quantum phase transitions in condensed matterqpt.physics.harvard.edu/talks/aps_onsager.pdf · paul...

Quantum phase transitions in condensed matter

Session L04: Lars Onsager PrizeAPS March Meeting, Los Angeles

Subir SachdevMarch 7, 2018

HARVARD

Talk online: sachdev.physics.harvard.edu

Thanks to students and postdocs, and many other collatorators

3/1/18, 4(56 PMSubir Sachdev - students and postdocs

Page 1 of 3http://qpt.physics.harvard.edu/students.html

Ph. D. Students

Jinwu Ye, Associate Professor, Department of Physics and Astronomy, Mississippi State UniversityThesis: Some Examples of Quantum Phase TransitionsT. Senthil, Professor, Department of Physics, Massachusetts Institute of Technology.Thesis: Quantum Phase Transitions in Random Spin SystemsKedar Damle, Department of Theoretical Physics, Tata Institute of Fundamental Research, Mumbai, India.Thesis: Turning on the Heat:Non-zero Temperature Dynamical Properties of Quantum Many-body SystemsChiranjeeb Buragohain, Microsoft Research.Thesis: Dynamical Properties of Quantum Antiferromagnets in One and Two DimensionsYing Zhang, Finisterre Capital, London.Thesis: Competing Orders in the Cuprate SuperconductorsAnatoli Polkovnikov, Associate Professor, Boston University.Thesis: Manifestation of Quantum Fluctuations in Strongly Correlated SystemsStephen Powell, Assistant Professor, University of Nottingham Thesis: Quantum phases and transitions of many-body systems realized using cold atomic gasesAdrian Del Maestro, Associate Professor, University of VermontThesis: The superconductor-metal quantum phase transition in ultra-narrow wiresEmily Dunkel (with David Coker, Boston University), NASA Jet Propulsion LaboratoryThesis: Quantum Phenomena in Condensed Phase SystemsYang Qi, Institute for Advanced Studies, Tsinghua UniversityThesis: Spin and Charge Fluctuations in Strongly Correlated Systems.Rudro Rana Biswas, Assistant Professor, Purdue University Thesis: Explorations in Dirac Fermions and Spin Liquids.Eun Gook Moon, Assistant Professor, Korea Advanced Institute of Science and Technology Thesis: Superfluidity in Strongly Correlated SystemsMax Metlitski, Assistant Professor, Department of Physics, Massachusetts Institute of Technology


of 3http://qpt.physics.harvard.edu/students.html

Thesis: Aspects of Critical Behavior of Two Dimensional Electron SystemsYejin Huh, Applied Scientist at AppleThesis: Quantum Phase Transitions in d-wave Superconductors and Antiferromagnetic Kagome LatticesSusanne Pielawa, Lyft, MunichThesis: Metastable Phases and Dynamics of Low-Dimensional Strongly-Correlated Atomic Quantum GasesDebanjan Chowdhury, Moore Foundation Postdoctoral Fellow, MITThesis: Interplay of Broken Symmetries and Quantum Criticality in Correlated Electronic SystemsJunhyun Lee, Postdoctoral fellow, University of MarylandThesis: Novel quantum phase transitions in low-dimensional systemsAndrew Lucas, Postdoctoral fellow, Stanford UniversityThesis: Transport and hydrodynamics in holography, strange metals and grapheneShubhayu Chatterjee, Harvard UniversityAavishkar Patel, Harvard UniversityWenbo Fu, Harvard UniversitySeth Whitsitt, Harvard UniversityAlex Thomson, Harvard UniversityJulia Steinberg, Harvard University

Postdocs

Pierre Le Doussal, Directeur de Recherche de Classe Exceptionnelle, Laboratoire de Physique Théorique de l'Ecole Normale Supérieure, Paris, France.Rodolfo Jalabert, Professeur à l'Université Louis Pasteur, Institut de Physique et Chimie des Matériaux deStrasbourg, France.Andrey Chubukov, William I. and Bianca M. Fine Chair in Theoretical Physics, University of Minnesota,Minneapolis.Satya Majumdar, Directeur de Recherche, Laboratoire de Physique Théorique et Modèles Statistiques,University of Paris XI, France.Matthias Vojta, Chair of Theoretical Solid State Physics, Technische Universität, Dresden, GermanyOleg Starykh, Professor, Department of Physics, University of Utah.Marcus Kollar, Theoretische Physik III, Institut für Physik, Universität Augsburg, Germany.Kwon Park, Professor, Korea Institute for Advanced Study, Seoul.Takao Morinari, Kyoto University, Kyoto, Japan.Adam Durst, Associate Professor, Hofstra University.

Students

Thanks to students and postdocs, and many other collatorators


Page 2 of 3http://qpt.physics.harvard.edu/students.html

Thesis: Aspects of Critical Behavior of Two Dimensional Electron SystemsYejin Huh, Applied Scientist at AppleThesis: Quantum Phase Transitions in d-wave Superconductors and Antiferromagnetic Kagome LatticesSusanne Pielawa, Lyft, MunichThesis: Metastable Phases and Dynamics of Low-Dimensional Strongly-Correlated Atomic Quantum GasesDebanjan Chowdhury, Moore Foundation Postdoctoral Fellow, MITThesis: Interplay of Broken Symmetries and Quantum Criticality in Correlated Electronic SystemsJunhyun Lee, Postdoctoral fellow, University of MarylandThesis: Novel quantum phase transitions in low-dimensional systemsAndrew Lucas, Postdoctoral fellow, Stanford UniversityThesis: Transport and hydrodynamics in holography, strange metals and grapheneShubhayu Chatterjee, Harvard UniversityAavishkar Patel, Harvard UniversityWenbo Fu, Harvard UniversitySeth Whitsitt, Harvard UniversityAlex Thomson, Harvard UniversityJulia Steinberg, Harvard University

Postdocs

Pierre Le Doussal, Directeur de Recherche de Classe Exceptionnelle, Laboratoire de Physique Théorique de l'Ecole Normale Supérieure, Paris, France.Rodolfo Jalabert, Professeur à l'Université Louis Pasteur, Institut de Physique et Chimie des Matériaux deStrasbourg, France.Andrey Chubukov, William I. and Bianca M. Fine Chair in Theoretical Physics, University of Minnesota,Minneapolis.Satya Majumdar, Directeur de Recherche, Laboratoire de Physique Théorique et Modèles Statistiques,University of Paris XI, France.Matthias Vojta, Chair of Theoretical Solid State Physics, Technische Universität, Dresden, GermanyOleg Starykh, Professor, Department of Physics, University of Utah.Marcus Kollar, Theoretische Physik III, Institut für Physik, Universität Augsburg, Germany.Kwon Park, Professor, Korea Institute for Advanced Study, Seoul.Takao Morinari, Kyoto University, Kyoto, Japan.Adam Durst, Associate Professor, Hofstra University.


of 3http://qpt.physics.harvard.edu/students.html

Krishnendu Sengupta, Professor, Indian Association for the Cultivation of Science, Kolkata, India.Lorenz Bartosch, Assistant Professor, University of Frankfurt.Predrag Nikolic, Associate Professor, George Mason UniversityRibhu Kaul, Associate Professor, University of KentuckyMarkus Müller, Scientist, Paul Scherrer Institute, Switzerland.Lars Fritz, Assistant Professor, University of UtrechtMichael Levin, Associate Professor, University of ChicagoCenke Xu, Associate Professor, University of California, Santa BarbaraSean Hartnoll, Associate Professor, Stanford UniversityErez Berg, Associate Professor, University of ChicagoLiang Fu, Lawrence C. (1944) and Sarah W. Biedenharn Career Development Associate Professor of Physics,Massachusetts Institute of TechnologyLiza Huijse, Software Engineer at Karius, Inc.Chris Laumann, Assistant Professor, Boston UniversityMatthias Punk, Faculty, LMU MunichPhilipp Strack, ZEISS GroupBrian Swingle, Assistant Professor, University of MarylandDmitry Abanin, Professor of Physics, University of GenevaLing-Yan (Janet) Hung, Professor of Physics, Fudan University, ShanghaiJay Sau, Assistant Professor, University of MarylandSarang Gopalakrishnan, Postdoctoral Fellow, CaltechAndrea Allais, Cruise Automation, San FranciscoJohannes Bauer, SCL Group, LondonPaul Chesler, Harvard UniversityAndreas Eberlein, Harvard UniversityWilliam Witczak-Krempa, Assistant Professor, University of MontrealRichard Davison, Harvard UniversityChong Wang, Harvard UniversityMathias Scheurer, Harvard University.

