Subir Sachdev

Subir Sachdev

Photo by Katherine Taylor for Quanta magazine
Residence Cambridge, Massachusetts
Fields Condensed matter theory
Alma mater Massachusetts Institute of Technology,
Harvard University,
Indian Institute of Technology, Delhi
Doctoral advisor D. R. Nelson
Known for Theories of critical and topological states of quantum matter;
SYK model of non-Fermi liquids and quantum black holes

Subir Sachdev is Herchel Smith Professor of Physics[1] at Harvard University specializing in condensed matter. He was elected to the U.S. National Academy of Sciences in 2014, and was awarded the Dirac Medal (UNSW) in 2015.

Sachdev's research describes the connection between physical properties of modern quantum materials and the nature of quantum entanglement in the many-particle wavefunction. Sachdev has made extensive contributions to the description of the diverse varieties of entangled states of quantum matter. These include states with topological order, with and without an energy gap to excitations, and critical states without quasiparticle excitations. Many of these contributions have been linked to experiments, especially to the rich phase diagrams of the high temperature superconductors.

Strange metals and black holes

Extreme examples of complex quantum entanglement arise in metallic states of matter without quasiparticle excitations, often called strange metals. Remarkably, there is an intimate connection between the quantum physics of strange metals found in modern materials (which can be studied in tabletop experiments), and quantum entanglement near black holes of astrophysics.

This connection is most clearly seen by first thinking more carefully about the defining characteristic of a strange metal: the absence of quasiparticles. In practice, given a state of quantum matter, it is difficult to completely rule out the existence of quasiparticles: while one can confirm that certain perturbations do not create single quasiparticle excitations, it is almost impossible to rule out a non-local operator which could create an exotic quasiparticle in which the underlying electrons are non-locally entangled. Sachdev argued[2][3] instead that it is better to examine how rapidly the system loses quantum phase coherence, or reaches local thermal equilibrium in response to general external perturbations. If quasiparticles existed, dephasing would take a long time during which the excited quasiparticles collide with each other. In contrast, states without quasiparticles reach local thermal equilibrium in the fastest possible time, bounded below by a value of order (Planck constant)/((Boltzmann constant) x (absolute temperature)).[2] Sachdev proposed[4][5] a solvable model of a strange metal (a variant of which is now called the Sachdev-Ye-Kitaev (SYK) model), which was shown to saturate such a bound on the time to reach quantum chaos.[6]

We can now make the connection to the quantum theory of black holes: quite generally, black holes also thermalize and reach quantum chaos in a time of order (Planck constant)/((Boltzmann constant) x (absolute temperature)),[7][8] where the absolute temperature is the black hole's Hawking temperature. And this similarity to quantum matter without quasiparticles is not a co-incidence: for the SYK models, Sachdev had argued[9] that the strange metal has a holographic dual description in terms of the quantum theory of black holes in a curved spacetime with 1 space dimension.

This connection, and other related work by Sachdev and collaborators, have led to valuable insights on the properties of electronic quantum matter, and on the nature of Hawking radiation from black holes. Solvable models of strange metals obtained from the gravitational mapping have inspired analyses of more realistic models of strange metals in the high temperature superconductors and other compounds. Such predictions have been connected to experiments, including some[10] that are in good quantitative agreement with observations on graphene.[11][12] These topics are discussed in more detail in Research.

Career

Sachdev attended school at St. Joseph's Boys' High School, Bangalore and Kendriya Vidyalaya, ASC, Bangalore. He attended college at Indian Institute of Technology, Delhi, Massachusetts Institute of Technology and Harvard University, and received his Ph.D. in theoretical physics. He held professional positions at Bell Labs (1985–1987) and at Yale University (1987–2005), where he was a Professor of Physics, before returning to Harvard, where he is now the Herchel Smith Professor of Physics. He also holds visiting positions as the Cenovus Energy James Clerk Maxwell Chair in Theoretical Physics [13] at the Perimeter Institute for Theoretical Physics, and the Dr. Homi J. Bhabha Chair Professorship[14] at the Tata Institute of Fundamental Research.

