Thermal Hall effect in the pseudogap

+ Bilocal field theory for

optimal dopingCIFAR Jouvence, October 31, 2019

HARVARD

Talk online: sachdev.physics.harvard.edu

Subir Sachdev

G. Grissonnanche, A. Legros, S. Badoux, E. Lefrancois, V. Zatko, M. Lizaire, F. Laliberte, A. Gourgout, J. Zhou, S. Pyon, T. Takayama, H. Takagi, S. Ono, N. Doiron- Leyraud, and L. Taillefer, Nature 571, 376 (2019)

Figure 1 a

b

LETTERhttps://doi.org/10.1038/s41586-019-1375-0

Giant thermal Hall conductivity in the pseudogap phase of cuprate superconductorsG. Grissonnanche1*, A. Legros1,2, S. Badoux1, E. Lefrançois1, V. Zatko1, M. Lizaire1, F. Laliberté1, A. Gourgout1, J.-S. Zhou3, S. Pyon4,5, T. Takayama4,6, H. Takagi4,6,7,8, S. Ono9, N. Doiron-Leyraud1 & L. Taillefer1,10*

The nature of the pseudogap phase of the copper oxides (‘cuprates’) remains a puzzle. Although there are indications that this phase breaks various symmetries, there is no consensus on its fundamental nature1. Fermi-surface, transport and thermodynamic signatures of the pseudogap phase are reminiscent of a transition into a phase with antiferromagnetic order, but evidence for an associated long-range magnetic order is still lacking2. Here we report measurements of the thermal Hall conductivity (in the x–y plane, κxy) in the normal state of four different cuprates—La1.6−xNd0.4SrxCuO4, La1.8−xEu0.2SrxCuO4, La2−xSrxCuO4 and Bi2Sr2−xLaxCuO6+δ. We show that a large negative κxy signal is a property of the pseudogap phase, appearing at its critical hole doping, p*. It is also a property of the Mott insulator at p ≈ 0, where κxy has the largest reported magnitude of any insulator so far3. Because this negative κxy signal grows as the system becomes increasingly insulating electrically, it cannot be attributed to conventional mobile charge carriers. Nor is it due to magnons, because it exists in the absence of magnetic order. Our observation is reminiscent of the thermal Hall conductivity of insulators with spin-liquid states4–6, pointing to neutral excitations with spin chirality7 in the pseudogap phase of cuprates.

Among the different families of unconventional superconductors, magnetism and superconductivity are often closely associated8. A nota-ble exception is the family of hole-doped cuprates, where superconduc-tivity mostly coexists instead with the pseudogap phase, which is an enigmatic state of matter whose nature remains unclear1. The critical doping p* (for the onset of the pseudogap phase) bears the hallmarks of an antiferromagnetic quantum critical point2, with a sharp drop in the carrier density n from n ≈ 1 + p above p* to n ≈ p below p*, a resistivity linear with temperature T, and a specific heat with a log(1/T) dependence. Yet, there is no evidence for long-range magnetic order appearing at p*. However, numerical solutions of the Hubbard model have shown that a pseudogap phase can arise from short-range antifer-romagnetic correlations9. It has been argued that an exotic state with topological order can account for such a pseudogap and for the drop in carrier density without breaking translational symmetry10, but the low-energy excitations of such a state have yet to be detected.

In recent years, the thermal Hall effect has emerged as a powerful probe of magnetic texture and topological excitations in insulators. On the theory side, a non-zero thermal Hall conductivity κxy was shown to arise even without long-range magnetic order, either from the spin chirality of a paramagnetic state7 or from fractionalized (topolog-ical) excitations in a spin liquid11. On the experimental side, a sizeable κxy has been measured in insulators without magnetic order, such as the spin-ice system Tb2Ti2O7 (ref. 12) and the spin-liquid systems RuCl3 (ref. 4), volborthite5 and Ca kapellasite6.

In cuprates, studies of κxy have so far been limited to the super-conducting state13–15, except for the case of YBa2Cu3Oy (YBCO) at p = 0.11, where κxy was measured in the field-induced normal state16,

which has charge-density-wave order2. See Methods for a discussion of this particular case.