Postdocs

Continuous quantum transitions

Broken symmetry No broken symmetry(A)



Exact solution by Onsager of the D=2 Ising model

Onsager Prizes:M.E. FisherL. KadanoffA. I. Larkin

V. L. Pokrovsky



Broken symmetry A different broken symmetry(D)

Topological order No topological order(B)

Topological order Broken symmetry(C)






Theory with emergent gauge fields: Higgs/confining phases and phase transitions

SDW LROReconstructed Fermi surface

SDW SROLarge Fermi surface.

Increasing SDW order

U/t

Antiferromagnetism in the Hubbard ModelH = �

X

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tijc†i↵cj↵ + U

X

i

✓ni" �

1

2

◆✓ni# �

1

2

◆� µ

X

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c†i↵ci↵

tij ! “hopping”. U ! local repulsion, µ ! chemical potential

Mean-field theory with a spin density wave (SDW)

order parameter ~�i = (�1)ix+iyDc†i↵~�↵�ci�

E/2

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(A)Symmetrybreakingphase

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U/t


X

i<j


X

i

✓ni" �

1

2

◆✓ni# �

1

2

◆� µ

X

i

c†i↵ci↵




E/2





Both states have Luttinger volume Fermi surfaces

H = �X

⇤⌧z⌧z⌧z⌧z � g

X

i

⌧x

⌧z⌧z

⌧z

⌧z

Gi =⌧x⌧x

⌧x

⌧x

Gauss’s Law: [H,Gi] = 0 , Gi = 1

(Wegner, 1971Onsager Prize, 2015)

Z2 lattice gauge theory

g

H = �X

⇤⌧z⌧z⌧z⌧z � g

X

i

⌧x

Z2 lattice gauge theory (Wegner, 1971Onsager Prize, 2015)

Deconfined phase.‘Perimeter law’ for

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Confined phase.‘Area law’ for

Wegner-Wilson loops<latexit sha1_base64="Fy9NKk0IJhRX9wayZX00B1bHh3Y=">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</latexit><latexit sha1_base64="Fy9NKk0IJhRX9wayZX00B1bHh3Y=">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</latexit><latexit sha1_base64="Fy9NKk0IJhRX9wayZX00B1bHh3Y=">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</latexit><latexit sha1_base64="Fy9NKk0IJhRX9wayZX00B1bHh3Y=">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</latexit>

C

WC =Y

C⌧z

<latexit sha1_base64="AE72rwZeLrXDyzpaQtOn0nFBaSw=">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</latexit><latexit sha1_base64="AE72rwZeLrXDyzpaQtOn0nFBaSw=">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</latexit><latexit sha1_base64="AE72rwZeLrXDyzpaQtOn0nFBaSw=">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</latexit><latexit sha1_base64="AE72rwZeLrXDyzpaQtOn0nFBaSw=">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</latexit>

g

H = �X

⇤⌧z⌧z⌧z⌧z � g

X

i

⌧x

(B)Topological

phase transition

Deconfined phase.Z2 flux expelled.Z2 (toric code)

topological order.<latexit sha1_base64="GJ5u8M98bDNBw8mEfkx5wlq/dCg=">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</latexit><latexit sha1_base64="GJ5u8M98bDNBw8mEfkx5wlq/dCg=">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</latexit><latexit sha1_base64="GJ5u8M98bDNBw8mEfkx5wlq/dCg=">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</latexit><latexit sha1_base64="GJ5u8M98bDNBw8mEfkx5wlq/dCg=">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</latexit>

Confined phase.Z2 flux proliferates.No topological order.

<latexit sha1_base64="9Qas7ap0+pF++uK1B+sXPiE8kYw=">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</latexit><latexit sha1_base64="9Qas7ap0+pF++uK1B+sXPiE8kYw=">AAAConicdVHbbtQwEHXCrYRLF3jkxWJbiQcUJVFb4K2iEkJCgq1gacVmtXKcya5Vx7ZsB7Gy8mN8Bm/8DZPtVuI6kq3RGZ+Zc8aVkcL5LPsRxdeu37h5a+d2cufuvfu7owcPPzndWQ5TrqW25xVzIIWCqRdewrmxwNpKwll1cTLUz76AdUKrj35tYN6ypRKN4MwjtBh9K2tokLvpFExnjYQVMOv7YJdVH7L0IH9Gs7TY3Ed5n5QVLIUKHJQH2ycnWg38mpoVykjLMgnl381o2aka7KAy7JUt86uqCp/7RbFHG9l9pcZqKRqwzINL+x67vNPUa4ONlqhVUm2RniYlqPpq9GI0RkVHh0VRoLRsE5jkGIcHNN8iY7KNyWL0vaw171qkc8mcm+WZ8fOA+gSXgM46B4bxC7aEGaaKteDmYWOmp/uI1LTRFo/ydIP+ygisdW7dVvhysOf+rA3gv2qzzjcv5kEo03lQ/HJQ00l0T4cPo7WwwL1cY8K4FaiV8hWzjOMOXIJLuHJK/59Mi/Rlmp8W4+NX223skMfkCXlKcvKcHJM3ZEKmhEc0eh29jybxfvw2Po0/XD6Noy3nEfkt4vInYR/P+A==</latexit><latexit sha1_base64="9Qas7ap0+pF++uK1B+sXPiE8kYw=">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</latexit><latexit sha1_base64="9Qas7ap0+pF++uK1B+sXPiE8kYw=">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</latexit>

Z2 lattice gauge theory

E. Fradkin and S. H. Shenker, PRD 19, 3682 (1979); N. Read and S. Sachdev, PRL 66, 1773 (1991); X.-G. Wen, PRB 44, 2664 (1991); A.Y. Kitaev, Annals of Physics 303, 2 (2003)




U/t


X

i<j


X

i

✓ni" �

1

2

◆✓ni# �

1

2

◆� µ

X

i

c†i↵ci↵




E/2








U/t


transition

SDW SROZ2 or U(1) topological order.Reconstructed Fermi surface.


