Quantum Monte Carlo Methods for Fermion-Boson Problems

Quantum Monte Carlo Methods for Fermion-Boson problems

Fakher F. Assaad (Autumn School on Correlated Electrons, Jülich 20th September 2018)

Outline: Ø Fermion-Boson problems Electron-Phonon, Su-Schrieffer-Heeger (SSH) Unconstrained Gauge theories “De-signer” Hamiltonians

Ø Auxiliary field Quantum Monte Carlo (QMC), Generalities

Ø Application of the Auxiliary field QMC to the SSH model Sign-free formulation Sampling, Hybrid Monte Carlo

Ø Conclusions

SFB1170 ToCoTronics Quantum Monte Carlo Methods for Fermion-Boson problems

Many thanks to

M. Hohenadler J. Hofmann S. Beyl M. Rackowski Z. Wang T. Sato

M. Ulybyshev F. Parisen Toldin J. Schwab E. Huﬀman Fermion-Boson problems: The Su-Schrieffer-Heeger (SSH) model Su-Schrieffer-Heeger model (SSH)

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cˆ†

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 i,

ˆ k ˆ ˆ X i,j !0 = Xb, Pb0 = i~ b,b0 AAACB3icbZDLSsNAFIYn9VbrLepSkMEiuJCSFEGXBTcuK9gLNCFMpift2MkkzEyEErpz46u4caGIW1/BnW/jtM1CW38Y+PjPOZw5f5hyprTjfFulldW19Y3yZmVre2d3z94/aKskkxRaNOGJ7IZEAWcCWpppDt1UAolDDp1wdD2tdx5AKpaIOz1OwY/JQLCIUaKNFdjH3pDovDsJco8TMeCA2fk99uScJ4FddWrOTHgZ3AKqqFAzsL+8fkKzGISmnCjVc51U+zmRmlEOk4qXKUgJHZEB9AwKEoPy89kdE3xqnD6OEmme0Hjm/p7ISazUOA5NZ0z0UC3WpuZ/tV6moys/ZyLNNAg6XxRlHOsET0PBfSaBaj42QKhk5q+YDokkVJvoKiYEd/HkZWjXa65Tc28vqo16EUcZHaETdIZcdIka6AY1UQtR9Iie0St6s56sF+vd+pi3lqxi5hD9kfX5A5H5mQo= h i 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 rm 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 h i

Limiting cases

AAAB83icbVDLSgNBEOyNrxhfUY9eBoPgKcyKoBch4MVjBBMD2SXMTmaTIfNYZmaFEPIbXjwo4tWf8ebfOEn2oIkFDUVVN91dSSa4dRh/B6W19Y3NrfJ2ZWd3b/+genjUtjo3lLWoFtp0EmKZ4Iq1HHeCdTLDiEwEe0xGtzP/8YkZy7V6cOOMxZIMFE85Jc5LUaQlG5AeRjcI96o1XMdzoFUSFqQGBZq96lfU1zSXTDkqiLXdEGcunhDjOBVsWolyyzJCR2TAup4qIpmNJ/Obp+jMK32UauNLOTRXf09MiLR2LBPfKYkb2mVvJv7ndXOXXscTrrLcMUUXi9JcIKfRLADU54ZRJ8aeEGq4vxXRITGEOh9TxYcQLr+8StoX9RDXw/vLWgMXcZThBE7hHEK4ggbcQRNaQCGDZ3iFtyAPXoL34GPRWgqKmWP4g+DzB1mJkIE= ! =0 !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 0 0 !1 Rev. Mod. Phys. 60, 781 (1988) Perfect nesting and Van-Hove singularity13.03.18 HQMC simulations of the SSH model 5 ˆ W H = t cî,† cˆj, +ˆcj,† cî, (QQ, i⌦ = 0) ln2 m ordered valence/ bondk statesT (VBS) i,j , 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 sha1_base64="hP+6LrUf2d3tZaldqaQQvEKMXyw=">AAAB2XicbZDNSgMxFIXv1L86Vq1rN8EiuCozbnQpuHFZwbZCO5RM5k4bmskMyR2hDH0BF25EfC93vo3pz0JbDwQ+zknIvSculLQUBN9ebWd3b/+gfugfNfzjk9Nmo2fz0gjsilzl5jnmFpXU2CVJCp8LgzyLFfbj6f0i77+gsTLXTzQrMMr4WMtUCk7O6oyaraAdLMW2IVxDC9YaNb+GSS7KDDUJxa0dhEFBUcUNSaFw7g9LiwUXUz7GgUPNM7RRtRxzzi6dk7A0N+5oYkv394uKZ9bOstjdzDhN7Ga2MP/LBiWlt1EldVESarH6KC0Vo5wtdmaJNChIzRxwYaSblYkJN1yQa8Z3HYSbG29D77odBu3wMYA6nMMFXEEIN3AHD9CBLghI4BXevYn35n2suqp569LO4I+8zx84xIo4 sha1_base64="E469Am/V+esbdjPBFF4sNuTXhCI=">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 sha1_base64="a22j3uBbepvqkCSkzfJ5mvCp9vw=">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 B hXi ⇣ ⌘ ✓ ◆ 2 g2 cˆ† cˆ +ˆc† cˆ 4k i, j, j, i, strong bonds weak bonds i,j ! hXi X g2 ˆ ˆ i,j S iS j +⌘î⌘ˆj

AAAD/nicvVPLbtNAFJ3aPEp4pcCOzYgIVGgb2RYSbJAqsemyCNJWyiTWeHztTGKPzcwYKRpZ4lfYsAAhtnwHO/6GieNWIemiK65l+fjcM/eeeUVlxpX2vD9bjnvt+o2b27c6t+/cvXe/u/PgRBWVZDBgRVbIs4gqyLiAgeY6g7NSAs2jDE6j2dtF/vQTSMUL8UHPSxjlNBU84YxqS4U7ziMSQcqFgY+CSknnL2oyodoc1RjjNxg/sw8+0ERVeWhIBokmGRVpBpjvT4nk6UQTuST2ieJpTutGtdtUYXVo+Dk/NiSmaQqyvshNz3N7m9Ql8otSy8bPrUVMyNJkx8JKxCAjSRmYA5LYr0nHQW1ezuor+G9941baNvoPsxgH2LIrfq9mFzcFSZlH5n1dh3ztf4r3VhgCmq6JWmqKFztNQMQrRyDs9ry+1wTeBH4LeqiN47D7m8QFq3IQmmVUqaHvlXpkqNScWa8dUikoKZvRFIYWCpqDGpnm+Nb4qWVinBTSvkLjhl0dYWiu1DyPrDKneqLWcwvystyw0snrkeGirDQItmyUVBnWBV7cBRxzCUxncwsok9x6xWxC7T5oe2M6dhH89SlvgpOg73t9/13QOwza5dhGj9ETtIt89AodoiN0jAaIOcb54nxzvruf3a/uD/fnUupstWMeon/C/fUXLnBZHw== k Valence Bond Solid h i | Quantum antiferromagnet/P {z SSC + CDW}

Tang & Hirsch, Phys. Rev. B 37, 9546 (1988) 13.03.18 HQMC simulations of the SSH model 9 Fermion-Boson problems: The Su-Schrieffer-Heeger (SSH) model Su-Schrieffer-Heeger model (SSH)

cˆ†

Limiting cases

Tang & Hirsch, Phys. Rev. B 37, 9546 (1988) 13.03.18 HQMC simulations of the SSH model 9 Fermion-Boson problems: The Su-Schrieffer-Heeger (SSH) model

Symmetries ˆ2 P i,j k ˆ ˆ h i ˆ 2 H = t cî,† cˆj, +ˆcj,† cî, + g X i,j cî,† cˆj, +ˆcj,† cî, + + X i,j h i 2m 2 h i i,j , i,j , i,j " # 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 hXi ⇣ ⌘ hXi ⇣ ⌘ hXi

Partial particle-hole symmetry ˆ 1 ˆ iQQ i P↵ cîi,† P↵ = ↵,e · cîi, +(1 ↵,)ˆcii,† 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 sha1_base64="hP+6LrUf2d3tZaldqaQQvEKMXyw=">AAAB2XicbZDNSgMxFIXv1L86Vq1rN8EiuCozbnQpuHFZwbZCO5RM5k4bmskMyR2hDH0BF25EfC93vo3pz0JbDwQ+zknIvSculLQUBN9ebWd3b/+gfugfNfzjk9Nmo2fz0gjsilzl5jnmFpXU2CVJCp8LgzyLFfbj6f0i77+gsTLXTzQrMMr4WMtUCk7O6oyaraAdLMW2IVxDC9YaNb+GSS7KDDUJxa0dhEFBUcUNSaFw7g9LiwUXUz7GgUPNM7RRtRxzzi6dk7A0N+5oYkv394uKZ9bOstjdzDhN7Ga2MP/LBiWlt1EldVESarH6KC0Vo5wtdmaJNChIzRxwYaSblYkJN1yQa8Z3HYSbG29D77odBu3wMYA6nMMFXEEIN3AHD9CBLghI4BXevYn35n2suqp569LO4I+8zx84xIo4 sha1_base64="rQ20K2/kGjcawcQHYn398eupReg=">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 sha1_base64="NiaOTykuxlLW3Jby+kQLz42VB8Y=">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 ˆ ˆ ˆ 1 ˆ ˆ P↵, H =0 P S iP = ⌘î AAACOnicbVC7SgNBFJ31bXxFLW0Gg2Bj2BVBGyFgYxnRRCG7Lncnk2TI7IOZu0pY9rts/Ao7CxsLRWz9ACebLdR4YOBwzrncuSdIpNBo28/WzOzc/MLi0nJlZXVtfaO6udXWcaoYb7FYxuomAM2liHgLBUp+kygOYSD5dTA8G/vXd1xpEUdXOEq4F0I/Ej3BAI3kVy/cAWDWzP3MTRNQKr7Pb7MDJ6eF7iZhkF3muS8onQ7SU/oj5nKEcdKv1uy6XYBOE6ckNVKi6Vef3G7M0pBHyCRo3XHsBL0MFAomeV5xU80TYEPo846hEYRce1lxek73jNKlvViZFyEt1J8TGYRaj8LAJEPAgf7rjcX/vE6KvRMvE1GSIo/YZFEvlRRjOu6RdoXiDOXIEGBKmL9SNgAFDE3bFVOC8/fkadI+rDt23bk4qjUOyzqWyA7ZJfvEIcekQc5Jk7QIIw/khbyRd+vRerU+rM9JdMYqZ7bJL1hf32CLr38= AAACF3icbVBNS8NAEN34bf2qevSyWAQPUhIR9CIUvHisYFuhCWGy3TSLmw92J0IJ+Rde/CtePCjiVW/+G7dtDtr6YODx3gwz84JMCo22/W0tLC4tr6yurdc2Nre2d+q7e12d5orxDktlqu4C0FyKhHdQoOR3meIQB5L3gvursd974EqLNLnFUca9GIaJCAUDNJJfb1JX8hD71I0Ai3bpFy7ILILyZKpcl9RVYhihR+kltf16w27aE9B54lSkQSq0/fqXO0hZHvMEmQSt+46doVeAQsEkL2turnkG7B6GvG9oAjHXXjH5q6RHRhnQMFWmEqQT9fdEAbHWozgwnTFgpGe9sfif188xvPAKkWQ58oRNF4W5pJjScUh0IBRnKEeGAFPC3EpZBAoYmihrJgRn9uV50j1tOnbTuTlrtE6rONbIATkkx8Qh56RFrkmbdAgjj+SZvJI368l6sd6tj2nrglXN7JM/sD5/ACE0now= " " h i

Time reversal symmetry (Survives finite doping and and hence allows simulations at finite chemical potential)

cˆ cˆ ˆ 1 ˆ with ˆ2 T ↵ " T = ↵ # AAAB+nicbVBNS8NAEN3Ur1q/Uj16WSyCF0tSBL0IBS8eK/QL2lg22027dLMJuxOlxPwULx4U8eov8ea/cdvmoK0PBh7vzTAzz48F1+A431ZhbX1jc6u4XdrZ3ds/sMuHbR0lirIWjUSkuj7RTHDJWsBBsG6sGAl9wTr+5Gbmdx6Y0jySTZjGzAvJSPKAUwJGGtjl/phA2szu01qGr/G5iwd2xak6c+BV4uakgnI0BvZXfxjRJGQSqCBa91wnBi8lCjgVLCv1E81iQidkxHqGShIy7aXz0zN8apQhDiJlSgKeq78nUhJqPQ190xkSGOtlbyb+5/USCK68lMs4ASbpYlGQCAwRnuWAh1wxCmJqCKGKm1sxHRNFKJi0SiYEd/nlVdKuVV2n6t5dVOq1PI4iOkYn6Ay56BLV0S1qoBai6BE9o1f0Zj1ZL9a79bFoLVj5zBH6A+vzB1gukq0= cˆ cˆ T = 1

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 ✓ #◆ ✓ " ◆ Fermion-Boson problems: The Su-Schrieffer-Heeger (SSH) model

U(N) symmetry is manifest cî,1 cî,1 . . . U . ,U†U =1 0 . 1 ! 0 . 1 cî,N cî,N 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 sha1_base64="hP+6LrUf2d3tZaldqaQQvEKMXyw=">AAAB2XicbZDNSgMxFIXv1L86Vq1rN8EiuCozbnQpuHFZwbZCO5RM5k4bmskMyR2hDH0BF25EfC93vo3pz0JbDwQ+zknIvSculLQUBN9ebWd3b/+gfugfNfzjk9Nmo2fz0gjsilzl5jnmFpXU2CVJCp8LgzyLFfbj6f0i77+gsTLXTzQrMMr4WMtUCk7O6oyaraAdLMW2IVxDC9YaNb+GSS7KDDUJxa0dhEFBUcUNSaFw7g9LiwUXUz7GgUPNM7RRtRxzzi6dk7A0N+5oYkv394uKZ9bOstjdzDhN7Ga2MP/LBiWlt1EldVESarH6KC0Vo5wtdmaJNChIzRxwYaSblYkJN1yQa8Z3HYSbG29D77odBu3wMYA6nMMFXEEIN3AHD9CBLghI4BXevYn35n2suqp569LO4I+8zx84xIo4 sha1_base64="j/guVYSPPNYdE2vfGJxZ4L/h430=">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 sha1_base64="du6ngX7CvuMdrE6W3SsrjC/wsjQ=">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 C B C @ A @ A Majorana fermions

i A : ii,1, = cîi, +ˆcii,† , ii,2, = i cîi, cîi,† 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 2 ˆ , ˆ =2

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 j,↵, i,, i,j ↵, , ⇣ ⌘ ⇣ ⌘ { 0 } 0 i B : ii,2, = cîi, +ˆcii,† , ii,1, = i cîi, cîi,† 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 2 N,⇣2 ⌘ ⇣ ⌘Pˆ2 ˆ 1 ˆ i,j k ˆ 2 H = t + gX i,j iî,↵,ˆj,↵, + h i + X i,j 2 h i 2m 2 h i i,j ,↵=1, i,j " # 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 h iX ⇣ ⌘ hXi î,1,1 î,1,1 . . O(2N) symmetry is now manifest 0 . 1 0 . 1 ˆ ˆ B i,i,N C O B i,i,N C ,OT O =1 B î,2,1 C ! B î,2,1 C B C B C B . C B . C B . C B . C B C B C Bî,2,N C Bî,2,N C 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 C B C @ A @ A Fermion-Boson problems: Unconstrained gauge theories