Honors

Research

Quantum phases of antiferromagnets

Sachdev has worked extensively on the quantum theory of antiferromagnetism, especially in two-dimensional lattices. Some of the spin liquid states of antiferromagnets can be described by examining the quantum phase transitions out of magnetically ordered states. Such an approach leads to a theory of emergent gauge fields and excitations in the spin liquid states. It is convenient to consider two classes of magnetic order separately: those with collinear and non-collinear spin order. For the case of collinear antiferromagnetism (as in the Néel state), the transition leads to a spin liquid with a U(1) gauge field, while non-collinear antiferromagnetism has a transition to a spin liquid with a Z2 gauge field.

These results agree with numerous numerical studies of model quantum spin systems in two dimensions. A particular spin liquid state proposed for the kagome lattice antiferromagnet[26] agrees well with a tensor network analysis,[33] and has been proposed[34] to describe neutron scattering and NMR experiments on herbertsmithite.[35][36]

Quantum criticality

Sachdev proposed that the anomalous dynamic properties of the cuprate superconductors, and other correlated electron compounds, could be understood by proximity to a quantum critical fixed point. In the quantum critical regime of a non-trivial renormalization group fixed point (in higher than one spatial dimension) the dynamics is characterized by the absence of quasiparticles, and a local equilibration time of order ħ/(kBT). This time was proposed to be the shortest possible such time in all quantum systems.[2] Transport measurements have since shown that this bound is close to saturation in many correlated metals.[37] Sachdev has made numerous contributions to quantum field theories of quantum criticality in insulators, superconductors, and metals.[2]

Deconfined criticality

Traditionally, classical and quantum phase transitions, have been described in terms of the Landau-Ginzburg-Wilson paradigm. The broken symmetry in one of the phases identifies as order parameter; the action for the order parameter is expressed as a field theory which controls fluctuations at and across the critical point. Deconfined critical points describe a new class of phase transitions in which the field theory is not expressed in terms of the order parameter, although there can be broken symmetry and order parameters present in one or both of the adjacent phases. Instead the critical field theory is expressed in terms of deconfined fractionalized degrees of freedom that cannot exist in isolation outside the sample. The fractionalized excitations can carry global symmetry quantum numbers, and the corresponding symmetry is broken in the confining phase where these excitations condense.

The first examples of such transitions appeared in studies of confinement transitions out of the Z2 spin liquid with topological order. On the square lattice, the phase transition driven by the condensation of the fractionalized m particle (the vison) leads to a confining state with broken Z4 square lattice rotational symmetry. By the rules of the LGW paradigm, the critical theory for the phase transition should have an exact Z4 symmetry; however, the deconfined critical theory for the visons turns out to have an exact Z8 symmetry [28][29][30] (which is further enlarged to U(1) after neglecting irrelevant terms). The phase transition driven by the condensation of the fractionalized m particle (the spinon) leads to a "Higgs" phase of the Z2 gauge theory with antiferromagnetic order broken spin rotation symmetry;[38] here the antiferromagnetic order parameter has SO(3) symmetry, and so should the LGW critical theory; but the deconfined critical theory for the spinons has an exact SU(2) symmetry (which is further enlarged to O(4) after neglecting irrelevant terms).

A more subtle class of deconfined critical points has confining phases on both sides, and the fractionalized excitations present only at the critical point.[39][40][41][42] The best studied examples of this class are quantum antiferromagnets with SU(N) symmetry on the square lattice. These exhibit a phase transition from a state with collinear antiferromagnetic order to a valence bond solid,[22][23] but the critical theory is expressed in terms of spinons coupled to an emergent U(1) gauge field.[43][39][40] The study of this transition involved the first computation[44] of the scaling dimension of a monopole operator in a conformal field theory in 2+1 dimensions; more precise computations[45][46] to order 1/N are in good accord with numerical studies[47] of the Néel-VBS transition.