(B) Topological

phase transition

g

h~�i = 0<latexit sha1_base64="cwmaWXVDli/9HiEe4deaP/XsRMQ=">AAACBnicdZDNSgMxFIUz9a/Wv6pLQYJFcFVmbLV1IRTduKxgbaEzlEx624ZmMkOSKZShOze+ihsXKm59Bne+jWk7gooeCBy+ey839/gRZ0rb9oeVWVhcWl7JrubW1jc2t/LbO7cqjCWFBg15KFs+UcCZgIZmmkMrkkACn0PTH15O680RSMVCcaPHEXgB6QvWY5Rogzr5fZcT0eeA3RHQxK0P2AS7co7Osd3JF+yifeJUyw42ZiZjTiulklPCTkoKKFW9k393uyGNAxCacqJU27Ej7SVEakY5THJurCAidEj60DZWkACUl8zumOBDQ7q4F0rzhMYz+n0iIYFS48A3nQHRA/W7NoV/1dqx7lW9hIko1iDofFEv5liHeBoK7jIJVPOxMYRKZv6K6YBIQrWJLmdC+LoU/28ax8WzonNdLtQu0jSyaA8doCPkoAqqoStURw1E0R16QE/o2bq3Hq0X63XemrHSmV30Q9bbJzuVmHU=</latexit><latexit sha1_base64="cwmaWXVDli/9HiEe4deaP/XsRMQ=">AAACBnicdZDNSgMxFIUz9a/Wv6pLQYJFcFVmbLV1IRTduKxgbaEzlEx624ZmMkOSKZShOze+ihsXKm59Bne+jWk7gooeCBy+ey839/gRZ0rb9oeVWVhcWl7JrubW1jc2t/LbO7cqjCWFBg15KFs+UcCZgIZmmkMrkkACn0PTH15O680RSMVCcaPHEXgB6QvWY5Rogzr5fZcT0eeA3RHQxK0P2AS7co7Osd3JF+yifeJUyw42ZiZjTiulklPCTkoKKFW9k393uyGNAxCacqJU27Ej7SVEakY5THJurCAidEj60DZWkACUl8zumOBDQ7q4F0rzhMYz+n0iIYFS48A3nQHRA/W7NoV/1dqx7lW9hIko1iDofFEv5liHeBoK7jIJVPOxMYRKZv6K6YBIQrWJLmdC+LoU/28ax8WzonNdLtQu0jSyaA8doCPkoAqqoStURw1E0R16QE/o2bq3Hq0X63XemrHSmV30Q9bbJzuVmHU=</latexit><latexit sha1_base64="cwmaWXVDli/9HiEe4deaP/XsRMQ=">AAACBnicdZDNSgMxFIUz9a/Wv6pLQYJFcFVmbLV1IRTduKxgbaEzlEx624ZmMkOSKZShOze+ihsXKm59Bne+jWk7gooeCBy+ey839/gRZ0rb9oeVWVhcWl7JrubW1jc2t/LbO7cqjCWFBg15KFs+UcCZgIZmmkMrkkACn0PTH15O680RSMVCcaPHEXgB6QvWY5Rogzr5fZcT0eeA3RHQxK0P2AS7co7Osd3JF+yifeJUyw42ZiZjTiulklPCTkoKKFW9k393uyGNAxCacqJU27Ej7SVEakY5THJurCAidEj60DZWkACUl8zumOBDQ7q4F0rzhMYz+n0iIYFS48A3nQHRA/W7NoV/1dqx7lW9hIko1iDofFEv5liHeBoK7jIJVPOxMYRKZv6K6YBIQrWJLmdC+LoU/28ax8WzonNdLtQu0jSyaA8doCPkoAqqoStURw1E0R16QE/o2bq3Hq0X63XemrHSmV30Q9bbJzuVmHU=</latexit><latexit sha1_base64="cwmaWXVDli/9HiEe4deaP/XsRMQ=">AAACBnicdZDNSgMxFIUz9a/Wv6pLQYJFcFVmbLV1IRTduKxgbaEzlEx624ZmMkOSKZShOze+ihsXKm59Bne+jWk7gooeCBy+ey839/gRZ0rb9oeVWVhcWl7JrubW1jc2t/LbO7cqjCWFBg15KFs+UcCZgIZmmkMrkkACn0PTH15O680RSMVCcaPHEXgB6QvWY5Rogzr5fZcT0eeA3RHQxK0P2AS7co7Osd3JF+yifeJUyw42ZiZjTiulklPCTkoKKFW9k393uyGNAxCacqJU27Ej7SVEakY5THJurCAidEj60DZWkACUl8zumOBDQ7q4F0rzhMYz+n0iIYFS48A3nQHRA/W7NoV/1dqx7lW9hIko1iDofFEv5liHeBoK7jIJVPOxMYRKZv6K6YBIQrWJLmdC+LoU/28ax8WzonNdLtQu0jSyaA8doCPkoAqqoStURw1E0R16QE/o2bq3Hq0X63XemrHSmV30Q9bbJzuVmHU=</latexit>



g

SDW SROLarge Fermi surface

with Luttinger volume.No topological order

SDW SROReconstructed Fermi surfacewith non-Luttinger volume.

Z2 vortices or hedgehogs expelled.Z2 or U(1) topological order.


Metallic states with non-Luttinger volume Fermi surfaces must have topological order

T. Senthil, M. Vojta, and S. Sachdev, PRB 69, 035111 (2004)S. Sachdev, M. A. Metlitski, Y. Qi, and C. Xu, PRB 80, 155129 (2009)

(B)Topological

phase transition

g

(B)Topological

phase transition;

phases of aa theory

with an emergentSU(2) gauge field.






Metallic states with non-Luttinger volume Fermi surfaces must have topological order

T. Senthil, M. Vojta, and S. Sachdev, PRB 69, 035111 (2004)S. Sachdev, M. A. Metlitski, Y. Qi, and C. Xu, PRB 80, 155129 (2009)

2 1 0 | N A T U R E | V O L 5 3 1 | 1 0 M A R C H 2 0 1 6

LETTERdoi:10.1038/nature16983

Change of carrier density at the pseudogap critical point of a cuprate superconductorS. Badoux1, W. Tabis2,3, F. Laliberté2, G. Grissonnanche1, B. Vignolle2, D. Vignolles2, J. Béard2, D. A. Bonn4,5, W. N. Hardy4,5, R. Liang4,5, N. Doiron-Leyraud1, Louis Taillefer1,5 & Cyril Proust2,5

The pseudogap is a partial gap in the electronic density of states that opens in the normal (non-superconducting) state of cuprate superconductors and whose origin is a long-standing puzzle. Its connection to the Mott insulator phase at low doping (hole concentration, p) remains ambiguous1 and its relation to the charge order2–4 that reconstructs the Fermi surface5,6 at intermediate doping is still unclear7–10. Here we use measurements of the Hall coefficient in magnetic fields up to 88 tesla to show that Fermi-surface reconstruction by charge order in the cuprate YBa2Cu3Oy ends sharply at a critical doping p = 0.16 that is distinctly lower than the pseudogap critical point p* = 0.19 (ref. 11). This shows that the pseudogap and charge order are separate phenomena. We find that the change in carrier density n from n = 1 + p in the conventional metal at high doping (ref. 12) to n = p at low doping (ref. 13) starts at the pseudogap critical point. This shows that the pseudogap and the antiferromagnetic Mott insulator are linked.

Electrons in cuprate materials go from a correlated metallic state at high p to a Mott insulator at p = 0. How the system evolves from one state to the other remains a fundamental question. At high doping, the Fermi surface of cuprates is well established. It is a large hole-like cylinder whose volume yields a carrier density n = 1 + p, as meas-ured, for example, by quantum oscillations14, in agreement with band structure calculations. The carrier density can also be measured using the Hall coefficient RH, because in the limit of T = 0 the Hall number nH of a single-band metal is such that nH = n. Indeed, in the cuprate Tl2Ba2CuO6+δ (Tl-2201), the normal-state Hall coefficient RH at p ≈ 0.3, measured at T → 0 in magnetic fields large enough to suppress superconductivity, is such that nH = V/(eRH) ≈ 1 + p, where e is the electron charge and V the volume per Cu atom in the CuO2 planes12,15.

By contrast, at low doping, measurements of RH in La2−xSrxCuO4 (LSCO) (ref. 13) and YBa2Cu3Oy (YBCO) (ref. 16) yield nH ≈ p, below p ≈ 0.08. Having a carrier density equal to the hole concentration, n = p, is known to be an experimental signature of the lightly doped cuprates. The question is: at what doping does the transition between those two limiting regimes take place? Specifically, does the transition from n = 1 + p to n = p occur at p* , the critical doping for the onset of the pseudogap phase? The pseudogap is a partial gap in the normal-state density of states that appears below p* ≈ 0.19 (ref. 11), and whose origin is a central puzzle in the physics of correlated electrons and the subject of much debate.

To answer this question using Hall measurements, one needs to reach low temperatures, which requires the use of large magnetic fields to suppress superconductivity. The only prior high-field study of cuprates that goes across p* was performed on LSCO (ref. 17), a cuprate super-conductor with a relatively low critical temperature (Tc < 40 K) and critical field (Hc2 < 60 T). For mainly two reasons, studies on LSCO were inconclusive on the transition from n = 1 + p to n = p. First, the

Fermi surface of overdoped LSCO undergoes a Lifshitz transition from a hole-like to an electron-like surface as its band structure crosses a saddle-point van Hove singularity at p ≈ 0.2 (ref. 18). This transition causes large changes in RH(T) (ref. 15) that can mask the effect of the pseudogap onset at p* ≈ 0.19. The second reason is the ill-defined impact of the charge-density-wave (CDW) modulations that develop at low temperature in a doping range near p ≈ 0.12 (ref. 19). Such CDW mod-ulations should cause a reconstruction of the Fermi surface, and hence change RH at low temperature6. Therefore, the anomalies in nH versus p observed below 60 K in LSCO (ref. 17)—and in Bi2La2−xSrxCuO6+δ (ref. 20)—between p ≈ 0.1 and p ≈ 0.2 are most likely to be the combined result of three effects that have yet to be disentangled: Lifshitz transition, Fermi-surface reconstruction (FSR) and pseudogap.

Here we turn to YBCO, a cuprate material with several advantages. First, it is one of the cleanest and best ordered of all cuprates, thereby ensuring a homogeneous doping ideal for distinguishing nearby crit-ical points. Second, the location of the pseudogap critical point is well established in YBCO, at p* = 0.19 ± 0.01 (ref. 11). Third, the Lifshitz transition in YBCO occurs at p > 0.29 (ref. 21), well above p* . Fourth, the CDW modulations in YBCO have been thoroughly characterized. They are detected by X-ray diffraction (XRD) between p ≈ 0.08 and p ≈ 0.16 (refs 22, 23), below a temperature TXRD (Fig. 1a). Above a threshold magnetic field, CDW order is detected by NMR (refs 2, 24) below a temperature TNMR (Fig. 1b). Fifth, the FSR caused by the CDW modulations has a well-defined signature in the Hall effect of YBCO: RH(T) decreases smoothly to become negative at low temperature6—the signature of an electron pocket in the reconstructed Fermi sur-face. Prior Hall measurements in magnetic fields up to 60 T show that the CDW-induced FSR begins sharply at p = 0.08 and persists up to p = 0.15, the highest doping reached so far6.

