Vladimir Korepin

Vladimir E. Korepin (born 1951) is a professor at the C. N. Yang Institute of Theoretical Physics of the Stony Brook University. Korepin made research contributions in several areas of mathematics and physics.

Vladimir Korepin
Born (1951-02-06) February 6, 1951
Russian SFSR
Alma materSaint Petersburg State University
Known forIzergin-Korepin model
Quantum determinant
Yang action
Scientific career
FieldsTheoretical Physics, Mathematics
InstitutionsStony Brook University
Doctoral advisorLudwig Faddeev
Notable studentsSamson Shatashvilli
Fabian Essler
Vitaly Tarasov

Educational background

Korepin completed his undergraduate study at Saint Petersburg State University, graduating with a diploma in theoretical physics in 1974.[1] In that same year he was employed by the Mathematical Institute of Academy of Sciences of Russia. He worked there until 1989, obtaining his PhD in 1977 under the supervision of Ludwig Faddeev. At the same institution he completed his postdoctoral studies. In 1985, he received a doctor of sciences degree in mathematical physics from the Council of Ministers of the Russian Soviet Federative Socialist Republic.

Contributions to physics

Korepin has made contributions to several fields of theoretical physics. Although he is best known for his involvement in condensed matter physics and mathematical physics, he significantly contributed to quantum gravity as well. In recent years, his work has focused on aspects of condensed matter physics relevant for quantum information.

Condensed matter

Among his contributions to condensed matter physics, we mention his studies on low-dimensional quantum gases. In particular, the 1D Hubbard model of strongly correlated fermions,[2] and the 1D Bose gas with delta potential interactions.[3]

In 1979, Korepin presented a solution of the massive Thirring model in one space and one time dimension using the Bethe ansatz, first published in Russian[4] and then translated in English.[5] In this work, he provided the exact calculation of the mass spectrum and the scattering matrix.

He studied solitons in the sine-Gordon model. He determined their mass and scattering matrix, both semiclassically and to one loop corrections.[6]

Together with Anatoly Izergin, he discovered the 19-vertex model (sometimes called the Izergin-Korepin model).[7]

In 1993, together with A. R. Its, Izergin and N. A. Slavnov, he calculated space, time and temperature dependent correlation functions in the XX spin chain. The exponential decay in space and time separation of the correlation functions was calculated explicitly.[8]

Quantum gravity

In this field, Korepin has worked on the cancellation of ultra-violet infinities in one loop on mass shell gravity.[9][10]

Contributions to mathematics

In 1982, Korepin introduced domain wall boundary conditions for the six vertex model, published in Communications in Mathematical Physics.[11] The result plays a role in diverse fields of mathematics such as algebraic combinatorics, alternating sign matrices, domino tiling, Young diagrams and plane partitions. In the same paper the determinant formula was proved for the square of the norm of the Bethe ansatz wave function. It can be represented as a determinant of linearized system of Bethe equations. It can also be represented as a matrix determinant of second derivatives of the Yang action.

The so-called "Quantum Determinant" was discovered in 1981 by A.G. Izergin and V.E. Korepin.[12] It is the center of the Yang–Baxter algebra.

The study of differential equations for quantum correlation functions led to the discovery of a special class of Fredholm integral operators. Now they are referred to as completely integrable integral operators.[13] They have multiple applications not only to quantum exactly solvable models, but also to random matrices and algebraic combinatorics.

Contributions to quantum information and quantum computation

Vladimir Korepin has produced results in the evaluation of the entanglement entropy of different dynamical models, such as interacting spins, Bose gases, and the Hubbard model.[14] He considered models with a unique ground states, so that the entropy of the whole ground state is zero. The ground state is partitioned into two spatially separated parts: the block and the environment. He calculated the entropy of the block as a function of its size and other physical parameters. In a series of articles,[15][16][17][18][19] Korepin was the first to compute the analytic formula for the entanglement entropy of the XX (isotropic) and XY Heisenberg models. He used Toeplitz Determinants and Fisher-Hartwig Formula for the calculation. In the Valence-Bond-Solid states (which is the ground state of the Affleck-Kennedy-Lieb-Tasaki model of interacting spins), Korepin evaluated the entanglement entropy and studied the reduced density matrix.[20][21] He also worked on quantum search algorithms with Lov Grover.[22][23] Many of his publications on entanglement and quantum algorithms can be found on ArXiv.[24]