SYK model of non-Fermi liquids and black holes

Sachdev, and his first graduate student Jinwu Ye, proposed[4] an exactly solvable model of a non-Fermi liquid, a variant of which is now called the Sachdev-Ye-Kitaev model. Sachdev first proposed[9] that this model is holographically dual to quantum gravity on AdS2, and provides a solvable realization of Bekenstein-Hawking entropy.[5] Unlike previous models of quantum gravity, it appears that the SYK model is solvable in a regime which accounts for the subtle non-thermal correlations in the Hawking radiation.

One-dimensional quantum systems with an energy gap

Sachdev and colloborators developed a formally exact theory for the non-zero temperature dynamics and transport of one-dimensional quantum systems with an energy gap.[48][49][50] The diluteness of the quasiparticle excitations at low temperature allowed the use of semi-classical methods. The results were in good quantitative agreement with NMR[51] and subsequent neutron scattering[52] observations on S=1 spin chains, and with NMR[53] on the Transverse Field Ising chain compound CoNb2O6

Quantum impurities

The traditional Kondo effect involves a local quantum degree of freedom interacting with a Fermi liquid or Luttinger liquid in the bulk. Sachdev described cases where the bulk was a strongly-interacting critical state without quasiparticle excitations.[54][55][56] The impurity was characterized by a Curie suspectibility of an irrational spin, and a boundary entropy of an irrational number of states.

Ultracold atoms

Sachdev predicted[57] density wave order and 'magnetic' quantum criticality in tilted lattices of ultracold atoms. This was subsequently observed in experiments.[58][59]

Metals with topological order

Sachdev and collaborators proposed[60][61] a new metallic state, the fractionalized Fermi liquid (FL*): this has electron-like quasiparticles around a Fermi surface, enclosing a volume distinct from that required by Luttinger's theorem. A general argument was given that any such state must have very low energy excitations on a torus, not related to the low energy quasiparticles: these excitations are generally related to the emergent gauge fields of an associated spin liquid state. In other words, a non-Luttinger Fermi surface volume necessarily requires topological order.[61][62] The FL* phase must be separated from the conventional Fermi liquid (FL) by a quantum phase transition: this transition need not involve any broken symmetry, and examples were presented involving confinement/Higgs transitions of the gauge field. Sachdev and collaborators also described a related metal,[63] the algebraic charge liquid (ACL), which also has a Fermi surface with a non-Luttinger volume, with quasiparticles carrying charge but not spin. The FL* and ACL are both metals with topological order. Evidence has been accumulating that the pseudogap metal of the hole-doped cuprates is such a state.[64][65]

Quantum critical transport

Sachdev developed the theory of quantum transport at non-zero temperatures in the simplest model system without quasiparticle excitations: a conformal field theory in 2+1 dimensions, realized by the superfluid-insulator transitions of ultracold bosons in an optical lattice. A comprehensive picture emerged from quantum-Boltzmann equations,[3] the operator product expansion,[66] and holographic methods.[67][68][69][70] The latter mapped the dynamics to that in the vicinity of the horizon of a black hole. These were the first proposed connections between condensed matter quantum critical systems, hydrodynamics, and quantum gravity. These works eventually led to the theory of hydrodynamic transport in graphene, and the successful experimental predictions[12] described below.

Quantum matter without quasiparticles

Sachdev developed the theory of magneto-thermoelectric transport in 'strange' metals: these are states of quantum matter with variable density without quasiparticle excitations. Such metals are found, most famously, near optimal doping in the hole-doped cuprates, but also appear in numerous other correlated electron compounds. For strange metals in which momentum is approximately conserved, a set of hydrodynamic equations were proposed in 2007,[71] describing two-component transport with momentum drag component and a quantum-critical conductivity. This formulation was connected to the holography of charged black holes, memory functions, and new field-theoretic approaches.[72] These equations are valid when the electron-electron scattering time is much shorter than the electron-impurity scattering time, and they lead to specific predictions for the density, disorder, temperature, frequency, and magnetic field dependence of transport properties. Strange metal behavior obeying these hydrodynamic equations was predicted in graphene,[10][73] in the 'quantum critical' regime of weak disorder and moderate temperatures near the Dirac density. The theory quantitatively describes measurements of thermal and electrical transport in graphene,[12] and points to a regime of viscous, rather than Ohmic, electron flow.