ˆ ˆ k !mX i,j iP i,j ! = ˆ† h i h i 0 Bosonic variables. b = 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 i,j p rm 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 h i 2!m

ˆ 1 ˆ ˆ ˆ ˆ H = g b†i,j + b i,j cˆi,† cˆj, +ˆcj,† cˆi, + !0 b†i,j b i,j 2!0m h i h i h i h i i,j b 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 sha1_base64="hP+6LrUf2d3tZaldqaQQvEKMXyw=">AAAB2XicbZDNSgMxFIXv1L86Vq1rN8EiuCozbnQpuHFZwbZCO5RM5k4bmskMyR2hDH0BF25EfC93vo3pz0JbDwQ+zknIvSculLQUBN9ebWd3b/+gfugfNfzjk9Nmo2fz0gjsilzl5jnmFpXU2CVJCp8LgzyLFfbj6f0i77+gsTLXTzQrMMr4WMtUCk7O6oyaraAdLMW2IVxDC9YaNb+GSS7KDDUJxa0dhEFBUcUNSaFw7g9LiwUXUz7GgUPNM7RRtRxzzi6dk7A0N+5oYkv394uKZ9bOstjdzDhN7Ga2MP/LBiWlt1EldVESarH6KC0Vo5wtdmaJNChIzRxwYaSblYkJN1yQa8Z3HYSbG29D77odBu3wMYA6nMMFXEEIN3AHD9CBLghI4BXevYn35n2suqp569LO4I+8zx84xIo4 sha1_base64="gurAvKVpNEBy9ziNqFumI0QMxv8=">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 sha1_base64="4Yswar4j5FyT7VQucq24bhmK+jY=">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 r hXi ⇣ ⌘ X ⇣ ⌘ X

Local Z2 conservation law

nˆb +ˆnb +ˆnb +ˆnb c i,i+a i,i a i,i+a i,i a nî Qˆ =( 1) h xi h xi h y i h y i ( 1) 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 i ˆ ˆ H,Qi =0 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 h i Boson parity on star Fermion parity on site F. F. ASSAAD and TARUN GROVER PHYS. REV. X 6, 041049 (2016) and velocity fluctuations in Dirac metals [26]. It consists of N species of fermions on the vertices of a square lattice interacting with quantum Ising spins that live on the links (d) (e) of the same square lattice. However, in contrast to the Hamiltonian in Ref. [25], our Hamiltonian has a local Z2 (a) invariance, which leads to LxLy conserved operators on a Lx × Ly lattice. Furthermore, we restrict ourselves to half- filling; thus, at low energies, instead of a Fermi surface, we obtain either Fermi points or no fermions (e.g., a Mott insulator). These differences completely alter the phase diagram of our model compared to Ref. [25]. The phase diagram for various values of N is summa- rized in Fig. 1. For N>1, we find that when the kinetic energy of the aforementioned Ising spins is small compared to the kinetic energy of the fermions, the ground state resembles the deconfined phase of the Z2 gauge theory coupled to 2N Dirac fermions (the N 1 case is special in that the deconfined fermions are unstable¼ to infinitesimal interactions). However, as we will discuss in detail, there is an important distinction between this phase and the (b) (c) deconfined phase of a conventional Z2 gauge matter theory (Refs. [27–29]). Namely, the local Z2 invariance in our FIG. 1. Schematic zero-temperature phase diagram of the model is an actual symmetry of the Hamiltonian, in model in Eq. (1) (a), as well as cartoons (b)–(e) of a selected contrast to the so-called “gauge symmetry” in gauge-matter number of phases. (a) We observe a Z2 Dirac deconfined phase theories, which just corresponds to redundancy in the (Z2D), a Néel antiferromagnet phase (AFM) [or a superconductor description of the physical states. This is because we do (SC), depending on the pattern of particle-hole symmetry break- not project the Hilbert space of our Hamiltonian to gauge- ing], a charge density wave (CDW) phase, and a valence-bond solid (VBS). For N 1, we do not find evidence for a Z D phase invariant states; that is, we do not impose Gauss’s law. ¼ 2 Instead, in our model, Gauss’s law in the ground state beyond h 0, consistent with the arguments in the main text. The phase¼ transitions from the Z D to AFM/SC (N 2) and emerges because of spontaneous symmetry breaking as the 2 VBS (N 3) are seemingly continuous. At N 3, we observe¼ a locally conserved operators order in a certain definite deconfined¼ quantum critical point (DCQP) between¼ the VBS and pattern. One physical significance of this result is that, AFM phases. (b)–(e) Cartoons of the corresponding phases. in contrast to regular Z2 gauge theory, in ourFermion model,-Boson equal- problems:Circles correspondUnconstrained to fermions, gauge and theories the color code corresponds space, unequal-time, non-gauge-invariant correlation func- to the flavor index. Circles with two colors represent a pair of tions do not vanish identically. Furthermore, there is a fermions on a site with corresponding flavors. The low-energy finite-temperature Ising transitionˆ corresponding1 ˆ to theˆ properties of the Z2D phase resembleˆ SUˆN fermions propa- H = g b†i,j + b i,j gating freelycî,† cˆj, in +ˆ space-timecj,† cî, and+ ! connected0 b†i,j byðb i,jÞ a Z gauge string restoration of the aforementioned symmetry.2!0m h i h i h i h i 2 i,j b AAADkHiclVJdb9MwFHUTPkZh0I1HXiyqSUOdqqQgbQ9MDPEy8TQkuk2qS+S4TuLNdjLbQaqs/B1+EG/8G5wP1rEOAVeydHTOvefea9244EybIPjR8/x79x883HjUf/xk8+mzwdb2qc5LReiU5DxX5zHWlDNJp4YZTs8LRbGIOT2LLz/U+tlXqjTL5WezLOhc4FSyhBFsHBVt9b6hDBt7XEEIDyHSpYgs4limnEK2dwGRarHTUydfKWNRojCxYWUnEOWCpjgKoIBV7YA4TcwubCzj6otFC5ymVFV/9BzdyC0yLE0u/lqDFEsz86pu146rWSpw1fZu7IgrZnsdv5riWrv4pY3WqTvSr61WnesYrdZvB4mrf9/8f9eOBsNgHDQB10HYgSHo4iQafEeLnJSCSkM41noWBoWZW6wMI5xWfVRqWmByiVM6c1BiQfXcNgdVwR3HLGCSK/ekgQ17s8JiofVSuIV3BDaZvq3V5F3arDTJwdwyWZSGStI2SkoOTQ7r64QLpigxfOkAJoq5WSHJsLs342647z4hvL3yOjidjMNgHH56MzyadN+xAV6Al2AXhGAfHIFjcAKmgHib3mvvrXfob/sH/jv/fZvq9bqa5+C38D/+BAGKJmU= sha1_base64="hP+6LrUf2d3tZaldqaQQvEKMXyw=">AAAB2XicbZDNSgMxFIXv1L86Vq1rN8EiuCozbnQpuHFZwbZCO5RM5k4bmskMyR2hDH0BF25EfC93vo3pz0JbDwQ+zknIvSculLQUBN9ebWd3b/+gfugfNfzjk9Nmo2fz0gjsilzl5jnmFpXU2CVJCp8LgzyLFfbj6f0i77+gsTLXTzQrMMr4WMtUCk7O6oyaraAdLMW2IVxDC9YaNb+GSS7KDDUJxa0dhEFBUcUNSaFw7g9LiwUXUz7GgUPNM7RRtRxzzi6dk7A0N+5oYkv394uKZ9bOstjdzDhN7Ga2MP/LBiWlt1EldVESarH6KC0Vo5wtdmaJNChIzRxwYaSblYkJN1yQa8Z3HYSbG29D77odBu3wMYA6nMMFXEEIN3AHD9CBLghI4BXevYn35n2suqp569LO4I+8zx84xIo4 sha1_base64="gurAvKVpNEBy9ziNqFumI0QMxv8=">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 sha1_base64="4Yswar4j5FyT7VQucq24bhmK+jY=">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 r As the kinetic energy of the IsinghXi degrees of⇣ freedom ⌘(b).X The⇣ symmetry-broken⌘ phasesX correspond to the confined is increased, we find that the aforementioned gapless phases of the model. At N 3, the AFM phase (e) has the fundamental (conjugate) representation¼ of SU(3) on sublattice Dirac deconfined phase enters a symmetry-broken confined A (B). The corresponding Young tableaux are included. The VBS phase. The nature of symmetry breaking depends on the phase (d) corresponds to a pattern of intersite SU(3) singlets. value of N. For N 1, it corresponds to a charge density Retain only one phonon excita/on per bond, float AAAC3nichVHJSgNBEK2Me9zicvPSGARPYUYEl5PgghdFwRjBiPRM2rHJbHR3AiqePYlXj171j/RbPPimnQgqYg09Vf2q6nUtfhZJbVz3teT09Q8MDg2PlEfHxicmK1PTxzrtqEDUgzRK1YnPtYhkIupGmkicZErw2I9Ew29v5v5GVygt0+TIXGXiLOZhIi9kwA2g88psU8sw5mydMY81g1ZqNNs/r1TdmmuF/Ta8wqhSIQdp5Y2a1KKUAupQTIISMrAj4qTxnZJHLmXAzugGmIIlrV/QLZWR20GUQAQH2sY/xO20QBPcc05tswO8EuEoZDJawNmxjD6i81cFbA39jnNtsfDPF24sc17hFbQPxhHLuAfc0CUi/suMi8heLf9n5l0ZuqBV241EfZlF8j6DL54teBSwtvUw2raRITh8e+9iAgl0HRXkU+4xMNtxC5pbLSxLUjBy8CnofPqoB2v2fi71t1Ffqq3VvMPl6sZSse9hmqN5WsRSV2iDdukAZQQY9hM904vDnTvn3nn4DHVKRc4MfRPn8QPpj5WD :1 N wave, while for N ¼ 2, because of an enlarged symmetry, ··· ¼ II. MODEL AND SYMMETRIES k the symmetry-broken phase can correspond to a super- !0 =

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 m conductor or a Néel AFM, depending on the sign of the Our model reads r infinitesimal symmetry-breaking field. The N 3 case is even richer. We find that after exiting the deconfined¼ phase, N the model enters a VBS phase, and as the tuning parameter ˆ ˆ † ˆ H Z i;j cˆi;αcˆj;α H:c: − Nh X i;j : 1 controlling the kinetic energy of Ising spins is tuned further, ¼ h i þ h i ð Þ i;j α 1 i;j the valence-bond-solid quantum melts and yields a Néel Xh i X¼ Xh i AFM phase. The phase transition between the VBS and † k AFM phase, if second order, corresponds to a deconfined Here, cˆi;α creates a fermion of flavor α inh a Wannier!0 = state 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 rm quantum critical point [30], reported earlier in the context centered aroundPHYSICAL lattice REVIEW site Xi 6,of041049 a square (2016) lattice, and the sum of quantum magnets [31,32]. Our results are consistent runs over bonds of nearest neighbors. Each bond, b i; j , Simple Fermionic Model of Deconfined Phases and Phase Transitions ¼ h i with a second-order transition. accommodates a two-levelF. F. Assaad1 and Tarun system Grover2,3 spanned by the Hilbert 1Institut für Theoretische Physik und Astrophysik, Universität Würzburg, 97074 Würzburg, Germany 2Department of Physics, University of California at San Diego, La Jolla, California 92093, USA 3Kavli Institute for Theoretical Physics, University of California, Santa Barbara, California 93106, USA (Received 7 August 2016; revised manuscript received 2 November 2016; published 12 December 2016) Using quantum Monte Carlo simulations, we study a series of models of fermions coupled to quantum Ising spins on a square lattice with N flavors of fermions per site for N 1, 2, and 3. The models have an ¼ extensive number of conserved quantities but are not integrable, and they have rather rich phase diagrams consisting of several exotic phases and phase transitions that lie beyond the Landau-Ginzburg paradigm. 041049-2 In particular, one of the prominent phases for N>1 corresponds to 2N gapless Dirac fermions coupled to

an emergent Z2 gauge field in its deconfined phase. However, unlike a conventional Z2 gauge theory, we do not impose “Gauss’s Law” by hand; instead, it emerges because of spontaneous symmetry breaking. Correspondingly, unlike a conventional Z2 gauge theory in two spatial dimensions, our models have a finite-temperature phase transition associated with the melting of the order parameter that dynamically imposes the Gauss’s law constraint at zero temperature. By tuning a parameter, the deconfined phase undergoes a transition into a short-range entangled phase, which corresponds to Néel antiferromagnet or superconductor for N 2 and a valence-bond solid for N 3. Furthermore, for N 3, the valence-bond ¼ ¼ ¼ solid further undergoes a transition to a Néel phase consistent with the deconfined quantum critical phenomenon studied earlier in the context of quantum magnets.

DOI: 10.1103/PhysRevX.6.041049 Subject Areas: Condensed Matter Physics, Strongly Correlated Materials

I. INTRODUCTION structure in their wave functions, leading to the infamous Monte Carlo fermion sign problem [15 17]. Similarly, Ground states of strongly interacting electronic systems – fractional quantum Hall phases again possess a nontrivial can exhibit an extremely rich variety of phases. One way sign structure to their wave functions [18]; therefore, they are to organize our understanding is to classify the zero- temperature phases by their entanglement structure at so far amenable only via techniques such as exact diago- nalization [19,20] and the density matrix renormalization long distances and low energies [1–9]. Gapped phases that possess a local order parameter are characterized by short- group [21], which are restricted to only small two- range entanglement; that is, the reduced density matrix of a dimensional systems because of exponential scaling of large subsystem A can be understood simply by patching numerical cost with system size. However, long-range entangled phases exist that do not suffer from the together density matrices of smaller subsystems Ai whose union is A [10]. This is no longer true for gapless phases Monte Carlo sign problem. Two prominent examples are (1) interacting Dirac fermions with an even number of such as Fermi liquids [11–13] or gapped topological phases such as a fractional quantum Hall liquid, and such flavors [22] and (2) a gapped Z2 topological ordered system phases are thus said to possess “long-range entanglement” such as a toric code Hamiltonian [14] in a magnetic field [7,8,14]. [23]. The absence of a sign problem for the former is related Experience as well as heuristic arguments suggest that to the positive fermion determinant, while in the latter Hamiltonians whose ground states possess long-range case, the Hamiltonian has nonpositive off-diagonal elements entanglement are relatively difficult to simulate on a in a local basis, allowing one to sample the corresponding classical computer. For example, even a phase as ubiqui- Boltzmann weight efficiently [24]. In this paper, we study a tous and as well understood as a Fermi liquid is rather hard sign-problem-free model that has a ground state with to simulate numerically because fermions at finite density features of both of the aforementioned long-range entangled with repulsive interactions tend to have an intricate sign phases together, namely, Dirac fermions coupled to a fluctuating Z2 gauge field. By tuning parameters in the Hamiltonian, we also study competition with symmetry- breaking short-range entangled phases, which leads to novel, Published by the American Physical Society under the terms of the Creative Commons Attribution 3.0 License. Further distri- strongly interacting, quantum critical phenomena. bution of this work must maintain attribution to the author(s) and Our Hamiltonian [Eq. (1)] is partially motivated by a the published article’s title, journal citation, and DOI. recent study of nematic instability of a Fermi surface [25]