In May 2003, Korepin helped organize a conference on quantum and reversible computations in Stony Brook.[25] Another conference was on November 15–18, 2010, entitled the Simons Conference on New Trends in Quantum Computation.[26]

Books
  • Essler, F. H. L.; Frahm, H., Goehmann, F., Kluemper, A., & Korepin, V. E., The One-Dimensional Hubbard Model. Cambridge University Press (2005).
  • V.E. Korepin, N.M. Bogoliubov and A.G. Izergin, Quantum Inverse Scattering Method and Correlation Functions, Cambridge University Press (1993).
  • Exactly Solvable Models of Strongly Correlated Electrons. Reprint volume, eds. F.H.L. Essler and V.E. Korepin, World Scientific (1994).

Honours

References
  1. "Cancellation of ultra-violet infinities in one loop gravity" (PDF). Retrieved August 28, 2010. (Korepin's graduation thesis)
  2. Essler, F. H. L.; Frahm, H.; Goehmann, F.; Kluemper, A.; Korepin, V. E. (2005). The One-Dimensional Hubbard Model. Cambridge University Press. ISBN 978-0-521-80262-8.]
  3. Korepin, V. E. (1993). Quantum Inverse Scattering Method and Correlation Functions. Cambridge University Press. doi:10.2277/0521586461. ISBN 978-0-521-58646-7. Retrieved January 12, 2012.
  4. "V. E. Korepin. Theoretical and Mathematical Physics, 41, 169 (1979)". Mathnet.ru. December 28, 1978. Retrieved January 12, 2012.
  5. Korepin, V. E. (1979). "Direct calculation of the S matrix in the massive thirring model". Theoretical and Mathematical Physics. 41 (2): 953–967. Bibcode:1979TMP....41..953K. doi:10.1007/BF01028501. S2CID 121527379.
  6. L. D. Faddeev & V. E. Korepin (1978). "Quantum theory of solitons". Physics Reports. 42 (1): 1–87. Bibcode:1978PhR....42....1F. doi:10.1016/0370-1573(78)90058-3.
  7. Izergin, A. G.; Korepin, V. E. (January 1, 1981). "The inverse scattering method approach to the quantum Shabat-Mikhaĭ lov model". Communications in Mathematical Physics. 79 (3): 303–316. Bibcode:1981CMaPh..79..303I. doi:10.1007/bf01208496. S2CID 119885983.
  8. Its, A.; Izergin, A.; Korepin, V.; Slavnov, N. (2009). "Temperature Correlation of Quantum Spins". Physical Review Letters. 70 (15): 1704–1708. arXiv:0909.4751. Bibcode:1993PhRvL..70.2357I. doi:10.1103/PhysRevLett.70.2357. S2CID 118375258.
  9. Feynman, R. P.; Morinigo, F. B.; Wagner, W. G.; Hatfield, B. (1995). Feynman lectures on gravitation. Addison-Wesley. ISBN 978-0-201-62734-3. See the web page
  10. Korepin, V. E. (May 13, 2009). "Cancellation of ultra-violet infinities in one loop gravity". arXiv:0905.2175 [gr-qc].
  11. Korepin, V. E. (January 1, 1982). "Calculation of norms of Bethe wave functions". Communications in Mathematical Physics. 86 (3): 391–418. Bibcode:1982CMaPh..86..391K. doi:10.1007/BF01212176. S2CID 122250890.
  12. Izergin, A. G.; Korepin, V. E. (October 2, 2009). "A lattice model related to the nonlinear Schroedinger equation". arXiv:0910.0295 [math.QA].
  13. Its, A.R.; Izergin, A.G.; Korepin, V.E.; Slavnov, N.A. (1990). "Differential Equations for Quantum Correlation Functions". International Journal of Modern Physics B. 04 (5): 1003. Bibcode:1990IJMPB...4.1003I. CiteSeerX 10.1.1.497.8799. doi:10.1142/S0217979290000504.