Here, we investigate the thermal Hall response of the pseu-dogap phase via measurements of κxy in four different cuprate

1Département de physique, Institut quantique, and RQMP, Université de Sherbrooke, Sherbrooke, Québec, Canada. 2SPEC, CEA, CNRS-UMR3680, Université Paris-Saclay, Gif-sur-Yvette, France. 3Materials Science and Engineering Program, Department of Mechanical Engineering, University of Texas at Austin, Austin, TX, USA. 4Department of Advanced Materials Science, University of Tokyo, Kashiwa, Japan. 5Department of Applied Physics, University of Tokyo, Tokyo, Japan. 6Max Planck Institute for Solid State Research, Stuttgart, Germany. 7Department of Physics, University of Tokyo, Tokyo, Japan. 8Institute for Functional Matter and Quantum Technologies, University of Stuttgart, Stuttgart, Germany. 9Central Research Institute of Electric Power Industry, Kanagawa, Japan. 10Canadian Institute for Advanced Research, Toronto, Ontario, Canada. *e-mail: gael.grissonnanche@usherbrooke.ca; louis.taillefer@usherbrooke.ca

0 0.1 0.2 0.3Doping, p

0

25

50

75

100

T (K

)

Nd-LSCOEu-LSCOLSCO

T*

p*

TN

Tm

0 30 60 90

T (K)

–3

–2

–1

0

1N x

y/T

(mW

K–2

m–1

)

Nd-LSCO

Eu-LSCO

LSCO

La2CuO4 –1

–0.5

0

2D(N

xy /T)/(k

B /")2

H = 15 T

p = 0.20

p = 0.08

p = 0.06

p = 0

a

b

Fig. 1 | Phase diagram and thermal Hall conductivity of cuprates. a, Temperature–doping phase diagram of Nd-LSCO, Eu-LSCO and LSCO, showing the antiferromagnetic phase below the Néel temperature TN and the pseudogap phase below T* (ref. 29), which ends at the critical doping p* = 0.23 for both Nd-LSCO (ref. 17) and Eu-LSCO (ref. 30). For LSCO, p* ≈ 0.18 (ref. 29). Short-range incommensurate spin order occurs below Tm, as measured by µSR on Nd-LSCO (squares21), Eu-LSCO (circles31) and LSCO (triangles32). The coloured vertical strips indicate the temperature range where the thermal Hall conductivity κxy/T at the corresponding doping decreases towards negative values at low temperature (see b). b, Thermal Hall conductivity κxy/T versus temperature in a field H = 15 T, for four materials and dopings as indicated, colour-coded with the vertical strips in a. On the right vertical axis, the magnitude of κxy/T is expressed in fundamental units of thermal conductance per plane (kB

2/ħ).

3 7 6 | N A T U R E | V O L 5 7 1 | 1 8 J U L Y 2 0 1 9

Figure 1 a

b

G. Grissonnanche, A. Legros, S. Badoux, E. Lefrancois, V. Zatko, M. Lizaire, F. Laliberte, A. Gourgout, J. Zhou, S. Pyon, T. Takayama, H. Takagi, S. Ono, N. Doiron- Leyraud, and L. Taillefer, Nature 571, 376 (2019)

LETTERhttps://doi.org/10.1038/s41586-019-1375-0

Giant thermal Hall conductivity in the pseudogap phase of cuprate superconductorsG. Grissonnanche1*, A. Legros1,2, S. Badoux1, E. Lefrançois1, V. Zatko1, M. Lizaire1, F. Laliberté1, A. Gourgout1, J.-S. Zhou3, S. Pyon4,5, T. Takayama4,6, H. Takagi4,6,7,8, S. Ono9, N. Doiron-Leyraud1 & L. Taillefer1,10*

The nature of the pseudogap phase of the copper oxides (‘cuprates’) remains a puzzle. Although there are indications that this phase breaks various symmetries, there is no consensus on its fundamental nature1. Fermi-surface, transport and thermodynamic signatures of the pseudogap phase are reminiscent of a transition into a phase with antiferromagnetic order, but evidence for an associated long-range magnetic order is still lacking2. Here we report measurements of the thermal Hall conductivity (in the x–y plane, κxy) in the normal state of four different cuprates—La1.6−xNd0.4SrxCuO4, La1.8−xEu0.2SrxCuO4, La2−xSrxCuO4 and Bi2Sr2−xLaxCuO6+δ. We show that a large negative κxy signal is a property of the pseudogap phase, appearing at its critical hole doping, p*. It is also a property of the Mott insulator at p ≈ 0, where κxy has the largest reported magnitude of any insulator so far3. Because this negative κxy signal grows as the system becomes increasingly insulating electrically, it cannot be attributed to conventional mobile charge carriers. Nor is it due to magnons, because it exists in the absence of magnetic order. Our observation is reminiscent of the thermal Hall conductivity of insulators with spin-liquid states4–6, pointing to neutral excitations with spin chirality7 in the pseudogap phase of cuprates.

Among the different families of unconventional superconductors, magnetism and superconductivity are often closely associated8. A nota-ble exception is the family of hole-doped cuprates, where superconduc-tivity mostly coexists instead with the pseudogap phase, which is an enigmatic state of matter whose nature remains unclear1. The critical doping p* (for the onset of the pseudogap phase) bears the hallmarks of an antiferromagnetic quantum critical point2, with a sharp drop in the carrier density n from n ≈ 1 + p above p* to n ≈ p below p*, a resistivity linear with temperature T, and a specific heat with a log(1/T) dependence. Yet, there is no evidence for long-range magnetic order appearing at p*. However, numerical solutions of the Hubbard model have shown that a pseudogap phase can arise from short-range antifer-romagnetic correlations9. It has been argued that an exotic state with topological order can account for such a pseudogap and for the drop in carrier density without breaking translational symmetry10, but the low-energy excitations of such a state have yet to be detected.