Phases of the high temperature superconductors

High temperature superconductivity appears upon changing the electron density away from a two-dimensional antiferromagnet. Much attention has focused on the intermediate regime between the antiferromagnet and the optimal superconductor, where additional competing orders are found at low temperatures, and a "pseudogap" metal appears in the hole-doped cuprates. Sachdev's theories for the evolution of the competing order with magnetic field,[74][75] density, and temperature have been successfully compared with experiments.[76][77] Sachdev and collaborators proposed[78] a sign-problem free Monte Carlo method for studying the onset of antiferromagnetic order in metals: this yields a phase diagram with high temperature superconductivity similar to that found in many materials, and has led to much subsequent work describing the origin of high temperature superconductivity in realistic models of various materials. Nematic order was predicted for the iron-based superconductors,[79] and a new type of charge density wave, a d-form factor density wave, was predicted[80] for the hole-doped cuprates; both have been observed in numerous experiments.[81][82][83][84][85] The pseudogap metal of the hole-doped cuprates was argued[86] to be a metal with topological order, as discussed above, based partly on its natural connection to the d-form factor density wave. Soon after, the remarkable experiments of Badoux et al.[87] displayed evidence for a small Fermi surface state with topological order near optimal doping in YBCO, consistent with the overall theoretical picture presented in Sachdev's work.[64][65][88]