2160-3308=16=6(4)=041049(19) 041049-1 Published by the American Physical Society F. F. ASSAAD and TARUN GROVER PHYS. REV. X 6, 041049 (2016) and velocity fluctuations in Dirac metals [26]. It consists of N species of fermions on the vertices of a square lattice interacting with quantum Ising spins that live on the links (d) (e) of the same square lattice. However, in contrast to the Hamiltonian in Ref. [25], our Hamiltonian has a local Z2 (a) invariance, which leads to LxLy conserved operators on a Lx × Ly lattice. Furthermore, we restrict ourselves to half- filling; thus, at low energies, instead of a Fermi surface, we obtain either Fermi points or no fermions (e.g., a Mott insulator). These differences completely alter the phase diagram of our model compared to Ref. [25]. The phase diagram for various values of N is summa- rized in Fig. 1. For N>1, we find that when the kinetic energy of the aforementioned Ising spins is small compared to the kinetic energy of the fermions, the ground state resembles the deconfined phase of the Z2 gauge theory coupled to 2N Dirac fermions (the N 1 case is special in that the deconfined fermions are unstable¼ to infinitesimal interactions). However, as we will discuss in detail, there is an important distinction between this phase and the (b) (c) deconfined phase of a conventional Z2 gauge matter theory (Refs. [27–29]). Namely, the local Z2 invariance in our FIG. 1. Schematic zero-temperature phase diagram of the model is an actual symmetry of the Hamiltonian, in model in Eq. (1) (a), as well as cartoons (b)–(e) of a selected contrast to the so-called “gauge symmetry” in gauge-matter number of phases. (a) We observe a Z2 Dirac deconfined phase theories, which just corresponds to redundancy in the (Z2D), a Néel antiferromagnet phase (AFM) [or a superconductor description of the physical states. This is because we do (SC), depending on the pattern of particle-hole symmetry break- not project the Hilbert space of our Hamiltonian to gauge- ing], a charge density wave (CDW) phase, and a valence-bond solid (VBS). For N 1, we do not find evidence for a Z D phase invariant states; that is, we do not impose Gauss’s law. ¼ 2 Instead, in our model, Gauss’s law in the ground state beyond h 0, consistent with the arguments in the main text. The phase¼ transitions from the Z D to AFM/SC (N 2) and emerges because of spontaneous symmetry breaking as the 2 VBS (N 3) are seemingly continuous. At N 3, we observe¼ a locally conserved operators order in a certain definite deconfined¼ quantum critical point (DCQP) between¼ the VBS and pattern. One physical significance of this result is that, AFM phases. (b)–(e) Cartoons of the corresponding phases. in contrast to regular Z2 gauge theory, in our model, equal- Circles correspond to fermions, and the color code corresponds space, unequal-time, non-gauge-invariant correlation func- to the flavor index. Circles with two colors represent a pair of tions do not vanish identically. Furthermore, there is a fermions on a site with corresponding flavors. The low-energy finite-temperature Ising transition corresponding to the properties of the Z2D phase resemble SU N fermions propa- ð Þ restoration of the aforementioned symmetry. gating freely in space-time and connected by a Z2 gauge string As the kinetic energy of the Ising degrees of freedom (b). The symmetry-broken phases correspond to the confined is increased, we find that the aforementioned gapless phases of the model. At N 3, the AFM phase (e) has the fundamental (conjugate) representation¼ of SU(3) on sublattice Dirac deconfined phase enters a symmetry-broken confined A (B). The corresponding Young tableaux are included. The VBS phase. The nature of symmetry breaking depends on the phase (d) corresponds to a pattern of intersite SU(3) singlets. value of N. For N 1, it corresponds to a charge density wave, while for N ¼ 2, because of an enlarged symmetry, the symmetry-broken¼ phase can correspond to a super- II. MODEL AND SYMMETRIES conductor or a Néel AFM, depending on the sign of the Our model reads infinitesimal symmetry-breaking field. The N 3 case is Fermion-Boson problems: Unconstrained gauge theories even richer. We find that after exiting the deconfined¼ phase, N the model enters a VBS phase, and as the tuning parameter ˆ ˆ † ˆ H Z i;j cî;αcˆj;α H:c: − Nh X i;j : 1 controlling the kinetic energy of Ising spins is tuned further, ¼ h i þ h i ð Þ i;j α 1 i;j the valence-bond-solid quantum melts and yields a Néel Xh i X¼ Xh i AFM phase. The phase transition between the VBSF. F. ASSAAD and and TARUN GROVER PHYS. REV. X 6, 041049 (2016) Sign free for allHere, values cofˆ† N.creates a fermion of flavor α in a Wannier state AFM phase, if second order, corresponds to a deconfinedand velocity fluctuations ini;α Dirac metals [26]. It consists of quantum critical point [30], reported earlier in theN=2nN contextspecies Antiunitary of fermionscenteredsymmetry on aroundthe vertices squaring lattice of ato square site -1 i of lattice a square lattice, and the sum of quantum magnets [31,32]. Our results are consistentinteracting withruns quantum over Ising bonds spins of that nearest live on neighbors. the links Each(d) bond, b i; j , (e) with a second-order transition. N=2n+1of the same Majonara squareaccommodatesrepresentation lattice. However, a à two-level inO(2N) contrast system to the spanned by the¼ Hilberth i Hamiltonian in Ref. [25], our Hamiltonian has a local Z2 (a) invariance, which leads to LxLy conserved operators on a Lx × Ly lattice. Furthermore, we restrict ourselves to half- filling; thus, at low energies, instead of a Fermi surface, we obtain041049-2 either Fermi points or no fermions (e.g., a Mott insulator). These differences completely alter the phase diagram of our model compared to Ref. [25]. The phase diagram for various values of N is summa- rized in Fig. 1. For N>1, we find that when the kinetic energy of the aforementioned Ising spins is small compared to the kinetic energy of the fermions, the ground state resembles the deconfined phase of the Z2 gauge theory coupled to 2N Dirac fermions (the N 1 case is special in that the deconfined fermions are unstable¼ to infinitesimal interactions). However, as we will discuss in detail, there F. F. Assaad and Tarun Grover, Phys. Rev. X 6 (2016), 041049. is an important distinction between this phase and the (b) (c) S.deconfined Gazit, F. F. Assaad, phase ofS. Sachdev, a conventional A. Vishwanath,Z gauge and matter C. Wang, theoryPNAS, (2018) 2 (Refs. [27–29]). Namely, the local Z2 invariance in our FIG. 1. Schematic zero-temperature phase diagram of the model is an actual symmetry of the Hamiltonian, in model in Eq. (1) (a), as well as cartoons (b)–(e) of a selected contrast to the so-called “gauge symmetry” in gauge-matter number of phases. (a) We observe a Z2 Dirac deconfined phase theories, which just corresponds to redundancy in the (Z2D), a Néel antiferromagnet phase (AFM) [or a superconductor description of the physical states. This is because we do (SC), depending on the pattern of particle-hole symmetry break- not project the Hilbert space of our Hamiltonian to gauge- ing], a charge density wave (CDW) phase, and a valence-bond solid (VBS). For N 1, we do not find evidence for a Z D phase invariant states; that is, we do not impose Gauss’s law. ¼ 2 Instead, in our model, Gauss’s law in the ground state beyond h 0, consistent with the arguments in the main text. The phase¼ transitions from the Z D to AFM/SC (N 2) and emerges because of spontaneous symmetry breaking as the 2 VBS (N 3) are seemingly continuous. At N 3, we observe¼ a locally conserved operators order in a certain definite deconfined¼ quantum critical point (DCQP) between¼ the VBS and pattern. One physical significance of this result is that, AFM phases. (b)–(e) Cartoons of the corresponding phases. in contrast to regular Z2 gauge theory, in our model, equal- Circles correspond to fermions, and the color code corresponds space, unequal-time, non-gauge-invariant correlation func- to the flavor index. Circles with two colors represent a pair of tions do not vanish identically. Furthermore, there is a fermions on a site with corresponding flavors. The low-energy finite-temperature Ising transition corresponding to the properties of the Z2D phase resemble SU N fermions propa- ð Þ restoration of the aforementioned symmetry. gating freely in space-time and connected by a Z2 gauge string As the kinetic energy of the Ising degrees of freedom (b). The symmetry-broken phases correspond to the confined is increased, we find that the aforementioned gapless phases of the model. At N 3, the AFM phase (e) has the fundamental (conjugate) representation¼ of SU(3) on sublattice Dirac deconfined phase enters a symmetry-broken confined A (B). The corresponding Young tableaux are included. The VBS phase. The nature of symmetry breaking depends on the phase (d) corresponds to a pattern of intersite SU(3) singlets. value of N. For N 1, it corresponds to a charge density wave, while for N ¼ 2, because of an enlarged symmetry, the symmetry-broken¼ phase can correspond to a super- II. MODEL AND SYMMETRIES conductor or a Néel AFM, depending on the sign of the Our model reads infinitesimal symmetry-breaking field. The N 3 case is even richer. We find that after exiting the deconfined¼ phase, N the model enters a VBS phase, and as the tuning parameter ˆ ˆ † ˆ H Z i;j cî;αcˆj;α H:c: − Nh X i;j : 1 controlling the kinetic energy of Ising spins is tuned further, ¼ h i þ h i ð Þ i;j α 1 i;j the valence-bond-solid quantum melts and yields a Néel Xh i X¼ Xh i AFM phase. The phase transition between the VBS and † AFM phase, if second order, corresponds to a deconfined Here, cî;α creates a fermion of flavor α in a Wannier state quantum critical point [30], reported earlier in the context centered around lattice site i of a square lattice, and the sum of quantum magnets [31,32]. Our results are consistent runs over bonds of nearest neighbors. Each bond, b i; j , with a second-order transition. accommodates a two-level system spanned by the¼ Hilberth i

041049-2 RESEARCH NEWS & VIEWS conditions. Both groups agree that, under Judith Berman is in the Department of 4. Hede, K. J. Natl Cancer Inst. 97, 87–89 (2005). conditions optimized by geneticists for growth Genetics, Cell Biology, and Development, and 5. Rancati, G. et al. Cell 135, 879–893 (2008). 6. Selmecki, A., Gerami-Nejad, M., Paulson, C., Forche, in conventional laboratories, aneuploid cells the Department of Microbiology, University of A. & Berman, J. Mol. Microbiol. 68, 624–641 (2008). usually divide less rapidly than cells with the Minnesota, Minneapolis, Minnesota 55455, 7. Torres, E. M. et al. Science 317, 916–924 (2007). normal chromosomal complement. USA. 8. Torres, E. M. et al. Cell 143, 71–83 (2010). An important distinction in the methods e-mail: [email protected] 9. Geiger, T., Cox, J. & Mann, M. PLoS Genet. 6, used to generate the aneuploid strains might e1001090 (2010). 2,7,8 1. http://8e.devbio.com/article.php?ch=4&id=24 10. Springer, M., Weissman, J. S. & Kirschner, M. W. explain the differences in the findings . 2. Pavelka, N. et al. Nature 468, 321–325 (2010). Mol. Syst. Biol. 368 (2010). 7,8 6, Torres et al. engineered yeast cells with a 3. Duesberg, P., Li, R., Fabarius, A. & Hehlmann, R. 11. Weaver, B. A. A. & Cleveland, D. W. Cancer Res. 67, haploid (single) set of chromosomes to carry Contrib. Microbiol. 13, 16–44 (2006). 10103–10105 (2007). one extra chromosome and then selected for faster growth using conventional lab conditions for 9–14 days — a time frame during HIGH-TEMPERATURE SUPERCONDUCTIVITY which mutations are expected to accumulate. By contrast, Pavelka et al. analysed strains that often carried multiple aneuploid chromosomes and, importantly, minimized the number of Mind the pseudogap generations before analysis. This illustrates a crucial truism of experimental genetics: you The discovery of predicted collective electronic behaviour in copper-oxide get what you select for. superconductors in the non-superconducting state provides clues to unlocking the Pavelka and co-workers also directly 24-year-old mystery of high-temperature superconductivity. SEE LETTER P.283 address a controversy concerning the role of excess proteins in aneuploid cells. Previously, Torres et al.7 proposed that there is a specific CHANDRA VARMA in the dome-shaped superconductivity region set of genes and proteins that are regulated in of the phase diagram (Fig. 1). This point would response to aneuploidy in general. In their he phenomenon of high-temperature occur at zero temperature and would involve more recent study8, they showed that some superconductivity is a beautiful and a change in the symmetry of the mater ials’ 20% of proteins exhibit levels that do not well-posed scientific problem with electronic structure. Because Tc is determined track with gene copy number, and that a large Tmany facets. On page 283 of this issue, Li et al.1 by the materials’ collective electronic excita- proportion of these proteins are members of report observing a special kind of intense col- tions in the non-superconducting state, it is macromolecular complexes. By contrast, other lective electronic fluctuation in the most un arguable that the coupling of electrons to groups9,10 have found that the levels of most mysterious phase of matter exhibited by high- such excitations in the strange-metal region proteins generally reflect changes in chromo- temperature superconducting copper-oxide and their modifications in the pseudogap some copy number and that less than 5% of materials (cuprates). Taken together with pre- region lead to high-FermionTc superconductivity.-Boson problems: ”De-signer” Hamiltonians the proteins exhibit ‘dosage compensation’ vious experimental2–6 and theoretical7 work, If it exists, a quantum critical point in the — whereby the relative protein level is inde- this observation significantly narrows the pendent of gene-copy number. Pavelka et al. range of directions likely to be fruitful in the La2-xSrxCuO4, HgBa2CuO4+δ specifically test this hypothesis by quantitative quest to understand high-temperature super- Insulator and mass spectrometry of about 2,000 proteins in conductivity. The authors performed their antiferromagnetic each of five aneuploid strains and do not find experiments in two samples of HgBa2CuO4+δ, Hˆ = t cˆ† cˆ +ˆc† cˆ compelling evidence for specific dosage com- which has one of the simplest crystal structures i, j, j, i, ? Strange Temperature i,j , pensation of protein-complex components. of any of the cuprate families and is ideal for T* metal Crossover hXi ⇣ ⌘ Overall, these studies2,7–10 are consistent with such studies. 1 + U (ˆni, 1/2) (ˆni, 1/2) µ nˆi, the idea that aneuploidy is not a single, unique Li and colleagues’ experiments pertain to " # i i, Fermi 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 state and that all aneuploid strains do not share the pseudogap region of the phase diagram of Pseudo- liquid X X a single, common phenotype or protein profile. the cuprates (Fig. 1), a sort of precursor state gap phase Tc Rather, different aneuploid strains use differ- to the superconducting phase that most con- ent mechanisms for optimal growth under densed-matter physicists regard as the Rosetta Doping different conditions. This conclusion may be Stone for discovering the physical principles Superconductivity QCP less satisfying than a single, simple answer, that underlie the cuprates’ behaviour. On C. Varma, Nature 468 (2010). especially given the crucial implications for entering the pseudogap region, at a temper- Figure 1 | Phase diagram of the cuprates. At very cancer cells: it remains unclear whether ature below T* but above the temperature à Frac%onaliza%onlow levels of electron–hole doping,Emergent cuprates are gauge theories cancer cells divide uncontrollably because below which superconductivity emerges (T ), c insulating and antiferromagneticSpin liquids (the (Z materials’2 gauge theory in deconﬁned phase) they are aneuploid and/or because they have all cuprates’ thermodynamic and electronic- neighbouring spins point in opposite directions). accumulated mutations that allow them to transport properties change by a large amount At increased doping levels, they become tolerate aneuploidy. But it should be remem- owing to the materials’ loss of low-energy Intertwinedconducting, and theorders exact temperature à Superconduc%vity and doping versus stripes bered that work on cancer cells themselves11 electronic excitations. level determine whichSuperconduc%vity phase of matter they will versus an%ferromagne%sm suggests that not all aneuploidies are equal: The pseudogap region is bounded on one be in. At temperatures below Tc, they become aneuploidy can either promote or inhibit side by a region of remarkably simple but superconducting, and at temperatures above Tc but tumorigenesis, depending on the context. unusual properties, which do not fit into the Quantumbelow T* they phasefall into the transi%ons pseudogap phase. à TheNema%c transi%ons Pavelka and colleagues’ work2 therefore Fermi-liquid-type model that has been used to boundary of the pseudogap region at low doping levels is unknown. The transition(See also between heavy the fermions) supports the idea that, whereas mutations describe metals at low temperatures for about Fermi-liquid phase and the strange-metal phase can facilitate the proliferation of aneuploid a hundred years. Some researchers got to grips occurs gradually (by crossover). QCP denotes the cells, aneuploidy itself can be sufficient to with understanding this ‘strange-metal’ region quantum critical point at which the temperature à Designer models: are numerically1 tractable and capture aspects of the above physics provide a growth advantage under a broad early in the history of high-Tc superconduct- T* goes to absolute zero. Li and colleagues’ study range of stress conditions. ivity, by hypothesizing a quantum critical point pertains to the pseudogap phase.