In recent years, the thermal Hall effect has emerged as a powerful probe of magnetic texture and topological excitations in insulators. On the theory side, a non-zero thermal Hall conductivity κxy was shown to arise even without long-range magnetic order, either from the spin chirality of a paramagnetic state7 or from fractionalized (topolog-ical) excitations in a spin liquid11. On the experimental side, a sizeable κxy has been measured in insulators without magnetic order, such as the spin-ice system Tb2Ti2O7 (ref. 12) and the spin-liquid systems RuCl3 (ref. 4), volborthite5 and Ca kapellasite6.

In cuprates, studies of κxy have so far been limited to the super-conducting state13–15, except for the case of YBa2Cu3Oy (YBCO) at p = 0.11, where κxy was measured in the field-induced normal state16,

which has charge-density-wave order2. See Methods for a discussion of this particular case.

Here, we investigate the thermal Hall response of the pseu-dogap phase via measurements of κxy in four different cuprate

1Département de physique, Institut quantique, and RQMP, Université de Sherbrooke, Sherbrooke, Québec, Canada. 2SPEC, CEA, CNRS-UMR3680, Université Paris-Saclay, Gif-sur-Yvette, France. 3Materials Science and Engineering Program, Department of Mechanical Engineering, University of Texas at Austin, Austin, TX, USA. 4Department of Advanced Materials Science, University of Tokyo, Kashiwa, Japan. 5Department of Applied Physics, University of Tokyo, Tokyo, Japan. 6Max Planck Institute for Solid State Research, Stuttgart, Germany. 7Department of Physics, University of Tokyo, Tokyo, Japan. 8Institute for Functional Matter and Quantum Technologies, University of Stuttgart, Stuttgart, Germany. 9Central Research Institute of Electric Power Industry, Kanagawa, Japan. 10Canadian Institute for Advanced Research, Toronto, Ontario, Canada. *e-mail: gael.grissonnanche@usherbrooke.ca; louis.taillefer@usherbrooke.ca

0 0.1 0.2 0.3Doping, p

0

25

50

75

100

T (K

)

Nd-LSCOEu-LSCOLSCO

T*

p*

TN

Tm

0 30 60 90

T (K)

–3

–2

–1

0

1N x

y/T

(mW

K–2

m–1

)

Nd-LSCO

Eu-LSCO

LSCO

La2CuO4 –1

–0.5

0

2D(N

xy /T)/(k

B /")2

H = 15 T

p = 0.20

p = 0.08

p = 0.06

p = 0

a

b

Fig. 1 | Phase diagram and thermal Hall conductivity of cuprates. a, Temperature–doping phase diagram of Nd-LSCO, Eu-LSCO and LSCO, showing the antiferromagnetic phase below the Néel temperature TN and the pseudogap phase below T* (ref. 29), which ends at the critical doping p* = 0.23 for both Nd-LSCO (ref. 17) and Eu-LSCO (ref. 30). For LSCO, p* ≈ 0.18 (ref. 29). Short-range incommensurate spin order occurs below Tm, as measured by µSR on Nd-LSCO (squares21), Eu-LSCO (circles31) and LSCO (triangles32). The coloured vertical strips indicate the temperature range where the thermal Hall conductivity κxy/T at the corresponding doping decreases towards negative values at low temperature (see b). b, Thermal Hall conductivity κxy/T versus temperature in a field H = 15 T, for four materials and dopings as indicated, colour-coded with the vertical strips in a. On the right vertical axis, the magnitude of κxy/T is expressed in fundamental units of thermal conductance per plane (kB

2/ħ).

3 7 6 | N A T U R E | V O L 5 7 1 | 1 8 J U L Y 2 0 1 9

LETTERRESEARCH

Extended Data Fig. 2 | Comparison of cuprates to other oxides. a, Thermal conductivity of two isostructural oxides, plotted as κxx/T versus T at H = 0, namely Y2Ti2O7 (red) and Tb2Ti2O7 (blue) (data points39). The presence of disordered magnetic moments in Tb2Ti2O7 produces a strong scattering of phonons, seen as a massive suppression of κxx (15-fold at T = 15 K). b, Field dependence of κxx, plotted as ∆κxx(H)/κxx(0) versus H, with ∆κxx = κxx(H) − κxx(0), at T = 15 K (blue data points12). The strong effect of field (30% in 8 T) is a direct signature of the strong coupling between phonons and spins in Tb2Ti2O7. Also shown is the transverse response in Tb2Ti2O7 at T = 15 K, plotted as κxy/T versus H (red data points12). Note that in Y2Ti2O7, κxy = 0 (ref. 12). c, Thermal conductivity of

