Quantum Phases of Matter

I present a selective survey of the phases of quantum matter with varieties of many-particle quantum entanglement. I classify the phases as gapped, conformal, or compressible quantum matter. Gapped quantum matter is illustrated by a simple discussion of the Z_2 spin liquid, and connections are made to topological field theories. I discuss how conformal matter is realized at quantum critical points of realistic lattice models, and make connections to a number of experimental systems. Recent progress in our understanding of compressible quantum phases which are not Fermi liquids is summarized. Finally, I discuss how the strongly-coupled phases of quantum matter may be described by gauge-gravity duality. The structure of the large N limit of SU(N) gauge theory, coupled to adjoint fermion matter at non-zero density, suggests aspects of gravitational duals of compressible quantum matter.

[1]  Read,et al.  Valence-bond and spin-Peierls ground states of low-dimensional quantum antiferromagnets. , 1989, Physical review letters.

[2]  Matthew P. A. Fisher,et al.  Z_2 Gauge Theory of Electron Fractionalization in Strongly Correlated Systems , 2000 .

[3]  Leon Balents,et al.  Identifying topological order by entanglement entropy , 2012, Nature Physics.

[4]  Shibaji Roy,et al.  Intersecting D-branes and Lifshitz-like space-time , 2012, 1204.4858.

[5]  L. Balents,et al.  Fractionalization in an easy-axis Kagome antiferromagnet , 2002 .

[6]  Leon Balents,et al.  Quantum criticality beyond the Landau-Ginzburg-Wilson paradigm , 2004 .

[7]  Michael Levin,et al.  Tensor renormalization group approach to two-dimensional classical lattice models. , 2006, Physical review letters.

[8]  Theory of finite-temperature crossovers near quantum critical points close to,or above, their upper-critical dimension , 1996, cond-mat/9606083.

[9]  S. Tung,et al.  Observation of Quantum Criticality with Ultracold Atoms in Optical Lattices , 2011, Science.

[10]  Philip W. Anderson,et al.  On the ground state properties of the anisotropic triangular antiferromagnet , 1974 .

[11]  W. Janke,et al.  Comprehensive quantum Monte Carlo study of the quantum critical points in planar dimerized/quadrumerized Heisenberg models , 2008, 0808.1418.

[12]  E. Kiritsis,et al.  Generalized holographic quantum criticality at finite density , 2011, 1107.2116.

[13]  Read,et al.  Spin-Peierls, valence-bond solid, and Néel ground states of low-dimensional quantum antiferromagnets. , 1990, Physical review. B, Condensed matter.

[14]  J. McGreevy,et al.  A controlled expansion for certain non-Fermi liquid metals , 2010, 1003.0894.

[15]  M. Matsumoto,et al.  Quantum magnets under pressure: controlling elementary excitations in TlCuCl3. , 2008, Physical review letters.

[16]  Yi Zhang,et al.  Topological entanglement entropy of Z 2 spin liquids and lattice Laughlin states , 2011, 1106.0015.

[17]  F. Haldane.  Luttinger's Theorem and Bosonization of the Fermi Surface , 2005, cond-mat/0505529.

[18]  S. Sachdev,et al.  Global Phase Diagrams of Frustrated Quantum Antiferromagnets in Two Dimensions: Doubled Chern-Simons Theory , 2008, 0811.1220.

[19]  Shibaji Roy,et al.  Lifshitz-like space-time from intersecting branes in string/M theory , 2012, 1203.5381.

[20]  G. Vidal.  Entanglement renormalization. , 2005, Physical review letters.

[21]  Linus Pauling,et al.  A resonating-valence-bond theory of metals and intermetallic compounds , 1949, Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences.

[22]  Lee,et al.  Gauge field, Aharonov-Bohm flux, and high-Tc superconductivity. , 1989, Physical review letters.

[23]  Finite Range Couplings in a Tensor Renormalization Group Approach to 2 D Classical Lattice Models , 2013 .

[24]  N. Trivedi,et al.  Weak Mott insulators on the triangular lattice: Possibility of a gapless nematic quantum spin liquid , 2009, 0907.1710.

[25]  S. Sachdev.  Model of a Fermi liquid using gauge-gravity duality , 2011, 1107.5321.

[26]  Quantum critical transport, duality, and M-theory , 2007, hep-th/0701036.

[27]  N=2 extremal black holes. , 1995, Physical review. D, Particles and fields.