YBCO has one disadvantage, however. Its orthorhombic structure contains conducting CuO chains along the b axis, which reduce the Hall signal coming from the CuO2 planes. While this has no impact on the qualitative features of RH(T) (such as its sign or its qualitative T depend-ence), it does modify the quantitative relation between the meas-ured Hall number nH and the inferred carrier density n. Specifically, n = (ρb/ρa)nH (ref. 16), where ρb and ρa are the in-plane resistivities parallel and perpendicular to the b axis, respectively (see Methods and Extended Data Fig. 1).

We have performed Hall measurements in YBCO up to 88 T, allowing us to extend the doping range upwards, and hence track the normal-state properties across p* , down to at least T = 40 K. Our com-plete data on four YBCO samples with dopings p = 0.16, 0.177, 0.19 and 0.205 are displayed in Extended Data Figs 2, 3, 4 and 5, respectively. In Fig. 2, we compare field sweeps of RH versus H at p = 0.15 (Fig. 2a; from ref. 6) and p = 0.16 (Fig. 2b), at various temperatures down to 25 K. The difference is striking. At p = 0.15, the high-field isotherms RH(H) drop monotonically with decreasing T until they become negative at low T.

1Département de physique, Regroupement Québécois sur les Matériaux de Pointe, Université de Sherbrooke, Sherbrooke, Québec J1K 2R1, Canada. 2Laboratoire National des Champs Magnétiques Intenses (CNRS, EMFL, INSA, UGA, UPS), Toulouse 31400, France. 3AGH University of Science and Technology, Faculty of Physics and Applied Computer Science, 30-059 Krakow, Poland. 4Department of Physics and Astronomy, University of British Columbia, Vancouver, British Columbia V6T 1Z1, Canada. 5Canadian Institute for Advanced Research, Toronto, Ontario M5G 1Z8, Canada.

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2 1 0 | N A T U R E | V O L 5 3 1 | 1 0 M A R C H 2 0 1 6

LETTERdoi:10.1038/nature16983

Change of carrier density at the pseudogap critical point of a cuprate superconductorS. Badoux1, W. Tabis2,3, F. Laliberté2, G. Grissonnanche1, B. Vignolle2, D. Vignolles2, J. Béard2, D. A. Bonn4,5, W. N. Hardy4,5, R. Liang4,5, N. Doiron-Leyraud1, Louis Taillefer1,5 & Cyril Proust2,5

The pseudogap is a partial gap in the electronic density of states that opens in the normal (non-superconducting) state of cuprate superconductors and whose origin is a long-standing puzzle. Its connection to the Mott insulator phase at low doping (hole concentration, p) remains ambiguous1 and its relation to the charge order2–4 that reconstructs the Fermi surface5,6 at intermediate doping is still unclear7–10. Here we use measurements of the Hall coefficient in magnetic fields up to 88 tesla to show that Fermi-surface reconstruction by charge order in the cuprate YBa2Cu3Oy ends sharply at a critical doping p = 0.16 that is distinctly lower than the pseudogap critical point p* = 0.19 (ref. 11). This shows that the pseudogap and charge order are separate phenomena. We find that the change in carrier density n from n = 1 + p in the conventional metal at high doping (ref. 12) to n = p at low doping (ref. 13) starts at the pseudogap critical point. This shows that the pseudogap and the antiferromagnetic Mott insulator are linked.

Electrons in cuprate materials go from a correlated metallic state at high p to a Mott insulator at p = 0. How the system evolves from one state to the other remains a fundamental question. At high doping, the Fermi surface of cuprates is well established. It is a large hole-like cylinder whose volume yields a carrier density n = 1 + p, as meas-ured, for example, by quantum oscillations14, in agreement with band structure calculations. The carrier density can also be measured using the Hall coefficient RH, because in the limit of T = 0 the Hall number nH of a single-band metal is such that nH = n. Indeed, in the cuprate Tl2Ba2CuO6+δ (Tl-2201), the normal-state Hall coefficient RH at p ≈ 0.3, measured at T → 0 in magnetic fields large enough to suppress superconductivity, is such that nH = V/(eRH) ≈ 1 + p, where e is the electron charge and V the volume per Cu atom in the CuO2 planes12,15.

By contrast, at low doping, measurements of RH in La2−xSrxCuO4 (LSCO) (ref. 13) and YBa2Cu3Oy (YBCO) (ref. 16) yield nH ≈ p, below p ≈ 0.08. Having a carrier density equal to the hole concentration, n = p, is known to be an experimental signature of the lightly doped cuprates. The question is: at what doping does the transition between those two limiting regimes take place? Specifically, does the transition from n = 1 + p to n = p occur at p* , the critical doping for the onset of the pseudogap phase? The pseudogap is a partial gap in the normal-state density of states that appears below p* ≈ 0.19 (ref. 11), and whose origin is a central puzzle in the physics of correlated electrons and the subject of much debate.

To answer this question using Hall measurements, one needs to reach low temperatures, which requires the use of large magnetic fields to suppress superconductivity. The only prior high-field study of cuprates that goes across p* was performed on LSCO (ref. 17), a cuprate super-conductor with a relatively low critical temperature (Tc < 40 K) and critical field (Hc2 < 60 T). For mainly two reasons, studies on LSCO were inconclusive on the transition from n = 1 + p to n = p. First, the

Fermi surface of overdoped LSCO undergoes a Lifshitz transition from a hole-like to an electron-like surface as its band structure crosses a saddle-point van Hove singularity at p ≈ 0.2 (ref. 18). This transition causes large changes in RH(T) (ref. 15) that can mask the effect of the pseudogap onset at p* ≈ 0.19. The second reason is the ill-defined impact of the charge-density-wave (CDW) modulations that develop at low temperature in a doping range near p ≈ 0.12 (ref. 19). Such CDW mod-ulations should cause a reconstruction of the Fermi surface, and hence change RH at low temperature6. Therefore, the anomalies in nH versus p observed below 60 K in LSCO (ref. 17)—and in Bi2La2−xSrxCuO6+δ (ref. 20)—between p ≈ 0.1 and p ≈ 0.2 are most likely to be the combined result of three effects that have yet to be disentangled: Lifshitz transition, Fermi-surface reconstruction (FSR) and pseudogap.

Here we turn to YBCO, a cuprate material with several advantages. First, it is one of the cleanest and best ordered of all cuprates, thereby ensuring a homogeneous doping ideal for distinguishing nearby crit-ical points. Second, the location of the pseudogap critical point is well established in YBCO, at p* = 0.19 ± 0.01 (ref. 11). Third, the Lifshitz transition in YBCO occurs at p > 0.29 (ref. 21), well above p* . Fourth, the CDW modulations in YBCO have been thoroughly characterized. They are detected by X-ray diffraction (XRD) between p ≈ 0.08 and p ≈ 0.16 (refs 22, 23), below a temperature TXRD (Fig. 1a). Above a threshold magnetic field, CDW order is detected by NMR (refs 2, 24) below a temperature TNMR (Fig. 1b). Fifth, the FSR caused by the CDW modulations has a well-defined signature in the Hall effect of YBCO: RH(T) decreases smoothly to become negative at low temperature6—the signature of an electron pocket in the reconstructed Fermi sur-face. Prior Hall measurements in magnetic fields up to 60 T show that the CDW-induced FSR begins sharply at p = 0.08 and persists up to p = 0.15, the highest doping reached so far6.

YBCO has one disadvantage, however. Its orthorhombic structure contains conducting CuO chains along the b axis, which reduce the Hall signal coming from the CuO2 planes. While this has no impact on the qualitative features of RH(T) (such as its sign or its qualitative T depend-ence), it does modify the quantitative relation between the meas-ured Hall number nH and the inferred carrier density n. Specifically, n = (ρb/ρa)nH (ref. 16), where ρb and ρa are the in-plane resistivities parallel and perpendicular to the b axis, respectively (see Methods and Extended Data Fig. 1).