Sec. I we define the lattice Hamiltonian and describe its dimensionless coupling strength between the pseudospin phase diagram; in the next section, we provide evidence and the fermion bond density. for the statementsIsing nematic above regarding quantum bosonic critical correlations point in a metal: a Monte Carlo study (III A), superconductivity (III C), and single-fermion cor- The pseudospins are not related to the physical spins relations (III D); finally, in Sec. 1,IV, we discuss various2, of the electrons; their2 ordering corresponds1 to a nematic Yoni Schattner, ⇤ Samuel Lederer, ⇤ Steven A. Kivelson, and Erez Berg caveats concerning1 the interpretation of our results, and transition. The model (3) should be viewed as an e↵ec- Department of Condensed Matter Physics, The Weizmanntive Institute lattice model, of Science, designed Rehovot, to give 76100, a nematic Israel QCP. Mi- their bearing on both2 priorDepartment theoretical of Physics, work and Stanford the in- University, Stanford, CA 94305, USA croscopically, the nematic degrees of freedom could rep- terpretation of experiments. (Dated: November 12, 2015) resent (via Hubbard-Stratonovich transformation) inter- 3 The Ising nematic quantum critical point (QCP) associatedactions withinvolving the zero the temperature same electrons transition that form the Fermi II.from MODEL a symmetric AND to PHASE a nematic DIAGRAMmetal is an exemplar ofsurface, metallic quantum or another, criticality. independent We have degree carried of freedom (such Sec. I we defineout a minus the lattice sign-free Hamiltonian quantum Monte and Carlo describe study its of thisasdimensionless a QCP phonon for a mode two coupling dimensional that becomes strength lattice soft betweenmodel at a structural the pseudospin tran- phase diagram;with sizes in the up nextto 24 section,24 sites. we The provide system evidence remains non-superconductingsition).and the So fermion long asdown bond the to properties density. the lowest of acces- the QCP are uni- Our modelsible is temperatures. illustrated in⇥ The Fig. results3, and exhibit is defined scaling on behavior over the accessible ranges of temperature, fora two-dimensional the statements square above lattice. regarding Every bosonic lattice correlations site hosts versal, the low-energy behavior does not depend on its (imaginary) time, and distance. This scaling behavior hasThe remarkable pseudospins similarities are not with related recently to the physical spins (aIII single A), superconductivity (spinful)measured fermionic properties ( degreeIII of C the), andof Fe-based freedom. single-fermion superconductors Each link cor- proximatemicroscopic to theirorigin. putative nematic QCP. relationshas a pseudospin-1 (III D); finally,/2 degree in Sec. of freedomIV, we that discuss couples various to of the electrons; their ordering corresponds to a nematic caveatsthe fermion concerning bond-density. the interpretation The system of is our described results, by and transition.For ↵ = 0, The the system model ( is3) composed should be of viewed two decoupled as an e↵ec- theirthe following bearing Hamiltonian: onI. both INTRODUCTION prior theoretical work and the in-maticsetstive order of latticedegrees [54– model,59 of] freedom: and designed nematic free quantum fermions, to give a critical and nematic pseudospins fluctu- QCP. Mi- terpretation of experiments. ationsgovernedcroscopically, [60–65 by] haveHb, the which been nematic hasobserved the degrees form in many of of a d freedom of= the 2 transverse mate- could2 rep- H = Hf + Hb + Hint, (3) rialsfieldresent mentioned Ising (via model. Hubbard-Stratonovichabove. At zero temperature, transformation) the pseudospins inter- A hallmark of strongly correlatedthe Matsubara electron frequency, systems and isA, b, and  are positive dimensionful constants. We find that theundergo exponents in a second-orderTABLE I: Critical quantum exponents for a phase 2d quantum transition Ising QCP from Ouractions simulations involvingwith z =1(classical3 are the limited samed Ising), to electrons finite and for the system Ising that nematic sizes form QCP – the up Fermi thewhere competition of ground statesthis with expression di↵erent take values kinds =1 of.0 0.1 and =1.0 0.1. II. MODEL AND PHASE DIAGRAM ± toa 24surface, paramagnet±24 lattice orin another, a to metal sites. an from “antiferromagnet” Withinindependent our DQMC our simulations. numerical degree In thethat limit of accuracy, breaks freedom of van- 90 (such order.[1] In this context, a central(See set Fig. of8.) unsettled theo- ⇥ ishing coupling to the fermions, exponents from our DQMC This implies, among other things, thatwerotational find theas uniform a evidence phonon symmetry for mode a continuous thatat h = becomeshc0 nematic. This soft transition quantum at a structural phase is in the tran- retical issuesHf concerns= t the characterci†cj µ of quantumci†ci, critical simulations are consistent with 3d classical Ising values.z ˜ is susceptibility D(h, T, 0, 0) has a Curie-Weiss3 dimensional form the classical “apparent dynamical Ising universality exponent” defined belowclass. Eq. At (1). T = 0, Our model is illustratedi,j ,Fermion in Fig. -Bosoni,3, and⌘ isproblems: defined on transition”De-signer”sition). with So Hamiltonians critical long as exponents the properties that are of the significantly QCP are uni- points (QCPs) in metalshX [i2–4]. Suchwith metallic anX e↵ective WeissQCPs temperature have whichthe varies pseudospin linearly degrees of freedom are gapped in both the with (hc h), as can be seen in Fig. di1.Eq.↵erentversal,1 can from be the those low-energy of a nematic behavior QCP does in an not insulator. depend on its beena two-dimensional identified in several square heavy lattice.z fermionz Every compounds[ latticex site5 hosts]; /⌫ Critical Exponents ⌫ ⌘ z˜ Hb = V ⌧i,jviewed⌧k,l ash a scaling⌧i,j relation,, D(h, T, q, 0) =nematic⇠ (x, y), and isotropic phases. At finite temperatures, ⌫ z˜ Specifically,microscopic in the2d Ising origin. disordered QCP phase0.63 near1.24 the0.04 QCP,1 where.0 evidencea single for (spinful) quantum fermionic critical degree behaviorwhere of⇠ freedom. hash hc also , x Each= beenT ⇠ , linky = q ⇠.InEq.1, Isingi,j ; k,l nematic quantum⇠ | i,j | critical point| | in a metal:2d metallic a MonteIsing ne- Carlo study is the conventionally defined susceptibilitythea exponent, dimensionless line of the second-order quantum classical control0.5 0.1 parameterd1.0=0. 21 0 Ising.0 h0.3 2 transitions.0hc0,the.3 has a pseudospin-1hh/2X degreei h ii of freedomhX thati couples to matic QCP found in the ruthenate Sr3Ru2O7 correlation[6, 7], and length the exponent cuprate⌫ /(2 ⌘), where ⌘ is ± ± ± ± z 1, 2,thermodynamicextendsFor from↵ = the0, (zero the2 QCP frequency) system in the ish nematic1 composedT plane. two-point of two decoupledcor- andthe iron-based fermionHint bond-density.= superconductors↵t ⌧Yonii,j c The† Schattner, [cthe8]. system anomalous. Quantum⇤ is dimensionSamuel described critical- of Lederer, the⌘(4) nematic by ⇤ Steven field, and A. we Kivelson, and Erez Berg i j relator (defined in Eq. 5) is consistent with the following Phys. Rev. X 6,1Department 031028i,j , (2016) of Condensedhave introduced Matter an “apparent Physics, dynamical The Weizmann exponent,”sets Institute ofz ˜ = degrees of Science, of freedom: Rehovot, free 76100, fermions, Israel and pseudospins itythe is followingalso often Hamiltonian: invokedh asi a possible1 explanation of the Forn non-zero ↵, the phase diagram is similar, but ex- X 2Department(⌫) . The values of Physics, of these Stanford exponents University,functional derivedgoverned from Stanford, our description: by H CA, 94305, which USA has the form of a d = 2 transverse “strange” or “bad” metal behaviorMonte seen Carlo in data a are variety given in of Table I. hibits quantitativeb and qualitative modifications. Fig. 1 Here, c† createsH a fermion= H + onH site+ToH illustratej with, the spin significance(Dated: = of, November(3)these, exponents,field 12, 2015) note Ising model. At zero temperature, the pseudospins such materialsj [9–16]. f b int shows the phase diagram, obtainedA by DQMC, for both The Ising nematicthat working quantum backwards critical from point the" # exponents (QCP) associated correspond- with the zero temperature transition i, j denotes a pair of nearest-neighboring to the d + z dimensional sites on classical the IsingD modelundergo(h, T, withq,i! an second-order= 0) quantum phase transition2 (1) from wherehTo date,i there existsfrom no satisfactory a symmetric to theory a nematic ofmetal metallicis an exemplar↵ of= metallic 0 and quantum↵ = criticality. 0.⇡ TheT We↵+ have=b(h 0 carried transitionhc)+ q between the square lattice, t and µ are thedynamical hopping exponent strengthz = 1 (also and listed in TableaI paramagnet) would 6 to an “antiferromagnet”6 | that| breaks 90 QCPs in d = 2 or 3 spatialout a dimensions minus sign-freeyielda although=1 quantum.96, =1 a Monte. number24, and Carlo 2 ⌘ study=1.96.nematic of Indeed,this QCP we and for a isotropic two dimensional phases lattice remains model second order, and chemical potential, respectively,with sizes† up tofind⌧ 24i,j critical(24a = sites.exponents†x, y, The z consistent) system denote remains withwhere this non-superconducting expectationrotationalT is the symmetry temperature, down to at thehq lowest=ish thec acces-0. wave-vector, This transition! isis in the of field theoreticHf = approachest ci havecj beenµ⇥ attemptedcici, [17– extrapolates to a new QCP, shifted relative to hc0.n More Pauli matrices that actsible on temperatures. theby pseudospin DQMC The when results we that set exhibit to lives zero scaling the on coupling behavior between3 dimensional over the the accessible classical ranges Isingof temperature, universality class. At T = 0, 34]. Among the unsettledi,j , issues are:i, a) the values of striking is the change in the slope with which the phase the bond connectingh(imaginary) theXi neighboring time,critical and fluctuations sitesX distance.i and and Thisj the, V> fermions. scaling0 behaviorthe has pseudospin remarkable degrees similarities of with freedom recently are gapped in both the critical exponents, andmeasured which properties propertiesz Includingz of can the results be Fe-based expressed atx finite superconductors (Matsubara)boundary, frequencies, proximate2 T toN their(h), putative approaches nematic the QCP. QCP. For ↵ = 0, the is the nearest-neighborHb = V Ising interaction⌧i,jwe⌧k,l find ah wide between intermediate⌧i,j , neigh- regime wherenematic the full and isotropic phases. At finite temperatures, as scaling functions involving thesedynamical exponents, fluctuation b) the spectrum ex- is wellslope described diverges by at low temperature, consistent with the ex- boring pseudospins (here,i,j ; k,l i, j ; k, l denotesi,j a pair of a line of second-order classical d = 2 Ising transitions tent to which metallic QCPs arethe preempted simple “functional by approximant”super- D(h,pectation T, q,i!n) for the transverse field Ising transition, where hh XiI.h hh INTRODUCTIONii i h ii hXi maticT order⇡ [54–59] and nematic quantum critical fluctu- nearest-neighbor bonds), h V is( theh, T, q strength,i!n), where of a trans- extends from⌫z the QCP in the h T plane. conducting [35–43] or other formsz A [38, 44, 45] of auxil- TN 1.5 h hc0 with ⌫ =0.63 and z = 1. In contrast, Hint = ↵t ⌧i,j· ci†cj. (4) ations [60–65] have been observed in many of the mate- verse field that acts on the pseudospins, and ↵ is theA / | | iary order, that gap out the Fermi surface,(h, T, q,i! ) c) whether forrials↵ mentioned. >(2)0, we above. find that TN (hc h). On the disor- A hallmarki,j of, strongly correlated electronn systems is 2 Quantum there exists a “non-FermihXi liquid” metalA in the⌘ quantumT + b(h hc)+ q + c !OurnForn simulations non-zero are↵ limited, the phase to finite/ diagram system sizes is similar, – up but ex- the competition of ground states with di↵erent kinds of| | dered| | side of the transition,critical we define a crossover line by critical regime. Controlled theoreticalHowever, methods, in contrast with that the can thermodynamictohibits correlation 24 1 24 quantitative lattice sites. Within and qualitative our numerical modifications. accuracy, Fig. 1 Here, c† order.[creates1] In a this fermion context, on a site centralj with set of spin unsettled = theo-, , identifying⇥ hcross(T ) as the value of h at which the static be usedj both to benchmark thecj†function, field theories as shown in and Fig. 10 for, the" # dynamicalweshows find response evidence the phase for a continuous diagram, nematic obtained quantum by DQMC, phase for both i, j denotesretical a issues pair concerns of nearest-neighbor theexhibits character noticeable of quantum deviations sites on critical from the this formsusceptibility for the few FIG. at 2: fixed FermionT Green’shas function, fallenG(k to, ⌧ = half/2), as of a func-its value lowest values of q ; q 2⇡/L where L istransition the system withtion critical of momentum exponentsk across thatthe Brillouin are significantly zone for three dif- comparisonh i withpoints experiments, (QCPs) in metals are [2 greatly–4]. Such needed. metallic Deter- QCPs have at↵h== 0hc and.NematicTcross↵ =(h), 0. the The inverse↵ = of 0 transitionhcross, also between vanishes the square lattice, t and µ are thewidth. hopping It is unclearx,y,z| | strength whether| | ⇠ these and deviationsdi↵ shoulderent0.5 be fromferent those values6 of of a the nematic transverse QCP6 field: inh =3 an.35 insulator. (in the nematic minant quantumbeen identified Monte Carlo in several (DQMC)[ heavy fermion46⌧ , 47]–in compounds[ par- 5]; nematic andphase), isotropich =3.525 (near phases the QCP), remains and h =3 second.7(inthedis- order, and chemical potential, respectively,dismissed⌧ a ( asa finitei,j= x, size y, e z↵ects,) denote or taken aslinearlySpecifically, indicators ofwith in theh upon disordered approaching phase near the theSymmetric QCP, QCP, where although its evidence for quantum criticaladi↵i,jerent behavior form of dynamical has also scaling been closer to criticality, ordered phase). In the nematically ordered phase, the data ticular, in cases whereV the fermion sign problem can slopetheextrapolates dimensionless is steeper to quantum than a new that control QCP, of T parameterN shifted(h). relativeh hc,the to hc0. More Pauli matricesfound in that the actruthenate on the Sri.e.Ru pseudospin atO lower[6, temperatures7], and that thelives and cuprate larger on system sizes than is averaged over both orientations of the order parameter. 3 2 7 G(k, ⌧ = /2) is proportional to the integral over the spec- be circumvented [48–50]–may serveare presently this purpose. accessible. Taking Such Eq. (2) at facethermodynamicstriking value,0 one is the (zero change frequency) in the nematic slope two-point with which cor- the phase the bond connectingand iron-based the superconductors neighboring sites[8]. Quantumi and j, critical-V>0 2 tral2.5 function A(k,3!) over an3.5 energy window4 of width T 4.5[see methods have been successfully appliedfinds the dynamical in several critical prob- exponent z relator2,The as would (defined linear behavior in Eq. 5) is of consistentT (h) for with small the followingh h is also ity is also often invoked as a possible explanation of the ⇡ boundary,Eq.TN (11()].h), The approaches temperatureN is T =1 the/8, QCP. and the systemc For size↵ = 0, the is the nearest-neighbor Ising interactionbe obtained from between a naive dynamical neigh- scalingfunctional assumption description: hc0 hc lems in which“strange” critical or bosonic “bad” metal fluctuationst behavior are seen coupled in a variety to of seenslope for diverges otheris L model= at20. Thelow parameters. data temperature, were taken from In systems consistentFig. with4hwe either show with pe- the ex- boring pseudospins (here, i, jz˜ =; zk,. This l lastdenotes result is at a odds pair with of theoretical expec- riodic or anti-periodic boundary conditions. The Fermi sur- fermions withsuch a materialsDirac-like [9– dispersion16hh]. tations,i h [51 [102ii–53] and]. so requires special analysis,phasepectation especially diagramfaces for for are the twoseen transverse as values peaks in ofG(Ak field the, ⌧ = fermion Ising/2). The transition, lowerdensity, right con- where given that there are reasons to question the validity of panel shows the Fermi surface of the bare band structure. nearest-neighbor bonds), h V is the strength of a trans-FIG.D 1:(h, Phase T, q,i! diagramn = 0)⌫z of the model obtained2 by(1) DQMC, arXiv:1511.03282v1 [cond-mat.supr-con] 10 Nov 2015 To date, there exists no satisfactory theory of metallic trolled by the chemical potential µ. As the fermion den- In this paper, we report a· DQMCnaive dynamical study scaling of a [3, two-16]. TN h hc0 ⇡withT +⌫b=0(h .h63c)+ and qz = 1. In contrast, At µ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 sha1_base64="I3qzNHs0O153gIiFHHCMQ4HedNo=">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 sha1_base64="ZGB5CT+PsS1f6/CwOpGBa+beq2c=">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 sha1_base64="gMuuCJSeRn+qX2ljhfq7jG2igVI=">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 =0 as , the a function model has of the the transversesame symmetries field h and as |the temperature| SSH modelT . verse fieldQCPs that in actsd = 2 on or 3 the spatial pseudospins, dimensionsIn the fermionic although and sector we↵ a number findis results the consistentsity is/ with reduced,| both| h h and the range over which dimensional sign problem-free lattice model that exhibits Here,forV ↵=>t, 0,↵ we=0 find.5, µ thatc= Tc00.N5t. The(hc solidh). line On marks the disor- of field theoretic approachesstrongly have renormalized been attempted FermiAt liquid, µ [ 17=0 “marginal – , theTwhere Fermi( theh)T is liq-O(2N)is linear thea smallsymmetry temperature, become power of smaller, Tis.q Fig.brokenis2 theshows indicating/ wave-vector, down maps of to the thatU(N) low-energy!n is the e↵ect an “Ising nematic” QCP in a metal at finite 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 Ndered side of the transition, we define a crossover line by 34]. Among the unsettleduid” issues [66], or are: weakly a) non-Fermi the values liquid6 of behaviorthe transition down to temperature,spectral weightT asN a,betweenthenematicandthe function of momentum in the dis- our lowest temperatures (T 0.02E ,whereof theE couplingis ordered between phase, at electrons the QCP, and near in the the ordered Fermi phase. surface density; incritical the nematic exponents, phase, and which the properties discrete can lattice be expressedmin ro-⇡ symmetricF identifyingF phases.hcross The(T line) as extrapolates the value oftoh theatT which=0QCP the static FIG. 3: Illustration of the latticec modelj†the Fermi (3 energy).). Spinful In particular, electrons at the QCP,and the the e↵2ec- nematicThe existence[ modes67] becomes of “cold-spots” weaker. along the The zone diago- fact that tational symmetryas scaling is functions spontaneously involvingtive quasiparticle thesebroken exponents, from weight,CZ b)4 ( thetoT ), defined ex- at inh termscsusceptibility. The of the dashednals, line where at marksfixed the quasiparticlesTTcrosshas, where are fallen relatively the to nematicweakly half scat- of sus- its value reside on the sites, while the Ising pseudospins livek onF the C . This istent a particularly to which metallic relevant QCPssingle-fermion QCP are preempted given Green’s that function by ne- super- in Eq. (ceptibility14),T remainsN (h) sub- appears reachestered, 50% linear is apparent. of atits low magnitude Finally, temperature no superconducting at h = forhc transition both3.525 values at 2 x,y,z atTh = hc. Tcross(h), the inverse of hcross⇡, also vanishes bonds. Theconducting pseudospins [35– interact43] or other withstantial. forms their However, [neighbors⌧38, 44 it monotonically, 45] antifer- of auxil- decreasestheof same on fermion cooling,1.5 temperature. densityis found down is The consistent to pale our lowest (grey) with temperatures, lines this show being although the a corre-theuniver- with downwardi,j curvature. In a Fermi liquid,linearlyZ (T ) withsuperconductingh upon approaching susceptibility in the the s-wave QCP, channel although is its romagnetically,iary order, and are that coupled gap out to the the Fermi fermion surface, bond c) density. whether kF would approach a positive limit as T spondingsal0, whileproperty temperaturesin a peaked of the about for metallich theQuantumh case. (See QCP.↵ Fig.= 013 (where.) the fermions V slope is steeper than⇡ thatc of TN (h). there exists a “non-Fermiweak liquid” non-Fermi metal liquid, in the it would quantum vanishand! in proportion pseudospins to areThe decoupled). remainder ofcritical the In paper this is case, organized the as QCP follows: occurs in critical regime. Controlled theoretical methods, that can 1 These authors have contributed equally to this work. at hc0 3.06. ⇤ be used both to benchmark the field theories and for ⇡The linear behavior of TN (h) for small hc h is also comparison with experiments, are greatly needed. Deter- t seen0.5 for otherNematic model parameters. In Fig. 4 we show the minant quantum Monte Carlo (DQMC)[46, 47]–in par- phase diagram for two values ofSymmetric the fermion density, con- ticular, in cases where the fermion sign problem can trolled by the chemical potential µ. As the fermion den- be circumvented [48–50]–may serve this purpose. Such 0 methods have been successfully applied in several prob- sity is2 reduced,2.5 both3 hc 3.5hc0 and4 the range4.5 over which hc0 hc lems in which critical bosonic fluctuations are coupled to TN (h) is linear become smaller, indicatingh that the e↵ect fermions with a Dirac-like dispersion [51–53]. of the coupling between electrons near the Fermi surface FIG. 1: Phase diagram of the model obtained by DQMC, FIG. 3:arXiv:1511.03282v1 [cond-mat.supr-con] 10 Nov 2015 IllustrationIn this ofpaper, the lattice we report model a DQMC (3). Spinful study ofelectrons a two- asand a function the nematic of the transverse modes field becomesh and temperature weaker. TheT . fact that reside on thedimensional sites, while sign problem-free the Ising pseudospins lattice model that live exhibits on the Here,T (Vh)= appearst, ↵ =0 linear.5, µ = at0 low.5t. temperature The solid line marks for both values an “Ising nematic” QCP in a metal at finite fermion N bonds. The pseudospins interact with their neighbors antifer- theof transition fermion temperature, density isTN consistent,betweenthenematicandthe with this being a univer- romagnetically,density; and in are the coupled nematic to phase, the fermion the discrete bond lattice density. ro- symmetric phases. The line extrapolates to the T =0QCP tational symmetry is spontaneously broken from C4 to atsalhc.property The dashed of line the marks metallicTcross, where QCP. the nematic sus- C2. This is a particularly relevant QCP given that ne- ceptibility reaches 50% of its magnitude at h = hc 3.525 at the same temperature. The pale (grey) lines show⇡ the corresponding temperatures for the case ↵ = 0 (where the fermions and pseudospins are decoupled). In this case, the QCP occurs at hc0 3.06. ⇤These authors have contributed equally to this work. ⇡ Quantum Monte Carlo Methods for Fermion-Boson problems