two Nd-LSCO samples, on either side of p* (red, p = 0.24; blue, p = 0.21), plotted as κxx/T versus T at H = 18 T (data points). We see that contrary to Tb2Ti2O7 (a), the appearance of the negative κxy signal in Nd-LSCO below p* is not accompanied by a large suppression of κxx (see Extended Data Fig. 3). d, Same as b but for LSCO p = 0.06, with the same x-axis and y-axis scales and data taken at (nearly) the same temperature (data points). We see that the situation in LSCO is very different to that found in Tb2Ti2O7 (b): instead of having a small κxy and a large ∆κxx (b), we now have a large κxy and a small ∆κxx. Quantitatively, κxy/∆κxx ≈ 1 in LSCO and approximately 0.01 in Tb2Ti2O7, at T = 15 K and H = 8 T (Table 1).

Model for the pseudogap

Fermionic ‘chargons’ of density � in hole pockets.The fermions carry electromagnetic gauge charge +e,and charges p = ±1 under an emergent U(1) gauge field.v is a valley index, vdis is an impurity potential.

Lf =X

v=1,2

X

p=±1

f†pv

✓@

@⌧� µ� ipa⌧ � (r� ipa� ieAem)2

2m⇤

◆fpv

+ vdis(r)f†pvfpv

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Increasing SDW order

Thermal Hall conductivity

Fermionic ‘chargons’ of density � in hole pockets.The fermions carry electromagnetic gauge charge +e,and charges p = ±1 under an emergent U(1) gauge field.v is a valley index, vdis is an impurity potential.

Lf =X

v=1,2

X

p=±1

f†pv

✓@

@⌧� µ� ipa⌧ � (r� ipa� ieAem)2

2m⇤

◆fpv

+ vdis(r)f†pvfpv

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Increasing SDW order

Leading order fermionic contribution is thatimplied by the Wiedemann-Franz law.

�xy =

✓�e2⌧

m⇤

◆!c⌧

0xy =

⇡2T

3

✓kBe

◆2

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Thermal Hall conductivity

Fermionic ‘chargons’ of density � in hole pockets.The fermions carry electromagnetic gauge charge +e,and charges p = ±1 under an emergent U(1) gauge field.v is a valley index, vdis is an impurity potential.

Lf =X

v=1,2

X

p=±1

f†pv

✓@

@⌧� µ� ipa⌧ � (r� ipa� ieAem)2

2m⇤

◆fpv

+ vdis(r)f†pvfpv

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Integrating out the fermions leads to an e↵ectiveaction for the emergent U(1) gauge field

Sa =

Zd2xd⌧

"K1(x)

2(r⇥ a)2 +

K2(x)

2(ra⌧ � @⌧a)

2

� i�xy(x)

2e2✏µ⌫�aµ@⌫a�

#+

Zd2kd!

8⇡3�k|!| [aT (k,!)]2

<latexit sha1_base64="5W9ByJg2r8r2FewV5yf+1iHNkAU=">AAAEO3icdVNNb9NAEHUTPoqh0II4cRkREaWCVrFD1V6QqnIBcSmiKZWySbS2x84q67Wx120i1/wYrvAX+CGcuSGu3BnbiUoL7Gn3zcebeTPrxFKkutv9ttJoXrt+4+bqLfP2nbW799Y37h+nUZa42HcjGSUnDk9RCoV9LbTEkzhBHjoS3zvTl6X9/SkmqYjUkZ7HOAx5oIQvXK4JGm80HjIVCeWh0uZrpTFIyKACiDINeoLgYxKSYwoSuZeCjoArQN9HV4tTBO6WacCPksobQ0wCSgX9jrUJAc8CyiBQemAyBwOhcvygeJLweWGykOuJy2X+rhhzaL9oAxMU6Y1smIEHTPMM2IEIAjlgfsLd/M3Ygg47JeZ8VmxCkdsFdKAGmOKO5AVFiRDTBUjvTcr2FJbx9t/xl8P5uKLdAhbzRAsu6/cyXZmN5FJZ6GACjJntj+2tRXYBLBVByMf5bF5c4cERUTGMUyFJ85yFGVMZkzQljxfEGWZ/EKqsLKM21gIkw7KHUhyz5iKNpuCxKMSAF/keBYtRjxieAQt4SDVM4by2npfgYFn/6Ag6U3KqTJvDkX3RjMlQeRfTGa+3utvdXs/atYEu1aHLTq/be74D1gJpGYtzON5YWWNe5GYhTd+VPE0HVjfWw7zsypVI485SjLk75QHm1doW8IQgr9odP6LmKvSSHw/TdB465FnuSnrVVoL/sg0y7e8Nc6HiTKNyayI/k+X2ln8APJGQIHJO6+tSvRnXVIc74aStpr9ishTpI6lAT3KmcabPhEdMeU8o6kPhmRuRyCRXpWsxsIY5cyLplbVGMm9ZRWGSgEuV4P+XY3vb6m3bb+3W/sFCylXjkfHY6BiWsWvsG6+MQ6NvuI3zxqfG58aX5tfm9+aP5s/atbGyiHlgXDrNX78B4BJkRQ==</latexit>

Thermal Hall conductivity

Fermionic ‘chargons’ of density � in hole pockets.The fermions carry electromagnetic gauge charge +e,and charges p = ±1 under an emergent U(1) gauge field.v is a valley index, vdis is an impurity potential.