[28]  R. Moessner,et al.  Resonating valence bond phase in the triangular lattice quantum dimer model. , 2001, Physical review letters.

[29]  Sarah M. Harrison,et al.  Aspects of holography for theories with hyperscaling violation , 2012, 1201.1905.

[30]  T. Hänsch,et al.  Quantum phase transition from a superfluid to a Mott insulator in a gas of ultracold atoms , 2002, Nature.

[31]  Lee,et al.  Theory of the half-filled Landau level. , 1993, Physical review. B, Condensed matter.

[32]  Sung-Sik Lee.  Low-energy effective theory of Fermi surface coupled with U(1) gauge field in 2+1 dimensions , 2009, 0905.4532.

[33]  Steven R White,et al.  Neél order in square and triangular lattice Heisenberg models. , 2007, Physical review letters.

[34]  P. Anderson,et al.  Gauge theory of high-temperature superconductors and strongly correlated Fermi systems. , 1988, Physical review. B, Condensed matter.

[35]  B. Swingle,et al.  Hidden Fermi surfaces in compressible states of gauge-gravity duality , 2011, 1112.0573.

[36]  M. Greiner,et al.  Quantum simulation of antiferromagnetic spin chains in an optical lattice , 2011, Nature.

[37]  Read,et al.  Large-N expansion for frustrated quantum antiferromagnets. , 1991, Physical review letters.

[38]  R. Jalabert,et al.  Spontaneous alignment of frustrated bonds in an anisotropic, three-dimensional Ising model. , 1991, Physical review. B, Condensed matter.

[39]  A. Houghton,et al.  Multidimensional bosonization , 1998, cond-mat/9810388.

[40]  K. Narayan.  Lifshitz scaling and hyperscaling violation in string theory , 2012, 1202.5935.

[41]  S. Trivedi,et al.  Holographic Fermi and non-Fermi liquids with transitions in dilaton gravity , 2011, 1105.1162.

[42]  A. Isidori,et al.  Functional renormalization group approach to the Ising-nematic quantum critical point of two-dimensional metals , 2012, 1203.2645.

[43]  Leon Balents,et al.  Deconfined Quantum Critical Points , 2003, Science.

[44]  T. Takayanagi,et al.  Soliton stars as holographic confined Fermi liquids , 2012, 1201.0764.

[45]  E. Kiritsis,et al.  Effective holographic theories for low-temperature condensed matter systems , 2010, 1005.4690.

[46]  T. Takayanagi,et al.  Holographic Derivation of Entanglement Entropy from AdS/CFT , 2006, hep-th/0603001.

[47]  M. Punk,et al.  Vison States and Confinement Transitions of \(Z_2\) Spin Liquids on the Kagome Lattice , 2011, 1106.3330.

[48]  Sachdev,et al.  Kagomé- and triangular-lattice Heisenberg antiferromagnets: Ordering from quantum fluctuations and quantum-disordered ground states with unconfined bosonic spinons. , 1992, Physical review. B, Condensed matter.

[49]  F. Alet,et al.  Impurity spin texture at a deconfined quantum critical point , 2010, 1002.1375.

[50]  Evidence for deconfined quantum criticality in a two-dimensional Heisenberg model with four-spin interactions. , 2006, Physical review letters.

[51]  A. Vishwanath,et al.  Entanglement Entropy of Gapped Phases and Topological Order in Three dimensions , 2011, 1108.4038.

[52]  John B. Shoven,et al.  I , Edinburgh Medical and Surgical Journal.

[53]  Dynamical properties of an antiferromagnet near the quantum critical point: Application to LaCuO2.5 , 1997 .

[54]  M. Vojta,et al.  Fractionalized fermi liquids. , 2002, Physical review letters.

[55]  R. Laughlin,et al.  Equivalence of the resonating-valence-bond and fractional quantum Hall states. , 1987, Physical review letters.

[56]  Dynamical properties of an antiferromagnet near the quantum critical point: Application to LaCuO 2.5 , 1997, cond-mat/9701202.

[57]  Fisher,et al.  Boson localization and the superfluid-insulator transition. , 1989, Physical review. B, Condensed matter.

[58]  Tao E. Li,et al.  Gapped spin-liquid phase in the J 1 -J 2 Heisenberg model by a bosonic resonating valence-bond ansatz , 2012, 1205.3838.

[59]  Castro Neto AH,et al.  Exact solution of the Landau fixed point via bosonization. , 1993, Physical review. B, Condensed matter.

