Application of the operator product expansion and sum

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Application of the operator product expansion and sum rules to the study of the

Application of the operator product expansion and sum rules to the study of the single-particle spectral density of the unitary Fermi gas ar. Xiv: 1501: 06053 [cond. mat. qunat-gas] Seminar at Yonsei University 12. 02. 2015 Philipp Gubler (ECT*) Collaborators: N. Yamamoto (Keio University), T. Hatsuda (RIKEN, Nishina Center), Y. Nishida (Tokyo Institute of Technology)

Contents n Introduction q The unitary Fermi gas n q n Recent theoretical developments

Contents n Introduction q The unitary Fermi gas n q n Recent theoretical developments The method q n Bertsch parameter, Tan’s Contact The operator product expansion Formulation of sum rules Results: the single-particle spectral density Conclusions and outlook

Introduction The Unitary Fermi Gas A dilute gas of non-relativistic particles with two species.

Introduction The Unitary Fermi Gas A dilute gas of non-relativistic particles with two species. Unitary limit: only one relevant scale Universality

Any relevance for the real world? It has only recently become possible to reach

Any relevance for the real world? It has only recently become possible to reach the unitary limit experimentally by tuning the scattering length by making use of a Feshbach resonance: Scattering length of Na atoms near a Feshbach resonance S. Inouye et al. , Nature 392, 151 (1998).

Example of universality M. Horikoshi et al. , Science, 327, 442 (2010). Measured using

Example of universality M. Horikoshi et al. , Science, 327, 442 (2010). Measured using the two lowest spin states of 6 Li atoms in a magnetic field. Ideal Fermi Gas This is a universal function applicable to all strongly interacting fermionic systems. Unitary Fermi Gas

Parameters charactarizing the unitary fermi gas (1) The “Bertsch problem” What are the ground

Parameters charactarizing the unitary fermi gas (1) The “Bertsch problem” What are the ground state properties of the many-body system composed of spin ½ fermions interacting via a zero-range, infinite scattering length contact interaction. (posed at the Many-Body X conference, Seattle, 1999) Or, more specifically: what is the value of ξ? Bertsch parameter

Values of ξ during the last few years ~ 0. 37 M. G. Endres,

Values of ξ during the last few years ~ 0. 37 M. G. Endres, D. B. Kaplan, J. -W. Lee and A. N. Nicholson, Phys. Rev. A 87, 023615 (2013).

Parameters charactarizing the unitary fermi gas (2) The “Contact” C interaction energy kinetic energy

Parameters charactarizing the unitary fermi gas (2) The “Contact” C interaction energy kinetic energy Tan relations S. Tan, Ann. Phys. 323, 2952 (2008); 323, 2971 (2008); 323, 2987 (2008).

What is C? : Number of pairs Number of atoms with wavenumber k larger

What is C? : Number of pairs Number of atoms with wavenumber k larger than K

What is the value of C? Experiment Theory S. Gandolfi, K. E. Schmidt and

What is the value of C? Experiment Theory S. Gandolfi, K. E. Schmidt and J. Carlson, Phys. Rev. A 83, 041601 (2011). Using Quantum Monte-Carlo simulation: 3. 40(1) J. T. Stewart, J. P. Gaebler, T. E. Drake, D. S. Jin, Phys. Rev. Lett. 104, 235301 (2010).

Deriving the Tan-relations with the help of the operator product expansion Zero-Range model: Operator

Deriving the Tan-relations with the help of the operator product expansion Zero-Range model: Operator corresponding to the contact density: E. Braaten and L. Platter, Phys. Rev. Lett. 100, 205301 (2008).

General OPE: Starting point: expression for momentum distribution: OPE (non-trivial) E. Braaten and L.

General OPE: Starting point: expression for momentum distribution: OPE (non-trivial) E. Braaten and L. Platter, Phys. Rev. Lett. 100, 205301 (2008).

take the expectation value After Fourier transformation: C E. Braaten and L. Platter, Phys.

take the expectation value After Fourier transformation: C E. Braaten and L. Platter, Phys. Rev. Lett. 100, 205301 (2008).

Further application of the OPE: the single-particle Green’s function location of pole in the

Further application of the OPE: the single-particle Green’s function location of pole in the large momentum limit Y. Nishida, Phys. Rev. A 85, 053643 (2012).

Comparison with Monte-Carlo simulations OPE results Quantum Monte Carlo simulation P. Magierski, G. Wlazłowski

Comparison with Monte-Carlo simulations OPE results Quantum Monte Carlo simulation P. Magierski, G. Wlazłowski and A. Bulgac, Phys. Rev. Lett. 107, 145304 (2011). Y. Nishida, Phys. Rev. A 85, 053643 (2012).

Our approach PG, N. Yamamoto, T. Hatsuda and Y. Nishida, ar. Xiv: 1501. 06053

Our approach PG, N. Yamamoto, T. Hatsuda and Y. Nishida, ar. Xiv: 1501. 06053 [cond-mat. quant-gas].

MEM PG, N. Yamamoto, T. Hatsuda and Y. Nishida, ar. Xiv: 1501. 06053 [cond-mat.

MEM PG, N. Yamamoto, T. Hatsuda and Y. Nishida, ar. Xiv: 1501. 06053 [cond-mat. quant-gas].

The sum rules The only input on the OPE side: Contact ζ Bertsch parameter

The sum rules The only input on the OPE side: Contact ζ Bertsch parameter ξ PG, N. Yamamoto, T. Hatsuda and Y. Nishida, ar. Xiv: 1501. 06053 [cond-mat. quant-gas].