References

  1. ↑ "Subir Sachdev. Herchel Smith Professor of Physics, Harvard University". Official website.
  2. 1 2 3 4 5 Sachdev, Subir (1999). Quantum phase transitions. Cambridge University Press. ISBN 0-521-00454-3.
  3. 1 2 Damle, Kedar; Sachdev, Subir (1997). "Nonzero-temperature transport near quantum critical points". Physical Review B. 56 (14): 8714–8733. arXiv:cond-mat/9705206Freely accessible. doi:10.1103/PhysRevB.56.8714. ISSN 0163-1829.
  4. 1 2 Sachdev, Subir; Ye, Jinwu (1993). "Gapless spin-fluid ground state in a random quantum Heisenberg magnet". Physical Review Letters. 70 (21): 3339–3342. arXiv:cond-mat/9212030Freely accessible. doi:10.1103/PhysRevLett.70.3339. ISSN 0031-9007.
  5. 1 2 Sachdev, Subir (2015). "Bekenstein-Hawking Entropy and Strange Metals". Physical Review X. 5 (4): 041025. arXiv:1506.05111Freely accessible. doi:10.1103/PhysRevX.5.041025. ISSN 2160-3308.
  6. ↑ Maldacena, Juan; Shenker, Stephen H.; Stanford, Douglas (2016). "A bound on chaos". Journal of High Energy Physics. 2016 (8). doi:10.1007/JHEP08(2016)106. ISSN 1029-8479.
  7. ↑ Dray, Tevian; 't Hooft, Gerard (1985). "The gravitational shock wave of a massless particle". Nuclear Physics B. 253: 173–188. doi:10.1016/0550-3213(85)90525-5. ISSN 0550-3213.
  8. ↑ Shenker, Stephen H.; Stanford, Douglas (2014). "Black holes and the butterfly effect". Journal of High Energy Physics. 2014 (3). doi:10.1007/JHEP03(2014)067. ISSN 1029-8479.
  9. 1 2 Sachdev, Subir (2010). "Holographic Metals and the Fractionalized Fermi Liquid". Physical Review Letters. 105 (15): 151602. arXiv:1006.3794Freely accessible. doi:10.1103/PhysRevLett.105.151602. ISSN 0031-9007.
  10. 1 2 Müller, Markus; Sachdev, Subir (2008). "Collective cyclotron motion of the relativistic plasma in graphene". Physical Review B. 78 (11): 115419. arXiv:0801.2970Freely accessible. doi:10.1103/PhysRevB.78.115419. ISSN 1098-0121.
  11. ↑ Bandurin, D. A.; Torre, I.; Kumar, R. K.; Ben Shalom, M.; Tomadin, A.; Principi, A.; Auton, G. H.; Khestanova, E.; Novoselov, K. S.; Grigorieva, I. V.; Ponomarenko, L. A.; Geim, A. K.; Polini, M. (2016). "Negative local resistance caused by viscous electron backflow in graphene". Science. 351 (6277): 1055–1058. doi:10.1126/science.aad0201. ISSN 0036-8075.
  12. 1 2 3 Crossno, J.; Shi, J. K.; Wang, K.; Liu, X.; Harzheim, A.; Lucas, A.; Sachdev, S.; Kim, P.; Taniguchi, T.; Watanabe, K.; Ohki, T. A.; Fong, K. C. (2016). "Observation of the Dirac fluid and the breakdown of the Wiedemann-Franz law in graphene". Science. 351 (6277): 1058–1061. arXiv:1509.04713Freely accessible. doi:10.1126/science.aad0343. ISSN 0036-8075.
  13. ↑ "Subir Sachdev, Perimeter Institute".
  14. ↑ "Endowment Chairs at TIFR".
  15. ↑ "Dirac Medal awarded to Professor Subir Sachdev".
  16. ↑ "Subir Sachdev NAS member".
  17. ↑ "Condensed matter physicist Subir Sachdev to deliver Salam Distinguished Lectures 2014".
  18. ↑ "Lorentz Chair".
  19. ↑ "Nine Leading Researchers Join Stephen Hawking as Distinguished Research Chairs at PI". Perimeter Institute for Theoretical Physics.
  20. ↑ "All Fellows - John Simon Guggenheim Memorial Foundation". John Simon Guggenheim Memorial Foundation. Retrieved 2010-01-26.
  21. ↑ "LeRoy Apker Award Recipient". American Physical Society. Retrieved 2010-06-30.