We have performed Hall measurements in YBCO up to 88 T, allowing us to extend the doping range upwards, and hence track the normal-state properties across p* , down to at least T = 40 K. Our com-plete data on four YBCO samples with dopings p = 0.16, 0.177, 0.19 and 0.205 are displayed in Extended Data Figs 2, 3, 4 and 5, respectively. In Fig. 2, we compare field sweeps of RH versus H at p = 0.15 (Fig. 2a; from ref. 6) and p = 0.16 (Fig. 2b), at various temperatures down to 25 K. The difference is striking. At p = 0.15, the high-field isotherms RH(H) drop monotonically with decreasing T until they become negative at low T.

1Département de physique, Regroupement Québécois sur les Matériaux de Pointe, Université de Sherbrooke, Sherbrooke, Québec J1K 2R1, Canada. 2Laboratoire National des Champs Magnétiques Intenses (CNRS, EMFL, INSA, UGA, UPS), Toulouse 31400, France. 3AGH University of Science and Technology, Faculty of Physics and Applied Computer Science, 30-059 Krakow, Poland. 4Department of Physics and Astronomy, University of British Columbia, Vancouver, British Columbia V6T 1Z1, Canada. 5Canadian Institute for Advanced Research, Toronto, Ontario M5G 1Z8, Canada.

© 2016 Macmillan Publishers Limited. All rights reserved

YBa2Cu3O6+x

g






S. Chatterjee, A. Eberlein, and S. Sachdev, PRB 96, 075103 (2017)

Can model the doping dependence of the Hall effect in the hole-doped cuprates

(B)Topological

phase transition;

phases of aa theory


g






M. S. Scheurer, S. Chatterjee, Wei Wu, M. Ferrero, A. Georges, and S. Sachdev, arXiv:1711.09925

SU(2) gauge theory fits the real and imaginary parts of the electron Green’s function computed by multi-site DMFT

F13.00008: Mathias Scheurer R04.00005: Antoine Georges, Thu 10:24 AM

(B)Topological

phase transition;

phases of aa theory





U/t


transition



(B) Topological

phase transition

g







U/t


transition


(B) Topological

phase transition

g




(C) Symmetry breaking and

topological phase transition


H = �X

⇤⌧z⌧z⌧z⌧z � g

X

i

⌧x

⌧z⌧z

⌧z

⌧z

Gi =⌧x⌧x

⌧x

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Gauss’s Law with background electric charges:[H,Gi] = 0 , Gi = �1

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Odd Z2 lattice gauge theory

gH = �

X

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R. Jalabert and S. Sachdev, PRB 44, 686 (1991)T. Senthil, L. Balents, S. Sachdev, A. Vishwanath, and M. P. A. Fisher, PRB 70, 144407 (2004)



Confined phase.Z2 flux proliferates.No topological order.Electric field lines lead

to symmetry breaking andvalence bond solid (VBS) order

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(C)Symmetry-breaking

andtopological

phase transition

gH = �

X

⇤⌧z⌧z⌧z⌧z � g

X

i

⌧x


andtopological

phase transition;

phases of a theory with an

emergentU(1) gauge field


Gi = �1<latexit sha1_base64="epK4xci9KM0TMQ/fSYcjfNSll/8=">AAACDnicdVBNSwMxFMz67fq16tFLsCpeLNmK1R6Eogc9KlgV2mXJpq9taDa7JFmhLP0HXvwrXjyoePXszX9jWiuo6EBgmHnDy5soFVwbQt6dsfGJyanpmVl3bn5hcclbXrnUSaYY1FgiEnUdUQ2CS6gZbgRcpwpoHAm4irrHA//qBpTmibwwvRSCmLYlb3FGjZVCb6sRQZvLnIE0oPruxknI8SHe8TfcBsjmlx56BVIsE7JX2sWkSIYYkIMyqZSxP1IKaISz0HtrNBOWxTbOBNW67pPUBDlVhjMBfbeRaUgp69I21C2VNAYd5MN7+njTKk3cSpR90uCh+j2R01jrXhzZyZiajv7tDcS/vHpmWgdBzmWaGZDsc1ErE9gkeFAObnIFzIieJZQpbv+KWYcqymwH2rUlfF2K/ye1UrFS9M9LherRqI0ZtIbW0Tby0T6qolN0hmqIoVt0jx7Rk3PnPDjPzsvn6JgzyqyiH3BePwCTbptS</latexit><latexit sha1_base64="epK4xci9KM0TMQ/fSYcjfNSll/8=">AAACDnicdVBNSwMxFMz67fq16tFLsCpeLNmK1R6Eogc9KlgV2mXJpq9taDa7JFmhLP0HXvwrXjyoePXszX9jWiuo6EBgmHnDy5soFVwbQt6dsfGJyanpmVl3bn5hcclbXrnUSaYY1FgiEnUdUQ2CS6gZbgRcpwpoHAm4irrHA//qBpTmibwwvRSCmLYlb3FGjZVCb6sRQZvLnIE0oPruxknI8SHe8TfcBsjmlx56BVIsE7JX2sWkSIYYkIMyqZSxP1IKaISz0HtrNBOWxTbOBNW67pPUBDlVhjMBfbeRaUgp69I21C2VNAYd5MN7+njTKk3cSpR90uCh+j2R01jrXhzZyZiajv7tDcS/vHpmWgdBzmWaGZDsc1ErE9gkeFAObnIFzIieJZQpbv+KWYcqymwH2rUlfF2K/ye1UrFS9M9LherRqI0ZtIbW0Tby0T6qolN0hmqIoVt0jx7Rk3PnPDjPzsvn6JgzyqyiH3BePwCTbptS</latexit><latexit sha1_base64="epK4xci9KM0TMQ/fSYcjfNSll/8=">AAACDnicdVBNSwMxFMz67fq16tFLsCpeLNmK1R6Eogc9KlgV2mXJpq9taDa7JFmhLP0HXvwrXjyoePXszX9jWiuo6EBgmHnDy5soFVwbQt6dsfGJyanpmVl3bn5hcclbXrnUSaYY1FgiEnUdUQ2CS6gZbgRcpwpoHAm4irrHA//qBpTmibwwvRSCmLYlb3FGjZVCb6sRQZvLnIE0oPruxknI8SHe8TfcBsjmlx56BVIsE7JX2sWkSIYYkIMyqZSxP1IKaISz0HtrNBOWxTbOBNW67pPUBDlVhjMBfbeRaUgp69I21C2VNAYd5MN7+njTKk3cSpR90uCh+j2R01jrXhzZyZiajv7tDcS/vHpmWgdBzmWaGZDsc1ErE9gkeFAObnIFzIieJZQpbv+KWYcqymwH2rUlfF2K/ye1UrFS9M9LherRqI0ZtIbW0Tby0T6qolN0hmqIoVt0jx7Rk3PnPDjPzsvn6JgzyqyiH3BePwCTbptS</latexit><latexit sha1_base64="epK4xci9KM0TMQ/fSYcjfNSll/8=">AAACDnicdVBNSwMxFMz67fq16tFLsCpeLNmK1R6Eogc9KlgV2mXJpq9taDa7JFmhLP0HXvwrXjyoePXszX9jWiuo6EBgmHnDy5soFVwbQt6dsfGJyanpmVl3bn5hcclbXrnUSaYY1FgiEnUdUQ2CS6gZbgRcpwpoHAm4irrHA//qBpTmibwwvRSCmLYlb3FGjZVCb6sRQZvLnIE0oPruxknI8SHe8TfcBsjmlx56BVIsE7JX2sWkSIYYkIMyqZSxP1IKaISz0HtrNBOWxTbOBNW67pPUBDlVhjMBfbeRaUgp69I21C2VNAYd5MN7+njTKk3cSpR90uCh+j2R01jrXhzZyZiajv7tDcS/vHpmWgdBzmWaGZDsc1ErE9gkeFAObnIFzIieJZQpbv+KWYcqymwH2rUlfF2K/ye1UrFS9M9LherRqI0ZtIbW0Tby0T6qolN0hmqIoVt0jx7Rk3PnPDjPzsvn6JgzyqyiH3BePwCTbptS</latexit>

R. Jalabert and S. Sachdev, PRB 44, 686 (1991)T. Senthil, L. Balents, S. Sachdev, A. Vishwanath, and M. P. A. Fisher, PRB 70, 144407 (2004)






g


R. Jalabert and S. Sachdev, PRB 44, 686 (1991)N. Read and S. Sachdev, PRL 66, 1773 (1991)






Similar phases and transitions in frustrated square lattice antiferromagnets with spin S=1/2


andtopological

phase transition;



(a)

(b)

(a)

