Built on

C. N. Yang, “Some Exact Results for the Many-Body Problem in one Dimension with Repulsive Delta-Function Interaction,” Physical Review Letters
1967
Earlier work this paper cites.
A. B. Zamolodchikov and A. B. Zamolodchikov, “Factorized S-matrices in two dimensions as the exact solutions of certain relativistic quantum field models,” Annals Phys
1979
Earlier work this paper cites.
E. K. Sklyanin, “Boundary conditions for integrable quantum systems,” J. Phys. A
1988
Earlier work this paper cites.
Springer Berlin Heidelberg, Berlin, Heidelberg, 1992
G. I. Olshanskii, “Twisted yangians and infinite-dimensional classical Lie algebras,” in Quantum Groups · 1992
Earlier work this paper cites.
H. J. de Vega and A. Gonzalez Ruiz, “Boundary K-matrices for the six vertex and the n(2n-1)A n-1
1993
Earlier work this paper cites.
S. Ghoshal and A. B. Zamolodchikov, “Boundary S matrix and boundary state in two-dimensional integrable quantum field theory,” Int. J. Mod. Phys
1994
Earlier work this paper cites.
A. R. Freedman, R. N. Gupta, and R. M. Guralnick, “Shirshov’s theorem and representations of semigroups,” Pacific Journal of Mathematics
1997
Earlier work this paper cites.
A. I. Molev, “Finite-dimensional irreducible representations of twisted Yangians,” Journal of Mathematical Physics
1998
Earlier work this paper cites.
G. W. Delius, “Soliton-preserving boundary condition in affine Toda field theories,” Physics Letters B
1998
Earlier work this paper cites.
A. Doikou, “Quantum spin chain with ‘soliton non-preserving’ boundary conditions,” Journal of Physics A Mathematical General
2000
Earlier work this paper cites.
M. Moriconi, “Integrable boundary conditions and reflection matrices for the O(N) nonlinear sigma model,” Nuclear Physics B
2001
Earlier work this paper cites.
A. Mudrov, “Characters of U q ( g l ( n ) ) U_{q}(gl(n)) -Reflection Equation Algebra,” Letters in Mathematical Physics
2002
Earlier work this paper cites.
North-Holland, 2003
A. Molev, “Yangians and their applications,” vol. 3 of Handbook of Algebra · 2003
Earlier work this paper cites.
J. A. Minahan and K. Zarembo, “The Bethe-ansatz for Script N = 4 super Yang-Mills,” Journal of High Energy Physics
2003
Earlier work this paper cites.
Y. A. Al’pin and K. D. Ikramov, “On the Unitary Similarity of Matrix Families,” Mathematical Notes
2003
Earlier work this paper cites.
D. Arnaudon, J. Avan, N. Crampé, A. Doikou, L. Frappat, and E. Ragoucy, “General boundary conditions for the sl(N) and sl(M|N) open spin chains,” Journal of Statistical Mechanics: Theory and Experiment
2004
Earlier work this paper cites.
O. DeWolfe and N. Mann, “Integrable Open Spin Chains in Defect Conformal Field Theory,” Journal of High Energy Physics
2004
Earlier work this paper cites.
M. Rigol, V. Dunjko, V. Yurovsky, and M. Olshanii, “Relaxation in a completely integrable many-body quantum system: An ab initio study of the dynamics of the highly excited states of 1D lattice hard-core bosons,” Physical Review Letters
2007
Earlier work this paper cites.
D. Perez-Garcia, F. Verstraete, M. M. Wolf, and J. I. Cirac, “Matrix Product State Representations,” Quantum Info. Comput
2007
Earlier work this paper cites.
D. Fioretto and G. Mussardo, “Quantum Quenches in Integrable Field Theories,” New Journal of Physics
Original
2009
Earlier work this paper cites.
M. Kormos and B. Pozsgay, “One-point functions in massive integrable QFT with boundaries,” JHEP
Original
2010
Earlier work this paper cites.
P. Calabrese, F. H. L. Essler, and M. Fagotti, “Quantum Quench in the Transverse-Field Ising Chain,” Phys. Rev. Lett
Original
2011
Earlier work this paper cites.
P. Calabrese, F. H. L. Essler, and M. Fagotti, “Quantum quench in the transverse field Ising chain: I. Time evolution of order parameter correlators,” Journal of Statistical Mechanics: Theory and Experiment
Original
2012
Earlier work this paper cites.

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2018

Integrable Matrix Product States from boundary integrability

Pozsgay, Balázs, Piroli, Lorenzo, Vernier, Eric

Understand

We consider integrable Matrix Product States (MPS) in integrable spin chains and show that they correspond to "operator valued" solutions of the so-called twisted Boundary Yang-Baxter (or reflection) equation.