Results Im Σ Re Σ A |k|=0. 0 |k|=0. 6 k. F |k|=1. 2

Results Im Σ Re Σ A |k|=0. 0 |k|=0. 6 k. F |k|=1. 2 k. F In a mean-field treatment, only these peaks are generated PG, N. Yamamoto, T. Hatsuda and Y. Nishida, ar. Xiv: 1501. 06053 [cond-mat. quant-gas].

Results pairing gap PG, N. Yamamoto, T. Hatsuda and Y. Nishida, ar. Xiv: 1501.

Results pairing gap PG, N. Yamamoto, T. Hatsuda and Y. Nishida, ar. Xiv: 1501. 06053 [cond-mat. quant-gas].

Comparison with other methods Monte-Carlo simulations Luttinger-Ward approach P. Magierski, G. Wlazłowski and A.

Comparison with other methods Monte-Carlo simulations Luttinger-Ward approach P. Magierski, G. Wlazłowski and A. Bulgac, Phys. Rev. Lett. 107, 145304 (2011). R. Haussmann, M. Punk and W. Zwerger, Phys. Rev. A 80, 063612 (2009). Qualitatively consistent with our results, except position of Fermi surface.

Conclusions n Unitary Fermi Gas is a strongly coupled system that can be studied

Conclusions n Unitary Fermi Gas is a strongly coupled system that can be studied experimentally q n n n Test + challenge for theory Operator product expansion techniques have been applied to this system recently We have formulated sum rules for the single particle self energy and have analyzed them by using MEM Using this approach, we can extract the whole singleparticle spectrum with just two inputparameters (Bertsch parameter and Contact)

Outlook n Generalization to finite temperature q n n Study possible existence of pseudo-gap

Outlook n Generalization to finite temperature q n n Study possible existence of pseudo-gap Role of so far ignored higher-dimensional operators Spectral density off the unitary limit Study other channels …

Backup slides

Backup slides

Taken from: M. Randeria, Nature Phys. 6, 561 (2010). How can the unitary limit

Taken from: M. Randeria, Nature Phys. 6, 561 (2010). How can the unitary limit be understood in a broader context? Key parameter: k. Fa well understood difficult well understood C. A. R. Sá de Melo, M. Randeria and J. R. Engelbrecht, Phys. Rev. Lett. 71, 3202 (1993).

OPERATOR Operator Operator Aritmatika Increment Decrement Operator LogikaOPERATOR Operator Operator Aritmatika Increment Decrement Operator Logika
Sum and Difference Formulas Sum Formulas Sum andSum and Difference Formulas Sum Formulas Sum and
SUM OF PRODUCT PRODUCT OF SUM DAN RANGKAIANSUM OF PRODUCT PRODUCT OF SUM DAN RANGKAIAN
SUM OF PRODUCT PRODUCT OF SUM DAN RANGKAIANSUM OF PRODUCT PRODUCT OF SUM DAN RANGKAIAN
Chapter 6 Product Operator Product operator is aChapter 6 Product Operator Product operator is a
8 3 sum sum i Lexer Parser sum8 3 sum sum i Lexer Parser sum
Riemann Sum Riemann Sum Formula Riemann Sum StepRiemann Sum Riemann Sum Formula Riemann Sum Step
Chapter 8 Operator Overloading Operator Overloading w OperatorChapter 8 Operator Overloading Operator Overloading w Operator
OPERATOR DAN UNGKAPAN Pengantar Operator dan Ungkapan OperatorOPERATOR DAN UNGKAPAN Pengantar Operator dan Ungkapan Operator
Operator Pengenalan Operator Letak operators pada aplikasi OperatorOperator Pengenalan Operator Letak operators pada aplikasi Operator
OPERATOR Ari Pambudi Penggunaan Operator Apa itu OperatorOPERATOR Ari Pambudi Penggunaan Operator Apa itu Operator
2 C arithmetic operator relational operator equality operator2 C arithmetic operator relational operator equality operator
PERTEMUAN 6 7 OPERATOR KONDISI Operator ARITMATIKA OperatorPERTEMUAN 6 7 OPERATOR KONDISI Operator ARITMATIKA Operator
Operator Aritmatika Operator Matematika yaitu operator yang digunakanOperator Aritmatika Operator Matematika yaitu operator yang digunakan
PRODUCT MANAGEMENT Product Decisions Product Development Product LifecyclePRODUCT MANAGEMENT Product Decisions Product Development Product Lifecycle
Inner Product Dot Product Outer Product Cross ProductInner Product Dot Product Outer Product Cross Product
Product Product Mix New Product Development Product LifeProduct Product Mix New Product Development Product Life
Counting Product and Sum Rules Product Rule AssumingCounting Product and Sum Rules Product Rule Assuming
Sum and Difference Identities Sum and Difference IdentitiesSum and Difference Identities Sum and Difference Identities
Common Loop Algorithm Sum and Average Sum keepCommon Loop Algorithm Sum and Average Sum keep
Sum Rules and Spin Structure Baldin Sum RuleSum Rules and Spin Structure Baldin Sum Rule
Warm Up Angle Sum Theorem The sum ofWarm Up Angle Sum Theorem The sum of
Sum Up lkj laksi Level Name Sum UpSum Up lkj laksi Level Name Sum Up
PROGRESS ON ENERGY SUM ELECTRONIC BOARD Energy SumPROGRESS ON ENERGY SUM ELECTRONIC BOARD Energy Sum