(b)or

Quantum phases of a S=1/2 square lattice antiferromagnet

Antiferromagnetwith broken spinrotation symmetry

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VBSwith broken latticerotation symmetry

<latexit sha1_base64="sA9sCrNMAhL1aBFFaMidZHYZxTk=">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</latexit><latexit sha1_base64="sA9sCrNMAhL1aBFFaMidZHYZxTk=">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</latexit><latexit sha1_base64="sA9sCrNMAhL1aBFFaMidZHYZxTk=">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</latexit><latexit sha1_base64="sA9sCrNMAhL1aBFFaMidZHYZxTk=">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</latexit>

g

N. Read and S. Sachdev, PRL 62, 1694 (1989)T. Senthil, A. Vishwanath, L. Balents, S. Sachdev, and M. P. A. Fisher, Science 303, 1490 (2004)

(D)Symmetry-breaking

tosymmetry-breaking transition

(a)

(b)

(a)

(b)or

Quantum phases of a S=1/2 square lattice antiferromagnet

Antiferromagnetwith broken spinrotation symmetry

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VBSwith broken latticerotation symmetry

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g

(D)Symmetry-breaking

tosymmetry-breaking transition;



N. Read and S. Sachdev, PRL 62, 1694 (1989)T. Senthil, A. Vishwanath, L. Balents, S. Sachdev, and M. P. A. Fisher, Science 303, 1490 (2004)

Phases described by Higgs/confining phases of a U(1) gauge theory, which is deconfined only at criticality

LETTERSPUBLISHED ONLINE: 17 JULY 2017 | DOI: 10.1038/NPHYS4190

4-spin plaquette singlet state in theShastry–Sutherland compound SrCu2(BO3)2M. E. Zayed1,2,3*, Ch. Rüegg2,4,5, J. Larrea J.1,6, A. M. Läuchli7, C. Panagopoulos8,9, S. S. Saxena8,M. Ellerby5, D. F. McMorrow5, Th. Strässle2, S. Klotz10, G. Hamel10, R. A. Sadykov11,12, V. Pomjakushin2,M. Boehm13, M. Jiménez–Ruiz13, A. Schneidewind14, E. Pomjakushina15, M. Stingaciu15, K. Conder15

and H. M. Rønnow1

The study of interacting spin systems is of fundamentalimportance for modern condensed-matter physics. On frus-trated lattices, magnetic exchange interactions cannot besimultaneously satisfied, and often give rise to compet-ing exotic ground states1. The frustrated two-dimensionalShastry–Sutherland lattice2 realized by SrCu2(BO3)2 (refs 3,4)is an important test case for our understanding of quantummagnetism. It was constructed to have an exactly solvable2-spin dimer singlet ground state within a certain range ofexchange parameters and frustration. While the exact dimerstate and the antiferromagnetic order at both ends of thephasediagram arewell known, the ground state and spin correlationsin the intermediate frustration range have been widely de-bated2,4–14. We report here the first experimental identificationof the conjectured plaquette singlet intermediate phase inSrCu2(BO3)2. It isobservedby inelasticneutronscatteringafterpressure tuning to 21.5 kbar. This gapped singlet state leadsto a transition to long-range antiferromagnetic order above40 kbar, consistentwith the existenceof a deconfinedquantumcritical point.

In the field of quantum magnetism, geometrically frustratedlattices generally imply major di�culties in analytical andnumerical studies. For very few particular topologies, however, ithas been shown that the ground state, at least, can be calculatedexactly as for the Majumdar–Ghosh model15 that solves the J1 � J2zigzag chain when J1 = 2J2. In two dimensions, the Shastry–Sutherland model2 consisting of an orthogonal dimer network ofspin S= 1/2 was developed to be exactly solvable. For an inter-dimer J 0 to intra-dimer J exchange ratio ↵ ⌘ J 0/J 0.5 the groundstate is a product of singlets on the strong bond J . Numericalcalculations have further shown that this remains valid up to↵⇠0.7 and for small values of three-dimensional (3D) couplingsJ 00 between dimer layers. At the other end, for ⇠0.9 ↵ 1the system approaches the well-known 2D square lattice, which

is antiferromagnetically (AFM) ordered, albeit with significantquantum fluctuations that are believed to include resonatingsinglet correlations resulting in fractional excitations16. The phasediagram of the Shastry–Sutherland model, both with and withoutapplied magnetic field, has been intensively studied by numeroustheoretical and numerical approaches4. In the presence of magneticfield, magnetization plateaus at fractional values of the saturationmagnetization corresponding to Mott insulator phases of dimerstates, as well as possible superfluid and supersolid phases have beenextensively studied7,17–19. At zero field, themain unsolved issue is theexistence and nature of an intermediate phase for⇠0.7↵⇠0.9.A variety of quantum phases and transitions between them havebeen predicted depending on the theoretical technique used: adirect transition from dimer singlet phase to AFM order2,6,7, or anintermediate phase with helical order5, columnar dimers11, valencebond crystal12 or resonating valence bond plaquettes9,10. Recentresults indicate that a plaquette singlet phase is favoured4,20. Fromsuch a phase, which would have an additional Ising-type orderparameter, a subsequent transition to AFM order could provide arealization of the so far elusive deconfined quantum critical point21.

The compound strontium copper borate SrCu2(BO3)2 is the onlyknown realization of the Shastry–Sutherland model with S= 1/2spins4 and has thus triggered considerable attention in the fieldof quantum magnetism. The spectrum of SrCu2(BO3)2 exhibitsan almost dispersionless � = 3meV gap, and a bound state oftwo triplets (BT) forms at EBT ' 5meV. The unusual size anddispersionless nature of the gap is an e�ect of the frustration thatprevents triplets from hopping up to sixth order4. The estimatedexchange parameters in the material J ⇠85K and ↵=0.635 (ref. 4)or J ⇠ 71K and ↵ = 0.603 (ref. 8) place the compound closeto an interesting regime ↵ ⇠ 0.7 where correlations may changedramatically at a critical point.

A precious means to tune a quantum magnet across a quantumphase transition is the application of hydrostatic pressure as

1Laboratory for Quantum Magnetism, Institute of Physics, École Polytechnique Fédérale de Lausanne (EPFL), 1015 Lausanne, Switzerland. 2Laboratory forNeutron Scattering and Imaging, Paul Scherrer Institut, 5232 Villigen PSI, Switzerland. 3Department of Physics, Carnegie Mellon University in Qatar,Education City, PO Box 24866, Doha, Qatar. 4Department of Quantum Matter Physics, University of Geneva, 1211 Geneva 4, Switzerland. 5London Centrefor Nanotechnology and Department of Physics and Astronomy, University College London, London WC1E 6BT, UK. 6Centro Brasileiro de Pesquisas Fisicas,Rua Doutor Xavier Sigaud 150, CEP 2290-180, Rio de Janeiro, Brazil. 7Institut für Theoretische Physik, Universität Innsbruck, 6020 Innsbruck, Austria.8Cavendish Laboratory, University of Cambridge, Cambridge CB3 0HE, UK. 9Division of Physics and Applied Physics, School of Physical and MathematicalSciences, Nanyang Technological University, Singapore 637371, Singapore. 10IMPMC; CNRS–UMR 7590, Université Pierre et Marie Curie, 75252 Paris,France. 11Institute for Nuclear Research, Russian Academy of Sciences, prospekt 60-letiya Oktyabrya 7a, Moscow 117312, Russia. 12Vereshchagin Institutefor High Pressure Physics, Russian Academy of Sciences, 108840, Moscow, Troitsk, Russia. 13Institut Laue-Langevin, 71 avenue des Martyrs - CS 20156-38042 Grenoble Cedex 9, France. 14Jülich Centre for Neutron Science (JCNS) at Heinz Maier-Leibnitz Zentrum (MLZ), Forschungszentrum Jülich GmbH,Lichtenbergstr. 1, 85748 Garching, Germany. 15Laboratory for Scientific Developments and Novel Materials, Paul Scherrer Institut, 5232 Villigen PSI,Switzerland. *e-mail: [email protected]

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© 2017 Macmillan Publishers Limited, part of Springer Nature. All rights reserved.