Fakher F. Assaad (Autumn School on Correlated Electrons, Jülich 20th September 2018)

Outline: Ø Fermion-Boson problems Electron-Phonon, Su-Schrieffer-Heeger (SSH) Unconstrained Gauge theories “De-signer” Hamiltonians

Ø Auxiliary field Quantum Monte Carlo (QMC), Generalities

Ø Application of the Auxiliary field QMC to the SSH model Sign-free formulation Sampling, Hybrid Monte Carlo

Ø Conclusions

SFB1170 ToCoTronics Quantum Monte Carlo: Formulation

Hˆ S (xx,⌧) Z =Tre = D (xx, ⌧) e { }

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(xx, ⌧) : Hubbard-Stratonovich AAAB+3icbVBNS8NAEJ34WetXrEcvwSJUkJIUQY8FLx4r2A9oQtlsN+3S3U3Y3UhLyF/x4kERr/4Rb/4bt20O2vpg4PHeDDPzwoRRpV3329rY3Nre2S3tlfcPDo+O7ZNKR8WpxKSNYxbLXogUYVSQtqaakV4iCeIhI91wcjf3u09EKhqLRz1LSMDRSNCIYqSNNLArfmtMa37Cw2yaX/kapZcDu+rW3QWcdeIVpAoFWgP7yx/GOOVEaMyQUn3PTXSQIakpZiQv+6kiCcITNCJ9QwXiRAXZ4vbcuTDK0IliaUpoZ6H+nsgQV2rGQ9PJkR6rVW8u/uf1Ux3dBhkVSaqJwMtFUcocHTvzIJwhlQRrNjMEYUnNrQ4eI4mwNnGVTQje6svrpNOoe27de7iuNhtFHCU4g3OogQc30IR7aEEbMEzhGV7hzcqtF+vd+li2bljFzCn8gfX5A2x5k/o= sha1_base64="hP+6LrUf2d3tZaldqaQQvEKMXyw=">AAAB2XicbZDNSgMxFIXv1L86Vq1rN8EiuCozbnQpuHFZwbZCO5RM5k4bmskMyR2hDH0BF25EfC93vo3pz0JbDwQ+zknIvSculLQUBN9ebWd3b/+gfugfNfzjk9Nmo2fz0gjsilzl5jnmFpXU2CVJCp8LgzyLFfbj6f0i77+gsTLXTzQrMMr4WMtUCk7O6oyaraAdLMW2IVxDC9YaNb+GSS7KDDUJxa0dhEFBUcUNSaFw7g9LiwUXUz7GgUPNM7RRtRxzzi6dk7A0N+5oYkv394uKZ9bOstjdzDhN7Ga2MP/LBiWlt1EldVESarH6KC0Vo5wtdmaJNChIzRxwYaSblYkJN1yQa8Z3HYSbG29D77odBu3wMYA6nMMFXEEIN3AHD9CBLghI4BXevYn35n2suqp569LO4I+8zx84xIo4 sha1_base64="NyQ6lcMEbx1DuYp1HCm3I93yZCw=">AAAB8HicbZBLSwMxFIXv+Ky16titm8EiVJAy40aXghuXFewDOkPJpJk2NMkMyR1pGfpX3LhQxF/jzn9j+lho64HAxzkJ9+bEmeAGff/b2dre2d3bLx2UDytHxyfuaaVt0lxT1qKpSHU3JoYJrlgLOQrWzTQjMhasE4/v53nnmWnDU/WE04xFkgwVTzglaK2+Ww2bI14PMxkXk9lViCS/7Ls1v+Ev5G1CsIIarNTsu1/hIKW5ZAqpIMb0Aj/DqCAaORVsVg5zwzJCx2TIehYVkcxExWL3mXdhnYGXpNoehd7C/f2iINKYqYztTUlwZNazuflf1ssxuY0KrrIcmaLLQUkuPEy9eRHegGtGUUwtEKq53dWjI6IJRVtX2ZYQrH95E9rXjcBvBI8+lOAMzqEOAdzAHTxAE1pAYQIv8Abvzsx5dT6WdW05q96q8EfO5w8Sc5Kk sha1_base64="GPYKF3WEHvyxzA9oOB1tKajuaLo=">AAAB+3icbVBNS8NAEJ3Ur1q/Yj16WSxCBSlJL3osePFYwX5AE8pmu2mXbjZhdyMtIX/FiwdFvPpHvPlv3LY5aOuDgcd7M8zMCxLOlHacb6u0tb2zu1ferxwcHh2f2KfVropTSWiHxDyW/QArypmgHc00p/1EUhwFnPaC6d3C7z1RqVgsHvU8oX6Ex4KFjGBtpKFd9doTVveSKMhm+bWncXo1tGtOw1kCbRK3IDUo0B7aX94oJmlEhSYcKzVwnUT7GZaaEU7zipcqmmAyxWM6MFTgiCo/W96eo0ujjFAYS1NCo6X6eyLDkVLzKDCdEdYTte4txP+8QarDWz9jIkk1FWS1KEw50jFaBIFGTFKi+dwQTCQztyIywRITbeKqmBDc9Zc3SbfZcJ2G++DUWs0ijjKcwwXUwYUbaME9tKEDBGbwDK/wZuXWi/VufaxaS1YxcwZ/YH3+AGs5k/Y= Multidimensional integral One body problem in external (or arbitrary field with à Monte Carlo predefined dynamics) field à Polynomial complexity Quantum Monte Carlo: Sign problem

Hˆ S (xx,⌧) Z =Tre = D (xx, ⌧) e { }

Sign problem. S is complex −S(){}Φ iϕ(){}Φ ∫ d {}Φ e e O(){}Φ i −S(){}Φ −S(){}Φ ϕ(){}Φ d {}Φ e O {}Φ ∫ d {}Φ e e O O = ∫ () = ≡ P P −S(){}Φ −S(){}Φ iϕ(){}Φ iϕ(){}Φ d {}Φ e d {}Φ e e e ∫ ∫ P −S(){}Φ d {}Φ e ∫

i ϕ(){}Φ −β N() e0 −e0 −βV Δ But in the low temperature limit: e = Z / Z ∝ e = e P

δ Z /Z 2βV Δ () << 1 but δ Z /Z ≈ 1 so that CPU >> e () CPU Z /Z Quantum Monte Carlo: Sign problem

Hˆ S (xx,⌧) Z =Tre = D (xx, ⌧) e { }

Sign problem. S is complex à Computa5onal eﬀort 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 sha1_base64="rNtFzNWJeSHVc8OvY3qyNf4Qs4Y=">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 sha1_base64="YL9dC69fRG7WmCSa+XDQLh4UK8Y=">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 sha1_base64="NRj3dDeCZLq8akQHyUzb5UMiPVs=">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 , > 0 , formula5on dependent

Ever growing class of interes'ng nega5ve sign free models à Polynomial complexity

C. Wu and S.-C. Zhang. Phys. Rev. B, 71, 155115, (2005). E. Huﬀman and S. Chandrasekharan, Phys. Rev. B 89 (2014), 111101. Zi-Xiang Li, Yi-Fan Jiang, and H. Yao Phys. Rev. LeX. 117 (2016), 267002. Z. C. Wei, C. Wu, Yi Li, Shiwei Zhang, and T. Xiang. Phys. Rev. LeX. 116 (2016), 250601. Z. C. Wei, arXiv:1712.09412

T T T A1 An Condi5ons on A 1 A n ,A i = A i Tr e e 0 ? 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 ··· | ··· ⇣ ⌘ Recent applica5on PHYSICAL REVIEW LETTERS 120, 107201 (2018)

Quantum Monte Carlo Simulation of Frustrated Kondo Lattice Models

Toshihiro Sato,1 Fakher F. Assaad,1 and Tarun Grover2 1Institut für Theoretische Physik und Astrophysik, Universität Würzburg, 97074 Würzburg, Germany 2Department of Physics, University of California at San Diego, La Jolla, California 92093, USA (Received 24 November 2017; published 8 March 2018)

The absence of the negative sign problem in quantum Monte Carlo simulations of spin and fermion systems has different origins. World-line based algorithms for spins require positivity of matrix elements whereas auxiliary field approaches for fermions depend on symmetries such as particle-hole symmetry. For negative-sign-free spin and fermionic systems, we show that one can formulate a negative-sign-free auxiliary field quantum Monte Carlo algorithm that allows Kondo coupling of fermions with the spins. Using this general approach, we study a half-filled Kondo lattice model on the honeycomb lattice with geometric frustration. In addition to the conventional Kondo insulator and antiferromagnetically ordered phases, we find a partial Kondo screened state where spins are selectively screened so as to alleviatefrustration,andthelatticerotationsymmetryis broken nematically.

