spectral parameters. The $R$-matrix is non-local, as it is expressed in terms of Majorana fermions. It is also non-regular. Nevertheless, we show that the quantum inverse scattering method can still be suitably adapted. We then recursively obtain the conserved quantities [in the infinite volume] by the boost operator method. Remarkably, among the conserved charges we also find the Kramers-Wannier duality and other non-invertible symmetries for the periodic transverse field Ising model.

hep-th↗

arXiv2025

Almost local integrable models from supersymmetry algebras

Supersymmetry algebras can be used to obtain algebraic expressions for constant Yang-Baxter solutions, also known as braid group generators. This was done for non-invertible braid operators in \cite{maity2025non}. In this work we extend this construction for the invertible ones. The resulting expressions are then shown to obey relations analogous to those satisfied by quotients of braid groups. Examples of the latter include the Iwahori-Hecke algebra and the Birman-Murakami-Wenzl (BMW) algebra. As a result, we can Baxterize the constant Yang-Baxter solutions to yield spectral parameter dependent $R$-matrices. The regularity of these $R$-matrices depends on the representation of SUSY generators. In some cases they are regular in the usual sense and in the remaining they are `almost' regular. In the latter case they are also non-invertible. Nevertheless, we show that they can still help us construct integrable models in all dimensions of the local Hilbert space. These models can be described by Hamiltonian densities that are either local or non-local, depending on the representation chosen for the SUSY generators. We demonstrate this for all constant $4\times 4$ invertible Yang-Baxter solutions. Apart from finding new nearest-neighbor interaction spin $\frac{1}{2}$ systems, we also find their higher spin analogs due to the algebraic [representation independent] approach.

hep-th↗

arXiv2025

Majorana fermions solve the tetrahedron equations as well as higher simplex equations

Yang-Baxter equations define quantum integrable models. The tetrahedron and higher simplex equations are multi-dimensional generalizations. Finding the solutions of these equations is a formidable task. In this work we develop a systematic method - constructing higher simplex operators [solutions of corresponding simplex equations] from lower simplex ones. We call it lifting. By starting from solutions of Yang-Baxter equations we can construct solutions of the tetrahedron equation and simplex equation in any dimension. We then generalize this by starting from a solution of any lower simplex equation and lifting it [construct solution] to another simplex equation in higher dimension. This process introduces several constraints among the different lower simplex operators that are lifted to form the higher simplex operators. We show that braided Yang-Baxter operators [solutions of Yang-Baxter equations independent of spectral parameters] constructed using Majorana fermions satisfy these constraints, thus solving the higher simplex equations. As a consequence these solutions help us understand the action of an higher simplex operator on Majorana fermions. Apart from these we show that solutions constructed using Dirac (complex) fermions and Clifford algebras also satisfy these constraints. Furthermore it is observed that the Clifford solutions give rise to positive Boltzmann weights resulting in the possibility of physical statistical mechanics models in higher dimensions. Finally we also show that anti-Yang-Baxter operators [solutions of Yang-Baxter-like equations with a negative sign on the right hand side] can also be lifted to higher simplex solutions.

hep-th↗

arXiv2025

Non-hermitian integrable systems from constant non-invertible solutions of the Yang-Baxter equation

We construct invertible spectral parameter dependent Yang-Baxter solutions ($R$-matrices) by Baxterizing constant non-invertible Yang-Baxter solutions. The solutions are algebraic (representation independent). They are constructed using supersymmetry (SUSY) algebras. The resulting $R$-matrices are regular leading to local non-hermitian Hamiltonians written in terms of the SUSY generators. As particular examples we Baxterize the $4\times 4$ constant non-invertible solutions of Hietarinta leading to nearest-neighbor Hamiltonians. On comparing with the literature we find two of the models are new. Apart from being non-hermitian, many of them are also non-diagonalizable with interesting spectrums. With appropriate representations of the SUSY generators we obtain spin chains in all local Hilbert space dimensions.

hep-th↗

arXiv2025

Non-invertible symmetry breaking in a frustration-free spin chain

A nearest-neighbor, frustration-free spin $\frac{1}{2}$ chain can be constructed {\it via} projectors of various ranks á la Bravyi-Gosset. We show that in the rank 1 case this system is gapped and has two ground states resembling ferromagnetic states. These states spontaneously break the non-invertible symmetry connecting them. The latter is proved using the machinery of algebraic quantum theory. The non-invertible symmetries of this system do not come from a duality.

hep-th↗

arXiv2025

Algebraic classification of Hietarinta's solutions of Yang-Baxter equations~:~invertible $4\times 4$ operators

In order to examine the simulation of integrable quantum systems using quantum computers, it is crucial to first classify Yang-Baxter operators. Hietarinta was among the first to classify constant Yang-Baxter solutions for a two-dimensional local Hilbert space (qubit representation). Including the one produced by the permutation operator, he was able to construct eleven families of invertible solutions. These techniques are effective for 4 by 4 solutions, but they become difficult to use for representations with more dimensions. To get over this limitation, we use algebraic ansätze to generate the constant Yang-Baxter solutions in a representation independent way. We employ four distinct algebraic structures that, depending on the qubit representation, replicate 10 of the 11 Hietarinta families. Among the techniques are partition algebras, Clifford algebras, Temperley-Lieb algebras, and a collection of commuting operators. Using these techniques, we do not obtain the $(2,2)$ Hietarinta class.

hep-th↗

arXiv2024

A stabilizer code model with non-invertible symmetries: Strange fractons, confinement, and non-commutative and non-Abelian fusion rules