[60]  Stuart E. Brown,et al.  Singular behavior in the pressure-tuned competition between Spin-Peierls and antiferromagnetic ground states of (TMTTF)2PF6. , 1999 .

[61]  M. Roček,et al.  Exact diagonalization of finite frustrated spin-(1/2 Heisenberg models. , 1990, Physical review. B, Condensed matter.

[62]  Alexei Kitaev,et al.  Anyons in an exactly solved model and beyond , 2005, cond-mat/0506438.

[63]  Low-energy dynamics of the spinon-gauge system , 1993, cond-mat/9303037.

[64]  I. Klebanov,et al.  Rényi entropies for free field theories , 2011, 1111.6290.

[65]  T. Takayanagi,et al.  Holographic Fermi surfaces and entanglement entropy , 2011, 1111.1023.

[66]  A. Sandvik,et al.  Lattice model for the SU(N) Néel to valence-bond solid quantum phase transition at large N. , 2011, Physical review letters.

[67]  L. Landau.  Fault-tolerant quantum computation by anyons , 2003 .

[68]  D. Tantillo,et al.  SINGULAR BEHAVIOR IN THE PRESSURE-TUNED COMPETITION BETWEEN SPIN-PEIERLS AND ANTIFERROMAGNETIC GROUND STATES OF (TMTTF)2PF6 , 1998 .

[69]  A. Sandvik.  Continuous quantum phase transition between an antiferromagnet and a valence-bond solid in two dimensions: evidence for logarithmic corrections to scaling. , 2010, Physical review letters.

[70]  S. Hartnoll,et al.  Fractionalization of holographic Fermi surfaces , 2011, 1111.2606.

[71]  S V Isakov,et al.  Spin-liquid phase in a spin-1/2 quantum magnet on the kagome lattice. , 2006, Physical review letters.

[72]  Phase diagram of Script N = 4 super-Yang-Mills theory with R-symmetry chemical potentials , 2006, hep-th/0602074.

[73]  Kai-Yu Yang,et al.  A phenomenological theory of the anomalous pseudogap phase in underdoped cuprates , 2011, Reports on progress in physics. Physical Society.

[74]  M. Punk,et al.  Antiferromagnetism in metals: from the cuprate superconductors to the heavy fermion materials , 2012, Journal of physics. Condensed matter : an Institute of Physics journal.

[75]  Kevin Walker,et al.  A class of P,T-invariant topological phases of interacting electrons , 2003, cond-mat/0307511.

[76]  Zee,et al.  Chiral spin states and superconductivity. , 1989, Physical review. B, Condensed matter.

[77]  M. Thouless.  Fluxoid quantization in the resonating-valence-bond model. , 1987, Physical review. B, Condensed matter.

[78]  Xiao-Gang Wen,et al.  Topological entanglement Rényi entropy and reduced density matrix structure. , 2009, Physical review letters.

[79]  E. Fradkin,et al.  SHORT RANGE RESONATING VALENCE BOND THEORIES AND SUPERCONDUCTIVITY , 1990 .

[80]  Quantum symmetries in discrete gauge theories , 1992, hep-th/9203046.

[81]  Simeng Yan,et al.  Spin-Liquid Ground State of the S = 1/2 Kagome Heisenberg Antiferromagnet , 2010, Science.

[82]  Harish-Chandra.  Black Hole Entropy Function and the Attractor Mechanism in Higher Derivative Gravity , 2005 .

[83]  S. Sachdev.  Exotic phases and quantum phase transitions: model systems and experiments , 2009, 0901.4103.

[84]  E. Fradkin,et al.  Nonperturbative behavior of the quantum phase transition to a nematic Fermi fluid , 2005, cond-mat/0508747.

[85]  S. Sachdev,et al.  Fermi surfaces and gauge-gravity duality , 2011, 1104.5022.

[86]  T. Senthil,et al.  Decohering the Fermi liquid: A dual approach to the Mott transition , 2011, 1107.4125.

[87]  Bom Soo Kim.  Schrödinger holography with and without hyperscaling violation , 2012, 1202.6062.

[88]  D-brane charges in five-brane backgrounds , 2001, hep-th/0108152.

[89]  Xiao-Gang Wen,et al.  Detecting topological order in a ground state wave function. , 2005, Physical review letters.