DOI: 10.1103/PhysRevLett.120.107201

Introduction.—Unconventional, highly entangled ˆ z ˆ z ˆ z ˆ ˆ − HSpin JijSi Sj Jij⊥ SiþSj H:c: ; states can arise if one starts from a system with a ¼ i;j ½ þ ð þ Þ large, perhaps infinite, ground state degeneracy, and X † 1 1 then perturb it slightly to lift the degeneracy. Fractional Hˆ cˆxσT cˆ U nˆ − nˆ − ; Fermion x;y yσ x;↓ 2 x;↑ 2 quantum Hall systems clearly fall in this category—at ¼ x;y;σ þ x X X any fractional filling the noninteracting problem of JK ˆ i;x † z ˆ z x ˆ − −ˆ electrons in Landau levels has an infinite number of H cˆx σ · S − −1 σþS σ Sþ cˆ . Kondo ¼ 2 ½ i ð Þ ð i þ i Þ x ground states in the thermodynamic limit. Perturbing i;x X this system with interactions leads to a particular 1 superposition of these groundstatesthatcorresponds ð Þ † to fractional quantum Hall states. Geometrically frus- Here the spin 1/2 local moments (electrons) Sˆ cˆx i ½ ¼ trated spin systems provide a different class of similar cˆ† ; cˆ† reside on a graph with sites labeled by i, j (x, y). phenomenon. As an example, consider a square lattice ð x;↑ x;↓Þ Jz , J defines the potentially frustrated spin model and where each link ij that connects vertices i, j hosts a ij ij⊥ T the hopping matrix elements of conduction electrons spin-1/2 spin Sˆ which interact via the Hamiltonian x;y i;j subject to electron correlations modeled by a Hubbard U Hˆ J Sˆ z Sˆ z Sˆ z Sˆ z . This model has an classical ¼ i;j;k;l∈□ ij jk kl li term [2]. The local moments and electrons interact via the extensive ground state entropy. Now consider a per- Kondo coupling JK . For the sake of generality we have P x i;x turbed model: Hˆ Hˆ ϵSˆ . For nonzero x quantum ¼ classical þ i included the phase factor −1 in the Kondo coupling. ϵ ≪ 1, the ground state of this new model is identical to This phase factor plays noð roleÞ if the transverse spin that of Kitaev’s celebrated toric code [1]:itcorresponds interaction is bipartite, or if the Kondo coupling includes to an equal weight superposition of the ground states conduction electron only on one sublattice. ˆ of Hclassical. Motivated by these examples, we ask what It is natural to ask when such Hamiltonians do not suffer phases emerge when a geometrically frustrated spin from the “sign problem” [3,4], which can make it impos- system is coupled to fermions. In this Letter, we will sible to simulate quantum systems using finite resources describe a quantum Monte Carlo (QMC) algorithm that [5]. There are two potential sources of the sign problem: allows one to study a large class of frustrated magnets the fermions and the geometrical frustration of spins. Kondo coupled to fermions, and demonstrate the algo- Conventionally, these difficulties are tackled in two very rithm by studying a specific model that exhibits new different ways. If the fermions were at half-filling on a partial Kondo screened (PKS) phases. bipartite lattice, then one can employ a determinantal For concreteness, consider the following Hamiltonian QMC approach to solve this problem [4,6–8], whereas ˆ ˆ ˆ of interacting fermions and spins, H HSpin HFermion for spins, if the condition Jij⊥ < 0 is met (which still allows ˆ ¼ þ þ HKondo, where for geometrical frustration [9,10]), then one can employ a

Hˆ S (xx,⌧) Z =Tre = D (xx, ⌧) e { }

Sampling (critical slowing down) à Global updates

Directed loops for bosonic models with retarded interactions week ending PRL 119, 097401 (2017) PHYSICAL REVIEW LETTERS 1 SEPTEMBER 2017

Directed-Loop Quantum Monte Carlo Method for Retarded Interactions

Manuel Weber, Fakher F. Assaad, and Martin Hohenadler Institut für Theoretische Physik und Astrophysik, Universität Würzburg, 97074 Würzburg, Germany Hybrid QMC à Global updates for(Received fermion-boson 28 April 2017; published systems 31 August 2017) The directed-loop quantum Monte Carlo method is generalized to the case of retarded interactions. Using the path integral,PHYSICAL fermion-boson REVIEW or spin-boson B 97 models,085144(2018) are mapped to actions with retarded interactions by analytically integrating out the bosons. This yields an exact algorithm that combines the highly efficient loop updates available in the stochastic series expansion representation with the advantages of avoiding a direct sampling of the bosons. The application to electron-phonon models reveals that the Revisitingmethod the hybrid overcomes quantum the previously Monte detrimental Carlo method issues of for long Hubbard autocorrelation and electron-phonontimes and exponentially models decreasing acceptance rates. For example, the resulting dramatic speedup allows us to investigate the Peierls quantum phase transitionStefan Beyl, on chains Florian of Goth, up to and 1282 Fakher sites. F. Assaad Institut für Theoretische Physik und Astrophysik, Universität Würzburg, Am Hubland, D-97074 Würzburg, Germany DOI: 10.1103/PhysRevLett.119.097401 Machine learning ... (Received 18 October 2017; revised manuscript received 22 January 2018; published 22 February 2018) A unique feature of the hybrid quantum Monte Carlo (HQMC) method is the potential to simulate negative sign Introduction.free— latticeIn the fermion absence models of with general subcubic exact scaling solutions in system size.of free Here bosons we will can revisit be the challenging algorithm for[28] various. In fact, for electron- for stronglymodels. correlated We will quantum show that systems, for the Hubbard the development model the HQMC suffersphonon from models, ergodicity the issues continuous-time and unbounded forcesinteraction-expansion of efficientin numerical the effective methods action. Solutions is a central to these objective. issues can be For found in(CT-INT) terms of a complexification QMC method of[29 the– auxiliary31] with fields. a βL 3 scaling and the one-dimensionalThis implementation (1D) case, of the the HQMC density-matrix that does not renorm- attempt to regularizelocal updates the fermionic produces matrix better so as resultsto circumvent forð correlatorsÞ at the alization groupthe aforementioned (DMRG) method singularities[1,2] doeshas not become outperform the single spinsame flip numerical determinantal cost methods[23].Ontheotherhand,directed-loop with cubic scaling. standard. However,On the other it hand is much we will less argue efficient that there is for a set higher of models formethods which the remain HQMC efficientis very efficient. for interactions This class is that are long- dimensions,characterized finite temperatures, by effective or actions long-range free of singularities.interactions, Usingranged the Majorana in space representation,[32,33]. we show that models so that quantumsuch as Monte the Su-Schrieffer-Heeger Carlo (QMC) Hamiltonian methods are at half often filling and onIn a this bipartite Letter, lattice we belong overcome to this class. these For this limitations by for- advantageous.specific The model latter subcubic yield scalinghigh-precision is achieved. results for mulating the problem in imaginary time, integrating out the rather generalDOI: 1D10.1103/PhysRevB.97.085144 fermion and spin Hamiltonians. In bosons analytically, and using directed-loop updates to particular, the cost for QMC simulations of path integrals efficiently sample the resulting problem with a retarded in the stochastic series expansion (SSE) representation [3] interaction. This novel approach combines the advantages I. INTRODUCTION The structure and main results of this paper are as follows. scales linearly with system size L and inverse temperature Inof Sec. globalII we updates will revisit available the ideas in of the Ref. SSE [25 representation]asappliedto and There has recently been tremendous progress in classifying β 1 =kBT.Autocorrelationtimesareshortduetotheuse thethe Hubbard analytical model. integration A number over of subsections the bosons will possible give us the in the fermionic¼ model Hamiltonians that one can solve with quan- of cluster updates (operator loops or directed loops) [4–6]. opportunityaction-based to introduce CT-INT notation method. and summarize important tum Monte Carlo (QMC) methods without encountering the While usually restricted to 1D fermionic models by the ideasMethod. of the HQMC.—The In SSE the final representation subsection,[3]II F,weshowthatcorresponds to a infamoussign problem, negative such sign problem QMC [ methods1– 4]. Remarkably, were successfully this class of models contains a number of extremely interesting phases thehigh-temperature algorithm is not ergodic, expansion even at of half the filling partition when function,the weight applied in higher dimensions to models of spins [7–9] and is always positive. The fermion determinant in each spin sector of matter and quantum phase transitions [5– 11]. Since in a ∞ n n lotbosons of these[10,11] models. the fermions are gapless, they cannot be has a strongly fluctuating signβ at low temperatures.ˆ Particle- Z α Ha ;b α : 1 integratedRetarded out, interactions and are at the (i.e., very nonlocal origin ofin novelimaginary quantum time) hole symmetry locks¼ the relativen ! signsh j of thep determinantsp j i ð Þ α n 0 Sn p 1 criticalimpose phenomena significant [12 limitations– 18]. From regarding the technical system point size, of view, tem- to unity and the weightX X¼ is positive.X Y Nevertheless,¼ at low thisperature, leads to and the necessityparameters. of developing They typically efficient arise algorithms from a temperaturesThe Hamiltonian the weightHˆ is has written many aszero a sum modes of and local different operators, regions of configuration space are separated by divergences forcoupling lattice fermions to bosonic so asmodes to unravel (e.g., facets phonons of fermion[12,13] criticalor spin Hˆ − Hˆ , where a specifies an operator type and b phenomena in two and higher spacial dimensions. in the¼ effectivea;b potentiala;b through which the molecular dy- fluctuations [14]), or from dynamical screening [15]. the bond between sites i b and j b . The expansion (1) is BecauseThe condensed such problems matter community are generically has focused nonintegrable, on the namics cannotP tunnel. In Sec.ð Þ III weð proceedÞ to describe acomplexificationofthealgorithmthatcircumventsthesesampled stochastically. A configuration with expansion Blankenbeclermuch of their Scalapino understanding Sugar comes (BSS) from auxiliary numerical field deter- inves- ergodicityorder n issues.corresponds In contrast to to a Ref. string [29 of]ourapproachisbasedn operators, specified minantaltigations. QMC While technique impurity [19– problems21]. This approach can be solved invariably very scales as the cubed of the volume since the fermionic determi- onby a complex the index Hubbard sequence StratonovichS a (HS);b transformation.; …; a ;b , In and a efficiently [16], lattice problems remain a challenge. In the n ¼ f½ 1 1 ½ n n g nant is explicitly calculated. Furthermore, since the determi- Sec.stateIII Cαwefrom show a complete that an ergodic basis algorithm in which H canˆ is be nonbranching, achieved DMRG [17–19], the large bosonic Hilbert space becomes a j i nant is nonlocal, cluster algorithms have remained elusive and andi.e., thatHˆ it canα reproduceα . If standardHˆ contains BSS resultsboth off-diagonal as obtained with (a 1 ) limiting factor, especially for finite-temperature or dynami- the ALF packagea;b ∼ [300 ]. However, for the Hubbard model, our single spin-flip updates are still the standard. Using machine and diagonalj i (ja i 2 ) operators, two types of updates¼ are learningcal properties. techniques For to SSE-based propose global QMC moves methods is an[20 ongoing–26], the implementation of the¼ HQMC does not provide an improved activeabsence research of global subject updates [22– 24]. for On the other bosons hand, (except in particle for a scalingsufficient and is to less achieve efficient ergodicity. in terms of For fluctuations. the diagonal Here updates, we physics,specific especially form of spin-phonon in the lattice gauge coupling theory[27] community,) makes simu- the noteitthat is convenient the BSS relies to fix on a the discrete length Hubbard-Stratonovich of the operator string hybridlations quantum significantly Monte less Carlo efficient method than (HQMC) for fermions was used or spins.and transformation,to N by inserting therebyN − circumventingn unit operators the (a above0 ). ergodicity Then, SN can ˆ¼ ˆ extendedAutocorrelation [25– 27]. times A glimpse increase at the strongly simulated near system phase sizes tran- issues.be traversed sequentially and updates H0 ;b↔H2 ;b be pro- fosterssitions the[21] hope.Extremelylongautocorrelationtimesalso to access larger lattice sizes, at least for a posed.To proceed The corresponding we will ask the configuration question if there weights is a class are directly of selected subset of models, by using HQMC. From a conceptual models in the solid state for which the HQMC canp beˆ the affect determinant QMC methods, where even the sampling obtained from the propagated state α p ∼ l 1 Hal;bl α . point of view, the HQMC method offers two main advantages method of choice. To do so, we willj followð Þi the idea¼ that j i in comparison to established methods in the condensed matter: the Hubbard model is hard, since the effective actionQ shows 0031-9007(i) A global=17= updating119(9)= scheme,097401(5) based on Hamilton’s equa-097401-1divergences and thereby generates© 2017 forces American that are Physical unbounded. Society tions of motion [28], that guarantees a good acceptance With the help of recent progress in our understanding of the rate. negative sign problem [2– 4], it can be shown that for a class (ii) Replacing determinants by Gaussian integrals is the of models the fermion determinant in a single spin sector is first key step to allow for subcubic scaling. positive semidefinite. This turns out to be the case for the Note that global update schemes such as Langevin dynamics so-called n-flavored Su-Schrieffer-Heeger (SSH) model [31] or hybrid Monte Carlo [28]canbeimplementedinschemes at half filling and on any bipartite lattice. We have implemented that explicitly retain the fermion determinant. the HQMC for this model and have benchmarked our results

Hˆ S (xx,⌧) Z =Tre = D (xx, ⌧) e { }

Eﬃcient, general and documented (!) implementa$ons

SciPost Phys. 3, 013 (2017)

The ALF (Algorithms for Lattice Fermions) project release 1.0 Documentation for the auxiliary ﬁeld quantum Monte Carlo code

Martin Bercx, Florian Goth, Johannes S. Hofmann and Fakher F. Assaad

Institut für Theoretische Physik und Astrophysik, Universität Würzburg, 97074 Würzburg, Germany

[email protected] hAps://alf.physik.uni-wuerzburg.de Abstract

The Algorithms for Lattice Fermions package provides a general code for the finite temperature auxiliary field quantum Monte Carlo algorithm. The code is engineered to be able to simulate any model that can be written in terms of sums of single-body operators, of squares of single-body operators and single-body operators coupled to an Ising field with given dynamics. We provide predefined types that allow the user to specify the model, the Bravais lattice as well as equal time and time displaced observables. The code supports an MPI implementation. Examples such as the Hubbard model on the honeycomb lattice and the Hubbard model on the square lattice coupled to a transverse Ising field are provided and discussed in the documentation. We furthermore discuss how to use the package to implement the Kondo lattice model and the SU(N)-Hubbard-Heisenberg model. One can download the code from our Git instance at https://alf.physik.uni-wuerzburg.de and sign in to file issues.