  22. 1 2 Read, N.; Sachdev, Subir (1989). "Valence-bond and spin-Peierls ground states of low-dimensional quantum antiferromagnets". Physical Review Letters. 62 (14): 1694–1697. doi:10.1103/PhysRevLett.62.1694. ISSN 0031-9007.
  23. 1 2 Read, N.; Sachdev, Subir (1990). "Spin-Peierls, valence-bond solid, and Néel ground states of low-dimensional quantum antiferromagnets". Physical Review B. 42 (7): 4568–4589. doi:10.1103/PhysRevB.42.4568. ISSN 0163-1829.
  24. ↑ Read, N.; Sachdev, Subir (1991). "Large-Nexpansion for frustrated quantum antiferromagnets". Physical Review Letters. 66 (13): 1773–1776. doi:10.1103/PhysRevLett.66.1773. ISSN 0031-9007.
  25. ↑ Sachdev, Subir; Read, N. (1991). "LARGE N EXPANSION FOR FRUSTRATED AND DOPED QUANTUM ANTIFERROMAGNETS". International Journal of Modern Physics B. 05 (01n02): 219–249. arXiv:cond-mat/0402109Freely accessible. doi:10.1142/S0217979291000158. ISSN 0217-9792.
  26. 1 2 Sachdev, Subir (1992). "Kagome and triangular-lattice Heisenberg antiferromagnets: Ordering from quantum fluctuations and quantum-disordered ground states with unconfined bosonic spinons". Physical Review B. 45 (21): 12377–12396. doi:10.1103/PhysRevB.45.12377. ISSN 0163-1829.
  27. ↑ Wen, X. G. (1991). "Mean-field theory of spin-liquid states with finite energy gap and topological orders". Physical Review B. 44 (6): 2664–2672. doi:10.1103/PhysRevB.44.2664. ISSN 0163-1829.
  28. 1 2 Jalabert, Rodolfo A.; Sachdev, Subir (1991). "Spontaneous alignment of frustrated bonds in an anisotropic, three-dimensional Ising model". Physical Review B. 44 (2): 686–690. doi:10.1103/PhysRevB.44.686. ISSN 0163-1829.
  29. 1 2 Sachdev, Subir, Duality mappings for quantum dimers (PDF)
  30. 1 2 Sachdev, S.; Vojta, M. (1999). "Translational symmetry breaking in two-dimensional antiferromagnets and superconductors" (PDF). J. Phys. Soc. Jpn. 69, Supp. B: 1. arXiv:cond-mat/9910231Freely accessible.
  31. ↑ Senthil, T.; Motrunich, O. (2002). "Microscopic models for fractionalized phases in strongly correlated systems". Physical Review B. 66 (20): 205104. arXiv:cond-mat/0201320Freely accessible. doi:10.1103/PhysRevB.66.205104. ISSN 0163-1829.
  32. ↑ Motrunich, O. I.; Senthil, T. (2002). "Exotic Order in Simple Models of Bosonic Systems". Physical Review Letters. 89 (27): 277004. arXiv:cond-mat/0205170Freely accessible. doi:10.1103/PhysRevLett.89.277004. ISSN 0031-9007.
  33. ↑ Mei, J.-W.; Chen, J.-Y.; He, H.; Wen, X.-G. (2016). "SU(2) spin-rotation symmetric tensor network state for spin-1/2 Heisenberg model on kagome lattice and its modular matrices". arXiv:1606.09639Freely accessible.
  34. ↑ Punk, Matthias; Chowdhury, Debanjan; Sachdev, Subir (2014). "Topological excitations and the dynamic structure factor of spin liquids on the kagome lattice". Nature Physics. 10 (4): 289–293. doi:10.1038/nphys2887. ISSN 1745-2473.
  35. ↑ Han, Tian-Heng; Helton, Joel S.; Chu, Shaoyan; Nocera, Daniel G.; Rodriguez-Rivera, Jose A.; Broholm, Collin; Lee, Young S. (2012). "Fractionalized excitations in the spin-liquid state of a kagome-lattice antiferromagnet". Nature. 492 (7429): 406–410. doi:10.1038/nature11659. ISSN 0028-0836.
