Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:1011.6386 · Journal of Statistical Mechanics Theory and Experiment

Ground-state properties of a supersymmetric fermion chain

Paul Fendley, Christian Hagendorf

Published Feb 7, 2011• 23 citations
arXivHTMLPDF

Abstract

We analyze the ground state of a strongly interacting fermion chain with a supersymmetry. We conjecture a number of exact results, such as a hidden duality between weak and strong couplings. By exploiting a scale-free property of the perturbative expansions, we find exact expressions for the order parameters, yielding the critical exponents. We show that the ground state of this fermion chain and another model in the same universality class, the XYZ chain along a line of couplings, are both written in terms of the same polynomials. We demonstrate this explicitly for up to N = 24 sites and provide consistency checks for large N . These polynomials satisfy a recursion relation related to the Painlevé VI differential equation and, using a scale-free property of these polynomials, we derive a simple and exact formula for their limit.

DOI
10.1088/1742-5468/2011/02/p02014

Related papers

A staggered fermion chain with supersymmetry on open intervals

Matteo Beccaria, Christian Hagendorf

A strongly interacting fermion chain with supersymmetry on the lattice and open boundary conditions is analysed. The local coupling constants of the model are staggered, and the properties of the ground states as a function of the staggering parameter are examined. In particular, a connection between certain ground-state components and solutions of nonlinear recursion relations associated with the Painlevé VI equation is conjectured. Moreover, various local occupation probabilities in the ground state have the so-called scale-free property, and allow for an exact resummation in the limit of infinite-system size.

arXiv:1206.4194•Journal of Physics A Mathematical and Theoretical•Aug 20, 2012•14 citations

Exact and simple results for the XYZ and strongly interacting fermion chains

Paul Fendley, Christian Hagendorf

We conjecture exact and simple formulas for some physical quantities in two quantum chains. A classic result of this type is Onsager, Kaufman and Yang's formula for the spontaneous magnetization in the Ising model, subsequently generalized to the chiral Potts models. We conjecture that analogous results occur in the XYZ chain when the couplings obey J x J y + J y J z + J x J z = 0, and in a related fermion chain with strong interactions and supersymmetry. We find exact formulas for the magnetization and gap in the former, and the staggered density in the latter, by exploiting the fact that certain quantities are independent of finite-size effects.

arXiv:1006.0237•Journal of Physics A Mathematical and Theoretical•Sep 9, 2010•30 citations

Nearest-Neighbour Correlation Functions for the Supersymmetric XYZ Spin Chain and Painlevé VI

Christian Hagendorf, Hjalmar Rosengren

Abstract We study nearest-neighbour correlation functions for the ground state of the supersymmetric XYZ spin chain with odd length and periodic boundary conditions. Under a technical assumption related to the Q-operator of the corresponding eight-vertex model, we show that they can be expressed exactly in terms of the Painlevé VI tau functions $$s_n$$ s n and $${{\bar{s}}}_n$$ s ¯ n introduced by Bazhanov and Mangazeev. Furthermore, we give an interpretation of the correlation functions in terms of the Painlevé VI Hamiltonian.

arXiv:2208.13533•Communications in Mathematical Physics•Apr 1, 2024•1 citation

The Eight-Vertex Model and Lattice Supersymmetry

Christian Hagendorf, Paul Fendley

We show that the XYZ spin chain along the special line of couplings J x J y +J x J z +J y J z =0 possesses a hidden $\\mathcal{N}=(2,2)$ supersymmetry. This lattice supersymmetry is non-local and changes the number of sites. It extends to the full transfer matrix of the corresponding eight-vertex model. In particular, it is shown how to derive the supercharges from Baxter's Bethe ansatz. This analysis leads to new conjectures concerning the ground state for chains of odd length. We also discuss a correspondence between the spectrum of this XYZ chain and that of a manifestly supersymmetric staggered fermion chain

•Jan 1, 2012•17 citations

Exact Ground State and Finite-Size Scaling in a Supersymmetric Lattice Model

Matteo Beccaria A, Gian Fabrizio De Angelis

We study a model of strongly correlated fermions in one dimension with extended N = 2 supersymmetry. The model is related to the spin S = 1/2 XXZ Heisenberg chain at anisotropy Delta = -1/2 with a real magnetic field on the boundary. We exploit the combinatorial properties of the ground state to determine its exact wave function on finite lattices with up to 30 sites. We compute several correlation functions of the fermionic and spin fields. We discuss the continuum limit by constructing lattice observables with well defined finite-size scaling behavior. For the fermionic model with periodic boundary conditions we give the emptiness formation probability in closed form.

arXiv:cond-mat/0407752•Physical Review Letters•Mar 14, 2005•33 citations

A novel translationally invariant supersymmetric chain with inverse-square interactions: partition function, thermodynamics and criticality