Nature Physics 13, 962–966 (2017)



and H. M. Rønnow1









LETTERS NATURE PHYSICS DOI: 10.1038/NPHYS4190

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Figure 1 | Phase diagram of SrCu2(BO3)2 as a function of pressure and temperature, including excitation energies. The blue region is the dimer phase, thered region the newly identified plaquette phase, and the green region is the antiferromagnetic phase where Q= (1, 0, 0) magnetic Bragg peaks, indicated bygreen squares, are observed only above 40 kbar. Circles are the triplet gap energy � at Q= (2, 0,L), diamonds are the corresponding two-triplet boundstate (BT) energy EBT and the star is a new low-energy excitation (LE) observed at Q= (1, 0, 1). The magenta line shows the tetragonal to monoclinicstructural transition. The procedure used to obtain it and its error bars is described in ref. 28. The corresponding monoclinic space groups areindicated29,30. The dashed line in the plaquette phase is the extrapolated energy gap using ref. 9. The insets depict the corresponding ground states. All ofthe experimental points are from this study.

it directly modifies the atomic distances and bridging angles,such as Cu–O–Cu and thus the magnetic exchange integrals.Quantum phase transitions were successfully discovered in dimermagnets following application of pressure22. However high-pressure measurements remain technically challenging. In thecase of SrCu2(BO3)2, magnetic susceptibility23 and electron spinresonance24 tomoderate pressures (p12 kbar) indicate a softeningof the gap, while the combined e�ect of pressure and field wasmeasured by susceptibility and NMR25. In the latter case, magneticorder occurring at 24 kbar and 7 T on a fraction of the dimers wasproposed. In an X-ray di�raction investigation, the temperaturedependence of the lattice parameters was analysed as an indirectproxy for the singlet–triplet gap leading to the suggestion that itcloses at 20 kbar26. At even higher pressures, neutron and X-raydi�raction experiments observed a transition above 45 kbar fromthe ambient I4̄2M tetragonal space group to monoclinic27–30.

Here we present neutron spectroscopy results, which directlydetermine the pressure dependence of the gap and throughthe dynamic structure factor allow us to address the natureof the correlations. Figure 1 summarizes the phase diagramof SrCu2(BO3)2, which we determined in this study. The exactdimer phase survives up to 16 kbar. The gap decreases from3meV to 2meV, but does not vanish. At 21.5 kbar, we discoverexperimentally a new, intermediate phase. We identify it by itsinelastic neutron scattering spectrum as the formation of 4-spinplaquette singlets. Above 40 kbar and below 117K we find byneutron di�raction that AFM order appears (Supplementary Fig. 6)while the compound probably still has tetragonal symmetry withorthogonal dimers. Above ⇠45 kbar, a structural distortion takesplace and the symmetry becomes monoclinic, implying non-orthogonal dimers28,29. SrCu2(BO3)2 is magnetically ordered afterthe distortion, but can no longer be described appropriately bythe original Shastry–Sutherland model. The transition from 2-spindimer to 4-spin plaquette singlets appears to be of first order,whereas the transition from the plaquette to the AFM phase couldbe of second order and concomitant with the continuous closure ofthe plaquette gap as sketched in Fig. 1 or of first order9,20.

To allow a quantitative comparison to theoretical predictions,we establish the pressure dependence of the exchange parametersJ� (p), J 0

�(p) and ↵(p) by measuring magnetic susceptibility �(p,T )

and fitting it using 20-site exact diagonalization. The peak insusceptibility shifts to lower temperature as pressure increases upto 10 kbar (Fig. 2a). This suggests a reduction of the spin gap.We parametrize the pressure dependence of J and J 0 by linear fits(Fig. 2b). J has the larger slope so that ↵ increases with pressure.Having established ↵(p) we see that the critical pressure lyingbetween 16 kbar and 21.5 kbar corresponds to 0.66< ↵c < 0.68, ingood agreement with theoretical predictions4,12,20.

A selection from the neutron spectra leading to the phasediagram is presented in Fig. 3; additional spectra at variousmomenta transfer Q are shown in the Supplementary Information.Up to 16 kbar an essentially Q-independent linear decrease of thegap energy is observed (Figs 1 and 3a). The measurement of thedispersion and of the structure factor in that pressure range showsthat the spin system is still in its original ‘exact dimer’ phase.The gap value and the integrated intensity decrease linearly withpressure. The dispersion increases slightly with pressure, whichcan be understood by the increase of ↵ (ref. 6). Interestingly, thebound triplet energy EBT softens twice as fast, implying that thetriplet binding energy, �=2��EBT =1.19(2)meV, remains quasipressure independent. This results in the unusual situation thatextrapolating the softenings, the bound triplet would reach zeroenergy before the single triplet, and hence that, before that point,exciting a bound state of two triplets would cost less energy thanexciting one triplet.

SrCu2(BO3)2 enters a new quantum phase between 16 and21.5 kbar, where a discontinuity in the gap softening occurs. Theinelastic neutron scattering peaks corresponding to the gap energy,�'2meV, at these two pressures remain unchanged (Fig. 3b). Thediscontinuity is also visible in the intensities (Fig. 3d), where thelinear decrease with pressure exhibits an abrupt halt above 16 kbar.

The transition to a new quantum phase is further asserted by anew type of excitation suddenly appearing at the higher pressure(Fig. 3b,c). We label this new low-energy excitation LE. LE is clearly

2


NATURE PHYSICS | ADVANCE ONLINE PUBLICATION | www.nature.com/naturephysics

Nature Physics 13, 962–966 (2017)



and H. M. Rønnow1










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Nature Physics 13, 962–966 (2017)

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Nature Physics 13, 962–966 (2017)

Broken lattice symmetry

C.H. Chung, J.B. Marston, and S. Sachdev,

PRB 64, 134407 (2001)



and H. M. Rønnow1










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Nature Physics 13, 962–966 (2017)

(D) Phasesdescribed by

U(1) gauge theory

C.H. Chung, J.B. Marston, and S. Sachdev,

PRB 64, 134407 (2001)

Strongly-coupled quantum criticality

S.Sachdev, Quantum Phase Transitions, 1991J. Zaanen, Nature 430, 512 (2004)

States of quantum matter with:

• No quasiparticle excitations.

• Strong interactions are at a universal RG fixed point,and this leads so fastest possible ‘dephasing’ and ‘localthermalization’ in a time of order ~/(kBT ).

• Eigenstate thermalization.

• Many-body quantum chaos (as measured by out-of-time-order correlations) in a time of order ~/(kBT ).

<latexit sha1_base64="hpPmW3T939/L1p0WwZAgGAGhxsE=">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</latexit><latexit sha1_base64="hpPmW3T939/L1p0WwZAgGAGhxsE=">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</latexit><latexit sha1_base64="hpPmW3T939/L1p0WwZAgGAGhxsE=">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</latexit><latexit sha1_base64="hpPmW3T939/L1p0WwZAgGAGhxsE=">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</latexit>

Strongly-coupled quantum criticalityStates of quantum matter with:






J. Maldacena, S. H. Shenker, and D. Stanford, JHEP 08, 106 (2016)









Onsager’s Ising criticality, other critical states in one spatial dimension, and quantum impurity models, do not have these properties