Lf =X

v=1,2

X

p=±1

f†pv

✓@

@⌧� µ� ipa⌧ � (r� ipa� ieAem)2

2m⇤

◆fpv

+ vdis(r)f†pvfpv

<latexit sha1_base64="lE7/nb7Ril4zsaNq4UHh2WXO4ps=">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</latexit>

The gauge field contributes a thermal Hall con-

ductivity, 1xy, which has the opposite sign from

the Wiedemann-Franz term determined from �xy.<latexit sha1_base64="HSYrXf9Yh5m2knwUtVfLvSfXAQY=">AAAC13icdVHLbtNAFB2bVwkUUtggsRmRILEokZ0ghW5QBRJkWaSmKYpDdD2+sUeZh+UZJw2WxQ6x5Vf4G/6GcRIWRXCl0Vydc5/nxrngxgbBL8+/cfPW7TsHd1v37h8+eNg+enRhdFkwHDMtdHEZg0HBFY4ttwIv8wJBxgIn8fJdw09WWBiu1bnd5DiTkCq+4Aysg+btn5HSXCWobOs8Q5pCmSJdcBQJZVrZgselRUOB2gwLCYKOQIiGSkpm+YrbzTHtRkvIc/gczqurTd09puuMs4xmYJosWkUoqc5zbbhFaniqarootNySE44JSlDq5fsC1BdqXReaYPO5jZJdYDdyWRJ25XvzdifoDcPhYDCkQS/YWuMMTl6d9Gm4Rzpkb2fzI+8wSjQrpduSCTBmGga5nVVQWM4E1q2oNJgDW0KK1VbSmj53kOuuC/eUpVv0WhxIYzYydpESbGb+5hrwX9y0tIvXs4qr3Omq2K7RohTUatrchya8QGbFhgJjbt4SrJuDZVAAc6K4WgbdkVVqsyqyeGXXPHGdqgFXbg+Fa6alkzOpopUrU0/DWRXFWiTNrFpUnbCuW07APyrR/zsX/V446PU/9junb/dSHpCn5Bl5QUIyJKdkRM7ImDDviffG++CN/E/+V/+b/30X6nv7nMfkmvk/fgPxLeQw</latexit>

Integrating out the fermions leads to an e↵ectiveaction for the emergent U(1) gauge field

Sa =

Zd2xd⌧

"K1(x)

2(r⇥ a)2 +

K2(x)

2(ra⌧ � @⌧a)

2

� i�xy(x)

2e2✏µ⌫�aµ@⌫a�

#+

Zd2kd!

8⇡3�k|!| [aT (k,!)]2

<latexit sha1_base64="5W9ByJg2r8r2FewV5yf+1iHNkAU=">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</latexit>

H =X

n.n.

J1Si ·Sj+X

n.n.n.

J2Si ·Sj+J�

X

4Si ·(Sj⇥Sk)�

X

i

BZ ·Si .

<latexit sha1_base64="pUDkRBoNqHLIyCr/fSKa7CEFmRY=">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</latexit>

J2/J1<latexit sha1_base64="G9P2QTuAClBk/GGfooSmq4LxCrc=">AAAB7nicbVBNSwMxEJ3Ur1q/qh69BIvgqe5WQY9FL9JTBfsB7bJk02wbms0uSVYoS3+EFw+KePX3ePPfmLZ70NYHA4/3ZpiZFySCa+M436iwtr6xuVXcLu3s7u0flA+P2jpOFWUtGotYdQOimeCStQw3gnUTxUgUCNYJxnczv/PElOaxfDSThHkRGUoeckqMlToNv3bR8F2/XHGqzhx4lbg5qUCOpl/+6g9imkZMGiqI1j3XSYyXEWU4FWxa6qeaJYSOyZD1LJUkYtrL5udO8ZlVBjiMlS1p8Fz9PZGRSOtJFNjOiJiRXvZm4n9eLzXhjZdxmaSGSbpYFKYCmxjPfscDrhg1YmIJoYrbWzEdEUWosQmVbAju8surpF2rupfV2sNVpX6bx1GEEziFc3DhGupwD01oAYUxPMMrvKEEvaB39LFoLaB85hj+AH3+APK5jqg=</latexit>