B. Basu-Mallick, Federico Finkel, Artemio González-López

Abstract We introduce a novel family of translationally invariant su ( m | n ) supersymmetric spin chains with long-range interaction not directly associated with a root system. We study the symmetries of this model, establishing in particular the existence of a boson–fermion duality characteristic of this type of system. Taking advantage of the relation of the new chains with an associated many-body supersymmetric spin dynamical model, we are able to compute their partition function in closed form for all values of m and n and for an arbitrary number of spins. When both m and n are even, we show that the partition function factorizes as the product of the partition functions of two supersymmetric Haldane–Shastry spin chains, which in turn leads to a simple expression for the thermodynamic free energy per spin in terms of the Perron eigenvalue of a suitable transfer matrix. We use this expression to study the thermodynamics of a large class of these chains, showing in particular that the specific heat presents a single Schottky peak at approximately the same temperature as a suitable k-level model. We also analyze the critical behavior of the new chains, and in particular, the ground-state degeneracy and the existence of low-energy excitations with a linear energy–momentum dispersion relation. In this way, we show that the only possible critical chains are those with m = 0 , 1 , 2 . In addition, using the explicit formula for the partition function we are able to establish the criticality of the su ( 0 | n ) and su ( 2 | n ) chains with even n, and to evaluate the central charge of their associated conformal field theory.

arXiv:2407.13827•Journal of Statistical Mechanics Theory and Experiment•Nov 8, 2024•1 citation
We analyze the ground state of a strongly interacting fermion chain with a supersymmetry. We conjecture a number of exact results, such as a hidden duality between weak and strong couplings. By exploiting a scale-free property of the perturbative expansions, we find exact expressions for the order parameters, yielding the critical exponents. We show that the ground state of this fermion chain and another model in the same universality class, the XYZ chain along a line of couplings, are both written in terms of the same polynomials. We demonstrate this explicitly for up to N = 24 sites and provide consistency checks for large N . These polynomials satisfy a recursion relation related to the Painlevé VI differential equation and, using a scale-free property of these polynomials, we derive a simple and exact formula for their limit.
A strongly interacting fermion chain with supersymmetry on the lattice and open boundary conditions is analysed. The local coupling constants of the model are staggered, and the properties of the ground states as a function of the staggering parameter are examined. In particular, a connection between certain ground-state components and solutions of nonlinear recursion relations associated with the Painlevé VI equation is conjectured. Moreover, various local occupation probabilities in the ground state have the so-called scale-free property, and allow for an exact resummation in the limit of infinite-system size.
We conjecture exact and simple formulas for some physical quantities in two quantum chains. A classic result of this type is Onsager, Kaufman and Yang's formula for the spontaneous magnetization in the Ising model, subsequently generalized to the chiral Potts models. We conjecture that analogous results occur in the XYZ chain when the couplings obey J x J y + J y J z + J x J z = 0, and in a related fermion chain with strong interactions and supersymmetry. We find exact formulas for the magnetization and gap in the former, and the staggered density in the latter, by exploiting the fact that certain quantities are independent of finite-size effects.
Abstract We study nearest-neighbour correlation functions for the ground state of the supersymmetric XYZ spin chain with odd length and periodic boundary conditions. Under a technical assumption related to the Q-operator of the corresponding eight-vertex model, we show that they can be expressed exactly in terms of the Painlevé VI tau functions sns_nsn​ s n and sˉn{{\bar{s}}}_nsˉn​ s ¯ n introduced by Bazhanov and Mangazeev. Furthermore, we give an interpretation of the correlation functions in terms of the Painlevé VI Hamiltonian.
We show that the XYZ spin chain along the special line of couplings J x J y +J x J z +J y J z =0 possesses a hidden mathcalN=(2,2)\\mathcal{N}=(2,2)mathcalN=(2,2) supersymmetry. This lattice supersymmetry is non-local and changes the number of sites. It extends to the full transfer matrix of the corresponding eight-vertex model. In particular, it is shown how to derive the supercharges from Baxter's Bethe ansatz. This analysis leads to new conjectures concerning the ground state for chains of odd length. We also discuss a correspondence between the spectrum of this XYZ chain and that of a manifestly supersymmetric staggered fermion chain
We study a model of strongly correlated fermions in one dimension with extended N = 2 supersymmetry. The model is related to the spin S = 1/2 XXZ Heisenberg chain at anisotropy Delta = -1/2 with a real magnetic field on the boundary. We exploit the combinatorial properties of the ground state to determine its exact wave function on finite lattices with up to 30 sites. We compute several correlation functions of the fermionic and spin fields. We discuss the continuum limit by constructing lattice observables with well defined finite-size scaling behavior. For the fermionic model with periodic boundary conditions we give the emptiness formation probability in closed form.
Abstract We introduce a novel family of translationally invariant su ( m | n ) supersymmetric spin chains with long-range interaction not directly associated with a root system. We study the symmetries of this model, establishing in particular the existence of a boson–fermion duality characteristic of this type of system. Taking advantage of the relation of the new chains with an associated many-body supersymmetric spin dynamical model, we are able to compute their partition function in closed form for all values of m and n and for an arbitrary number of spins. When both m and n are even, we show that the partition function factorizes as the product of the partition functions of two supersymmetric Haldane–Shastry spin chains, which in turn leads to a simple expression for the thermodynamic free energy per spin in terms of the Perron eigenvalue of a suitable transfer matrix. We use this expression to study the thermodynamics of a large class of these chains, showing in particular that the specific heat presents a single Schottky peak at approximately the same temperature as a suitable k-level model. We also analyze the critical behavior of the new chains, and in particular, the ground-state degeneracy and the existence of low-energy excitations with a linear energy–momentum dispersion relation. In this way, we show that the only possible critical chains are those with m = 0 , 1 , 2 . In addition, using the explicit formula for the partition function we are able to establish the criticality of the su ( 0 | n ) and su ( 2 | n ) chains with even n, and to evaluate the central charge of their associated conformal field theory.