`Solvable’ models with these properties: the SYK models


The Sachdev-Ye-Kitaev (SYK) model

Pick a set of random positions

H =1

(2N)3/2

NX

i,j,k,`=1

Uij;k` f†i f

†j fkf` � µ

X

i

f†i fi

fifj + fjfi = 0 , fif†j + f

†j fi = �ij

Q =1

N

X

i

f†i fi

<latexit sha1_base64="zZK3iAiDzEMzAN3FV7Ac6SGwCFI=">AAADlXicjVLbbtpAEF3sXii9kfalUl9GRY1SxXFtCiFRRRW1VZUnmkiliZQFtKwXWFhf6l0jIcv5l35WH/ovXRsiEUSkzoN99syc3ZmjGUaCS+U4f0qGee/+g4flR5XHT54+e17defFThklMWZeGIowvh0QywQPWVVwJdhnFjPhDwS6Gsy95/mLOYsnD4IdaRKznk3HAR5wSpanBTun37im0AY9iQlM3S/fq0HnXTz+8r2cZYJ n4g5RbU2tmYSZE2836HehqavpxlhO6xILRgPexR8ZjFms8XcOzforn0YQEKvRhRWeaz6VbU5UDwH5y8262cTXfpsG4sgvXReiKvAHYL775qQ0O4F8J8azl765LbvW9vzHFdoX2zGNCkdyMoonrLYF9oiaUiPQ8Wze5s7KW/8+AMKjWHNs5bjXcFji2e+g0DusaNBvNZrMOru0UUUOrOBtU/2IvpInPAkUFkfLKdSLVS0msOBUsq+BEsojQGRmzKw0D4jNpeXMeyQL20mKtMnirk9qyULcWBgoKdl2cEl/KhT/UlfmccjOXk9tyV4kaHfVSHkSJYgFdPjRKBKgQ8h0Fj8eMKrHQgNCY67aBTog2T+lNrmg/boaGu0G3bh/bznmjdvJ5ZUwZvUZv0B5yUQudoFN0hrqIGmXDNlrGkfnK/GR+Nb8tS43SSvMS3Qrz+z9wPSe8</latexit><latexit sha1_base64="zZK3iAiDzEMzAN3FV7Ac6SGwCFI=">AAADlXicjVLbbtpAEF3sXii9kfalUl9GRY1SxXFtCiFRRRW1VZUnmkiliZQFtKwXWFhf6l0jIcv5l35WH/ovXRsiEUSkzoN99syc3ZmjGUaCS+U4f0qGee/+g4flR5XHT54+e17defFThklMWZeGIowvh0QywQPWVVwJdhnFjPhDwS6Gsy95/mLOYsnD4IdaRKznk3HAR5wSpanBTun37im0AY9iQlM3S/fq0HnXTz+8r2cZYJ n4g5RbU2tmYSZE2836HehqavpxlhO6xILRgPexR8ZjFms8XcOzforn0YQEKvRhRWeaz6VbU5UDwH5y8262cTXfpsG4sgvXReiKvAHYL775qQ0O4F8J8azl765LbvW9vzHFdoX2zGNCkdyMoonrLYF9oiaUiPQ8Wze5s7KW/8+AMKjWHNs5bjXcFji2e+g0DusaNBvNZrMOru0UUUOrOBtU/2IvpInPAkUFkfLKdSLVS0msOBUsq+BEsojQGRmzKw0D4jNpeXMeyQL20mKtMnirk9qyULcWBgoKdl2cEl/KhT/UlfmccjOXk9tyV4kaHfVSHkSJYgFdPjRKBKgQ8h0Fj8eMKrHQgNCY67aBTog2T+lNrmg/boaGu0G3bh/bznmjdvJ5ZUwZvUZv0B5yUQudoFN0hrqIGmXDNlrGkfnK/GR+Nb8tS43SSvMS3Qrz+z9wPSe8</latexit>

S. Sachdev and J. Ye, Phys. Rev. Lett. 70, 3339 (1993); A. Kitaev (2015)

Place electrons randomly on some sites

The SYK model

H =1

(2N)3/2

NX

i,j,k,`=1

Uij;k` f†i f

†j fkf` � µ

X

i

f†i fi


†j fi = �ij

Q =1

N

X

i

f†i fi



Entangle electrons pairwise randomly

The SYK model

H =1

(2N)3/2

NX

i,j,k,`=1

Uij;k` f†i f

†j fkf` � µ

X

i

f†i fi


†j fi = �ij

Q =1

N

X

i

f†i fi



This describes both a strange metal and a black hole!

The SYK model

H =1

(2N)3/2

NX

i,j,k,`=1

Uij;k` f†i f

†j fkf` � µ

X

i

f†i fi


†j fi = �ij

Q =1

N

X

i

f†i fi



Many-bodylevel spacing ⇠2�N = e�N ln 2

W. Fu and S. Sachdev, PRB 94, 035135 (2016)

Non-quasiparticleexcitations withspacing ⇠ e�Ns0

There are 2N many body levelswith energy E, which do not

admit a quasiparticledecomposition. Shown are all

values of E for a single cluster ofsize N = 12. The T ! 0 statehas an entropy SGPS = Ns0

with

s0 =G

⇡+

ln(2)

4= 0.464848 . . .

< ln 2

where G is Catalan’s constant,for the half-filled case Q = 1/2.

The SYK model

GPS: A. Georges, O. Parcollet, and S. Sachdev, PRB 63, 134406 (2001)

The SYK model

• The last property indicates ⌧eq ⇠ ~/(kBT ), and thishas been found in a recent numerical study.

• Low energy, many-body density of states⇢(E) ⇠ eNs0 sinh(

p2(E � E0)N�)

• Low temperature entropy S = Ns0 +N�T + . . ..

• T = 0 fermion Green’s function G(⌧) ⇠ ⌧�1/2 atlarge ⌧ . (Fermi liquids with quasiparticles have G(⌧) ⇠1/⌧)

• T > 0 Green’s function has conformal invarianceG ⇠ (T/ sin(⇡kBT ⌧/~))1/2 A. Georges and O. Parcollet PRB 59, 5341 (1999)

A. Eberlein, V. Kasper, S. Sachdev, and J. Steinberg, arXiv:1706.07803

D. Stanford and E. Witten, 1703.04612A. M. Garica-Garcia, J.J.M. Verbaarschot, 1701.06593

D. Bagrets, A. Altland, and A. Kamenev, 1607.00694

S. Sachdev and J. Ye, Phys. Rev. Lett. 70, 3339 (1993)

Black holes

Black holes share the properties of strongly-coupled quantum criticality

• Black holes have an entropy and a temperature, TH =~c3/(8⇡GMkB).

• Black holes relax to thermal equilibrium in a time⇠ ~/(kBTH) = 8⇡GM/c3.

• The entropy of black holes is proportional to theirsurface area. So they can only be equivalent to quantum-critical systems in one lower dimension.

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The SYK model

• The last property indicates ⌧eq ⇠ ~/(kBT ), and thishas been found in a recent numerical study.

• Low energy, many-body density of states⇢(E) ⇠ eNs0 sinh(

p2(E � E0)N�)

• Low temperature entropy S = Ns0 +N�T + . . ..

• T = 0 fermion Green’s function G(⌧) ⇠ ⌧�1/2 atlarge ⌧ . (Fermi liquids with quasiparticles have G(⌧) ⇠1/⌧)

• T > 0 Green’s function has conformal invarianceG ⇠ (T/ sin(⇡kBT ⌧/~))1/2 A. Georges and O. Parcollet PRB 59, 5341 (1999)

A. Eberlein, V. Kasper, S. Sachdev, and J. Steinberg, arXiv:1706.07803

D. Stanford and E. Witten, 1703.04612A. M. Garica-Garcia, J.J.M. Verbaarschot, 1701.06593

D. Bagrets, A. Altland, and A. Kamenev, 1607.00694

S. Sachdev and J. Ye, Phys. Rev. Lett. 70, 3339 (1993)

⇣ ~x

SYK and black holesBlack holehorizon

Black holes with a near-horizon AdS2 geometry (described by quantum gravity in 1+1 spacetime

dimensions) match the properties of the 0+1 dimensional SYK model in the previous

slide: Ns0 is the Bekenstein-Hawking entropyS. Sachdev, PRL 105, 151602 (2010); A. Kitaev (2015); J. Maldacena, D. Stanford, and Zhenbin Yang, arXiv:1606.01857

⇣ ~x

SYK and black holesBlack holehorizon

A. Kitaev (2015); J. Maldacena and D. Stanford, arXiv:1604.07818

Both the SYK model and the black holesaturate the lower bound on the Lyapunovtime to quantum chaos: ⌧L = ~/(2⇡kBT ).

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fc

SYK building blocks for strange metals

A. Georges and O. Parcollet PRB 59, 5341 (1999); Xue-Yang Song, Chao-Ming Jian, L. Balents, PRL 119, 216601 (2017); A. A. Patel, J. McGreevy, D. P. Arovas, S. Sachdev, arXiv:1712.05026; D. Chowdhury, Y. Werman, E. Berg, T. Senthil, arXiv:1801.06178

Yields solvable models with linear-in-T resistivity(possibly ‘bad metals’ with ⇢ > h/e2) and

linear-in-B magnetoresistance with B/T scaling.<latexit sha1_base64="khXY4s+E7sycFlFWt7UKKQjy3EI=">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</latexit><latexit sha1_base64="khXY4s+E7sycFlFWt7UKKQjy3EI=">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</latexit><latexit sha1_base64="khXY4s+E7sycFlFWt7UKKQjy3EI=">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</latexit><latexit sha1_base64="khXY4s+E7sycFlFWt7UKKQjy3EI=">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</latexit>

Looking ahead:

• Experimental and theoretical stud-ies of metals with bulk topologicalorder and Fermi surfaces, possiblywith non-Luttinger volumes.

• Experimental and theoretical stud-ies of ‘strange’, ‘bad’, ‘incoherent’,‘ultra-quantum’ metals: microscopicbasis for SYK building blocks.

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quantum phase transitions in condensed matterqpt.physics.harvard.edu/talks/aps_onsager.pdf · paul...

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