Neel<latexit sha1_base64="cHZ+hdHQ7ZaQctcccFVVh0hrSw4=">AAACAHicdVDLSsNAFJ34rPVVdelmsIiuQpoqxp3oxpVUsFUwpUymN+3YySRkbtQSuvEL3OoXuBO3/okf4H84rXVR0QMDl3POcM89QSKFRsf5sKamZ2bn5gsLxcWl5ZXV0tp6Q8dZyqHOYxmnVwHTIIWCOgqUcJWkwKJAwmXQOxnql7eQahGrC+wn0IxYR4lQcIaGapz5OwCyVSo79r5b9fY96tjOCGZwD6sH3iGtjJkyGaPWKn367ZhnESjkkml9XXESbOYsRcElDIp+piFhvMc6kI8yDui2odo0jFPzFNIRO+Fjkdb9KDDOiGFX/9aG5F/adYah18yFSjIExb8XhZmkGNPhwbQtUuAo+5RxbvJmDE0O3mUp42iKKfoaTGuqg93cR7jHO9E2m/I9oQaml5/j6f9Dw7UrVds9d8tHx+OGCmSTbJFdUiEH5IickhqpE05uyCN5Is/Wg/VivVpv39Ypa/xng0zAev8CCPiXmA==</latexit> VBS

<latexit sha1_base64="uPoEJMJvjPXtwbG1x8MCLr1Pg+o=">AAAB/XicdVDLTsJAFJ3iC/GFunQzkZi4Ii2g1h3BjUsM8kgoIdPpABOm06Zzq5KG+AVu9QvcGbd+ix/gfzhAXWD0JpOcnHMm99zjhoIrMM1PI7Oyura+kd3MbW3v7O7l9w9aKogjypo0EEHUcYligkvWBA6CdcKIEd8VrO2Or2Z6+45FigfyFiYh6/lkKPmAUwKaarRqjX6+YBbN80vbLmMN5qOBfWZXrDK2UqaA0qn381+OF9DYZxKoIEp1LTOEXkIi4FSwac6JFQsJHZMhS+YBp/hEUx4eBJF+EvCcXfIRX6mJ72qnT2Ckfmsz8i+tG8PA7iVchjEwSReLBrHAEODZtdjjEaMgJphQqvPGBHQOOiIRoaBbyTmK6crkEEaJA+wB7rmnNyUVLqe6l5/j8f+gVSpa5WLpplSo1tKGsugIHaNTZKELVEXXqI6aiKIhekLP6MV4NF6NN+N9Yc0Y6Z9DtDTGxzfKGpZY</latexit>

VBS+CSL

<latexit sha1_base64="Kay7A7ZG06EIBFNGY8n99G1pMWY=">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</latexit>

Neel+CSL

<latexit sha1_base64="deBLsLlfkarDesqPgWgmvXs0940=">AAACJHicdVDLSgMxFM34tr6qLt0EiygIZVoLbXeiGxciilYFp5RM5rYNzWSG5I5ahv6Cn+EXuNUvcCcu3LjzP8zUulD0QOByzn3kHD+WwqDrvjlj4xOTU9Mzs7m5+YXFpfzyyrmJEs2hwSMZ6UufGZBCQQMFSriMNbDQl3Dh9/Yz/eIatBGROsN+DM2QdZRoC87QUq38ludDR6iUg0LQg9yRtwkgPW97//Qw54EKvpVWvuAWK/VqrexSt+gOYYtKtV6v1WlpxBTICMet/IcXRDwJ7TiXzJirkhtjM2UaBZcwyHmJgZjxHutAOvQxoBuWCmg70vYppEP2Rx8LjemHvu0MGXbNby0j/9KuEmzXmqlQcYKg+NehdiIpRjQLhQZCA0fZp4xndhOG9h+8yzTj1rvdZcAmqzrYTT2EW7wRgb2UVoTKcvk2T/8vzsvF0k6xfFIu7O6NEpoha2SdbJESqZJdckCOSYNwckceyCN5cu6dZ+fFef1qHXNGM6vkB5z3T994peM=</latexit>

Chiral spin liquid (CSL)with

semion topological order<latexit sha1_base64="WPfAOeQx8IQ0QpdxAcUvqDZbBzo=">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</latexit>