  36. ↑ Fu, M.; Imai, T.; Han, T.-H.; Lee, Y. S. (2015). "Evidence for a gapped spin-liquid ground state in a kagome Heisenberg antiferromagnet". Science. 350 (6261): 655–658. doi:10.1126/science.aab2120. ISSN 0036-8075.
  37. ↑ Bruin, J. A. N.; Sakai, H.; Perry, R. S.; Mackenzie, A. P. (2013). "Similarity of Scattering Rates in Metals Showing T-Linear Resistivity". Science. 339 (6121): 804–807. doi:10.1126/science.1227612. ISSN 0036-8075.
  38. ↑ Chubukov, Andrey V.; Senthil, T.; Sachdev, Subir (1994). "Universal magnetic properties of frustrated quantum antiferromagnets in two dimensions". Physical Review Letters. 72 (13): 2089–2092. doi:10.1103/PhysRevLett.72.2089. ISSN 0031-9007.
  39. 1 2 Senthil, T.; Vishwanath, Ashvin; Balents, Leon; Sachdev, Subir; Fisher, Matthew P. A. (2004). "Deconfined Quantum Critical Points". Science. 303 (5663): 1490–1494. arXiv:cond-mat/0311326Freely accessible. doi:10.1126/science.1091806. ISSN 0036-8075.
  40. 1 2 Senthil, T.; Balents, Leon; Sachdev, Subir; Vishwanath, Ashvin; Fisher, Matthew P. A. (2004). "Quantum criticality beyond the Landau-Ginzburg-Wilson paradigm". Physical Review B. 70 (14): 144407. arXiv:cond-mat/0312617Freely accessible. doi:10.1103/PhysRevB.70.144407. ISSN 1098-0121.
  41. ↑ Fradkin, Eduardo; Huse, David A.; Moessner, R.; Oganesyan, V.; Sondhi, S. L. (2004). "Bipartite Rokhsar–Kivelson points and Cantor deconfinement". Physical Review B. 69 (22): 224415. arXiv:cond-mat/0311353Freely accessible. doi:10.1103/PhysRevB.69.224415. ISSN 1098-0121.
  42. ↑ Vishwanath, Ashvin; Balents, L.; Senthil, T. (2004). "Quantum criticality and deconfinement in phase transitions between valence bond solids". Physical Review B. 69 (22): 224416. arXiv:cond-mat/0311085Freely accessible. doi:10.1103/PhysRevB.69.224416. ISSN 1098-0121.
  43. ↑ Chubukov, Andrey V.; Sachdev, Subir; Ye, Jinwu (1994). "Theory of two-dimensional quantum Heisenberg antiferromagnets with a nearly critical ground state". Physical Review B. 49 (17): 11919–11961. arXiv:cond-mat/9304046Freely accessible. doi:10.1103/PhysRevB.49.11919. ISSN 0163-1829.
  44. ↑ Murthy, Ganpathy; Sachdev, Subir (1990). "Action of hedgehog instantons in the disordered phase of the (2 + 1)-dimensional CPN−1 model". Nuclear Physics B. 344 (3): 557–595. doi:10.1016/0550-3213(90)90670-9. ISSN 0550-3213.
  45. ↑ Dyer, Ethan; Mezei, Márk; Pufu, Silviu S.; Sachdev, Subir (2015). "Scaling dimensions of monopole operators in the CPN-1 theory in 2 + 1 dimensions". Journal of High Energy Physics. 2015 (6). arXiv:1504.00368Freely accessible. doi:10.1007/JHEP06(2015)037. ISSN 1029-8479.
  46. ↑ Dyer, Ethan; Mezei, Márk; Pufu, Silviu S.; Sachdev, Subir (2016). "Erratum to: Scaling dimensions of monopole operators in the CPN-1 theory in 2 + 1 dimensions". Journal of High Energy Physics. 2016 (3). doi:10.1007/JHEP03(2016)111. ISSN 1029-8479.
  47. ↑ Block, Matthew S.; Melko, Roger G.; Kaul, Ribhu K. (2013). "Fate of CPN-1 Fixed Points with q Monopoles". Physical Review Letters. 111 (13). arXiv:1307.0519Freely accessible. doi:10.1103/PhysRevLett.111.137202. ISSN 0031-9007.
  48. ↑ Sachdev, Subir; Young, A. P. (1997). "Low Temperature Relaxational Dynamics of the Ising Chain in a Transverse Field". Physical Review Letters. 78 (11): 2220–2223. doi:10.1103/PhysRevLett.78.2220. ISSN 0031-9007.