We introduce a stabilizer code model with a qutrit at every edge on a square lattice and with non-invertible plaquette operators. The degeneracy of the ground state is topological as in the toric code, and it also has the usual deconfined excitations consisting of pairs of electric and magnetic charges. However, there are novel types of confined fractonic excitations composed of a cluster of adjacent faces with vanishing flux. They manifest confinement, and even larger configurations of these fractons are fully immobile although they acquire emergent internal degrees of freedom. Deconfined excitations change their nature in presence of these fractonic defects. As for instance, fractonic defects can absorb magnetic charges making magnetic monopoles exist while electric charges acquire restricted mobility. Furthermore, some generalized symmetries can annihilate any ground state and also the full sector of fully mobile excitations. All these properties can be captured via a novel type of \textit{non-commutative} and \textit{non-Abelian} fusion category in which the product is associative but does not commute, and can be expressed as a sum of (operator) equivalence classes. Generalized non-invertible symmetries give rise to the feature that the fusion products form a non-unital category without a proper identity. We show that a variant of this model features a deconfined fracton liquid phase and a phase where the dual (magnetic) strings have condensed.

hep-th↗

arXiv2024

Unitary tetrahedron quantum gates

Quantum simulations of many-body systems using 2-qubit Yang-Baxter gates offer a benchmark for quantum hardware. This can be extended to the higher dimensional case with $n$-qubit generalisations of Yang-Baxter gates called $n$-simplex operators. Such multi-qubit gates potentially lead to shallower and more efficient quantum circuits as well. Finding them amounts to identifying unitary solutions of the $n$-simplex equations, the building blocks of higher dimensional integrable systems. These are a set of highly non-linear and over determined system of equations making it notoriously hard to solve even when the local Hilbert spaces are spanned by qubits. We systematically overcome this for higher simplex operators constructed using two methods: from Clifford algebras and by lifting Yang-Baxter operators. The $n=3$ or the tetrahedron case is analyzed in detail. For the qubit case our methods produce 13 inequivalent families of unitary tetrahedron operators. 12 of these families are obtained by appending the 5 unitary families of 4 by 4 constant Yang-Baxter operators of Dye-Hietarinta, with a single qubit operator. As applications, universal sets of single, two and three qubit gates are realized using such unitary tetrahedron operators. The ideas presented in this work can be naturally extended to the higher simplex cases.

quant-ph↗

arXiv2024

Toffoli gates solve the tetrahedron equations

The circuit model of quantum computation can be interpreted as a scattering process. In particular, factorised scattering operators result in integrable quantum circuits that provide universal quantum computation and are potentially less noisy. These are realized through Yang-Baxter or 2-simplex operators. A natural question is to extend this construction to higher qubit gates, like the Toffoli gates, which also lead to universal quantum computation but with shallower circuits. We show that unitary families of such operators are constructed by the 3-dimensional generalizations of the Yang-Baxter operators known as tetrahedron or 3-simplex operators. The latter satisfy a spectral parameter-dependent tetrahedron equation. This construction goes through for $n$-Toffoli gates realized using $n$-simplex operators.

quant-ph↗

arXiv2024

Solving the Yang-Baxter, tetrahedron and higher simplex equations using Clifford algebras

Bethe Ansatz was discoverd in 1932. Half a century later its algebraic structure was unearthed: Yang-Baxter equation was discovered, as well as its multidimensional generalizations [tetrahedron equation and $d$-simplex equations]. Here we describe a universal method to solve these equations using Clifford algebras. The Yang-Baxter equation ($d=2$), Zamalodchikov's tetrahedron equation ($d=3$) and the Bazhanov-Stroganov equation ($d=4$) are special cases. Our solutions form a linear space. This helps us to include spectral parameters. Potential applications are discussed.

hep-th↗

arXiv2024

Yang-Baxter solutions from commuting operators

We construct $R$-matrices (with a multidimensional spectral parameter) that include additive as well as non-additive parameters. They satisfy the colored Yang-Baxter equation. The solutions depend on a set of commuting operators. They change with the representation of the latter. The associated Yang-Baxter algebra and the spectrum of the transfer matrix are also studied.

hep-th↗

arXiv2024

Disorder-free localisation in continuous-time quantum walks : Role of symmetries

We investigate the phenomenon of disorder-free localisation in quantum systems with global permutation symmetry. We use permutation group theory to systematically construct permutation symmetric many-fermion Hamiltonians and interpret them as generators of continuous-time quantum walks. When the number of fermions is very large we find that all the canonical basis states localise at all times, without the introduction of any disorder coefficients. This time-independent localisation is not the result of any emergent disorder distinguishing it from existing mechanisms for disorder-free localisation. Next we establish the conditions under which the localisation is preserved. We find that interactions that preserve and break the global permutation symmetry sustains localisation. Furthermore the basis states of systems with reduced permutation symmetry, localise even for a small number of fermions when the symmetry-reducing parameters are tuned accordingly. We show that similar localisation also occurs for a permutation symmetric Heisenberg spin chain and permutation symmetric bosonic systems, implying that the localisation is independent of the superselected symmetry. Finally we make connections of the Hamiltonians studied here to the adjacency matrices of graphs and use this to propose a prescription for disorder-free localisation in continuous-time quantum walk systems. Many of the models proposed here feature all-to-all connectivity and can be potentially realised on superconducting quantum circuits, trapped ion systems and ultracold atoms.

quant-ph↗