Copyright the ALF Project. Received 06-04-2017 This work is licensed under the Creative Commons Accepted 17-07-2017 Check for Attribution-ShareAlike 4.0 International License. Published 16-08-2017 updates Published by the SciPost Foundation. doi:10.21468/SciPostPhys.3.2.013 You are free to share and beneﬁt from this documentation as long as this license is preserved and proper attribution to the authors is given. For details, see the ALF project homepage alf.physik.uni-wuerzburg.de. Contact address: [email protected].

Contents

1 Introduction 1 1.1 Motivation 1 1.2 Deﬁnition of the Hamiltonian 3 1.3 Outline 5

2 Auxiliary Field Quantum Monte Carlo 6 2.1 Formulation of the method 6 2.1.1 The partition function 7 2.1.2 Observables 8 2.1.3 Reweighting and the sign problem 9 2.2 Updating schemes 10 2.2.1 The default: sequential single spin ﬂips 10

1 Quantum Monte Carlo: ALF@ http://alf.physik.uni-wuerzburg.de

Kinetic Potential (sum of perfect squares) 2 MT Ncol Nfl Ndim MV Ncol Nfl Ndim ˆ (ks) (ks) H = cˆx†sTxy cˆys + Uk cˆx†sVxy cˆys + ↵ks 2 MT Ncol Nfl Ndim MV k=1Ncol=1Nsfl=1 x,yNdim k=1 (=1 s=1 " x,y ! #) ˆ (ks) X X X X (ks) X X X X H = cˆx†sTxy cˆys + Uk cˆx†sVxy cˆys + ↵ksMI Ncol Nfl Ndim (ks) k=1 =1 s=1 x,y k=1 (Coupling=1 s=1 " ofx,y fermions to Ising! field with#)ˆ predefined dynamics ˆ X X X X X X X X + Zk cˆx†sIxy cˆys + HIsing MI Ncol Nfl Ndim k=1 =1 s=1 x,y ! ˆ (ks) Xˆ X X X + Zk cˆx†sIxy cˆys + HIsing =1 s=1 x,y ! kX=1 X X X

Ø Block diagonal in flavors, Nfl

Ø SU(Ncol) symmetric in colors Ncol F. Goth M. Bercx J. Hoffmann M. Ulybyshev Ø Arbitrary Bravais lattice for d=1,2

Ø Model can be specified at minimal programming cost

Ø Fortran 2003 standard Wissenschaftliche Ø MPI implementation Literaturversorgungs Ø Parallel tempering, projective and finite T approaches und Informationssysteme (LIS)

Ø Long range Coulomb M. Bercx, F. Goth, J. S. Hofmann and F. F. Assaad, SciPost Phys. 3 (2017), 013 Quantum Monte Carlo Methods for Fermion-Boson problems

Fakher F. Assaad (Autumn School on Correlated Electrons, Jülich 20th September 2018)

Outline: Ø Fermion-Boson problems Electron-Phonon, Su-Schrieffer-Heeger (SSH) Unconstrained Gauge theories “De-signer” Hamiltonians

Ø Auxiliary field Quantum Monte Carlo (QMC), Generalities

Ø Application of the Auxiliary field QMC to the SSH model Sign-free formulation Sampling, Hybrid Monte Carlo

Ø Conclusions

SFB1170 ToCoTronics SSH: Integrating out bosons

Integrate out the phonons in favor of a retarded interaction

⎡ β β ⎤ Z = ⎡dc+dc⎤ exp −S + dτ dτ ' K (τ ) D0 (b − b',τ −τ ') K (τ ') ∫ ⎣ ⎦ ⎢ 0 ∫ ∫ ∑ b b' ⎥ ⎣⎢ 0 0 b,b' ⎦⎥ g 2 D0 (b − b',τ −τ ') = δ P(τ −τ ') b,b' 2k

Phonon propagator ω | | ( | |) 0 ⎡ −τ ω0 − β −τ ω0 ⎤ P(τ ) = e + e , ω 0 = k / M −βω0 ⎣ ⎦ 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 2(1− e )

Simulation: Expand in the interaction and use MC to sum up all connected and disconnected Feynman diagrams. CT-INT: A. N. Rubtsov, V. V. Savkin, and A. I. Lichtenstein, Phys. Rev. B 72 (2005), 035122.

Computational effort (Vβ)3. No sign problem in 1+1 Dimension. Sign problem in higher dimensions.

F. F. Assaad and T. C. Lang Phys. Rev. B76, 035116 (2007). http://www.cond-mat.de/events/correl14/manuscripts/assaad.pdf Continuous-time QMC Solvers for Electronic Systems in Fermionic and Bosonic Baths SSH: Real space path integral

ˆ2 P i,j k ˆ ˆ h i ˆ 2 H = t + gX i,j cˆi,† cˆj, +ˆcj,† cˆi, + + X i,j h i 2m 2 h i i,j , i,j " # hXi ⇣ ⌘⇣ ⌘ hXi ˆ ˆ ˆ H0( Xb ) H1 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 { } | {z } | {z }

L⌧ ˆ ˆ ˆ ˆ H ⌧H1 ⌧H0( Xb ) 2 Aim: Z =Tr ˆ e =Tr ˆ e e + (⌧ ) Xb , cˆi, Xb , cˆi, { } { } { } " # O AAAENnictVFNb9NAFHyO+WjLVwpHLisipFSUyK6Q4FKpEgX1AKJITRuRTaK1s3GWOLZlbyqK5f/F3+AIB26IKzcuHBhvnBJaQU+stX5v582bnd31klBl2nE+WTX70uUrV1dW165dv3HzVn399mEWz1Jftv04jNOOJzIZqki2tdKh7CSpFFMvlEfe5GlZPzqWaabi6ECfJLI3FUGkRsoXGtBg3Xr7hm0zruU7nR+kxSDnoRxpnvOx0HkHa69gPFXBWPNiky0XfRTVJs9UMBW/OcWc02Wynz/kntSCGfYeKqxi9ZD9lz15ksZDyGkx23aLfv5iUKbF3MuuDEszABaOBm5xtnJayp1Ktcn+4a5KNlixdLQHLOe+CNmrorkk3d/aGNQbTssxg51P3CppUDX24/pn4jSkmHya0ZQkRaSRhyQow9cllxxKgPUoB5YiU6YuqaA19M7AkmAIoBP8A6y6FRphXWpmptvHLiFmik5G9zGfG0UP7HJXiTxD/In53mDBX3fIjXLp8ATRg+KqUXwJXNMYjIs6pxVz4eXizvJUmkb0xJxGwV9ikPKc/qnOLiopsImpMHpmmAE0PLM+xg1EiG04KG95ocDMiYeIwkRpVKJKUUAvRSxvH37wzO7ZRz2fHG61XKflvn7U2HGqB1+hu3SPmnjVx7RDe7QPH7710fpRo5plf7C/2F/tb3Nqzap67tAfw/7+C+6VFxg= { } { } ⌧=1 h i Y LAAACBXicbVA9SwNBEJ3z2/gVtdRiMQhW4c5GG0HQwsJCwRghF8LcZqKLex/szgnhSGPj37C0sVDE1v9g579x81Go8cHA2/dm2JkXZVpZ9v0vb2Jyanpmdm6+tLC4tLxSXl27tGluJNVkqlNzFaElrRKqsWJNV5khjCNN9ej2qO/X78hYlSYX3M2oGeN1ojpKIjupVd48bRUhY94T4TFpRtF/iAMRRsTYKlf8qj+AGCfBiFQO5x/ajwBw1ip/hu1U5jElLDVa2wj8jJsFGlZSU68U5pYylLd4TQ1HE4zJNovBFT2x7ZS26KTGVcJioP6cKDC2thtHrjNGvrF/vb74n9fIubPfLFSS5UyJHH7UybXgVPQjEW1lSLLuOoLSKLerkDdoULILruRCCP6ePE4ud6uBXw3OXRo+DDEHG7AFOxDAHhzCCZxBDSTcwxO8wKv34D17b977sHXCG82swy94H9+SoJlr sha1_base64="OvZ/nE7ka4ziCTx7cP8FlfzYpZQ=">AAACBXicbVA9SwNBEN3zO/EraqnFoggWEu5stBECWlhYRDAq5EKY20ySJXsf7M4J4UiTJn/FxkIRW/+Dnb9G9xILNT4YePveDDvzgkRJQ6774czMzs0vLC4Vissrq2vrpY3NGxOnWmBNxCrWdwEYVDLCGklSeJdohDBQeBv0znL/9h61kXF0Tf0EGyF0ItmWAshKzdLOZTPzCdIB989REfD8wU+5HyBBs7Tnlt0x+DTxvslepTBqPX4OD6vN0rvfikUaYkRCgTF1z02okYEmKRQOin5qMAHRgw7WLY0gRNPIxlcM+L5VWrwda1sR8bH6cyKD0Jh+GNjOEKhr/nq5+J9XT6l90shklKSEkZh81E4Vp5jnkfCW1ChI9S0BoaXdlYsuaBBkgyvaELy/J0+Tm6Oy55a9K5uGyyZYYttslx0wjx2zCrtgVVZjgg3ZA3tiz87IeXRenNdJ64zzPbPFfsF5+wLiJ5sl sha1_base64="UWT0X7E/NCGZAkxY8GigivVdlE4=">AAACBXicbVA9SwNBEN2LXzF+RS21WAyCVbiz0UYIaGFhEcHEQC6Euc0kWbL3we6cEI40Nv4VGwtFbP0Pdv4bNx+FJj4YePveDDvzgkRJQ6777eSWlldW1/LrhY3Nre2d4u5e3cSpFlgTsYp1IwCDSkZYI0kKG4lGCAOF98HgcuzfP6A2Mo7uaJhgK4ReJLtSAFmpXTy8aWc+QTri/hUqAj5+8AvuB0jQLpbcsjsBXyTejJTYDNV28cvvxCINMSKhwJim5ybUykCTFApHBT81mIAYQA+blkYQomllkytG/NgqHd6Nta2I+ET9PZFBaMwwDGxnCNQ3895Y/M9rptQ9b2UySlLCSEw/6qaKU8zHkfCO1ChIDS0BoaXdlYs+aBBkgyvYELz5kxdJ/bTsuWXv1i1V3FkceXbAjtgJ89gZq7BrVmU1Jtgje2av7M15cl6cd+dj2ppzZjP77A+czx8bFZeX ⌧ ⌧ = Real space path integral Xˆ x = x x 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 sha1_base64="I3qzNHs0O153gIiFHHCMQ4HedNo=">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 sha1_base64="M0anq4viWgYeQJFosx0kpaP0kGE=">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 sha1_base64="hNK+Re73IPeTJ2krxXIQgKEQI4o=">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 b| bi b| bi

Insert the resolution of unity at each time step:

dpb ipx 1ˆ = dx x x = p p , x p = e B b| bih b| 2⇡ | bih b| h b| bi 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 sha1_base64="I3qzNHs0O153gIiFHHCMQ4HedNo=">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 sha1_base64="R0wht0BeVaLWaE78mlCLaijlp/g=">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 sha1_base64="ARu3IFRDYI1qHyuG6FQtZfXbLJM=">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 Z Yb Z Yb to obtain

L⌧ ⌧Hˆ1 ⌧Hˆ0( xb(⌧) ) 2 Z = dxb(⌧)Tr cˆ xb(⌧ + 1) e xb(⌧) e { } + (⌧ ) { i, } h{ }| |{ }i O ⌧,b "⌧=1 # 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 Z Y Y with boundary condition

x (L + 1) = x (1) |{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 sha1_base64="J+Z7LUuYmLnwynHtmNP9DE2exho=">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 sha1_base64="sD7GT4D3oyPcBIAGwdThD0oSRXM=">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 b ⌧ }i |{ b }i SSH: Real space path integral

2 ⌧Hˆ dpb ⌧ Pˆ ⌧ k Xˆ 2 x (⌧ + 1) e 1 x (⌧) x (⌧ + 1) e 2m p p e 2 b x (⌧) h{ b }| { b }i' 2⇡ h b | | bih b| | b i Z b Y 2 pb k 2 dp ⌧ + xb (⌧) b ipb[xb(⌧+1) xb(⌧)] 2m 2 = e e 2⇡  Z Yb x (⌧+1) x (⌧) 2 ⌧ m b b + k x (⌧)2 b 2 ⌧ 2 b 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 sha1_base64="I3qzNHs0O153gIiFHHCMQ4HedNo=">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 sha1_base64="h0r5mZWnn+2/8S521yYYw75qIuo=">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 sha1_base64="/X3pM5UWUL01l03X+2DFODCg7F0=">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 = e ✓ ◆ P h i SSH: Real space path integral

Real space path integral Xˆ x = x x 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 sha1_base64="I3qzNHs0O153gIiFHHCMQ4HedNo=">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 sha1_base64="M0anq4viWgYeQJFosx0kpaP0kGE=">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 sha1_base64="hNK+Re73IPeTJ2krxXIQgKEQI4o=">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 b| bi b| bi Ncol

S0(x) ⌧( t+gxxb,⌧ )(cˆ†cˆj +ˆc†cˆi) Z = D x e Tr e i j { } 2 3 ⌧,b

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 Z Y 2 4 5 m xb(⌧ + 1) xb(⌧) k 2 S0(x)=⌧ + xb(⌧) 2 ⌧ 2 ! 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 sha1_base64="I3qzNHs0O153gIiFHHCMQ4HedNo=">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 sha1_base64="3cnqCvo6cYPn72KFK2Lh5dFhRhA=">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 sha1_base64="LENwCg0FxGWp4FSYMgns334u4EQ=">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 Xb,⌧ 

Integrating out the fermions. Use determinant formula (see lecture notes)

cˆ†A(x )cˆ cˆ†A(x )cˆ cˆ†A(x )cˆ A(x ) A(x ) A(x ) Tr e NbL⌧ e 2 e 1 =det 1+e NbL⌧ e 2 e 1

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 ··· ··· h i ⇣ ⌘ with cˆ†A(xb,⌧ )cˆ = ⌧ ( t + gxxb,⌧ ) cˆi†cˆj +ˆcj†cˆi