We argue that the integrability condition is equivalent to a new linear intertwiner relation, which we call the "square root relation", because it involves half of the steps of the reflection equation.
It is then shown that the square root relation leads to the full Boundary Yang-Baxter equations.
We provide explicit solutions in a number of cases characterized by special symmetries.

Built on

C. N. Yang, “Some Exact Results for the Many-Body Problem in one Dimension with Repulsive Delta-Function Interaction,” Physical Review Letters
1967
Earlier work this paper cites.
A. B. Zamolodchikov and A. B. Zamolodchikov, “Factorized S-matrices in two dimensions as the exact solutions of certain relativistic quantum field models,” Annals Phys
1979
Earlier work this paper cites.
E. K. Sklyanin, “Boundary conditions for integrable quantum systems,” J. Phys. A
1988
Earlier work this paper cites.
Springer Berlin Heidelberg, Berlin, Heidelberg, 1992
G. I. Olshanskii, “Twisted yangians and infinite-dimensional classical Lie algebras,” in Quantum Groups · 1992
Earlier work this paper cites.
H. J. de Vega and A. Gonzalez Ruiz, “Boundary K-matrices for the six vertex and the n(2n-1)A n-1
1993
Earlier work this paper cites.
S. Ghoshal and A. B. Zamolodchikov, “Boundary S matrix and boundary state in two-dimensional integrable quantum field theory,” Int. J. Mod. Phys
1994
Earlier work this paper cites.
A. R. Freedman, R. N. Gupta, and R. M. Guralnick, “Shirshov’s theorem and representations of semigroups,” Pacific Journal of Mathematics
1997
Earlier work this paper cites.
A. I. Molev, “Finite-dimensional irreducible representations of twisted Yangians,” Journal of Mathematical Physics
1998
Earlier work this paper cites.
G. W. Delius, “Soliton-preserving boundary condition in affine Toda field theories,” Physics Letters B
1998
Earlier work this paper cites.
A. Doikou, “Quantum spin chain with ‘soliton non-preserving’ boundary conditions,” Journal of Physics A Mathematical General
2000
Earlier work this paper cites.
M. Moriconi, “Integrable boundary conditions and reflection matrices for the O(N) nonlinear sigma model,” Nuclear Physics B
2001
Earlier work this paper cites.
A. Mudrov, “Characters of U q ( g l ( n ) ) U_{q}(gl(n)) -Reflection Equation Algebra,” Letters in Mathematical Physics
2002
Earlier work this paper cites.
North-Holland, 2003
A. Molev, “Yangians and their applications,” vol. 3 of Handbook of Algebra · 2003
Earlier work this paper cites.
J. A. Minahan and K. Zarembo, “The Bethe-ansatz for Script N = 4 super Yang-Mills,” Journal of High Energy Physics
2003
Earlier work this paper cites.
Y. A. Al’pin and K. D. Ikramov, “On the Unitary Similarity of Matrix Families,” Mathematical Notes
2003
Earlier work this paper cites.
D. Arnaudon, J. Avan, N. Crampé, A. Doikou, L. Frappat, and E. Ragoucy, “General boundary conditions for the sl(N) and sl(M|N) open spin chains,” Journal of Statistical Mechanics: Theory and Experiment
2004
Earlier work this paper cites.
O. DeWolfe and N. Mann, “Integrable Open Spin Chains in Defect Conformal Field Theory,” Journal of High Energy Physics
2004
Earlier work this paper cites.
M. Rigol, V. Dunjko, V. Yurovsky, and M. Olshanii, “Relaxation in a completely integrable many-body quantum system: An ab initio study of the dynamics of the highly excited states of 1D lattice hard-core bosons,” Physical Review Letters
2007
Earlier work this paper cites.
D. Perez-Garcia, F. Verstraete, M. M. Wolf, and J. I. Cirac, “Matrix Product State Representations,” Quantum Info. Comput
2007
Earlier work this paper cites.
D. Fioretto and G. Mussardo, “Quantum Quenches in Integrable Field Theories,” New Journal of Physics
Original
2009
Earlier work this paper cites.
M. Kormos and B. Pozsgay, “One-point functions in massive integrable QFT with boundaries,” JHEP
Original
2010
Earlier work this paper cites.
P. Calabrese, F. H. L. Essler, and M. Fagotti, “Quantum Quench in the Transverse-Field Ising Chain,” Phys. Rev. Lett
Original
2011
Earlier work this paper cites.
P. Calabrese, F. H. L. Essler, and M. Fagotti, “Quantum quench in the transverse field Ising chain: I. Time evolution of order parameter correlators,” Journal of Statistical Mechanics: Theory and Experiment
Original
2012
Earlier work this paper cites.