Operator spectrum and spontaneous symmetry breaking in SYK-like models

Grisha Tarnopolsky Carnegie Mellon University

Strings 2021 ICTP-SAIFR, São Paulo June 24, 2021 Talk based on:

J. Kim, I. R. Klebanov, GT, W. Zhao, PRX, 9, (2019)

I. R. Klebanov, A. Milekhin, GT, W. Zhao, JHEP, 162, (2020)

M. Tikhanovskaya, H. Guo, S. Sachdev, GT, PRB, 103, (2021) Plan:

• Introduction to Random SYK Models

• Operators in SYK models and their effects

• Coupled Majorana and complex SYK models and symmetry breaking Introduction to Random SYK Models

• A general non-relativistic Hamiltonian of interacting particles:

where

• In the second quantization representation it has the form (assume fermions)

where

And , eigenvalues and eigenfunctions of

• If are irregular and “random” we can assume that are random Complex SYK model in Nuclear physics Volume 33B, number 7 PHYSICS LETTERS 7 December 1970 [ J. French and S. Wong ‘70, product structure of the space being ignore d in • Two-body random interaction model O. Bohigas and J. Flores ’71 ] the random matrix calculations, it is like wis e was introduced in Nuclear Physics in 70s [S. Sachdev ’15] not surprising that the spectrum which emerges (a) is altogether different. 400 Fig. la gives the random-matrix spectrum for a set of 100 50-dimensional matrices, the ensem- 7

ble being one in which the individua l matrix ele- 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 ments H~.~ (i --< j) are, inde pe nde ntly, uniformly ~j N distributed over the inte rva l (-1,1). This en- N~ semble satisfies the Wigner conditions [4] for a semi-circular spectrum and, as found in similar [ J. French and S. Wong ‘70] cases by other authors [2], this is found here •-8 Energy-6 -4 -2 histogram0 2 4 6 8 • Reproduction of it ENERGY also. We now "re a lize " the 50-dimensional space orbitals in terms of the states of several particles con- ] 1200 strained to a set of single-particle states, there- by introducing a direct-product structure. At the 1000 same time we restrict H to be a (0 + 1 + 2)-body 300 operator and to have the standard inva ria nce s . 800 The realization which we choose is (f 7/2, f 5/2) for ide ntica l particles, which, for 7 particles ~mc 600 (= atomicf 7) with J = ½, has dimensionality 50. There are 21 two-particle matrix elements, as 400 well as a zero-body energy and two one-body energies Cone for each j-orbit), so that the prop- erly restricted H resides in a space of 24 dimen- 200 sions ins te a d of the 1275 of the conventional t -16 -12 -8 -4 0 4 8 12 [6 0 theory. Fig. lb shows the spectrum for an en- -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 semble of 100 Hamiltonians with inde pe nde nt ENERGy two-body matrix elements chosen uniformly in Fig. i. The cumulative spectra for two 100-member sets 7 identical particles (-1, 1) inte rva l, the zero- and one-body parameters of 50-dimensional matrices. (a) is for a conventional being chosen zero (a choice which should have ensemble while (b) is for an ensemble of two-body fixed total momentum no significant effect on the results); the Hamil- operators acting in the 7-particle space f7 with T=J =3" tonian spaceis now 21-dimensional.We seean excellentnormal spectrum**. • The main conclusion at that time – it is not the Wigner semicircle! We shouldconsider whether the spectrum shapesare characteristicfor typical members of the Hamiltonian ensembles or whether they emerge only after ensemble averaging. It is con- venient for this purpose to define finite -dime n- # For a one-bodyH we may choosea single-particle sional semi-circular and normal spectra, and a basiswhich diagonalizesH, and then the total energy correlation function which measures how close in is a sumof single-particleenergies to which, if shape(ignoring energytranslations and scaleex- Pauli principle effectsare ignored, we may apply pansions)two spectraare to eachother. For the the usuallimit theorem.For two-bodyinteract'ions, spectrawe use a procedureof Rateliff [6]; if things are by no meanswell understood.It can be shownthat, in spaceswith a la rge numberN of f(E) is the fre que ncyfunction, then the (cumulative) single-particlestates, the dominantpart of a given distributionfunction Hamiltonianwill, with high probability, tend, as the E fermion numberincre a s e sto , a number-dependent FCE) = / fCE')dE' , one-bodyH (an operator of lower than maximum symmetrywith respectto the group U(N) of trans- forma tionsin the single-particlespace), so that the where sameresult shouldemerge in this case; but the F(~o) = d , resultsare clearly more generalthan that. ** A similar analysis,with similar re s ults ,has been the matrix dimensionality, is a multi-step function madefor setsof 500 50-dimensionaltra ce le s sfixed- with unit jumps at the eigenvalues. A natural cor- strengthmatrices, in which extra conditionshave beenimpos e dthat the centroid of the eigenvaluesbe respondencethen with any continuous distribution zero and the trace of H2 havea fixed value. The function is to define the eigenvalue as those points resultantspectra are reproducedin ref. [7]. where the function takes on half-integral values,

450 Random models for quantum dots

• Hamiltonian of interacting particles in quantum dot: Quantum dot

is a random quantity with zero average

How does this interaction affect free particles?