AAADi3ichVFdT9RAFL2lKrCKLvroy8SNZgmyaYUEYzTBCMYHTTBxFxKKm+nsbHfYfmU6S8Sm/82/ob/FB0+HQnDRMM303jn33HPvzA3zWBXG8346C+6t23cWl5Zbd++t3H/QXn04KLKZFrIvsjjThyEvZKxS2TfKxPIw15InYSwPwum7On5wKnWhsvSLOcvlccKjVI2V4AbQsP0jmHBTBnkSlqKqvpbBiEeR1BV727Xgt2pYhs8Dw2fVGpvjVuwN2wh2ZWx4TWAsiOXYdNmGYesRm09ngVbRxKydk6yUQFRdKXoJnlTrV/x/EFTVqLWG7Y7X8+xi1x2/cTrUrP2s/YsCGlFGgmaUkKSUDPyYOBX4jsgnj3Jgx1QC0/CUjUuqqIXcGVgSDA50in+E01GDpjjXmoXNFqgSY2tkMnqK/d4qhmDXVSX8AvY39neLRf+tUFrlusMz2BCKy1bxE3BDEzBuykwa5kUvN2fWtzI0ppf2Ngr95Rap7ykudXYR0cCmNsJozzIjaIT2fIoXSGH76KB+5QsFZm88guXWSquSNoocehq2fn30gzH780O97gxe9Hyv53/e6ux4zcCX6DE9oS6muk079IH20Ydwnjkfnb4zcFfcTfeV+/qcuuA0OY/or+Xu/QGib9cZ ⇣ ⌘ SSH: Real space path integral

(S (x) N log det(M(x)) S(x) Z = D x e 0 col D x e { } ⌘ { } Z Z A(x ) A(x ) A(x ) M(x)=1+e NbL⌧ e 2 e 1 AAAD+nicjVHLbtNAFL2uebThlcKSzRURUQJqZGcDG6QiCmJBUVFJW9Epke1MXCuO7dqTqMH4M/gBdogtO7bwEfAtLDgenEKoUBlrfO+ce+6ZMzNuEgaZsqxvxpJ57vyFi8srtUuXr1y9Vl+9vpPFk9STPS8O43TPdTIZBpHsqUCFci9JpTN2Q7nrjh6V9d2pTLMgjl6qWSIPxo4fBcPAcxSg/qrREa70gyiXR5GTps7sTvGKm/yAmyKIFG+IUA6VyFkkYzc/LlikgX+oRMHydc5rLd7uW8ytk3Kbmdc4f97PhZLHKofBoihEGPssBlK1NlsVsw1mwSzk0SSY8v/stb2wTdkrGE6bXOPfqjDONt8tOx7OwT7suPysL5QzKdqQ9QaxyniR020XC2sbRD2EjAZ/XE6/3rA6lh58OrGrpEHV2Irr30nQgGLyaEJjkhSRQh6SQxm+fbLJogTYAeXAUmSBrksqqIbeCVgSDAfoCH8fq/0KjbAuNTPd7WGXEDNFJ9NtzCda0QW73FUizxB/YL7RmP/PHXKtXDqcIbpQXNGKm8AVHYJxVue4Ys69nN1ZnkrRkO7r0wTwl2ikPKd3orOBSgpspCtMjzXTh4ar11PcQITYg4PylucKrE88QHR0lFolqhQd6KWI5e3DD57Z/vtRTyc73Y5tdewX3cZ6t3rwZbpJt6iFV71H6/SUtuDDM94Zn40vxlfzrfne/GB+/EVdMqqeG7QwzE8/Ab209ds= ··· 2 m xb(⌧ + 1) xb(⌧) k 2 S0(x)=⌧ + xb(⌧) 2 ⌧ 2 ! 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 sha1_base64="I3qzNHs0O153gIiFHHCMQ4HedNo=">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 sha1_base64="3cnqCvo6cYPn72KFK2Lh5dFhRhA=">AAADcXichVLbbtNAEB0nXNpSaOCFB15WFKREoZGdF3iphEQvvCAVQdpKcWrtuhvXim9aryuK5Y/q58C3oIqzWxdBKtS11jNzZubMzO6KIolL7bo/nE733v0HD1dW1x6tP36y0Xu6fljmlQrlJMyTXB0LXsokzuRExzqRx4WSPBWJPBKLD8Z/dC5VGefZV31RyFnKoyyexyHXgILe5ZfA7ftFKupvzYCxbebvyERz5mteMb+s0qAWb4zRMD+Rc91n/lzxsE6betxYZNoirGUJRN/ED5k3YFtsCR2wpv6rQgO/iqMzPTsZs2FLtDDUy3kn4+vAAQt6m+7ItYvdVrxW2aR2HeS9n+TTKeUUUkUpScpIQ0+IU4lvSh65VACbUQ1MQYutX1JDa8itECURwYEu8I9gTVs0g204S5sdokqCrZDJ6DX2nmUUiDZVJfQS8hf2d4tF/61QW2bT4QWkAOOqZfwEXNMZIu7KTNvIm17uzjRTaZrTOztNjP4Ki5g5wz88O/AoYAvrYbRrIyNwCGuf4wQyyAk6MKd8w8DsxKeQ3EppWbKWkYNPQZrTRz+4Zm/5Um8rh+OR5468zy6t0At6SX1c5lt6Tx/pAOVD57mz7ew5+52r7qvu8PpBdJz2ZTyjf1Z36zfZtssM sha1_base64="LENwCg0FxGWp4FSYMgns334u4EQ=">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 Xb,⌧  SSH: Real space path integral

S0(x) ⌧( t+gxxb,⌧ )(cˆ†cˆj +ˆc†cˆi) Z = D x e Tr e i j { } 2 3 ⌧,b

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2 L⌧ L⌧ ⌧ ⌧ ( t+gxxb,⌧ )(cˆ†cˆj +ˆc†cˆi) i ( t+gxxb,⌧ )ij Tr e b i j = Tr e 4 b 0 ⌧=1 P " ⌧=1 P # Y Y Z. C. Wei, Congjun Wu, Yi Li, Shiwei Zhang, and T. Xiang, Phys. Rev. Lett. 116, 250601, (2016). Z.-X. Li, Y.-F. Jiang, and H. Yao. Phys. Rev. B, 91, 241117, (2015). E. F. Huffman and S. Chandrasekharan. Phys. Rev. B, 89, 111101, (2014). SSH: Real space path integral

The action is real for any number of fermion flavors !

(At finite chemical potential there is no sign problem for even number of flavors!) SSH: Observables

t2 ⌧( t+gx ))Kˆ Def t =(⌧,b) Uˆ (t ,t )= e t bt for t t . 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TrF Uˆx(L⌧ Nb, 0)Oˆ 1 S(x) Oˆ = Dxxxe h i Dxxxe S(x) h ˆ i Z TrF Ux(L⌧ Nb, 0) 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 R h i

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Gi ,i (t ,t0 ) Gi ,i (t ,t0 ) Tr Uˆ (L N , 0)Oˆ 1 10 1 1 1 n0 1 n F x ⌧ b . ··· . Wick’s theorem =det . .. . . holds. h i 0 . . . 1 TrF Uˆx(L⌧ Nb, 0) Gi ,i (t ,t0 ) Gin,i (t ,t0 ) 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 sha1_base64="wZPmztHdJccAvisrqUZSuupN1eI=">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 sha1_base64="zJbWPZDUqDw5DnT6mW4p9vVFpp8=">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 sha1_base64="/ENMB1CfIyqZrdVUIrASARh4Tm8=">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 B n 10 n 1 ··· n0 n n C h i @ A

Equal time Green functions t2 1 ⌧( t+gxt))Kb G(t0,t0)=[1+B (t0, 0)B (L N ,t0)] , with B (t ,t )= e t for t t x x ⌧ b x 2 1 2 1 t=t +1 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 Y1 SSH: Sampling 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

m x (⌧ + ⌧) x (⌧) 2 k S (x)=⌧ b b + x (⌧)2 0 2 ⌧ 2 b b,⌧  ! 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 X

Metropolis-Hastings (C is a configuration in Monte Carlo space) 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

Aim: Find T0 such that acceptance is high and moves allow efficient sampling of configuration space.

Possible solutions:

Single spin-flip à Very long autocorrelations

Machine learning

Hybrid molecular dynamics SSH: Sampling AAAEQXicjVFNb9NAEJ0kfLRpgRaOXEZUSInaRE6FBBJCqkRBHCgq6qfotpHtbN0ljm3sddRi7a/jV4D4A9wQVzjwvHUKpaCyyXpm38y8ebPrJaHKtON8rNUbV65euz413ZyZvXHz1tz87e0szlNfbvlxGKe7npvJUEVySysdyt0kle7IC+WON3xaxnfGMs1UHG3qk0Tuj9wgUofKdzWg/nwtf8NPWKhI86oI5aEWBYtk5BXHhkWqgiMtDMuDgjst3ug7zK2zcJuZO1y86hdCy2NdQIwxRoRxwGIgdWutVWW2kWmYhXyXq/F/9do418YssXg8+YOZfzGX2m3zzRQcSRoPSjVuDrxnDoqXpydjuGl5xaoMtVtCIstH/cJDUSmkxR3Ni8GkJwJLtq6S1T5NEkcupkRUgVoM3CCQqTkD35rF3/y/JChTseEy+nMLTtexiy86vcpZoGqtx3OfSNCAYvIppxFJikjDD8mlDL896pFDCbB9KoCl8JSNSzLURG2OLIkMF+gQ3wCnvQqNcC45M1vto0uInaKS6T72c8voIbvsKuFnsD+w31ss+GeHwjKXCk9gPTBOW8Y14JqOkHFZ5ajKnGi5vLKcStMhPbLTKOhLLFLO6Z/xrCKSAhvaCNMzmxmAw7PnMW4ggt2CgvKWJwxsJx7AutZKyxJVjC74Utjy9qEHz9z781EvOtvL3Z7T7b1+sLCyXD34FN2le9TCqz6kFXpB69Dh1z7X6/WZ+mzjQ+NL42vj22lqvVbV3KFzq/H9JwXvFZY=

Hybrid Molecular dynamics. Basics.

Introduce conjugate momentum and sample 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 Start with random x

Step 1. Sample p from Gauss distribution

Step 2. Hamiltonian dynamics with H over time T

m 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

Time reversal (Leapfrog) Finite time step Step 3. à Step1. Liouville SSH: Sampling 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

Hybrid Molecular dynamics. Key questions

Hubbard (2D, fields couple OO(2Ncol) symmetric to z-component of models such as SSH. magnetization) Most of the time. Are the forces bounded ? No Each determinant is positive AAAC/HichVFNT9tAEH24tAVa2gDHXqxGleglslGl9hipBXFBooIkSEkUrc0mrOIvrTdRQxT+Rv8At6rX3nqF39D+Fg48bw0Iooq11jP75s3bmZ0gi1RuPO/PgvNk8emz50vLKy9err56XVlbb+bpSIeyEaZRqo8CkctIJbJhlInkUaaliINItoLh5yLeGkudqzQ5NJNMdmMxSFRfhcIQ6lX8Tl+LcNrJhDZKRO6Bu+l2sjiYfpu572d3+C0461WqXs2zy513/NKpolz7aeUvOjhGihAjxJBIYOhHEMj5teHDQ0asiykxTU/ZuMQMK8wdkSXJEESH/A94apdownOhmdvskLdE3JqZLt5x71jFgOziVkk/p73iPrXY4L83TK1yUeGENqDislXcI25wQsZjmXHJvKnl8cyiK4M+PtluFOvLLFL0Gd7qfGFEExvaiIttyxxQI7DnMV8goW2wguKVbxRc2/ExrbBWWpWkVBTU07TF67Mejtl/ONR5p7lV872a//VDtb5VDnwJb/AWm5zqR9Sxi33WEeI7fuMCl86Zc+78cH7+ozoLZc4G7i3n1zWt56Km semidefinite Does the configuration space split into regions split by potential barriers Yes No

See: Stefan Beyl, Florian Goth, and Fakher F. Assaad, Revisiting the hybrid quantum Monte Carlo method for Hubbard and electron-phonon models, Phys. Rev. B 97 (2018), 085144. Deconfined quantum critical points in the SSH model Su-Schrieffer-HeegerSu-Schrieffer-Heeger model (SSH) model (SSH)

cˆ†

ˆ k Q i,j !0 = 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 h i 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 rm

Limiting cases Rev. Mod. Phys. 60, 781 (1988) 13.03.18 HQMC simulations of the SSH model 5

orderedValence valence Bond bond Solid states (VBS)

!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 0 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 ! 0 0 ! !1 Quantum an4ferromagnet strong bonds weak bonds

Rev. Mod. Phys. 60, 781 (1988) 13.03.18 HQMC simulations of the SSH model 5

Tang & Hirsch, Phys. Rev. B 37, 9546 (1988) 13.03.18 HQMC simulations of the SSH model 9 Deconfined quantum critical points in the SSH model

S. Beyl, F. Goth, M. Hohenadler and FFA, (Unpublished)

(Q + Q) c 1/⌫ z ! R =1 = F (! ! )L ,L /,L (Q) 0 0

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 sha1_base64="rNtFzNWJeSHVc8OvY3qyNf4Qs4Y=">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 sha1_base64="A/hO6L5kfEltK3GPFBWV/ukP8yE=">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 sha1_base64="OElI4gszvEJC/GFXKXy/qLVht9Q=">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 ⇣ ⌘ SDW

VBS

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 = L

Direct and con0nuous transi0on between VBS and SDW ! Deconfined quantum critical points in the SSH model

Order parameter theory, Ginzburg-Landau

Fine tuning à not generic Deconfined quantum critical points in the SSH model

Deconﬁned quantum cri0cality T. Senthil, A. Vishwanath, L. Balents, S. Sachdev, and M. P. A. Fisher, Deconﬁned quantum cri0cal points, Science 303 (2004), 1490–1494.

Topological defects of one phase carry the charge of the other phase.

Possible realization:

SrCu2(BO3)2

M. Zayed et al. Nat. Phys. 13, 962 (2017).

C4 VBS vortex carries a spinon. Prolifera0on of vor0ces destroys the VBS and generates SDW order. Quantum Monte Carlo Methods for Fermion-Boson problems

Fakher F. Assaad (Autumn School on Correlated Electrons, Jülich 20th September 2018)

Conclusion:

Ø Fermion-Boson problems cover an extremely rich class of phenomena

Ø Quantum Monte Carlo simulations are challenging

Ø Ever growing class of negative sign free problems

Ø Progress in sampling methods.

SFB1170 ToCoTronics Quantum Monte Carlo Methods for Fermion-Boson problems

Fakher F. Assaad (Autumn School on Correlated Electrons, Jülich 20th September 2018)

Review: World-line and determinantal quantum monte carlo methods for spins, phonons and electrons, F.F.A and H.G. Evertz. Lecture Notes in Physics, vol. 739, Springer, Berlin Heidelberg, 2008, pp. 277–356

Continuous-time QMC Solvers for Electronic Systems in Fermionic and Bosonic Baths, F. F. Assaad, DMFT at 25: Infinite dimensions: Lecture notes of the autumn school on correlated electrons, vol. 4, ch. 7. Verlag des Forschungszentrum Jülich, Jülich, 2014, ISBN 978-3-89336-953-9

Lecture notes of this school.

ALF: General Implementation: Martin Bercx, Florian Goth, Johannes S. Hofmann, and Fakher F. Assaad The ALF (Algorithms for Lattice Fermions) project release 1.0. Documentation for the auxiliary field quantum Monte Carlo code, SciPost Phys. 3 (2017), 013.