  49. ↑ Sachdev, Subir; Damle, Kedar (1997). "Low Temperature Spin Diffusion in the One-Dimensional QuantumO(3)NonlinearσModel". Physical Review Letters. 78 (5): 943–946. doi:10.1103/PhysRevLett.78.943. ISSN 0031-9007.
  50. ↑ Damle, Kedar; Sachdev, Subir (1998). "Spin dynamics and transport in gapped one-dimensional Heisenberg antiferromagnets at nonzero temperatures". Physical Review B. 57 (14): 8307–8339. doi:10.1103/PhysRevB.57.8307. ISSN 0163-1829.
  51. ↑ Takigawa, M.; Asano, T.; Ajiro, Y.; Mekata, M.; Uemura, Y. J. (1996). "Dynamics in theS=1One-Dimensional Antiferromagnet AgVP2S6 via 31P and 51V NMR". Physical Review Letters. 76 (12): 2173–2176. doi:10.1103/PhysRevLett.76.2173. ISSN 0031-9007.
  52. ↑ Xu, G.; Broholm, C.; Soh, Y.-A.; Aeppli, G.; DiTusa, J. F.; Chen, Y.; Kenzelmann, M.; Frost, C. D.; Ito, T.; Oka, K.; Takagi, H. (2007). "Mesoscopic Phase Coherence in a Quantum Spin Fluid". Science. 317 (5841): 1049–1052. doi:10.1126/science.1143831. ISSN 0036-8075.
  53. ↑ Kinross, A. W.; Fu, M.; Munsie, T. J.; Dabkowska, H. A.; Luke, G. M.; Sachdev, Subir; Imai, T. (2014). "Evolution of Quantum Fluctuations Near the Quantum Critical Point of the Transverse Field Ising Chain System CoNb2O6". Physical Review X. 4 (3). doi:10.1103/PhysRevX.4.031008. ISSN 2160-3308.
  54. ↑ Sachdev, S.; Buragohain, C.; Vojta, M. (1999). "Quantum Impurity in a Nearly Critical Two-Dimensional Antiferromagnet". Science. 286 (5449): 2479–2482. doi:10.1126/science.286.5449.2479. ISSN 0036-8075.
  55. ↑ Kolezhuk, Alexei; Sachdev, Subir; Biswas, Rudro R.; Chen, Peiqiu (2006). "Theory of quantum impurities in spin liquids". Physical Review B. 74 (16). doi:10.1103/PhysRevB.74.165114. ISSN 1098-0121.
  56. ↑ Kaul, Ribhu K.; Melko, Roger G.; Metlitski, Max A.; Sachdev, Subir (2008). "Imaging Bond Order near Nonmagnetic Impurities in Square-Lattice Antiferromagnets". Physical Review Letters. 101 (18). doi:10.1103/PhysRevLett.101.187206. ISSN 0031-9007.
  57. ↑ Sachdev, Subir; Sengupta, K.; Girvin, S. M. (2002). "Mott insulators in strong electric fields". Physical Review B. 66 (7): 075128. arXiv:cond-mat/0205169Freely accessible. doi:10.1103/PhysRevB.66.075128. ISSN 0163-1829.
  58. ↑ Simon, Jonathan; Bakr, Waseem S.; Ma, Ruichao; Tai, M. Eric; Preiss, Philipp M.; Greiner, Markus (2011). "Quantum simulation of antiferromagnetic spin chains in an optical lattice". Nature. 472 (7343): 307–312. arXiv:1103.1372Freely accessible. doi:10.1038/nature09994. ISSN 0028-0836.
  59. ↑ Meinert, F.; Mark, M. J.; Kirilov, E.; Lauber, K.; Weinmann, P.; Daley, A. J.; Nägerl, H.-C. (2013). "Quantum Quench in an Atomic One-Dimensional Ising Chain". Physical Review Letters. 111 (5): 053003. doi:10.1103/PhysRevLett.111.053003. ISSN 0031-9007.
  60. ↑ Senthil, T.; Sachdev, Subir; Vojta, Matthias (2003). "Fractionalized Fermi Liquids". Physical Review Letters. 90 (21): 216403. arXiv:cond-mat/0209144Freely accessible. doi:10.1103/PhysRevLett.90.216403. ISSN 0031-9007.
  61. 1 2 Senthil, T.; Vojta, Matthias; Sachdev, Subir (2004). "Weak magnetism and non-Fermi liquids near heavy-fermion critical points". Physical Review B. 69 (3): 035111. arXiv:cond-mat/0305193Freely accessible. doi:10.1103/PhysRevB.69.035111. ISSN 1098-0121.
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