J�J1

/ B?<latexit sha1_base64="yDkwGhRdqmdJZQ5UdhppVuR05Vo=">AAACInicbVDLSgMxFM3UV62vUZdugkUQF2WmCrosdSOuKtha6AxDJs20oclMSDJCGeYf/Ai/wa2u3YkrwY1/YtrOwrYeCDmccy/33hMKRpV2nC+rtLK6tr5R3qxsbe/s7tn7Bx2VpBKTNk5YIrshUoTRmLQ11Yx0hSSIh4w8hKPrif/wSKSiSXyvx4L4HA1iGlGMtJEC+8yLJMLZbeDhIc3N7+bQEzIROoGZF3LYzIPME0SKPLCrTs2ZAi4TtyBVUKAV2D9eP8EpJ7HGDCnVcx2h/QxJTTEjecVLFREIj9CA9AyNESfKz6Y35fDEKH0YJdK8WMOp+ rcjQ1ypMQ9NJUd6qBa9ifif10t1dOVnNBapJjGeDYpSBs3Bk4Bgn0qCNRsbgrCkZleIh8iEpE2Mc1NCPsnEXUxgmXTqNfe8Vr+7qDaaRTplcASOwSlwwSVogBvQAm2AwRN4Aa/gzXq23q0P63NWWrKKnkMwB+v7F/p4pRM=</latexit>

Sf =

Zd2rd⌧

2X

↵=1

f↵�

µ(@µ � iAµ)f↵ +m� f↵f↵ +mN f↵�f↵

�

<latexit sha1_base64="v/rmpgtscV7GTPTOWwWYwJQAZ3I=">AAADCnicdVLNbtQwEHbCX1ko3cKRy4gVUhGwStJquz1UKnCAEyqCbSutt5HjOIlV24liB1hF+wY8AVd4Am6IKy/BA/Ae2Ltb0KJ2JMvjmc+ebz5PUgmuTRD88vwrV69dv7F2s3Pr9vqdje7m3SNdNjVlI1qKsj5JiGaCKzYy3Ah2UtWMyESw4+Tshcsfv2e15qV6Z6YVm0iSK55xSowNxZveOlYlVylTpoMTlnPVplxXgkwlMc Wsg91GiWjfzuIM9gEwVwbS0whqSAEb0gDWjYxbTERVkP1wZlM44XkuxoBLW9oxa7NZvAAAzomU5BTLBrZwRWrDiYjd6SlweOa8R5Cdg+ExyBjTggN+cuFrf5EO+PoyFH7pav4DL/hNoIOZSlf6jbu9oB/tDIeDPQj6YRhGu84JtgeD4R6E/WBuPbS0w7j7G6clbaTVjwqi9TgMKjNpXV9UMKtfo1lF6BnJWTv/rRk8tKEUsrK2y2o5j67giNR6KhOLdJz0/7k50Qty48Zkw0nLVdUYpuiiUNYIMCW4r4eU14waMQVCqeXbEGN50ILUhBo7Ih2smZ0flZuixYZ9NB94aiu1O1w5Xc6bh8udo6gfbvejN1Hv4PlSoTV0Hz1AWyhEu+gAvUKHaISoZ7zP3hfvq//J/+Z/938soL63vHMPrZj/8w8TBPc9</latexit>

R. Samajdar, M. S. Scheurer, S. Chatterjee, Haoyu Guo, Cenke Xu, and S. Sachdev, Nature Physics (2019), arXiv:1903.01992

Metal with large Fermi

surface

p

h�ai = 0<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>

h�ai 6= 0<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>

h�ai = 0<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>

Metal with electron and hole pockets

Metal with electron and hole pockets

Antiferromagneticmetal

Confiningphase of

SU(2) gauge theory

Higgs phase of SU(2) gauge theory

CDW, Ising-nematic, and/or

Z2 topological order

hHi 6= 0hRi = 0

<latexit sha1_base64="ZYUK2MqqnoXtaxNDL+tRqgnbCZY=">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</latexit>

hHi 6= 0hRi 6= 0

<latexit sha1_base64="VyT/j2tQRFSpbLCp7wbRqmtWndA=">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</latexit>

hHi = 0hRi 6= 0

<latexit sha1_base64="B0Bmt/FA3GK8qPJIhsIfApMiZzw=">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</latexit>

�a ) Incommensurate antiferromagnetism<latexit sha1_base64="FJXxh6ciuJ0FRrBqFoY7Hy3AMaI=">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</latexit>

S. Sachdev, H. Scammell, M. Scheurer, G. Tarnopolsky, PRB 99, 054516 (2019)

Gauge theory of fluctuating SDWcoupled to large Fermi surface

SU(2) gauge theory for Higgs field Ha`

a = 1, 2, 3 is a SU(2) gauge index

` = 1 . . . Nh is a flavor index.

S =

Zd2xd⌧

1

2

�@µH

a` � ✏abcA

bµH

c`

�2+

1

4g2F aµ⌫F

aµ⌫ + V (H

a` )

�

+Sf–coupling to electrons with large Fermi surface<latexit sha1_base64="70PYerwdN1EIsJxT9QdNNsakoCc=">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</latexit>

Gauge theory of fluctuating SDWcoupled to large Fermi surface

Gauge-invariant order parameters at x, ⌧ couple to gaugeinvariant order parameters at x, ⌧ 0 with Jf (⌧) ⇠ 1/⌧2

via Fermi surface excitations.