[90]  O. Motrunich.  Variational study of triangular lattice spin-1/2 model with ring exchanges and spin liquid state in kappa-(ET)2Cu2(CN)3 , 2004, cond-mat/0412556.

[91]  S. Sachdev,et al.  Quantum phase transitions of metals in two spatial dimensions: I. Ising-nematic order , 2010, 1001.1153.

[92]  S. Kachru,et al.  Holography of charged dilaton black holes , 2009, 0911.3586.

[93]  John Preskill,et al.  Topological entanglement entropy. , 2005, Physical Review Letters.

[94]  S. Kivelson,et al.  Exact spin liquid ground states of the quantum dimer model on the square and honeycomb lattices. , 2011, Physical review letters.

[95]  E. Fradkin,et al.  Topological Order and Conformal Quantum Critical Points , 2003, cond-mat/0311466.

[96]  Immanuel Bloch,et al.  Single-spin addressing in an atomic Mott insulator , 2011, Nature.

[97]  Reizer.  Effective electron-electron interaction in metals and superconductors. , 1989, Physical review. B, Condensed matter.

[98]  R. Moessner,et al.  Short-ranged resonating valence bond physics, quantum dimer models, and Ising gauge theories , 2001, cond-mat/0103396.

[99]  F. Verstraete,et al.  Possible spin liquid state in the spin 1/2 J1-J2 antiferromagnetic Heisenberg model on square lattice: A tensor product state approach , 2011 .

[100]  L. Balents,et al.  Spin Liquid Ground State of the Spin-1/2 Square $J_1$-$J_2$ Heisenberg Model , 2011, 1112.2241.

[101]  Matthew B. Hastings,et al.  Topological entanglement entropy of a Bose-Hubbard spin liquid , 2011, 1102.1721.

[102]  S. Sachdev.  What Can Gauge-Gravity Duality Teach Us about Condensed Matter Physics? , 2011, 1108.1197.

[103]  B. Swingle.  Entanglement entropy and the Fermi surface. , 2009, Physical review letters.

[104]  T. Takayanagi,et al.  Holographic derivation of entanglement entropy from the anti-de Sitter space/conformal field theory correspondence. , 2006, Physical review letters.

[105]  P. Anderson.  The Resonating Valence Bond State in La2CuO4 and Superconductivity , 1987, Science.

[106]  Universal quantum-critical dynamics of two-dimensional antiferromagnets. , 1992, Physical review letters.

[107]  S. Tung,et al.  Observation of Quantum Criticality with Ultracold Atoms in Optical Lattices , 2012, Science.

[108]  G. Hooft.  A Planar Diagram Theory for Strong Interactions , 1974 .

[109]  Dimitri Gioev,et al.  Entanglement entropy of fermions in any dimension and the Widom conjecture. , 2006, Physical review letters.

[110]  Wen,et al.  Mean-field theory of spin-liquid states with finite energy gap and topological orders. , 1991, Physical review. B, Condensed matter.

[111]  B. Swingle,et al.  Entanglement Renormalization and Holography , 2009, 0905.1317.

[112]  G Misguich,et al.  Quantum dimer model on the kagome lattice: solvable dimer-liquid and ising gauge theory. , 2002, Physical review letters.

[113]  Edgar Shaghoulian.  Holographic entanglement entropy and Fermi surfaces , 2011, 1112.2702.

[114]  Read,et al.  Statistics of the excitations of the resonating-valence-bond state. , 1989, Physical review. B, Condensed matter.

[115]  Yuji Matsuda,et al.  Highly Mobile Gapless Excitations in a Two-Dimensional Candidate Quantum Spin Liquid , 2010, Science.

[116]  D. Rokhsar,et al.  Superconductivity and the quantum hard-core dimer gas. , 1988, Physical review letters.

[117]  Patrick A. Lee,et al.  U(1) gauge theory of the Hubbard model: spin liquid states and possible application to kappa-(BEDT-TTF)2Cu2(CN)3. , 2005, Physical review letters.

[118]  C. Lhuillier,et al.  Kagome antiferromagnet: a chiral topological spin liquid? , 2011, Physical review letters.

[119]  Harvendra Singh.  Lifshitz/Schrödinger Dp-branes and dynamical exponents , 2012, 1202.6533.

[120]  C. Fuertes,et al.  Entanglement Entropy in the O(N) Model , 2009, 0904.4477.

[121]  Albert Einstein,et al.  Can Quantum-Mechanical Description of Physical Reality Be Considered Complete? , 1935 .