[B. Altshuler, Y. Gefen, A. Kamenev, L. Levitov, ‘96 Y. Alhassid, Ph. Jacquod, A. Wobst, ’00 A. Lunkin, A. Kitaev, M. Feigel’man, ’20, …] Non-random large N models

• There are non-random models with the same large N limit

[R. Gurau’10, S. Carrozza, A. Tanasa’15, E. Witten ‘16, I. R. Klebanov, GT’16]

• Models of these types are called tensor models

• Randomness is replaced by addition of two “orbital” indices with a specific “tetrahedron” contraction: The Sachdev-Ye-Kitaev model

[ S. Sachdev, J.Ye ‘93, A.Georges, • Hamiltonian: O. Parcollet, S. Sachdev ’01, A.Kitaev ’15 ]

• N Majorana fermions:

• are antisymmetric Gaussian random

• The Hilbert space dimension is 2N/2

• One can compute numerically spectrum using Jordan-Wigner representation: 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 Spectrum of the SYK model • Energy levels for N=32 Majorana q=4 SYK model: 65536 energy levels

• s0 is zero temperature entropy • Most of the random SYK-like models admit scale invariant solution in IR for the two-point functions at large N limit

• This scale invariance is related to high density of energy levels near the ground state

• To check theoretically if such a scaling invariant solution is stable and find corrections to it one has to analyze spectrum of two-particle operators Operators in the SYK model • For the two-particle operators we consider the 3pt function and can derive the Bethe-Salpeter equation using large N:

[D. Gross, V. Rosenhaus ’16]

• This Bethe-Salpeter equation determines and reads

• Assume that in IR 3pt function has conformal form

the Bethe-Salpeter equation reduces to Operators in the SYK model

• The Bethe-Salpeter equation:

• Graphical solution of this equation gives operator spectrum Scaling dimensions

• The first solution is ; breaks conformal invariance. • The higher scaling dimensions are [J. Maldacena, D. Stanford ’16 approaching D. Gross, V. Rosenhaus ’16] SYK as Nearly CFT • We can think about SYK as CFT perturbed by irrelevant operators

[A. Kitaev ’15, J. Maldacena, D. Stanford ‘16] [D. Gross, V. Rosenhaus’17] where are couplings

Scaling dimensions

• Next order corrections to the Green’s function are given by conformal perturbation theory: SYK two-point function • General formula for the full SYK two-point function

• Comparison with numerics at zero temperature: Scaling dimensions

• We can detect irrational scaling exponent numerically Thermodynamics of the SYK model • We can think about SYK as CFT perturbed by irrelevant operators

[A. Kitaev ’15, J. Maldacena, D. Stanford ‘16] [D. Gross, V. Rosenhaus’17] where are couplings

• Free energy for the SYK model using conformal perturbation theory: Scaling dimensions

• Computing the integrals we find Schwarzian coupling

specific heat ground energy zero temperature entropy Thermodynamics of the complex SYK model • In the complex SYK we have two sets of operators and

• Perturbation by conserved U(1) charge is a marginal Scaling dimensions deformation of the theory

• The free energy reads Schwarzian coupling

specific heat ground energy zero temperature entropy

• In both SYK and complex SYK the small temperature thermodynamics is governed by operator Coupled Majorana SYK models

• Consider two types of Majorana fermions and with Hamiltonian:

• are Gaussian random, exactly as in the SYK models

• is the interaction strength. Using symmetry one can show

• We assume that

• Under this assumption we find that coincides with SYK two-point function and we can find anomalous dimensions of various bilinear operators Operator spectrum

• Hamiltonian:

• For the model is similar to the SYK Scaling dimensions The operator is relevant. If we perturbed by it the spectrum will be gapped [J. Maldacena, X. Qi’18]

• For the operator has complex anomalous dimension and acquires a vev

The Z2 symmetry is spontaneously broken Coupled complex SYK models • Hamiltonian of the coupled complex SYK models

[see also S.Sahoo, E. Lantagne-Hurtubise, S. Plugge, M. Franz ’20 E. Lantagne-Hurtubise, S. Sahoo, M. Franz, 20]

• Random couplings

• The Hamiltonian has symmetry

• For coupling we find symmetric Hamiltonian Operator spectrum

• Hamiltonian:

Scaling dimensions • For the model is similar to the cSYK The operators and are relevant. If we perturbed the Hamiltonian by them the spectrum will be gapped

• For there is a U(2) global symmetry and all quadratic operators are marginal

• For and the operators and have complex anomalous dimension and acquire vev The U(1) symmetry is spontaneously broken Operator spectrum

• Hamiltonian:

Scaling dimensions • For there is an operator with the scaling dimension

• This operator is not generated in the effective action, but we can add it. In the Hamiltonian formalism this corresponds to changing commutation relations between fermions

[A. Milekhin ‘21]

• In this case the low-energy effective action is not Schwarzian, but

[J. Maldacena, D. Stanford, Z. Yang ’16, A. Milekhin ‘21] Conclusions

• There is a variety of random models with similar large N physics, described by conformal solution

• The operator spectrum of such models can be very different

• In some models the operator spectrum implies instability of the conformal solution and spontaneous symmetry breaking

• In some models there exists an operator with scaling dimension 1 < h < 3/2 which leads to the non-local IR effective action instead of the Schwarzian action Thank you for your attention!