  14. Korepin, V. E. (2004). "Universality of Entropy Scaling in One Dimensional Gapless Models". Physical Review Letters. 92 (9): 096402. arXiv:cond-mat/0311056. Bibcode:2004PhRvL..92i6402K. doi:10.1103/PhysRevLett.92.096402. PMID 15089496. S2CID 20620724.
  15. Jin, B.-Q.; Korepin, V. E. (2004). "Quantum Spin Chain, Toeplitz Determinants and the Fisher–Hartwig Conjecture". Journal of Statistical Physics. 116 (1–4): 79–95. arXiv:quant-ph/0304108. Bibcode:2004JSP...116...79J. doi:10.1023/B:JOSS.0000037230.37166.42. S2CID 15965139.
  16. Its, A R; Jin, B-Q; Korepin, V E (2005). "Entanglement in the XY spin chain". Journal of Physics A: Mathematical and General. 38 (13): 2975. arXiv:quant-ph/0409027. Bibcode:2005JPhA...38.2975I. doi:10.1088/0305-4470/38/13/011. S2CID 118958889.
  17. Its, A. R.; Jin, B. -Q.; Korepin, V. E. (2006). "Entropy of XY Spin Chain and Block Toeplitz Determinants". In I. Bender; D. Kreimer (eds.). Fields Institute Communications, Universality and Renormalization. 50. p. 151. arXiv:quant-ph/0606178. Bibcode:2006quant.ph..6178I.
  18. Franchini, F; Its, A R; Jin, B-Q; Korepin, V E (2007). "Ellipses of constant entropy in theXYspin chain". Journal of Physics A: Mathematical and Theoretical. 40 (29): 8467. arXiv:quant-ph/0609098. Bibcode:2007JPhA...40.8467F. doi:10.1088/1751-8113/40/29/019. S2CID 119628346.
  19. Franchini, F; Its, A R; Korepin, V E (2008). "Renyi entropy of the XY spin chain". Journal of Physics A: Mathematical and Theoretical. 41 (2): 025302. arXiv:0707.2534. Bibcode:2008JPhA...41b5302F. doi:10.1088/1751-8113/41/2/025302. S2CID 119672750.
  20. Fan, Heng; Korepin, Vladimir; Roychowdhury, Vwani (2004). "Entanglement in a Valence-Bond Solid State". Physical Review Letters. 93 (22): 227203. arXiv:quant-ph/0406067. Bibcode:2004PhRvL..93v7203F. doi:10.1103/PhysRevLett.93.227203. PMID 15601113. S2CID 28587190.
  21. Korepin, Vladimir E.; Xu, Ying (2009). "Entanglement in Valence-Bond-Solid States". International Journal of Modern Physics B. 24 (11): 1361–1440. arXiv:0908.2345. Bibcode:2010IJMPB..24.1361K. doi:10.1142/S0217979210055676. S2CID 115174731.
  22. Korepin, Vladimir E.; Grover, Lov K. (2005). "Simple Algorithm for Partial Quantum Search". Quantum Information Processing. 5 (1): 5–10. arXiv:quant-ph/0504157. Bibcode:2005quant.ph..4157K. doi:10.1007/s11128-005-0004-z. S2CID 31236849.
  23. Korepin, Vladimir E.; Vallilo, Brenno C. (2006). "Group Theoretical Formulation of Quantum Partial Search Algorithm". Progress of Theoretical Physics. 116 (5): 783. arXiv:quant-ph/0609205. Bibcode:2006PThPh.116..783K. doi:10.1143/PTP.116.783. S2CID 1750374.
  24. https://arxiv.org/find/quant-ph/1/au:+Korepin/0/1/0/all/0/1?skip=0&query_id=47279949c7a17e00
  25. "Simons Conference on Quantum and Reversible Computation". Retrieved August 28, 2010.
  26. "Simons Conference on New Trends in Quantum Computation". Retrieved August 28, 2010.
  27. "Faculty Page". Stony Brook University. Retrieved August 28, 2010.
  28. "The 5th Asia Pacific workshop on Quantum Information Science in conjunction with the Korepin Festschriff".
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