This coupling is irrelevant by naive power-counting.<latexit sha1_base64="WKUGdUZBr3CqKZy/9zxSZXbPXGc=">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</latexit>

SU(2) gauge theory for Higgs field Ha`

a = 1, 2, 3 is a SU(2) gauge index

` = 1 . . . Nh is a flavor index.

S =

Zd2xd⌧

1

2

�@µH

a` � ✏abcA

bµH

c`

�2+

1

4g2F aµ⌫F

aµ⌫ + V (H

a` )

�

+Sf–coupling to electrons with large Fermi surface<latexit sha1_base64="70PYerwdN1EIsJxT9QdNNsakoCc=">AAAEHXicdVNNbxMxEN0mUEr4auHIZURDlaptlN2GtpdKBaSqJ1RU0laKk8jrnd1Y9XqD7W0TrfZXcAXxa7ghrohfA94kpF/g03jm+c3Mm7E/EFybRuPXXKl85+78vYX7lQcPHz1+srj09FgnqWLYYolI1KlPNQousWW4EXg6UEhjX+CJf/a2iJ+co9I8kR/MaICdmEaSh5xRY129pdI88THiMmMoDaq8ctSqeasQ0TRCMH1M1AjCRMEBjyINIUcRQJXE1PQZFdlB3qU9gkJUCalUKeyCu+6tb1aBa6BwlYrLAIcFqEAXOCAiSIyGd73+FB4Kem4zjZH1CkEZzIqa1ogfJVWKjvIKzEo4ymFldwWgQrg0EHS9IQRADE1tAgxNm4SKsszNMy8fO2pkQJXhVPRInMLtTmADCA40F1adjPosh9dd/xaWTbBE8ahvVrserMEsUROirpfDviXM7Dsi0xuXNTiuXSXLaD6mW53ydYDIRKaxjwoIgRXbXcF/2XEvtDc/GWYbGyxJ7ZrICEwCKJAZlUgNF9z0QVBlhd9HFXPQqQopw3xGPNH3UtDe4nKj3tzaarpb0Kg3xscazcarbXcH3Kln2Zmew97ibxIkLI3thJigWrfdxsB0skJbJtCOLNU4oOyMRti2pqQx6k42XtgcXlpPMN6rMLFTG3uvvshorPUo9i2y6FrfjBXOf8XaqQl3OhmXg9SgZJNEYSoKdYrth4Arq5EYWYMyxW2twPrUDs6u2fUsfmx70Gg/lIxMPyMGh+aCBzZv1qx7XOZWsb+ywP+NY6/ubta9997y3pupdgvOc+eFU3NcZ9vZcw6cQ6flsFJc+lT6XPpS/lr+Vv5e/jGBluamb54510755x+ZwVbH</latexit>

Sf = �1

2Nh

Zd2xd⌧d⌧ 0Ha

` (x, ⌧)Ham(x, ⌧)Jf (⌧ � ⌧ 0)Hb

`(x, ⌧0)Hb

m(x, ⌧ 0)<latexit sha1_base64="UzSuxBgdR0T82Bbs3mNEAjwICuY=">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</latexit>

Gauge theory of fluctuating SDWcoupled to large Fermi surface

In the large Nh limit, we are required to decouple Sf by introducinga bilocal field Cab(x, ⌧, ⌧ 0)

Sf =

Zd2xd⌧d⌧ 0

"Nh

2

[Cab(x, ⌧, ⌧ 0)]2

Jf (⌧ � ⌧ 0)� Cab(x, ⌧, ⌧

0)Ha` (x, ⌧)H

b`(x, ⌧

0)

#

At the large Nh saddle point, we have Cab(x, ⌧, ⌧ 0) = �abC(⌧ � ⌧ 0).Saddle point equations show that Cab displays strong scaling, and Sf

is not irrelevant.. . .<latexit sha1_base64="Za2MxjYgBuYBthzPTCiPpIyZj8Y=">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</latexit>

Bilocal quantum field theory

The path integral for the SYK model is abilocal field theory

Z =

ZDG(⌧1, ⌧1)D⌃(⌧1, ⌧2)e

�NS[G,⌃]

for a known action S[G,⌃]. The saddle point,Gs(⌧1�⌧2), ⌃s(⌧1�⌧2), depends only on time dif-ferences, and obeys Planckian !/T scaling. Thefluctuations

G(⌧1, ⌧2) = Gs(⌧1 � ⌧2) + �G(⌧1, ⌧2)

⌃(⌧1, ⌧2) = ⌃s(⌧1 � ⌧2) + �⌃(⌧1, ⌧2)

require bilocal fields.Similar remarks apply to other random quan-

tum systems, and to DMFT.<latexit sha1_base64="CoxwpgSs4ODHF4/ECkI+VV6Z7Tc=">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</latexit>